Ordering Decisions with an Unreliable Supplier under the Carbon Cap-and-Trade System

Abstract

The global focus on carbon reduction has intensified, prompting numerous high-energy-consuming enterprises to venture into the carbon cap-and-trade system. However, in recent years, the emergence of destabilizing factors has introduced disruptions to supply chains. The study addresses the two-stage ordering problem for a manufacturer under the carbon cap-and-trade system. In the first stage, the manufacturer engages in green investments and places orders with both an unreliable and a reliable supplier. After updating demand forecast information in the second stage, orders are placed with the backup supplier, and carbon allowances are settled at the end of the period. Under these conditions, three supply scenarios of the unreliable supplier are considered: time-varying supply with imperfect demand updates, all-or-nothing supply with imperfect demand updates, and time-varying supply with perfect demand updates. Optimal ordering decisions are provided for each scenario. We find that when demand updates are imperfect, the manufacturer will invariably engage with the unreliable supplier. However, when demand updates are perfect, the manufacturer may choose to forgo the unreliable supplier. Next, we analyze the influence of carbon trading prices on ordering decisions in these scenarios. We find that when the probability of disruption is substantial, dual sourcing must exist in the first stage under the all-or-nothing supply. Finally, we conduct numerical analysis by utilizing parameters, such as carbon trading prices, as referenced in the existing literature. Through numerical analysis, we find that opting for the all-or-nothing supplier becomes economically advantageous for the manufacturer when the backup supplier is profitable. Conversely, when the backup supplier is not profitable, the manufacturer tends to opt for the unreliable supplier with time-varying supply. Moreover, optimal profit for the manufacturer is not achieved when demand updates are perfect.

1. Introduction

The issue of global warming is becoming increasingly serious, garnering significant attention from nations worldwide. Several countries have successively set their carbon neutrality targets, such as China’s commitment to achieve carbon neutrality by 2060. To effectively reduce carbon dioxide emissions, various regulations about carbon emissions, such as the Kyoto Protocol and The Paris Agreement, have been successively enacted. As one widely applied regulatory policy, multiple countries and regions have adopted the carbon cap-and-trade system, exemplified by the European Union Emission Trading Scheme (EU-ETS) and China Carbon Emission Trade Exchange (CCETE). Under the carbon cap-and-trade system, emission-reducing enterprises first acquire a certain quantity of carbon allowances. Subsequently, enterprises can purchase additional allowances to increase their output or to sell surplus allowances after production concludes. For high-energy-consuming enterprises, such as metallurgical enterprises and power generation enterprises, which constitute one of the primary sources of carbon emissions, undertaking a low-carbon transformation has emerged as an effective means to reduce their carbon emissions. Furthermore, participation in the carbon cap-and-trade system also offers the possibility of additional revenue for high-energy-consuming enterprises, thereby greatly incentivizing their proactive efforts to reduce emissions.

However, the market is fraught with numerous unstable factors, such as policy, emergency, terrorist attacks, and natural disasters. These instabilities have the potential to cause disruptions in the supply chain of businesses, thereby posing a threat to their low-carbon transformation efforts. In the event of a supply disruption, participating emission-reducing enterprises would lack sufficient raw materials for production, leading to significant disruptions in the supply chain. For instance, the COVID-19 pandemic swiftly paralyzed the supply chain of companies such as Apple and Tesla [1]. To mitigate the risk of disruptions, diversifying sourcing has become the primary approach for businesses, as exemplified by Apple procuring critical components for the iPhone 7 from multiple suppliers [2]. Furthermore, the international political landscape is intricate, and geopolitical factors may have a substantial impact on supply chain security. For example, disruptions in the food supply chain occurred due to the Russia–Ukraine conflict [3]. While pursuing low-carbon transformation, businesses must also consider the repercussions of potential supply disruptions. Yet, in practical operations, driven by the pursuit of cost-effectiveness, companies may opt for unreliable suppliers that are vulnerable to disruption [4]. Meanwhile, the delivery quantities from the unreliable supplier may be influenced by the duration of his disruptions. In the event of disruptions from the unreliable supplier, only a partial order quantity may be delivered—or none at all. To mitigate this uncertainty and ensure their production as much as possible, companies may also place orders with reliable suppliers [5]. At this juncture, companies will need to make decisions regarding how to optimally allocate orders to maximize their profitability.

Additionally, postponing some orders close to the sales period to obtain more perfect supply and demand information is an effective strategy for supply–demand alignment [6]. Under this approach, companies not only gain access to the latest demand forecasts but also monitor the production status of the unreliable supplier and the reliable supplier. Armed with this more precise information, companies can place emergency orders with the backup supplier to ensure uninterrupted production. However, ordering from the backup supplier may entail higher ordering costs. Therefore, when all three types of suppliers are viable options for the company, it must weigh how to formulate a rational ordering plan to maximize its profitability.

In the above context, we consider the following problems. Firstly, in the context of random disruptions under the carbon cap-and-trade system, should companies place orders with the unreliable supplier? Secondly, how can companies judiciously allocate orders among three suppliers to maximize profitability while mitigating the impact of random disruptions? Thirdly, what alterations in ordering decisions occur for companies in response to fluctuations in carbon trading prices? Lastly, how do various scenarios of disruptions and levels of demand information updates influence companies’ ordering decisions?

To address the aforementioned issues, we adopt a two-stage ordering model similar to ref. [6]. The manufacturer has the option to place orders with three suppliers. At the onset of the first stage, the manufacturer undertakes green investments to reduce the carbon emissions per unit of product. The manufacturer places orders with both unreliable and reliable suppliers. Upon commencement of the second stage, the manufacturer updates demand forecasts and acquires production information from the unreliable supplier and the reliable supplier, culminating in the decision of whether to place orders with the backup supplier. At the end of the sales period, the manufacturer settles his carbon allowances through the carbon cap-and-trade system.

The study makes several contributions. Firstly, it addresses the ordering problem for multiple suppliers with demand and production information updates in the context of carbon cap-and-trade system. Unlike most literature on supply disruption, this research considers the updating of demand and production information, along with the scenarios of the unreliable supplier supplying over time-varying and all-or-nothing supply. Additionally, the incorporation of carbon cap-and-trade system as a new source of profit is explored, which can effectively mitigate the bankruptcy risk faced by enterprises in the event of disruptions. Many studies on the ordering problem with demand and production information updates overlook the benefits brought by the carbon cap-and-trade system to businesses. Hence, this paper integrates carbon cap-and-trade system with demand and production information updates, constructing a two-stage ordering model for a manufacturer to address this issue. Based on this foundation, this paper provides insights into how enterprises facing disruptions can formulate ordering decisions under a carbon cap-and-trade system. This research not only complements existing literature on supply chain management under carbon cap-and-trade system but also offers practical guidance for enterprises in devising ordering strategies.

The structure of this paper is outlined as follows. Section 2 provides a comprehensive review of relevant literature. Section 3 encompasses the model description and the definition of variables. In Section 4, optimal ordering decisions are analyzed in various scenarios, along with an examination of the influence of carbon trading prices on the manufacturer’s optimal ordering decisions. Section 5 presents the numerical analysis. Finally, Section 6 concludes with a summary of pertinent findings.

2. Literature Review

To effectively reduce carbon dioxide emissions, various carbon reduction policies have been introduced. The carbon cap-and-trade system, as an effective emission reduction approach, has been extensively analyzed by scholars in its impact on emission reduction through factors such as carbon trading prices [7,8,9,10,11] and carbon allowances [9]. As research progresses, scholars have integrated the carbon cap-and-trade mechanism with supply chain issues such as inventory management [12,13], supply chain financing [14,15], remanufacturing [8,16,17], channel structure [17,18,19], and competition [20]. Additionally, studies have compared the impact of carbon tax and carbon trading policies on the supply chain, such as refs. [21,22] comparing the emission reduction effects of the carbon tax and the carbon cap-and-trade mechanism. Unlike these, ref. [21] argued that the carbon cap-and-trade mechanism leads to better emission reduction results. Furthermore, ref. [22] further considered the combined impact of carbon tax and carbon trading policies. Building upon the comparison of the two policies, refs. [23,24] introduced a third policy for comparison. Ref. [23], in addition to comparing carbon tax and carbon trading, also assessed carbon intensity-based target strategies. Meanwhile, ref. [24] compared carbon tax, carbon trading, and government subsidies, noting that carbon tax may reduce supply chain profits.

In actual production operations, the production of the upstream supplier is often subject to significant uncertainty due to various factors, such as COVID-19 pandemics [25], earthquakes, floods, hurricanes, and terrorist attacks [26]. Under the influence of these uncertainties, downstream the manufacturer typically faces substantial risks of disruptions. To address these risks, scholars have explored various solutions, including the design of resilient supply chain networks [25,27], enhancing reconfigurable resource capabilities [28], dynamic pricing strategies [26], setting up forward inventory [29] and devising effective ordering decisions [28,30]. Formulating effective ordering decisions, as one of the effective approaches to mitigate disruption risks, has been the focus of extensive research by scholars. For instance, ref. [6] employed an information-updating approach to formulate appropriate strategies. Ref. [31] considered sustainability and resilience in the order allocation problem. Additionally, refs. [32,33,34,35] analyzed disruption issues from a supply chain perspective and formulated ordering decisions. What sets apart refs. [32,34] is their emphasis on formulating optimal pricing strategies concurrently. Ref. [33] devised ordering schemes by establishing dual wholesale price contracts. Ref. [35], based on addressing ordering and inventory issues, further discussed the impact on carbon emissions reduction.

