Risk-Averse Co-Decision for Lower-Carbon Product Family Configuration and Resilient Supplier Selection

Abstract

Due to global pandemics, political unrest and natural disasters, the stability of the supply chain is facing the challenge of more uncertain events. Although many scholars have conducted research on improving the resilience of the supply chain, the research on integrating product family configuration and supplier selection (PCSS) under disruption risks is limited. In this paper, the centralized supply chain network, which contains only one major manufacturer and several suppliers, is considered, and one resilience strategy (i.e., the fortified supplier) is used to enhance the resilience level of the selected supply base. Then, an improved stochastic bi-objective mixed integer programming model is proposed to support co-decision for PCSS under disruption risks. Furthermore, considering the above risk-neutral model as a benchmark, a risk-averse mixed integer program with Conditional Value-at-Risk (CVaR) is formulated to achieve maximum potential worst-case profit and minimum expected total greenhouse gases (GHG) emissions. Then, NSGA-II is applied to solve the proposed stochastic bi-objective mixed integer programming model. Taking the electronic dictionary as a case study, the risk-neutral solutions and risk-averse solutions that optimize, respectively, average and worst-case objectives of co-decision are also compared under two different ranges of disruption probability. The sensitivity analysis on the confidence level indicates that fortifying suppliers and controlling market share in co-decision for PCSS can effectively reduce the risk of low-profit/high-cost while minimizing the expected GHG emissions. Meanwhile, the effects of low-probability risk are more likely to be ignored in the risk-neutral solution, and it is necessary to adopt a risk-averse solution to reduce potential worst-case losses.

1. Introduction

Facing global economic volatility and growing customer demands under mass customization, quickly responding to customer needs and dealing with supply chain uncertainties become the two most important decision-making aspects [1]. Therefore, the integrating product family and supply chains (PFSC) are receiving extensive attention and research [2]. However, as the epidemic spreads, the climate changes, and unilateralism prevails, the supply chain is more easily disrupted, resulting in material shortage, production disruption, delivery delays, and even severe economic losses. It is becoming a challenge to address disruption risks and bring sustainability to the supply chain [3]. Although supply risk has been mentioned in previous PFSC research [4], in co-decision for PCSS considering disruption risks, the research on improving the resilience of the supply chain and measuring the worst-case losses is limited.

Generally, the resiliency can be defined as the ability to respond to disruption caused by unexcepted events, and supplier resilience reflects the supplier’s ability to manage risks and quick recovery capacity. Because the resiliency can be explored and analyzed by paying attention to the strength of the supply chain and its recovery speed, and resilient practices in critical conditions contribute to sustainability of the supply chain [5]. Therefore, selecting the resilience supplier and providing suppliers with a reliable level of resilience is essential to protect the organization against defects and disruption [6]. Despite the growing concerns about resilient supplier selection and resilient strategies, few scholars have introduced the resilience features in co-decision for PCSS. In addition, continued environmental degradation and frequent extreme climates force organizations to increase their attention to green [7]. Since the scheme of component sharing among product variants also influences the GHG emissions of the product family [8], the GHG emissions is also explored in co-decision for PCSS.

Furthermore, the attitude of decision-makers in most of the literature on co-decision for PCSS is supposed to be risk-neutral. However, in a global stochastic supply setting, the increasing uncertainties and risks of supply chains bring challenges to the feasibility and reliability of decision-making schemes, which results in the assumption that risk neutrality is inadequate for contemporary supply chain management [9,10]. Although robust optimization can be used to represent the uncertainty by introducing an uncertainty set addressing the data ambiguity [11], the robust optimization is criticized because it always generates an overly conservative solution in the worst-case scenario [12,13]. The risk-averse objectives have higher reliability in uncertain environments than traditional risk-neutral models and are also less conservative than robust optimization [14,15]. In conclusion, it is necessary to adopt a risk-averse policy for fewer worst-case scenarios in co-decision of PCSS [16].