In studies addressing inventory issues under carbon trading, refs. [12,36] considered the impact of competition among multiple retailers. The distinction lies in ref. [12] taking into account the influence of altruism on profit distribution, whereas ref. [36] compared different replenishment methods. Additionally, scholars have begun to concurrently consider various factors influencing supply chain decisions, such as ref. [37] considered the location and inventory issues under uncertainties in both carbon prices and demand. For the carbon emissions reduction problem under supply disruptions, ref. [35] addressed it by formulating optimal ordering and inventory decisions. On the other hand, ref. [38] tackled low-carbon disruptions through the formulation of a resilience strategy. Unlike this paper, refs. [35,38] approach the carbon emissions reduction issue under supply disruptions from a supply chain perspective. This paper focuses on how the manufacturer undergoing a low-carbon transition under supply disruptions can formulate order decisions to achieve emissions reduction while ensuring the sustainability of his operations.

The aforementioned literature has conducted extensive research on both the carbon cap-and-trade system and ordering decisions under disruptions. However, there has been relatively less research focused on ordering decisions considering supply disruptions under the carbon cap-and-trade system. In today’s intricate landscape, supply chains, amidst their low-carbon transformation, cannot afford to overlook the disruptions they might face. Unlike existing studies, this paper explores the ordering decision problem considering the updates of demand and production information, along with scenarios involving the unreliable suppliers supplying over time-varying and all-or-nothing supply. Moreover, it highlights the carbon cap-and-trade system as a novel source of profit that can effectively mitigate the bankruptcy risk for enterprises facing disruptions. However, most literature addressing the ordering problem with demand and production information updates tends to overlook the benefits brought by the carbon cap-and-trade system to businesses. Therefore, this paper focuses on solving the ordering decision problem with the consideration of carbon cap-and-trade system and the presence of an unreliable supplier. It aims to investigate how manufacturers can ensure a secure low-carbon transformation by formulating appropriate ordering strategies.

3. Model Description

This paper examines the two-stage ordering problem for the manufacturer constrained by emission reduction under the carbon cap-and-trade system. In this paper, we consider three types of suppliers:

(1)

Unreliable Supplier (S1): A disruption will occur., but the cost of placing orders with him is low for the manufacturer;

(2)

Reliable Supplier (S2): No disruption will occur, but the cost of placing orders with him is higher than with the unreliable supplier;

(3)

Backup Supplier (S3): No disruption will occur, but the cost of placing orders with him is the highest for the manufacturer.

In the first stage, the manufacturer places orders with both an unreliable supplier (S1) and a reliable supplier (S2), while also undertaking green investments. In the second stage, utilizing updated demand information before the sales period, orders are placed with a backup supplier (S3). When the sales period begins, the manufacturer sells the products and settles his carbon allowances. The parameters are described in

Table 1. Parameters and definitions.

Parameter	Definition	Parameter	Definition
r	Unit product price	$c_i$	Unit ordering cost from Supplier Si, $i\in \{1,2,3\}$
s	Unit product salvage value	$p_e$	Carbon trading price
$\alpha$	The disruption probability of S1	T	Planning time
$\Delta t$	Start time of Stage 2	t	Time of disruption occurrence
$\epsilon$	Actual demand uncertainty	${\epsilon }_j$	Forecasted demand uncertainty for Stage j, $j\in \{1,2\}$
$q_i$	Order quantity from supplier Si, $i\in \{1,2,3\}$	K	Green investment
E	Carbon allowance	$\delta$	Increase in demand per unit of green investment
$\theta$	The unit carbon emission reduction rate from green investment	e	Initial carbon emissions per unit of product
${\pi }_j$	The manufacturer’s expected profit for Stage j, $j\in \{1,2\}$

.

We assume that the manufacturer’s demand D is stochastic, comprising both deterministic and uncertain components. Furthermore, the deterministic portion is contingent upon green investment. Therefore, the demand function is given by $D=\delta \sqrt{K}+\epsilon$ [39]. $\delta$ represents the increase in demand per unit of green investment, K denotes the level of green investment, and $\epsilon$ represents the uncertainty in demand. To ensure that the products can be sold during the sales season, the manufacturer makes ordering decisions at time T before the start of the sales season. At this point, the manufacturer’s forecast of the uncertain demand is represented by ${\epsilon }_1$. The manufacturer then places orders $q_1$ with Supplier 1 (S1) at a unit cost of $c_1$ and places orders $q_2$ with Supplier 2 (S2) at a unit cost of $c_2$. Meanwhile, the manufacturer undertakes green investment K. Since S1 is unreliable and S2 is completely reliable, the unit cost satisfies $c_1<c_2$. For S1, a disruption may occur within the planning time T. The probability of a disruption occurring is given by $\alpha \in [0,1]$. The disruption time t follows a density function represented by equation $g\left(t\right)$, $t\in [0,T]$. After $\Delta t$, the manufacturer updates the forecast of uncertain demand ${\epsilon }_2$, as well as production information for S1 and S2. The manufacturer then places an order $q_3$ with Supplier 3 (S3) at a unit cost of $c_3$. The manufacturer receives the orders before the sales season. During the sales season, each unit is sold at a retail price of r, and the salvage value for each unsold unit is s. At the end of the sales period, his carbon allowances are settled through the carbon cap-and-trade system. To eliminate irrelevant scenarios, we assume: $s<c_1<c_2<c_3<r$, $t<T$ [6]. Additionally, $p_e$ is known before ordering occurs. Furthermore, to prevent the total order quantity from being zero, we assume $r-c_1-p_e(e-\theta K)>0$.

If the manufacturer places an order with Supplier 1 (S1), Figure 1 illustrates the different time of production disruptions for S1. Figure 1a presents the scenario where a disruption occurs in the first stage, denoted as $t\in [0,\Delta t]$. At the beginning of the second stage, both the disruption status of S1 and the available quantity of products are known, and additional orders are placed with S3 if necessary. Figure 1b depicts the scenario where no disruption occurs in the first stage. At the onset of the second stage, it remains uncertain when a disruption might occur with S1 and the final quantity delivered by S1 is also unknown.

We adopt the demand forecasting updating modeling approach proposed by ref. [6]. Given the forecast of uncertain demand ${\epsilon }_1$ in the first stage, the uncertain demand ${\epsilon }_2$ in the second stage is represented by the distribution function $F_1(.|{\epsilon }_1)$ and the density function $f_1(.|{\epsilon }_1)$. With the forecast of uncertain demand ${\epsilon }_2$ in the second stage, the actual uncertain demand $\epsilon$ is characterized by distribution and density functions $F_2(.|{\epsilon }_2)$ and $f_2(.|{\epsilon }_2)$, respectively [40]. We assume that the forecasts of uncertain demand in each stage are non-negative, and all distributions are continuous, differentiable, and integrable. Additionally, we posit that $F_2(.|{\epsilon }_2)$ strictly increases with ${\epsilon }_2$.

We use the reverse inductive approach to address the two-stage ordering problem. Given $q_1$ and $q_2$, we first determined the optimal ordering quantity $q_3^{1\ast }$ for the second stage when S1 experiences a disruption in the first stage. The optimal ordering quantity $q_3^{2\ast }$ for the second stage when S1 does not encounter a disruption in the first stage was then determined. Subsequently, we derived the optimal ordering quantities $q_1^{\ast }$ and $q_2^{\ast }$ for S1 and S2 in the first stage. We assume that in the first stage, the manufacturer can place orders simultaneously with both S1 and S2, referred to as Strategy D. To gain meaningful insights, we compared this strategy with two single-source procurement strategies, as outlined below:

(1)

Strategy A: In the first stage, the manufacturer undertakes green investment and exclusively places orders with S1. In the second stage, after information is updated, the decision is made whether to place orders with S3. At the onset of the sales period, the manufacturer sells products and settles the carbon allowance through the carbon cap-and-trade system;

(2)

Strategy B: In the first stage, the manufacturer undertakes green investment and exclusively places orders with S2. In the second stage, after information is updated, the decision is made whether to place orders with S3. At the onset of the sales period, the manufacturer sells products and settles carbon allowance through the carbon cap-and-trade system;

(3)

Strategy D: In the first stage, the manufacturer undertakes green investment and exclusively places orders with both S1 and S2. In the second stage, after information is updated, the decision is made whether to place orders with S3. At the onset of the sales period, the manufacturer sells products and settles carbon allowance through the carbon cap-and-trade system.

4. Model Analysis

4.1. Time-Varying Supplier under Imperfect Demand Information Updating

When a disruption occurs at time t, S1 can only supply the quantity produced before the disruption $q_1^{{}^{\prime }}=q_1\frac{t}{T}$ in this situation. First, we consider the scenario where S1 encounters a disruption in the first stage. We assume that the manufacturer will settle the ordering costs with S1 only after receiving the products. Additionally, the quantity of products paid for by S1 in the second stage is known. Given the current forecasted demand uncertainty ${\epsilon }_2$ the disruption time $t\in [0,\Delta t]$, and the ordered quantities $q_1$ and $q_2$ to S1 and S2 in the first stage. The manufacturer’s expected profit in the second stage is calculated as follows:

$$\begin{array}{c}{\pi }_2^1=-c_3{q}_3-c_1{q}_1\frac{t}{T}+r\min \{q_1\frac{t}{T}+q_2+q_3,\delta \sqrt{K}+\epsilon \}\hfill \\ +s(q_1\frac{t}{T}+q_2+q_3-\delta \sqrt{K}{-\epsilon )}^{+}+p_e[E-(e-\theta K)(q_1\frac{t}{T}+q_2+q_3)]\hfill \end{array}$$

(1)

In Equation (1), the first two terms correspond to the ordering costs from S3 and S1. The third term represents the manufacturer’s sales revenue. The fourth term accounts for the salvage value of unsold products. The fifth term represents the revenue from carbon trading.

Lemma 1.

When S1 occurs disruption in the first stage, the manufacturer’s expected profit ${\pi }_2^1$ in the second stage is a concave function of $q_3$. (All proofs in Supplementary Material).

Given Lemma 1, when a disruption occurs at S1 in the first stage, the manufacturer’s order quantity to S3 in the second stage is provided by Proposition 1.

Proposition 1.

When S1 occurs disruption in the first stage, given $q_1$, $q_2$, ${\epsilon }_2$ and t, the optimal order quantity $q_3^{1\ast }$ to S3 in the second stage satisfies the following: (1) when $r-c_3-p_e(e-\theta K)<0$, then $q_3^{1\ast }=0$; (2) when $r-c_3-p_e(e-\theta K)\ge 0$, then $q_3^{1\ast }$ satisfied

$$q_3^{1\ast }=\left\{\begin{array}{cc}0& {\epsilon }_2<{\epsilon }_M(q_1,q_2,t)\\ q_3^1(q_1,q_2,{\epsilon }_2,t)& {\epsilon }_2\ge {\epsilon }_M(q_1,q_2,t)\end{array}\right.$$

where $q_3^1$ and ${\epsilon }_M$ are satisfied:

$$q_3^1=F_2^{-1}(\frac{r-c_3-p_e(e-\theta K)}{r-s}|{\epsilon }_2)-q_1\frac{t}{T}-q_2+\delta \sqrt{K}$$

$$F_2(q_1\frac{t}{T}+q_2-\delta \sqrt{K}|{\epsilon }_M(q_1,q_2,t))=\frac{r-c_3-p_e(e-\theta K)}{r-s}$$

Next, we consider the scenario where S1 does not occur a disruption in the first stage. In this case, the absence of a disruption for S1 in the first stage is a prerequisite. Therefore, the probabilities of S1 experiencing a disruption or not in the second stage remain consistent with Ref. [6], denoted as $\alpha {\int }_{\Delta t}^{T}g\left(t\right)dt/[1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt]$ and $(1-\alpha )/[1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt]$ respectively. Hence, when S1 does not encounter a disruption in the first stage, the expected profit for the manufacturer in the second stage is determined by:

$$\begin{array}{c}\hfill {\pi }_2^2=\frac{1}{1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt}[\alpha {\int }_{\Delta t}^T{\pi }_2^1(q_1,q_2,q_3,t,{\epsilon }_2)g\left(t\right)dt+(1-\alpha ){\pi }_2^1(q_1,q_2,q_3,T,{\epsilon }_2)]\hfill \end{array}$$

(2)

Lemma 2.

When S1 does not experience a disruption in the first stage, the manufacturer’s expected profit ${\pi }_2^2$, is a concave function of $q_3$.

Based on Lemma 2, when S1 does not experience a disruption in the first stage, the manufacturer’s ordering quantity to S3 in the second stage is provided by Proposition 2.

Proposition 2.

When S1 does not experience an disruption in the first stage, given $q_1$, $q_2$, and ${\epsilon }_2$, the optimal order quantity $q_3^{2\ast }$ to S3 in the second stage satisfies the following: (1) when $r-c_3-p_e(e-\theta K)<0$, then $q_3^{2\ast }=0$; (2) when $r-c_3-p_e(e-\theta K)\ge 0$, then $q_3^{2\ast }$ satisfied

$$q_3^{2\ast }=\left\{\begin{array}{cc}0& {\epsilon }_2<{\epsilon }_N(q_1,q_2)\\ q_3^2(q_1,q_2,{\epsilon }_2)& {\epsilon }_2\ge {\epsilon }_N(q_1,q_2)\end{array}\right.$$

where $q_3^2$ and ${\epsilon }_N$ are satisfied:

$$\begin{array}{c}\alpha {\int }_{\Delta t}^T{F}_2(q_1\frac{t}{T}+q_2+q_3^2-\delta \sqrt{K}\left|{\epsilon }_2\right)g\left(t\right)dt+(1-\alpha )F_2(q_1+q_2+q_3^2-\delta \sqrt{K}|{\epsilon }_2)\hfill \\ =[1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt]\frac{r-c_3-p_e(e-\theta K)}{r-s}\hfill \end{array}$$

$$\begin{array}{c}\alpha {\int }_{\Delta t}^T{F}_2(q_1\frac{t}{T}+q_2-\delta \sqrt{K}\left|{\epsilon }_N\right)g\left(t\right)dt+(1-\alpha )F_2(q_1+q_2-\delta \sqrt{K}|{\epsilon }_N)\hfill \\ =[1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt]\frac{r-c_3-p_e(e-\theta K)}{r-s}\hfill \end{array}$$

Propositions 1 and 2 indicate that when the carbon trading price is too high, it renders ordering from S3 unprofitable for the manufacturer. However, when the carbon trading price is relatively low, regardless of whether S1 experiences a disruption in the first stage, as long as the updated forecast of uncertain demand ${\epsilon }_2$ in the second stage is large enough, ordering from S3 is a must option. Furthermore, the ordering quantity $q_3$ increases with ${\epsilon }_2$. This is because the manufacturer can reduce shortages by placing orders with S3 when the updated forecast is relatively large.

Based on Propositions 1 and 2, we understand that the carbon trading price affects the manufacturer’s ordering decision. Given the forecast of uncertain demand ${\epsilon }_1$, we first consider $r-c_3-p_e(e-\theta K)<0$. In this case, the manufacturer’s expected profit ${\pi }_1^1$ in the first stage is given by:

$$\begin{array}{c}{\pi }_1^1=-c_2{q}_2-K+\alpha {\int }_0^{\Delta t}[{\int }_0^{\infty }{\pi }_2^1(q_1,q_2,0,t,{\epsilon }_2)f_1\left({\epsilon }_2\right|{\epsilon }_1)d{\epsilon }_2]g\left(t\right)dt\hfill \\ +[1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt]{\int }_0^{\infty }{\pi }_2^2(q_1,q_2,0,t,{\epsilon }_2)f_1\left({\epsilon }_2\right|{\epsilon }_1)d{\epsilon }_2\hfill \end{array}$$

(3)

In Equation (3), the first term represents the ordering cost from S2. The second term is the green investment. The third term (or the fourth term) is the expected profit in the second stage when S1 experiences a disruption (or does not experience a disruption) in the first stage.

Lemma 3.

When $r-c_3-p_e(e-\theta K)<0$, the manufacturer’s expected profit ${\pi }_1^1$ in the first stage is jointly concave concerning $q_1$ and $q_2$.

Lemma 3 demonstrates that the optimal order quantity can be obtained by taking the derivative with respect to ${\pi }_1^1$. Let $M_1=\frac{\partial {\pi }_1^1}{\partial q_1}$, $M_2=\frac{\partial {\pi }_1^1}{\partial q_2}$. Thus, when $r-c_3-p_e(e-\theta K)<0$, the optimal ordering quantities for Strategy A and Strategy B are given by Lemma 4.

Lemma 4.

When $r-c_3-p_e(e-\theta K)<0$, given ${\epsilon }_1$: (1) in the first stage, for Strategy A, the optimal ordering quantity $q_1^{1A}$ satisfies $M_1(q_1^{1A},0,{\epsilon }_1)=0$. (2) if $r-c_2-p_e(e-\theta K)<0$, for Strategy B in the first stage, the optimal ordering quantity $q_2^{1B}$ satisfies $q_2^{1B}=0$; otherwise, $q_2^{1B}$ satisfies $M_2(0,q_2^{1B},{\epsilon }_1)=0$.

Next, we analyze the optimal ordering quantities $q_1^{1\ast }$ and $q_2^{1\ast }$ for S1 and S2 in Strategy D. Given ${\epsilon }_1$, the optimal ordering quantities for S1 and S2 in the first stage can be determined by Proposition 3.

Proposition 3.

Given ${\epsilon }_1$, if $M_2(q_1^{1A},0,{\epsilon }_1)\le 0$, the optimal ordering quantity in the first stage is $(q_1^{1\ast },q_2^{1\ast })=(q_1^{1A},0)$; otherwise, both optimal ordering quantities $(q_1^{1\ast },q_2^{1\ast })$ in the first stage are positive and satisfy $M_1(q_1^{1\ast },q_2^{1\ast },{\epsilon }_1)=0$ and $M_2(q_1^{1\ast },q_2^{1\ast },{\epsilon }_1)=0$.

Next, we consider the case $r-c_3-p_e(e-\theta K)\ge 0$. In this scenario, the expected profit ${\pi }_2^1$ in the first stage is given by:

$$\begin{array}{c}{\pi }_1^2=-c_2{q}_2-K+\alpha {\int }_0^{\Delta t}[{\int }_0^{{\epsilon }_M}{\pi }_2^1(q_1,q_2,0,t,{\epsilon }_2)f_1\left({\epsilon }_2\right|{\epsilon }_1)d{\epsilon }_2\hfill \\ +{\int }_{{\epsilon }_M}^{\infty }{\pi }_2^1(q_1,q_2,q_3^1,t,{\epsilon }_2)f_1\left({\epsilon }_2\right|{\epsilon }_1)d{\epsilon }_2]g\left(t\right)dt\hfill \\ +[1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt][{\int }_0^{{\epsilon }_N}{\pi }_2^2(q_1,q_2,0,t,{\epsilon }_2)f_1\left({\epsilon }_2\right|{\epsilon }_1)d{\epsilon }_2\hfill \\ +{\int }_{{\epsilon }_N}^{\infty }{\pi }_2^2(q_1,q_2,q_3^2,t,{\epsilon }_2)f_1\left({\epsilon }_2\right|{\epsilon }_1)d{\epsilon }_2]\hfill \end{array}$$

(4)

The expression in Equation (4) is analogous to Equation (3). In Equation (4), the first term represents the ordering cost from supplier S2. The second term signifies the investment in green initiatives. The third term (or the fourth term) accounts for the manufacturer’s expected profits in the second stage if there is a disruption (or not a disruption) in the first stage with supplier S1.

Lemma 5.

When $r-c_3-p_e(e-\theta K)\ge 0$, the expected profits ${\pi }_1^2$ in the first stage is jointly concave concerning $q_1$ and $q_2$.