Overall, considering disruption risks, we developed an improved bi-objective stochastic mixed integer programming model to support co-decision for low-carbon product family configuration and resilient supplier selection. The major contribution of this paper is that (i) the low-carbon performance of the product family and the resilience of the supply chain are considered while taking into account the profit; (ii) the fortified supplier is taken to improve supply chain resiliency; (iii) by introducing two popular financial engineering percentile measures of risk, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), a risk-averse model is established to measure the potential worst-case losses caused by supply disruptions; (iv) the sensitivity analysis on the confidence level indicates that fortifying suppliers and controlling market share in co-decision for PCSS can effectively reduce the risk of low-profit/high-cost while minimizing the expected GHG emissions. Meanwhile, the effects of low-probability risk are more likely to be ignored in the risk-neutral solution, and it is necessary to adopt a risk-averse solution to reduce potential worst-case losses.

The rest of this paper is organized as follows: Section 2 positions this paper in the literature and highlights the gap that is intended to be filled by in this research. Section 3 describes the details of the problem considered in this paper; moreover, the improved bi-objective stochastic mixed integer programs are proposed for risk-neutral and risk-averse solutions of PCSS. The solution procedure is presented in Section 4. Section 5 illustrates a case illustration of an electronic dictionary along with their optimization results. Finally, Section 6 draws some conclusions from this study.

2. Related Work

Based on the research content of this paper, a review of the related literature is presented below in three distinct sections: Product family configuration and supplier selection (PCSS), supply risks and resiliency, and risk preference and measurement.

2.1. Product Family Configuration and Supplier Selection

At present, the popular practice treating co-decision for PCSS has effectively aggregated these two different problems as either one single- or multi-objective optimization model. Considering consumer purchase behavior and supplier availability, Luo et al. [17] proposed a mixed-integer nonlinear programming model that integrates product family configuration into supplier selection. By introducing multinomial logit (MNL) choice rule, Cao et al. [18] extended the product family model integrating supplier selection decision and compared the two consumer choice rules by sensitivity analysis. However, only profit of the product family is reckoned as the optimization objective in their research, which is not suitable for the multi-criteria attributes of the product family design optimization or supplier selection. Generally, the disruption risk and risk management of supply chains have a considerable influence on the decision-making process and should be further considered.

Considering customer demands and technological progress, Liu et al. [19] studied the co-evolution of product family configuration and supplier selection (PCSS) across product generations from a system perspective and revealed the inherent distributed collaborative decision-making in the co-evolution process of PCSS. By mapping and co-development with the product platform, Olivares-Aguila and ElMaraghy [20] proposed a novel methodology for the co-development of the product and supplier platform and illustrated that strategically defined selection criteria and choosing decision-makers is impactive for defining the platform and non-platform suppliers. Besides, considering profit and GHG emissions, Wang, Tang, Yin, Ullah, Salido, and Giret [6] proposed a simultaneous optimization for low-carbon product family configuration and supplier selection and found that co-decision for PCSS can reduce GHG emission and increase profit. However, disruption risks and risk management are not considered in their research. Although the disruption risks and multiple sourcing based on price discount are introduced into integrated optimization for PCSS by Luo, Li, Kwong, and Cao [3], nevertheless, one of the assumptions in their paper is that orders from failed suppliers can be allocated by other selected suppliers at a certain replenishment price. The fact is that the key parts are not always available from other suppliers in practice. Assuming suppliers are capacity-constrained, Liu and Li [21] proposed a joint decision-making model that integrates supplier selection and order allocation into product family configuration (PCOA) in which risk mitigation strategies (e.g., protected supplier and multi-source procurement) are adopted to reduce the impact of disruption risks. Nevertheless, there is no further analysis from the perspective of risk aversion and GHG emissions.