Lemma 5 demonstrates that the optimal order quantity can be obtained by taking the derivative with respect to ${\pi }_1^2$. Let $G_1=\frac{\partial {\pi }_1^2}{\partial q_1}$, $G_2=\frac{\partial {\pi }_1^2}{\partial q_2}$. Thus, when $r-c_3-p_e(e-\theta K)\ge 0$, the optimal ordering quantities for Strategy A and Strategy B are given by Lemma 6.

Lemma 6.

When $r-c_3-p_e(e-\theta K)\ge 0$, given ${\epsilon }_1$: (1) in the first stage, for Strategy A, the optimal ordering quantity $q_1^{2A}$ satisfies $G_1(q_1^{2A},0,{\epsilon }_1)=0$. (2) in the first stage, the optimal order quantity $q_2^{2B}$ satisfies $G_2(0,q_2^{2B},{\epsilon }_1)=0$.

Next, we analyze the optimal ordering quantities $q_1^{2\ast }$ and $q_2^{2\ast }$ for S1 and S2 in Strategy D. Given ${\epsilon }_1$, the optimal ordering quantities for S1 and S2 in the first stage can be determined by Proposition 4.

Proposition 4.

Given ${\epsilon }_1$, if $M_2(q_1^{1A},0,{\epsilon }_1)\le 0$ holds, then the optimal ordering quantity in the first stage is $(q_1^{2\ast },q_2^{2\ast })=(q_1^{2A},0)$; otherwise, both optimal ordering quantities $(q_1^{2\ast },q_2^{2\ast })$ in the first stage are positive and satisfy $M_1(q_1^{2\ast },q_2^{2\ast },{\epsilon }_1)=0$ and $G_2(q_1^{2\ast },q_2^{2\ast },{\epsilon }_1)=0$.

Propositions 3 and 4 indicate that employing a dual-sourcing strategy in the first stage is not always advantageous when the manufacturer can order from both S1 and S2 simultaneously. In certain scenarios, S2 may be of no use, and the manufacturer will opt for S1 in the first stage. This implies that while the supply uncertainty introduced by the unreliable supplier affects the expected profits, cost savings influence the supplier’s choice in the first stage. Therefore, the unreliable supplier with cost advantages becomes the manufacturer’s preferred choice. We further explore the conditions under which carbon trading prices $p_e$ affect supplier selection in the first stage, and Proposition 5 provides insights.

Proposition 5.

(1) When $p_e\in (\frac{r-c_2}{e-\theta K},\frac{r-c_1}{e-\theta K})$, ordering from S2 is not a viable option; (2) When $p_e\in (\frac{r-c_3}{e-\theta K},\frac{r-c_2}{e-\theta K}]$, if $p_e$ ensures $M_2(q_1^{1\ast }(p_e),0,{\epsilon }_1)>0$, then ordering from both S1 and S2 simultaneously; otherwise, ordering solely from S1 is optimal; (3) When $p_e\in [0,\frac{r-c_3}{e-\theta K}]$, if $p_e$ ensures $G_2(q_1^{2\ast }(p_e),0,{\epsilon }_1)>0$, then ordering from both S1 and S2 simultaneously; otherwise, ordering solely from S1 is optimal.

Proposition 5 delineates the optimal ordering strategy when both S1 and S2 are available in various scenarios of carbon trading prices. If the carbon trading price exceeds $\frac{r-c_2}{e-\theta K}$, ordering from the reliable supplier becomes unprofitable. When the carbon trading price is at $[\frac{r-c_3}{e-\theta K},\frac{r-c_2}{e-\theta K})$, ordering from the reliable supplier is no longer unprofitable. However, the decrease in carbon trading price leads to an increase in order quantity when exclusively ordering from the unreliable supplier, resulting in higher benefits. If the benefits of ordering exclusively from the unreliable supplier outweigh those of ordering exclusively from the reliable supplier, the manufacturer will abandon the reliable supplier. When the carbon trading price is at $[0,\frac{r-c_3}{e-\theta K})$, ordering from the backup supplier also becomes profitable. At this point, the decrease in carbon trading price leads to an increase in order quantity when exclusively ordering from the unreliable supplier, resulting in higher benefits. Although using the backup supplier reduces the benefits of exclusively ordering from the unreliable supplier, if the benefits at this point exceed those of exclusively ordering from the reliable supplier, the manufacturer will opt to forgo the reliable supplier.

Next, we will compare the optimal order quantities and expected profits in the first stage between Strategy D and Strategies A and B in Propositions 6 and 7.

Proposition 6.

Within the range of $p_e\in [0,\frac{r-c_1}{e-\theta K})$, in the first stage, the optimal order quantity of Strategy D is always greater than that of Strategy B and less than or equal to that of Strategy A. Furthermore, the order quantities placed with S1 and S2 separately in Strategy D do not exceed those in Strategy A and Strategy B.

Proposition 6 illustrates that as long as the manufacturer engages in ordering behavior, regardless of the carbon emission price, the order quantity in the first stage using Strategy D will always be no more than Strategy A, but it will not be less than Strategy B. The reason is that compared to using Strategy A, when using Strategy D, the manufacturer can place orders with the reliable supplier, reducing the risk of shortages. This not only decreases the order quantity from the unreliable supplier but also reduces the total order quantity. In contrast to using Strategy B, when employing Strategy D, the manufacturer, to save costs, utilizes the unreliable supplier. While this behavior can lower the order quantity from the reliable supplier, it will increase the manufacturer’s risk of shortages. Therefore, to mitigate the risk of shortages, the total order quantity will increase.

Proposition 7.

Under any carbon emission price, Strategy D consistently leads to better profits for the manufacturer.

Proposition 7 demonstrates that as long as the manufacturer engages in ordering behavior, under any carbon trading price, the expected profit under Strategy D will always be optimal. The reason is that compared to Strategy A, in the first stage, Strategy D can place orders with the reliable supplier, thereby reducing losses due to shortages. In contrast to Strategy B, although in the first stage, Strategy D ordering from the unreliable supplier increases the cost of shortages, the lower ordering cost from the unreliable supplier leads to greater profits.

4.2. All-or-Nothing Supply under Imperfect Demand Information Updating

Many studies often assume that an unreliable supplier is either fully supplied or completely disrupted. The supply of S1 is 0 regardless of when the disruption occurs in this situation. In this section, we discuss the manufacturer’s ordering decisions in the all-or-nothing supply scenario. Finally, we compare these results with those in Section 4.1 to explore the impact of changing supplies over time.

In the all-or-nothing scenario, when S1 experiences a disruption in the first stage, its supply quantity is zero. Therefore, the expected profit ${\pi }_2^{1a}$ in the second stage is given by:

$$\begin{array}{c}{\pi }_2^{1a}=-c_3{q}_3-c_1{q}_1\frac{t}{T}+r\min \{q_2+q_3,\delta \sqrt{K}+\epsilon \}\hfill \\ +s(q_2+q_3-\delta \sqrt{K}{-\epsilon )}^{+}+p_e[E-(e-\theta K)(q_2+q_3)]\hfill \end{array}$$

(5)

In Proposition 8, we provide the conditions under which the optimal order quantity for S3 in the second stage is satisfied when S1 experiences a disruption in the first stage.

Proposition 8.

When S1 experiences a disruption in the first stage, given $q_2$ and ${\epsilon }_2$, the optimal order quantity $q_3^{1a\ast }$ for S3 in the second stage is determined by: (1) when $r-c_3-p_e(e-\theta K)<0$, then $q_3^{1a\ast }=0$; (2) when $r-c_3-p_e(e-\theta K)\ge 0$, then $q_3^{1a\ast }$ satisfied

$$q_3^{1a\ast }=\left\{\begin{array}{cc}0& {\epsilon }_2<{\epsilon }_M^a(0,q_2,t)\\ q_3^{1a}(0,q_2,{\epsilon }_2,t)& {\epsilon }_2\ge {\epsilon }_M^a(0,q_2,t)\end{array}\right.$$

where $q_3^{1a}$ and ${\epsilon }_M^a$ are satisfied:

$$q_3^{1a}=F_2^{-1}(\frac{r-c_3-p_e(e-\theta K)}{r-s}|{\epsilon }_2)-q_2+\delta \sqrt{K}$$

$$F_2(q_2-\delta \sqrt{K}|{\epsilon }_M^a(0,q_2,t))=\frac{r-c_3-p_e(e-\theta K)}{r-s}$$

When S1 does not experience a disruption in the first stage, the expected profit ${\pi }_2^{2a}$ in the second stage is given by:

$${\pi }_2^{2a}=\frac{1}{1-\alpha {\int }_0^{\Delta t}g\left(t\right)dt}[\alpha {\int }_{\Delta t}^T{\pi }_2^{1a}(0,q_2,q_3,t,{\epsilon }_2)g\left(t\right)dt+(1-\alpha ){\pi }_2^{1a}(0,q_2,q_3,T,{\epsilon }_2)]$$

(6)

Proposition 9 provides the condition for the optimal order quantity from S3 in the second stage when S1 does not experience a disruption in the first stage.

Proposition 9.