2.2. Supply Risks and Resiliency

Generally, the risks that exist in the supply chain can be divided into operational risks and disruption risks [22]. Operational risks refer to inherent uncertainties (i.e., customer demand and cost rate uncertainty, and also supply uncertainty due to operational difficulties) in supply chains. Disruption risks refer to the major disruptions caused by unexpected events (i.e., earthquakes, floods, terrorist attacks or employee strikes) from natural, man-made or technological threats. Generally, operational risks are caused by normal events with medium to high likelihood and low impact, which have only short-term negative impact, while disruption risks are caused by disruptive events with low likelihood and high impact, which may have short- or long-term negative impact. Therefore, the rest of this section will focus on how to mitigate the effects of disruption risks.

To mitigate the effects of disruption risks, several practical strategies have been adopted by some scholars. (i) Dual/multiple sourcing. By comparing the expected profit between the single sourcing and dual sourcing, Yu et al. [23] studied the sourcing decision alternatives in the context that the demand is price-sensitive, and the market scale increases when a supply disruption occurs. Finally, a numerical example was used to prove that single source or dual source depends on the probability of failure. (ii) Back-up supplier. Supposing a buyer has two candidate suppliers, one of which is a cheaper but unreliable supplier, while the other is reliable but more expensive, Hou et al. [24] studied co-decision for supplier selection and order allocation (SSOA) with a buy-back contact between a buyer and its backup supplier when the main supplier is disrupted. In addition, they also discussed the common nature of contracts and the differences under demand uncertainty and the recurrent supply uncertainty of major suppliers. (iii) Fortified suppliers. Assuming that the remaining capacity of a fortified supplier is unchanged after any disruptive event, Sawik [25] adopted fortified supplier strategy to address the SSOA problem under disruption risks. However, this assumption may not be realistic. For example, fortifying a supplier against flood risk may not protect it against other disruptive events such as earthquakes. In an effort to be more realistic, Torabi et al. [26] extended this assumption so that a fortified supplier has multi-level of remaining capacity based on the level of protection and the type of disruption event. This extended assumption is adopted in this paper. (iv) Pre-positioning inventory. Esmaeili-Najafabadi et al. [27] applied pre-positioning inventory strategy in a simultaneous optimization model for SSOA under disruption risks for reducing the impact of disruption and found that the factors which impact the SSOA-related decisions change with the increase of the disruption probability. To sum up, although the risk mitigation strategy has been well applied in the SSOA field, few scholars have considered the risk mitigation strategy to solve the PCSS problem.

2.3. Risk Preference and Measurement

Risk preference reflects the attitude of decision-makers towards risks and can be divided into the following three categories. (1) Risk-neutral. Decision-makers with this attitude are neither aggressive nor conservative in an uncertain environment and believe that the benefit of the decisions they make are the same as those under risk-free situations. Generally, in the risk-neutral model related to the supply chain, the expected benefit or expected cost is used as the optimization goal [28]. (2) Risk-seeking. Decision-makers with this attitude tend to pursue risks in order to achieve greater benefit with low probability [29]. (3) Risk-averse. Decision-makers with this attitude are seeking reliable solutions when faced with uncertain risks. Risk-averse decision-making is the most commonly used strategy for risk management and control and has been applied in supply chain design [30].

Furthermore, due to the uncertainty of the decision-makers’ subjective conscious-ness, several risk preference measurement methods have been introduced to accurately measure the preferences of the decision-makers in some of the literature. (1) Mean-variance (MV). Wang et al. [31] investigated the robust equilibrium control-measure policy for an ambiguity-averse and risk-averse fund manager under the mean-variance (MV) criterion. (2) Mean absolute deviation. Using the average absolute deviation as a risk measure, Deva and Bogataj [32] established a mathematical model to determine the optimal portfolio of securities. (3) Conditional Value-at-Risk (CVaR). CVaR is used in conjunction with Value-at-Risk (VaR). By introducing CVaR and the confidence degree theory, Tan et al. [33] proposed a risk aversion scheduling model with the minimum CVaR objective considering the maximum operation revenue. (4) Fuzzy chance constrained. Khishtandar [34] proposed a fuzzy chance-constrained programming model to include uncertainty in the biogas supply chain design problem and studied impacts of uncertain parameters on the overall cost of the biogas supply chain through sensitivity analysis.

In addition, the risk preference of decision-makers in an uncertain environment has an important impact on the feasibility and reliability of decision-making schemes [35]. In order to enhance the feasibility and reliability of supply chains, the CVaR-based risk-averse approach has attracted the attention of some scholars. Taking the CVaR criterion to model risk aversion of the manufacturer, Huang, He and Lei [8] investigated coordination and risk-sharing issues of the supply chain consisting of a dominant retailer and a risk-averse manufacturer. Considering the uncertainty and risk aversion in the sustainable supply chain network, Rahimi et al. [36] established a CVaR-based risk-averse sustainable multi-objective mathematical model to optimize the supply chain network and discussed the impact of some important risk-aversion parameters on the pareto solution through sensitivity analyses. By introducing two different service level measures (i.e., the expected worst-case demand fulfillment rate and the expected worst-case order fulfillment rate), Sawik [37] studied a combinatorial stochastic optimization problem with conditional service-at-risk as a worst-case service level measure. The results of numerical examples indicate that the worst-case order fulfillment rate has a higher service performance than the worst-case demand fulfillment rate. While maximization of the expected worst-case fraction of fulfilled customer orders better mitigates the impact of disruption risks. In addition, other scholars have also conducted valuable research on CVaR-based risk-averse supply chains design [38,39,40]. To the best of our knowledge, however, there is a critical lack of research that apply the CVaR-based risk-averse approach in co-decision for PCSS considering disruption risks

3. Problem Descriptions and Model Development

3.1. Problem Descriptions

The optimization problem of this study is described as follows: A modular product platform architecture containing a series of function modules has been developed. By selecting one and only one module instance from each function module, various product variants can be configured to quickly respond the requirement of different customer groups. In our problem setting, it was supposed that module instances are provided by external suppliers and that the main manufacturer assembles the product. Normally, each supplier can provide several types of module instances and its capacity can meet the demand of manufacturer. However, under disruption risks, capacity constraints exist for each supplier. In order to encourage business, different suppliers offer different fortified levels to improve worst-case supply volume. In this study, therefore, to account for supply risks, scenario-based modeling and Conditional Value-at-Risk (CVaR) are used to include a number of discrete scenarios and measure worst-case losses, respectively. The research problem of this study was to decide which module instances to order for product variants and from which suppliers, and which fortified level to adopt for maximizing the potential worst-case profit and minimizing the expected GHG emissions of the product family.

In order to simplify the model and better reflect the problem, the following assumptions are used in this research. (1) The module instances, produced by different suppliers in the same module, have the same internal interface and, thus, are fully replaceable. (2) Each product variant can be configured by selecting one and only one module instance from each basic module. (3) All candidate suppliers are qualified through the early process of evaluation and sifting. (4) Each supplier is independent of each other and has an independent and stochastic disruption event. (5) Without supply disruption, each supplier can provide the entire order volume to manufacturer. With supply disruption, each supplier can only provide the manufacturer with the order volume corresponding to the fortified level.

Achievements of this paper can be used in any co-decision for PCSS; the main limiting feature of this problem is that there is only one main manufacturer and several suppliers, and all module instances are provided by the supplier. There are many renowned international manufacturers including only one main manufacturer. For instance, Zara, as a leading clothing and accessories retailer, still depends on the main manufacturing plant in Zaragoza, Spain. In the case where the manufacturer provides some module instances, the manufacturer can be regarded as a special supplier.

3.2. Symbols

To formulate the optimization model, all indexes, parameters and variables in-volved in this paper are presented as shown below, in which lowercases, uppercases and italics represent different meanings.

1.