When S1 experiences a disruption in the first stage, given $q_1$, $q_2$, and ${\epsilon }_2$, the optimal order quantity $q_3^{2a\ast }$ for S3 in the second stage is determined by: (1) when $r-c_3-p_e(e-\theta K)<0$, then $q_3^{2a\ast }=0$; (2) when $r-c_3-p_e(e-\theta K)\ge 0$, then $q_3^{2a\ast }$ satisfied

$$q_3^{2a\ast }=\left\{\begin{array}{cc}0& {\epsilon }_2<{\epsilon }_N^a(q_1,q_2)\\ q_3^{2a}(q_1,q_2,{\epsilon }_2)& {\epsilon }_2\ge {\epsilon }_N^a(q_1,q_2)\end{array}\right.$$

where $q_3^{2a}$ and ${\epsilon }_N^a$ are satisfied:

$$\begin{array}{c}\alpha {\int }_{\Delta t}^T{F}_2(q_1+q_2+q_3^{2a}-\delta \sqrt{K}|{\epsilon }_2)g(t)dt+(1-\alpha )F_2(q_1+q_2+q_3^{2a}-\delta \sqrt{K}|{\epsilon }_2)\hfill \\ =[1-\alpha {\int }_0^{\Delta t}g(t)dt]\frac{r-c_3-p_e(e-\theta K)}{r-s}\hfill \end{array}$$

$$\begin{array}{c}\alpha {\int }_{\Delta t}^T{F}_2(q_1+q_2-\delta \sqrt{K}|{\epsilon }_N^a)g(t)dt+(1-\alpha )F_2(q_1+q_2-\delta \sqrt{K}|{\epsilon }_N^a)\hfill \\ =[1-\alpha {\int }_0^{\Delta t}g(t)dt]\frac{r-c_3-p_e(e-\theta K)}{r-s}\hfill \end{array}$$

Based on Propositions 8 and 9, we can derive the expected profit ${\pi }_1^{1a}$ when $r-c_3-p_e(e-\theta K)<0$ and ${\pi }_1^{2a}$ when $r-c_3-p_e(e-\theta K)\ge 0$ in the first stage.

$$\begin{array}{c}{\pi }_1^{1a}=-c_2{q}_2-K+\alpha {\int }_0^{\Delta t}[{\int }_0^{\infty }{\pi }_2^{1a}(0,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2]g(t)dt\hfill \\ +[1-\alpha {\int }_0^{\Delta t}g(t)dt]{\int }_0^{\infty }{\pi }_2^{2a}(q_1,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2\hfill \end{array}$$

(7)

$$\begin{array}{c}{\pi }_1^{2a}=-c_2{q}_2-K+\alpha {\int }_0^{\Delta t}[{\int }_0^{{\epsilon }_M^a}{\pi }_2^{1a}(q_1,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2+{\int }_{{\epsilon }_M^a}^{\infty }{\pi }_2^{1a}(q_1,q_2,q_3^{1a},t,\hfill \\ {\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2]g(t)dt+[1-\alpha {\int }_0^{\Delta t}g(t)dt][{\int }_0^{{\epsilon }_N^a}{\pi }_2^{2a}(q_1,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2\hfill \\ +{\int }_{{\epsilon }_N^a}^{\infty }{\pi }_2^2(q_1,q_2,q_3^{2a},t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2]\hfill \end{array}$$

(8)

We denote $M_1^a=\frac{\partial {\pi }_1^{1a}}{\partial q_1}$, $M_2^a=\frac{\partial {\pi }_1^{1a}}{\partial q_2}$, $G_1^a=\frac{\partial {\pi }_1^{2a}}{\partial q_1}$ and $G_2^a=\frac{\partial {\pi }_1^{2a}}{\partial q_2}$. Therefore, the optimal order quantities for Strategy A and Strategy B are provided by Lemma 7.

Lemma 7.

Given ${\epsilon }_1$, (1) when $r-c_3-p_e(e-\theta K)<0$, the optimal order quantity $q_1^{1Aa}$ for Strategy A in the first stage satisfies $M_1^a(q_1^{1Aa},0,{\epsilon }_1)=0$; if $p_e\in [\frac{r-c_2}{e-\theta K},\frac{r-c_1}{e-\theta K}]$, the optimal order quantity $q_2^{1B}$ for Strategy B in the first stage satisfies $q_2^{1Ba}=0$; otherwise, it satisfies $M_2^a(0,q_2^{1Ba},{\epsilon }_1)=0$; (2) when $r-c_3-p_e(e-\theta K)\ge 0$, the optimal order quantity $q_1^{2Aa}$ for Strategy A in the first stage satisfies $G_1^a(q_1^{2Aa},0,{\epsilon }_1)=0$; the optimal order quantity $q_2^{2B}$ for Strategy B in the first stage satisfies $G_2^a(0,q_2^{2Ba},{\epsilon }_1)=0$;

Based on the above analysis, under the all-or-nothing supply conditions of S1, the optimal order quantities for Strategy D in the first stage, for both S1 and S2, are provided by Proposition 10.

Proposition 10.

Given ${\epsilon }_1$, (1) If $M_2^a(q_1^{1Aa},0,{\epsilon }_1)\le 0$, the optimal order quantity in the first stage is $(q_1^{1a\ast },q_2^{1a\ast })=(q_1^{1Aa},0)$; otherwise, the optimal order quantities $(q_1^{1a\ast },q_2^{1a\ast })$ for the first stage are positive, and satisfy $M_1^a(q_1^{1a\ast },q_2^{1a\ast },{\epsilon }_1)=0$, $M_2^a(q_1^{1a\ast },q_2^{1a\ast },{\epsilon }_1)=0$; (2) If $G_2^a(q_1^{2Aa},0,{\epsilon }_1)\le 0$, the optimal order quantity in the first stage is $(q_1^{2a\ast },q_2^{2a\ast })=(q_1^{2Aa},0)$; otherwise, the optimal order quantities $(q_1^{2a\ast },q_2^{2a\ast })$ for the first stage are positive, and satisfy $G_1^a(q_1^{2a\ast },q_2^{2a\ast },{\epsilon }_1)=0$, $G_2^a(q_1^{2a\ast },q_2^{2a\ast },{\epsilon }_1)=0$;

Proposition 10 also illustrates that when the manufacturer can order from both S1 and S2 simultaneously, adopting a dual-source procurement strategy in the first stage is not always advantageous. In certain scenarios, S2 may be of no use, and the manufacturer will definitely choose S1 in the first stage. We further explore the conditions under which carbon trading prices $p_e$ affect the choice of supplier in the first stage, as presented in Proposition 11.

Proposition 11.

Under the all-or-nothing supply condition:

(1) 

When $p_e\in (\frac{r-c_2}{e-\theta K},\frac{r-c_1}{e-\theta K}]$, ordering from S2 is never an option;

(2) 

When $p_e\in (\frac{r-c_3}{e-\theta K},\frac{r-c_2}{e-\theta K}]$, if $\alpha >\frac{c_2-c_1}{c_3-c_1}$ holds, there exists ${\tilde{p}}_e^a$. If $p_e<{\tilde{p}}_e^a$, ordering from both S1 and S2 is optimal; otherwise, ordering from only S1 is optimal. If $\alpha \le \frac{c_2-c_1}{c_3-c_1}$, ordering from only S1 is optimal;

(3) 

When $p_e\in [0,\frac{r-c_3}{e-\theta K}]$, if $p_e$ satisfies $G_2^a(q_1^{2a\ast }(p_e),0,{\epsilon }_1)>0$, ordering from both S1 and S2 is optimal; otherwise, ordering from only S1 is optimal.

where ${\tilde{p}}_e^a$ satisfies $M_2^a(q_1^{1Aa}({\tilde{p}}_e^a),0,{\epsilon }_1)=0$.

Proposition 11 provides the optimal ordering strategy when both S1 and S2 are available in the first stage under different carbon trading prices. When the carbon trading price exceeds $\frac{r-c_2}{e-\theta K}$ or falls below $\frac{r-c_3}{e-\theta K}$, the underlying reasons align with Proposition 5 (1) and (3). If the carbon trading price exceeds $\frac{r-c_2}{e-\theta K}$, ordering from the reliable supplier becomes unprofitable. When the carbon trading price is at $[0,\frac{r-c_3}{e-\theta K})$, ordering from the backup supplier also becomes profitable. At this point, the decrease in carbon trading price leads to an increase in order quantity when exclusively ordering from the unreliable supplier, resulting in higher benefits. Although using the backup supplier reduces the benefits of exclusively ordering from the unreliable supplier, if the benefits at this point exceed those of exclusively ordering from the reliable supplier, the manufacturer will opt to forgo the reliable supplier. However, when the carbon trading price is within $(\frac{r-c_3}{e-\theta K},\frac{r-c_2}{e-\theta K}]$, the underlying reasons differ from Proposition 5 (2). From Proposition 11, we observe that the decision to use a reliable supplier is influenced by the disruption probability. When the disruption probability is low, the unreliable supplier has a low chance of experiencing stockouts and offers a significant price advantage, resulting in more cost savings. On the other hand, when the disruption probability is high, the unreliable supplier is more likely to face stockouts. In this scenario, if the carbon trading price is low, purchasing from the reliable supplier yields greater profits.

Building upon Proposition 11, we arrive at Corollary 1.

Corollary 1.

When it is profitable to order from S2, as long as the disruption probability $\alpha >\frac{c_2-c_1}{c_3-c_1}$, there will always be a scenario where the manufacturer orders from S2.

Corollary 1 illustrates that when the carbon trading price is small ($p_e<\frac{r-c_2}{e-\theta K}$), excessively high disruption probabilities can lead the manufacturer to engage the reliable supplier. This is because opting for the reliable supplier results in greater profits.

Next, in Proposition 12, we compare the optimal ordering quantities for the all-or-nothing supply versus the time-varying supply under imperfect demand information updates.

Proposition 12.

(1) When $r-c_3-p_e(e-\theta K)<0$, the optimal ordering in the second stage remains constant and is always zero. When ordering is exclusively from S1 in the first stage, it follows that $q_1^{1A}>q_1^{1Aa}$. In the scenario where both S1 and S2 are ordered from in the first stage, there exist $q_1^{1\ast }>q_1^{1a\ast }$, $q_2^{1\ast }<q_2^{1a\ast }$, or $q_1^{1\ast }<q_1^{1a\ast }$, $q_2^{1\ast }>q_2^{1a\ast }$;

(2) When $r-c_3-p_e(e-\theta K)\ge 0$, the optimal ordering in the second stage always fulfills $q_3^{1\ast }\le q_3^{1a\ast }$ and $q_3^{2\ast }\le q_3^{2a\ast }$. When ordering is exclusively from S1 in the first stage, it follows that $q_1^{2A}>q_1^{2Aa}$. In the scenario where both S1 and S2 are ordered from in the first stage, there exist $q_1^{1\ast }>q_1^{2a\ast }$, $q_2^{2\ast }<q_2^{2a\ast }$, or $q_1^{2\ast }<q_1^{2a\ast }$, $q_2^{2\ast }>q_2^{2a\ast }$.

Proposition 12 demonstrates that under the conditions of all-or-nothing supply from the unreliable supplier, the second-stage ordering quantity is always greater than or equal to the ordering quantity in the case of time-varying supply. The reason is that when it is profitable to order from the backup supplier, in the event of a disruption, the unreliable supplier in the all-or-nothing supply scenario cannot provide the ordered products, and the manufacturer must increase their orders from the backup supplier to ensure profit. When ordering is exclusively from the unreliable supplier in the first stage, the time-varying supply can still provide products in the event of a disruption. In this case, the manufacturer may obtain more favorable terms by increasing their orders.

4.3. Time-Varying Supplier under Perfect Demand Information Updating

In this section, we will consider the scenario where the second stage has perfect forecasting of the uncertain demand. When a disruption occurs at time t, S1 can also only supply the quantity produced before the disruption $q_1^{{}^{\prime }}=q_1\frac{t}{T}$ in this situation. This is a special case where the predicted uncertain demand ${\epsilon }_2$ in the second stage equals the actual value of uncertain demand $\epsilon$.

When there is an disruption at S1 in the first stage, the expected profit ${\pi }_2^{1p}$ for the second stage is given by:

$$\begin{array}{c}{\pi }_2^{1p}=-c_3{q}_3-c_1{q}_1\frac{t}{T}+r\min \{q_1\frac{t}{T}+q_2+q_3,\delta \sqrt{K}+{\epsilon }_2\}\hfill \\ +s(q_1\frac{t}{T}+q_2+q_3-\delta \sqrt{K}-{\epsilon }_2{)}^{+}+p_e[E-(e-\theta K)(q_1\frac{t}{T}+q_2+q_3)]\hfill \end{array}$$

(9)

When there is no disruption at S1 in the first stage, the expected profit ${\pi }_2^{2p}$ for the second stage is given by:

$$\begin{array}{c}\hfill {\pi }_2^{2p}=\frac{1}{1-\alpha {\int }_0^{\Delta t}g(t)dt}[\alpha {\int }_{\Delta t}^T{\pi }_2^{1p}(q_1,q_2,q_3,t,{\epsilon }_2)g(t)dt+(1-\alpha ){\pi }_2^{1p}(q_1,q_2,q_3,T,{\epsilon }_2)]\hfill \end{array}$$

(10)

In Proposition 13, we provide the optimal order quantity from S3 for the manufacturer, regardless of whether a disruption occurs in the first stage.

Proposition 13.

Given $q_1$, $q_2$, and ${\epsilon }_2$, the optimal order quantity $q_3^{1p\ast }$ and $q_3^{2p\ast }$ from S3 in the second stage satisfies:

(1) 

When there is a disruption in the first stage,

$$q_3^{1p\ast }=\left\{\begin{array}{cc}0& r-c_3-p_e(e-\theta K)<0\\ \max \{0,{\epsilon }_2+\delta \sqrt{K}-q_1\frac{t}{T}-q_2\}& r-c_3-p_e(e-\theta K)\ge 0\end{array}\right.$$

(2) 

When there is no disruption in the first stage,

$$q_3^{2p\ast }=\left\{\begin{array}{cc}0& r-c_3-p_e(e-\theta K)<0\\ \max \{0,{\epsilon }_2+\delta \sqrt{K}-q_1\frac{t_0^{\ast }}{T}-q_2\}& r-c_3-p_e(e-\theta K)\ge 0\end{array}\right.$$

when $r-c_3-p_e(e-\theta K)\ge 0$, if $\alpha {\int }_{\Delta t}^{T}g(t)dt[r-c_3-p_e(e-\theta K)]-(1-\alpha )[c_3-s+p_e(e-\theta K)]>0$, $t_0^{\ast }$ satisfies $\frac{\partial {\pi }_2^{2p}}{\partial q_3}=0$; otherwise, $t_0^{\ast }=T$.

Given the predicted demand uncertainty ${\epsilon }_1$, we first consider the scenario when $r-c_3-p_e(e-\theta K)<0$. In this case, the expected profit ${\pi }_1^{1p}$ in the first stage is given by:

$$\begin{array}{c}{\pi }_1^{1p}=-c_2{q}_2-K+\alpha {\int }_0^{\Delta t}[{\int }_0^{\infty }{\pi }_2^{1p}(q_1,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2]g(t)dt\hfill \\ +[1-\alpha {\int }_0^{\Delta t}g(t)dt]{\int }_0^{\infty }{\pi }_2^{2p}(q_1,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2\hfill \end{array}$$

(11)

We let $M_1^p=\frac{\partial {\pi }_1^{1p}}{\partial q_1}$, $M_2^p=\frac{\partial {\pi }_1^{1p}}{\partial q_2}$. Therefore, when $r-c_3-p_e(e-\theta K)<0$, the optimal order quantities for Strategy A and Strategy B are provided by Lemma 8.

Lemma 8.

When $r-c_3-p_e(e-\theta K)<0$, given ${\epsilon }_1$,

(1) 

In the first stage, if ${\tilde{p}}_e^{1p}\le \frac{r-c_3}{e-\theta K}$, the optimal order quantity $q_1^{1Ap\ast }$ for Strategy A is $q_1^{1Ap\ast }=q_1^{1Ap}$. If ${\tilde{p}}_e^{1p}>\frac{r-c_3}{e-\theta K}$, the optimal order quantity for Strategy A satisfies:

$$q_1^{1Ap\ast }=\left\{\begin{array}{cc}\infty & p_e\le {\tilde{p}}_e^{1p}\\ q_1^{1Ap}& p_e>{\tilde{p}}_e^{1p}\end{array}\right.$$

(2) 

If $r-c_2-p_e(e-\theta K)<0$, in the first stage, the optimal order quantity $q_2^{1Bp}$ for Strategy B satisfies $q_2^{1Bp}=0$. Otherwise, it satisfies $q_2^{1Bp\ast }=F_1^{-1}(\frac{r-c_2-p_e(e-\theta K)}{(r-s)\alpha {\int }_0^{\Delta t}g(t)dt})$.

where, $q_1^{1Ap}$ satisfies $M_1^p(q_1^{1Ap},0,{\epsilon }_1)=0$ and ${\tilde{p}}_e^{1p}=\frac{r-c_1}{e-\theta K}-\frac{(r-s)\alpha {\int }_0^{\Delta t}g(t)dt}{(e-\theta K)\alpha {\int }_0^{\infty }\frac{t}{T}g(t)dt}$.

In the case of $r-c_3-p_e(e-\theta K)\ge 0$, the expected profit ${\pi }_1^{2p}$ in the first stage is:

$$\begin{array}{c}{\pi }_1^{2p}=-c_2{q}_2-K+\alpha {\int }_0^{\Delta t}[{\int }_0^{q_1\frac{t}{T}+q_2-\delta \sqrt{K}}{\pi }_2^{1p}(q_1,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2\hfill \\ +{\int }_{q_1\frac{t}{T}+q_2-\delta \sqrt{K}}^{\infty }{\pi }_2^{1p}(q_1,q_2,q_3^{1p},t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2]g(t)dt+(1-\alpha {\int }_0^{\Delta t}g(t)dt)\hfill \\ [{\int }_0^{q_1\frac{t_0^{\ast }}{T}+q_2-\delta \sqrt{K}}{\pi }_2^{2p}(q_1,q_2,0,t,{\epsilon }_2)f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2+{\int }_{q_1\frac{t_0^{\ast }}{T}+q_2-\delta \sqrt{K}}^{\infty }{\pi }_2^{2p}(q_1,q_2,q_3^{2p},t,{\epsilon }_2)\hfill \\ f_1({\epsilon }_2|{\epsilon }_1)d{\epsilon }_2]\hfill \end{array}$$

(12)

We define $G_1^p=\frac{\partial {\pi }_1^{2p}}{\partial q_1}$, $G_2^p=\frac{\partial {\pi }_1^{2p}}{\partial q_2}$. Therefore, when $r-c_3-p_e(e-\theta K)\ge 0$, the optimal order quantities for Strategy A and Strategy B are determined by Lemma 9.

Lemma 9.

When $r-c_3-p_e(e-\theta K)\ge 0$, given ${\epsilon }_1$,

(1) 

In the first stage, there exists ${\tilde{p}}_e^{2p}\in [0,\frac{r-c_3}{e-\theta K}]$, the optimal order quantity $q_1^{2Ap\ast }$ for Strategy A satisfies:

$$q_1^{2Ap\ast }=\left\{\begin{array}{cc}0& p_e\le {\tilde{p}}_e^{2p}\\ q_1^{2Ap}& p_e>{\tilde{p}}_e^{2p}\end{array}\right.$$

(2) 

In the first stage, the optimal order quantity $q_2^{2Bp}$ for Strategy B satisfies:

$$q_2^{2Bp\ast }=F_1^{-1}(\frac{c_3-c_2}{c_3-s+p_e(e-\theta K)-(r-s)\alpha {\int }_{\Delta t}^{T}g(t)dt})$$

where, $q_1^{2Ap}$ satisfies $G_1^p(q_1^{2Ap},0,{\epsilon }_1)=0$, and ${\tilde{p}}_e^{2p}=-\frac{1}{(e-\theta K)\alpha {\int }_{\Delta t}^T(1-\frac{t}{T})g(t)dt}[(c_3-c_1)(1-\alpha +\alpha {\int }_{\Delta t}^T\frac{t}{T}g(t)dt)+\alpha {\int }_{\Delta t}^T(r-c_1)\frac{t}{T}g(t)dt-\alpha {\int }_{\Delta t}^T(r-c_3)g(t)dt]$.