Indices

• $i$: Index of market segment, $i=\left\{1,\dots ,I\right\}$
• $j$: Index of variant, $j=\left\{1,\dots ,J\right\}$
• $k$: Index of module, $k=\left\{1,\dots ,K\right\}$
• $l$: Index of module instance, $l=\left\{1,\dots ,L_k\right\}$
• $v$: Index of supplier, $v=\left\{1,\dots ,V\right\}$
• $s$: Index of scenario, $s=\left\{1,\dots ,S\right\}$
• $t$: Index of fortified level, t$=\left\{0,\dots ,T_{klv}\right\}$

2.

Abbreviations

• ${\mathrm{M}}_{kl}$: Module instance l of module k
• ${\mathrm{M}}_{klv}$: Module instance l of module k supplied by the supplier v

3.

Parameters

• $u_{ikl}$: Part-worth utility of module instance ${\mathrm{M}}_{kl}$ in market segment i
• ${\eta }_j$: Constant related to the derivation of the utility from part utilities of variant j
• $d_i$: Demand for products in market segment i
• $P_j$: Selling price of variant j
• $N^e,N^c$: Number of products from competitive companies and number of products that have been launched in the market by this company, respectively
• ${\lambda }_{ij},{\lambda }_{ij}^e,{\lambda }_{ij}^c$: Utility surplus of the new, existing and competitive variant j in segment i
• $c_J^{in\left(fix\right)}$: Fixed cost part of the intra-manufacturer for a product family which has J product variants
• $c_{kl}^{in\left(var\right)}$: Variable unit cost of intra-manufacturer for module instance ${\mathrm{M}}_{kl}$
• $c_{klv}^{out\left(fix\right)}$: Outsourcing-related fixed cost paid to the supplier v for module instance ${\mathrm{M}}_{kl}$
• $c_{klv}^{pur}$: Unit purchase cost (including transported cost) for module instance ${\mathrm{M}}_{klv}$
• $c_j^{sho}$: Unit shortage cost of product variant j due to disruption
• ${\delta }_{klvt}$: Percentage of unit fortified cost for module instance ${\mathrm{M}}_{klv}$ fortified to level t to its unit purchase cost
• ${\alpha }_{klvt}$: Percentage of the supply quantity of module instance ${\mathrm{M}}_{klv}$ corresponding to fortified level t to its order quantity while supplier v fails
• $g_J^{fix}$: Fixed GHG emission of intra-manufacturer for a product family which has J product variants
• $g_{kl}^{var}$: Variable unit GHG emission of intra-manufacturer for module instance ${\mathrm{M}}_{kl}$
• $r_{vs}$: Binary variable to indicate whether the supplier v to disruption occurs in scenario s
• $VaR$: The targeted profit based on the $\left(1-\zeta \right)$-percentile of total profit, i.e., in $100\left(1-\zeta \right)\%$ of scenarios, the outcome does not exceed $VaR$ (Value-at-risk of profit)

4.

Decision variables

• $x_{jkl}$: Binary decision variable to indicate whether module instance ${\mathrm{M}}_{kl}$ has been selected in variant j
• $y_{klv }$: Binary decision variable to indicate whether supplier v is preferred supplier for module instance ${\mathrm{M}}_{kl}$
• ${\gamma }_{klvt }$: Binary decision variable to indicate whether the module instance $M_{klv}$ is fortified to level t

3.3. Customer Preference and Demand Analysis

For a market where products from various companies compete, the choice probability of a customer can be estimated via the probabilistic choice rules (e.g., MNL or BTL), which assume that utility is a random variable, and the choice process follows random utility maximization criterion [41]. According to the MNL rule, the most widely applied probabilistic choice rule, the choice probability towards a product variant can be formulated as follows:

$${\rho }_{ij}=\frac{e^{\mu {\lambda }_{ij}}}{{\sum }_{j=1}^J{e}^{\mu {\lambda }_{ij}}+{\sum }_{j=1}^{N^e}{e}^{\mu {\lambda }_{ij}^e}+{\sum }_{j=1}^{N^c}{e}^{\mu {\lambda }_{ij}^c}}$$