From Lemma 8 and Lemma 9, we observe that when the manufacturer orders exclusively from the unreliable supplier in the first stage, the order quantity from this supplier varies with changes in carbon trading prices. This is because when the carbon trading price satisfies $p_e\le {\tilde{p}}_e^{2p}$, the manufacturer gains more profit by ordering from the backup supplier in the second stage, leading them to forsake the unreliable supplier. When the carbon trading price satisfies ${\tilde{p}}_e^{2p}<p_e\le \frac{r-c_3}{e-\theta K}$, although there is still profit in ordering from the backup supplier in the second stage, the profit margin decreases. Therefore, the manufacturer orders from the unreliable supplier. As the carbon trading price continues to rise, if ${\tilde{p}}_e^{1p}\le \frac{r-c_3}{e-\theta K}$, the manufacturer will no longer utilize the backup supplier, and the profit margin from ordering from the unreliable supplier is also relatively small. Thus, to maximize his profit, the manufacturer will set the optimal order quantity from the unreliable supplier. If ${\tilde{p}}_e^{1p}>\frac{r-c_3}{e-\theta K}$, when the carbon trading price satisfies $p_e\le {\tilde{p}}_e^{1p}$, there is a large profit margin in ordering from the unreliable supplier, so the manufacturer increases the order quantity as much as possible. However, when the carbon trading price is ${\tilde{p}}_e^{1p}<p_e$, the profit margin in ordering from the unreliable supplier decreases. Therefore, the manufacturer sets the optimal order quantity to ensure the maximization of their profit. Based on Lemma 8 and Lemma 9, we present in Proposition 14 the optimal order quantities for the manufacturer from both S1 and S2 in the first stage under Strategy D.

Proposition 14.

Given ${\epsilon }_1$,

(1) 

If $M_2^p(q_1^{1Ap},0,{\epsilon }_1)\le 0$, the optimal order quantity in the first stage is $(q_1^{1p\ast },q_2^{1p\ast })=(q_1^{1Ap},0)$. If $M_1^p(0,q_2^{1Ap},{\epsilon }_1)\le 0$, then the optimal order quantity in the first stage is $(q_1^{1p\ast },q_2^{1p\ast })=(0,q_2^{1Bp})$. Otherwise, the optimal order quantity $(q_1^{1p\ast },q_2^{1p\ast })$ in the first stage is positive and satisfies $M_1^p(q_1^{1p\ast },q_2^{1p\ast },{\epsilon }_1)=0$ and $M_2^p(q_1^{1p\ast },q_2^{1p\ast },{\epsilon }_1)=0$;

(2) 

If $G_2^p(q_1^{2Ap},0,{\epsilon }_1)\le 0$, the optimal order quantity in the first stage is $(q_1^{2p\ast },q_2^{2p\ast })=(q_1^{2Ap},0)$. If $G_1^p(0,q_2^{2Ap},{\epsilon }_1)\le 0$, then the optimal order quantity in the first stage is $(q_1^{2p\ast },q_2^{2p\ast })=(0,q_2^{2Bp})$. Otherwise, the optimal order quantity $(q_1^{1p\ast },q_2^{1p\ast })$ in the first stage is positive and satisfies $G_1^p(q_1^{2p\ast },q_2^{2p\ast },{\epsilon }_1)=0$ and $G_2^p(q_1^{2p\ast },q_2^{2p\ast },{\epsilon }_1)=0$.

We observe that placing orders with both S1 and S2 simultaneously in the first stage may not be profitable. However, in contrast to Section 4.1, the unreliable supplier may also be eliminated. This is because the unreliable supplier could lead to a significant shortage, requiring an increase in order quantities in the second stage for better supply-demand matching. Yet, an increase in order quantity in the second-stage would entail higher costs for the manufacturer. To ensure the maximization of their profit, the manufacturer may choose to forego the unreliable supplier. Next, we analyze the conditions under which the carbon trading price $p_e$ affects the choice of supplier.

Proposition 15.

When the second stage demand uncertainty can be perfectly predicted:

(1) 

When $p_e\in [\frac{r-c_2}{e-\theta K},\frac{r-c_1}{e-\theta K}]$, it is certain that no orders will be placed with S2;

(2) 

When $p_e\in [\frac{r-c_3}{e-\theta K},\frac{r-c_2}{e-\theta K})$, for S1, if $p_e$ leads to $M_1^p(0,q_2^{1p\ast }(p_e),{\epsilon }_1)>0$, orders will be placed with both S1 and S2; otherwise, only purchases from S2 will be made. For S2, if $c_3-c_2>\alpha (r-s){\int }_0^{\Delta t}{F}_1(q_1^{1p\ast }\frac{t}{T}-\delta \sqrt{K}|{\epsilon }_1)g(t)dt$, there exists ${\tilde{p}}_e^{3p}$, and when $p_e<{\tilde{p}}_e^{3p}$ is satisfied, orders will be placed with both S1 and S2; otherwise, only orders from S1 will be placed.

(3) 

When $p_e\in [0,\frac{r-c_3}{e-\theta K})$, for S1, if $p_e$ leads to $G_1^p(0,q_2^{2p\ast }(p_e),{\epsilon }_1)>0$, orders will be placed with both S1 and S2; otherwise, only purchases from S2. For S2, if $p_e>{\tilde{p}}_e^{2p}$ and $c_3-c_2>\alpha (r-s){\int }_0^{\Delta t}{F}_1(q_1^{2p\ast }\frac{t}{T}-\delta \sqrt{K}|{\epsilon }_1)g(t)dt+(r-s)F_1(q_1^{2p\ast }\frac{t_0^{\ast }}{T}-\delta \sqrt{K}|{\epsilon }_1)g(t)dt(1-\alpha {\int }_0^{t_0^{\ast }}g(t)dt)$ are satisfied, then orders will definitely be placed with both S1 and S2; otherwise, there exists ${\tilde{p}}_e^{4p}$, and when $p_e<{\tilde{p}}_e^{4p}$ is satisfied, orders will be placed with both S1 and S2; otherwise, only orders from S1 will be placed. If $p_e\le {\tilde{p}}_e^{2p}$, orders will be placed with only S2.

Proposition 15 provides the optimal ordering strategy when both S1 and S2 are available in the first stage under different carbon trading prices. In contrast to Proposition 5, when the demand uncertainty in the second stage can be perfectly predicted, there are scenarios where the manufacturer may choose to abandon the unreliable supplier. This is because the unreliable supplier may result in significant shortages due to disruptions, and ordering from him may not yield higher profits. In these scenarios, in the first stage, the manufacturer opts for the reliable supplier alone, aiming for a better supply-demand match to maximize profits.

5. Numerical Example

In this section, we continue our exploration with numerical analysis to uncover further insights. Taking China Baowu Steel Group Corporation Limited as an example, to mitigate the impact of supply disruptions, the company procures iron ore from multiple countries, including Australia and Brazil. Additionally, with the growing awareness of carbon reduction, metallurgical enterprises are gradually being incorporated into the carbon cap-and-trade system. Therefore, by examining iron ore prices and relevant settings in literature on supply disruptions and carbon trading, we assume that ${\epsilon }_1$ is 100, ${\epsilon }_2$ follows a uniform distribution specified by $[{\epsilon }_1-50,{\epsilon }_1+50]$, $\epsilon$ follows a uniform distribution specified by $[{\epsilon }_2-20,{\epsilon }_2+20]$, t follows a uniform distribution specified by $[0,T]$, with r is 5, s is 1, $c_1$ is 1.5, $c_2$ is 2, $c_3$ is 2.5, $\alpha$ is 0.6, $\delta$ is 0.5, $\Delta t$ is 15, T is 30, E is 10,000, e is 1, $\theta$ is 0.0005, and K is 1000. To obtain results for $r-c_3-p_e(e-\theta K)<0$ and $r-c_3-p_e(e-\theta K)\ge 0$, we assume $p_e$ to be 3 and 6, respectively. In this section, we represent ${\pi }^{\ast }$, ${\pi }^{a\ast }$, and ${\pi }^{p\ast }$ as the manufacturer’s profits under three scenarios: time-varying supply with imperfect demand updates, all-or-nothing supply with imperfect demand updates, and time-varying supply with perfect demand updates.

First, we observe the impact of $c_1$ on the expected profits of the three scenarios in Section 4. We keep the other parameters as previously specified, assuming $c_1\in (1.1,1.9)$, and the results are displayed in Figure 2. We can see that regardless of $r-c_3-p_e(e-\theta K)<0$ or $r-c_3-p_e(e-\theta K)\ge 0$, ${\pi }^{\ast }$, ${\pi }^{a\ast }$, and ${\pi }^{p\ast }$ all decrease with the increase of $c_1$. This is because the increase of reduces the cost savings of ordering from the unreliable supplier. Additionally, From Figure 2a, we observe that when $r-c_3-p_e(e-\theta K)<0$, ${\pi }^{\ast }>{\pi }^{p\ast }>{\pi }^{a\ast }$. This is due to disruptions occurring under the all-or-nothing scenario, preventing the manufacturer from receiving products from the unreliable supplier. Additionally, due to excessively high carbon trading prices, the manufacturer ceases to place orders with the backup supplier, resulting in substantial losses from shortages. Consequently, this situation yields the lowest expected profits. In the case of perfect demand information, while the manufacturer can place orders with greater accuracy, they may inadvertently overlook the economic value obtained through carbon trading. As a result, expected profits are lower compared to situations with imperfect demand information. From Figure 2b we observe that when $r-c_3-p_e(e-\theta K)\ge 0$, ${\pi }^{a\ast }>{\pi }^{p\ast }>{\pi }^{\ast }$. This is because under the all-or-nothing condition, the unreliable supplier no longer supplies products to the manufacturer when a disruption occurs. Although the total output of the manufacturer will decrease, the manufacturer can obtain larger profits through carbon trading. Therefore, under the all-or-nothing condition, the manufacturer can achieve higher profits. When demand information is updated more perfectly, the manufacturer can order more precisely, thereby avoiding wastage of ordering costs or losses due to shortages.

We maintained the remaining parameters as described above and assumed $c_2\in (2,2.4)$. In Figure 3, we illustrate the variations in profits with changing $c_2$ under three scenarios. From Figure 3a, we also observe that ${\pi }^{\ast }>{\pi }^{p\ast }>{\pi }^{a\ast }$. This is also due to a disruption occurring under the all-or-nothing scenario, preventing the manufacturer from receiving products from the unreliable supplier. Additionally, due to excessively high carbon trading prices, the manufacturer ceases to place orders with the backup supplier, resulting in substantial losses from shortages. Consequently, this situation yields the lowest expected profits. From Figure 3b, we note that ${\pi }^{\ast }$, ${\pi }^{a\ast }$, and ${\pi }^{p\ast }$ remain unchanged by changes in $c_2$. The reason is that none of the three scenarios order from the reliable supplier. Furthermore, for the same reasons as in Figure 2b, this is because when a disruption occurs with the unreliable supplier under the all-or-nothing condition, he no longer supplies products to the manufacturer. Although the overall production of the manufacturer will decrease, the manufacturer can obtain greater revenue through the carbon cap-and-trade system. Therefore, ${\pi }^{a\ast }>{\pi }^{p\ast }>{\pi }^{\ast }$.

Next, keeping the remaining parameters as previously stated, We assume $p_e$ is 3. When $r-c_3-p_e(e-\theta K)<0$, we set $c_3\in (3.6,4.4)$. From Figure 4a, we observe ${\pi }^{\ast }>{\pi }^{p\ast }>{\pi }^{a\ast }$. Similarly, when a disruption occurs under the all-or-nothing condition, the manufacturer does not receive products from the unreliable supplier. Moreover, due to the excessively high carbon trading prices, the manufacturer no longer places orders with the backup supplier. This results in significant losses due to shortages, making the expected profit in this scenario the minimum. When $r-c_3-p_e(e-\theta K)\ge 0$, we set $c_3\in (2.1,2.9)$. In Figure 4b, we can see ${\pi }^{a\ast }>{\pi }^{p\ast }>{\pi }^{\ast }$. This is because under the all-or-nothing condition, if the unreliable supplier experiences a disruption, he no longer supplies the manufacturer. With reduced production from the manufacturer, the carbon trading revenue increases. When demand information is updated perfectly, the manufacturer can make more precise orders to reduce order shortage costs.

We analyze the impact of $p_e$ on the expected profits under the three scenarios. Keeping the remaining parameters as previously stated, we assume $c_3$ is 2.5. When $r-c_3-p_e(e-\theta K)<0$, we set $p_e\in (5,7)$. From Figure 5a, we observe that ${\pi }^{\ast }$, ${\pi }^{a\ast }$, and ${\pi }^{p\ast }$ all increase with the increase of $p_e$. This is because the increase in carbon trading price leads to a reduction in the manufacturer’s order quantity. However, reducing the order quantity can result in more carbon trading revenue. Additionally, ${\pi }^{\ast }>{\pi }^{p\ast }>{\pi }^{a\ast }$ still exists. Due to the disruption when the all-or-nothing condition is in place, the manufacturer cannot receive products from the unreliable supplier. Furthermore, due to excessively high carbon trading prices, the manufacturer no longer places orders with the backup supplier. This results in substantial losses due to shortages, making the expected profit in this scenario the minimum. When $r-c_3-p_e(e-\theta K)\ge 0$, we set $p_e\in (2.5,4.5)$. In Figure 5b, we can see that ${\pi }^{\ast }$, ${\pi }^{a\ast }$, and ${\pi }^{p\ast }$ all increase with the increase of $p_e$. The reasons behind this result are the same as when $r-c_3-p_e(e-\theta K)<0$. Similarly, ${\pi }^{a\ast }>{\pi }^{p\ast }>{\pi }^{\ast }$ still exists in this situation.

Finally, we analyze the impact of K on the expected profits under the three scenarios. Keeping the other parameters as previously stated, assuming $c_3$ is 2.5 and $p_e$ is 3. When $r-c_3-p_e(e-\theta K)<0$ is satisfied, we let $K\in (100,300)$. From Figure 6a, we observe that ${\pi }^{\ast }$, ${\pi }^{a\ast }$, and ${\pi }^{p\ast }$ all increase with the increment of K. This is because the increase in green investment leads to a reduction in carbon emission costs per unit product, resulting in more revenue from the carbon cap-and-trade system. Additionally, ${\pi }^{\ast }>{\pi }^{p\ast }>{\pi }^{a\ast }$ exists for the same reasons mentioned earlier. When a disruption occurs under the all-or-nothing condition, the manufacturer cannot receive goods from the unreliable supplier, and due to excessively high carbon trading prices, the manufacturer no longer places orders with the backup supplier, resulting in significant losses due to shortages, making the expected profit in this scenario the minimum. When $r-c_3-p_e(e-\theta K)\ge 0$ is satisfied, we let $K\in (350,550)$. From Figure 6b, we observe that ${\pi }^{a\ast }$ increases with the increase in K, while ${\pi }^{\ast }$ and ${\pi }^{p\ast }$ decrease with the increase in K. This is because, under the all-or-nothing condition, when a disruption occurs with the unreliable supplier, it cannot provide products. However, due to the increase in green investment, the carbon trading revenue per unit product increases. At this point, carbon trading revenue under the all-or-nothing condition is sufficient to offset the losses from shortages. Therefore, ${\pi }^{a\ast }$ increases with the increase in K. However, for the other two scenarios, since the unreliable supplier can provide a partial order quantity, ordering costs must be paid to the unreliable supplier. However, the incurred ordering costs will exceed the revenue from carbon trading, so ${\pi }^{\ast }$ and ${\pi }^{p\ast }$ decrease with the increase in K. Similar to the previous cases, ${\pi }^{a\ast }>{\pi }^{p\ast }>{\pi }^{\ast }$ still exists.

The above numerical example consistently demonstrate the following result: if $r-c_3-p_e(e-\theta K)<0$, ${\pi }^{\ast }>{\pi }^{p\ast }>{\pi }^{a\ast }$; if $r-c_3-p_e(e-\theta K)\ge 0$, ${\pi }^{a\ast }>{\pi }^{p\ast }>{\pi }^{\ast }$. This outcome indicates that when it is not profitable to use a backup supplier, the manufacturer opting for an unreliable supplier with time-varying supply can achieve higher profits. Conversely, when using a backup supplier is profitable, the manufacturer opting for an unreliable supplier with all-or-nothing supply can attain better profits.

6. Conclusions

This paper analyzes the two-stage ordering problem for the manufacturer under the carbon cap-and-trade system. At the onset of the first stage, the manufacturer undertakes green investments and places orders with both unreliable and reliable suppliers. At the beginning of the second stage, the manufacturer updates his demand forecasts and places orders with the backup supplier. When the sales period begins, the manufacturer sells the products and settles the carbon allowance. We consider three supply scenarios for the unreliable supplier: time-varying supply with imperfect demand updates, all-or-nothing supply with imperfect demand updates, and time-varying supply with perfect demand updates. We derive the conditions for the manufacturer’s optimal ordering quantity and provide the optimal ordering quantity when single sourcing is conducted in the first stage. We find that in cases of imperfect demand information updates, the manufacturer will invariably opt for the unreliable supplier in the first stage and will not solely rely on the reliable supplier. However, in the situation where demand information updates are perfect, the manufacturer might choose to abandon the unreliable supplier and exclusively utilize the reliable supplier. Alternatively, the manufacturer might maximize his orders from the unreliable supplier. Additionally, under imperfect demand information updates and time-varying supply from the unreliable supplier, the scenario where the manufacturer can place orders with both suppliers in the first stage is guaranteed to yield better profits.

Building on this, we further analyze the impact of carbon trading prices on the manufacturer’s ordering decisions. Furthermore, we provide insights into how the manufacturer should choose suppliers and formulate ordering decisions as carbon trading prices change. We find that under the all-or-nothing supply condition, there is always a carbon trading price that leads to dual sourcing for the manufacturer if the disruption probability is too high. Finally, through the numerical examples, we find that when ordering from the backup supplier is profitable, the unreliable supplier providing all-or-nothing supply can bring more revenue to the manufacturer. On the other hand, when ordering from the backup supplier is not profitable, unreliable supplier time-varying supply with imperfect demand updates can lead to higher profits for the manufacturer. This result indicates that perfect demand information does not lead to better outcomes. Moreover, the use of the carbon cap-and-trade system provides new sources of profit for businesses, offering both incentives for emissions reduction and reducing the disruption caused by supply disruption. This significantly enhances the sustainability of the supply chain.

This paper has certain limitations. We assume that the manufacturer’s green investment is fixed, while in actual operations, the amount of green investment may affect the manufacturer’s optimal decisions. Treating the green investment amount as a decision variable could be a new research direction. Additionally, the paper assumes that the manufacturer only settles carbon allowances through the carbon cap-and-trade system at the end of the period. However, the carbon cap-and-trade system offers flexibility in trading, and carbon prices can fluctuate at different times. Therefore, considering the introduction of carbon trading activities in different periods into this ordering problem is an interesting research question. Finally, we assume that the reliable supplier remains reliable throughout the entire period. However, in reality, a reliable supplier may also become unreliable due to external uncertainties. Considering the scenario where the reliable supplier becomes unreliable in the second stage is a new research question to consider.