(1)

where μ is the scaling parameter in the MNL and can be calibrated by the survey data of consumer choice [42]. The perceived utility $U_{ij}$ of a product variant and the utility surplus ${\lambda }_{ij}$ can be calculated by Equations (2) and (3), respectively:

$$U_{ij}={\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{u}_{ikl}{x}_{jkl}+{\eta }_j$$

(2)

$${\lambda }_{ij}=U_{ij}-P_j$$

(3)

Then, according to the above customer preference towards different product variants, the total demand (i.e., the market share) for product variant j can be formulated as shown in Equation (4). Furthermore, based on the second assumption, the demand for module instance ${\mathrm{M}}_{kl}$ in variant j is formulated as shown in Equation (5). The demand for module instance ${\mathrm{M}}_{kl}$ in all product variants can be calculated by Equation (6):

$$d_j={\displaystyle \sum }_{i=1}^I{\rho }_{ij}{d}_i$$

(4)

$$d_{jkl}=x_{jkl}{d}_j$$

(5)

$$d_{kl}={\displaystyle \sum }_{j=1}^J{d}_{jkl}$$

(6)

3.4. Disruption and Delivery

Random disruption of the material supply adversely impacts the flow of material in a supply chain network. The failure of a supplier implies that all module instances ordered from that supplier for the manufacturer cannot be produced at full capacity for the lead time. Because each supplier is either ‘on’ or ‘off’, the total number of different possible scenarios, S, is equal to $2^n$, where n is equal to the number of selected suppliers. The probability, ${\beta }_s$, of a scenario, s, occurring can be given by Equation (7) according to the literature [43] where V_sel is the set of selected suppliers:

$${\beta }_s={\displaystyle \prod }_{v\in V_{sel}}\max \left\{\left(1-{\theta }_v\right)\left(1-r_{vs}\right), {\theta }_v{r}_{vs}\right\}$$

(7)

Generally, the manufacturer does not need to pay for ordered and undelivered parts. However, it is charged with a much higher cost of unfulfilled customer orders due to supply disruptions. In order to mitigate the potential worst-case losses due to disruption, some resilience strategies can be used to improve the resilience of supply chains by managers. In this research, the fortified supplier, one of the resilience strategies, is taken into account. Additionally, suppliers can be fortified at different levels, each of which has its own the fortified cost percentage and supply volume percentage.

$$Q_{sklv}=\left(1-r_{vs}\right)d_{kl}{y}_{klv}+r_{vs}{\displaystyle \sum }_{t=1}^{T_v}{\alpha }_{klvt}{d}_{kl}{y}_{klv}{\gamma }_{klvt }$$

(8)

$$Q_{skl}={\displaystyle \sum }_{v=1}^V{Q}_{sklv}$$

(9)

$$Q_{sj}=\underset{k\in K; l\in L_k; x_{jkl}=1}{\min }\left\{\frac{d_{jkl}}{d_{kl}}{Q}_{skl}\right\}$$

(10)

According to forth assumption, the supply quantity of the module instance M_klv in scenario s can be calculated by Equation (8). Then the total delivered quantity of module instance M_kl in scenario s is formulated as shown in Equation (9). Theoretically, the smallest delivered quantity among all selected module instance (i.e., the delivery bottleneck) determines the final quantity of product variants, so the quantity of product variants in scenario s is formulated as shown in Equation (10).

3.5. Risk-Neutral Model

3.5.1. Modeling Product Family Cost

Generally, the cost model of the product family includes two parts: intra-manufacturer cost and outsourcing-related cost. Each part can be divided into the fixed costs and the variable costs, which has been applied in much of the literature [44]. The fixed cost of intra-manufacturer mainly includes the cost of product development, management and so on. It is related to the number of product variant types, and can be expressed by Equation (11) in scenario s. The variable cost of intra-manufacturer mainly refers to the assembly cost and packaging cost and can be modeled by Equation (12) in scenario s according to linear-additive cost models [45]:

$$C_s^{in\left(fix\right)}=c_J^{in\left(fix\right)}$$

(11)

$$C_s^{in\left(var\right)}={\displaystyle \sum }_{j=1}^J{\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{c}_{kl}^{in\left(var\right)}{Q}_{sj}{x}_{jkl}$$

(12)

The outsourcing-related fixed cost is associated with supplier selection, such as the cost used for negotiation communication and contract-signing. It can be calculated by Equation (13) in scenario s. The outsourcing-related variable costs in this research are the aggregation result of the fortified cost, the purchase cost and the shortage cost due to disruption. The fortified cost based on increasing block tariffs is expressed by Equation (14) in scenario s, and the purchase cost and the shortage cost can be formulated as shown in Equations (15) and (16) in scenario s, respectively:

$$C_s^{out\left(fix\right)}={\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{\displaystyle \sum }_{v=1}^V{c}_{klv}^{out\left(fix\right)}{y}_{klv}$$

(13)

$$C_s^{for}={\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{\displaystyle \sum }_{v=1}^V{\displaystyle \sum }_{t=1}^{T_{klv}}{\displaystyle \sum }_{t^{\prime }=1}^t{c}_{klv}^{pur}{\delta }_{klv{t}^{\prime }}\left({\alpha }_{klv{t}^{\prime }}-{\alpha }_{klv\left(t^{\prime }-1\right)}\right)d_{kl}{y}_{klv}{\gamma }_{klvt }$$

(14)

$$C_s^{pur}={\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{\displaystyle \sum }_{v=1}^V{c}_{klv}^{pur}{Q}_{sklv}$$

(15)

$$C_s^{sho}={\displaystyle \sum }_{j=1}^J{c}_j^{sho}\left(d_j-Q_{sj}\right)$$

(16)

To summarize, the total cost of all product variants is equal to the sum of the fixed cost of the intra-manufacturer, the variable cost of the intra-manufacturer, the outsourcing-related fixed cost and the outsourcing-related variable cost. It can be calculated by Equation (17) in scenario s. The total profit in scenario s can be formulated as shown in Equation (18):

$$T{C}_s=C_s^{in\left(fix\right)}+C_s^{in\left(var\right)}+C_s^{out\left(fix\right)}+C_s^{pur}+C_s^{for}+C_s^{sho}$$

(17)

$$T{P}_s={\displaystyle \sum }_{j=1}^J{Q}_{sj}{P}_j-T{C}_s$$

(18)

3.5.2. GHG Emission Model of Product Family

Since CO₂ emissions are used as a proxy for air pollution [46], and are the main body of GHG, the CO₂ emissions are used to measure the GHG emissions in this paper. Similar to the cost model, the total GHG emission model of the product family also includes two parts: intra GHG emission and outsourcing-related GHG emission. Each part can be divided into fixed GHG emission and variable GHG emission. The fixed GHG emission of the intra-manufacturer comes from the product development process and is related to the number of product variant types. The variable GHG emission of the intra-manufacturer is mainly generated from the product assembly and so on. Then, the outsourcing-related fixed GHG emission reflects the carbon emissions of the supplier selection process such as investigations and negotiation communication. The outsourcing-related variable GHG emission is mainly generated by product transportation and storage. Based on the research [6], in summary, the above-mentioned carbon emissions in scenario s can be modeled by Equations (19)–(22), respectively. Finally, the total GHG emissions model of the product family in scenario s is equal to the sum of all the above carbon emissions as shown in Equation (23).

$$G_s^{in\left(fix\right)}=g_J^{in\left(fix\right)}$$

(19)

$$G_s^{in\left(var\right)}={\displaystyle \sum }_{j=1}^J{\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{g}_{kl}^{in\left(var\right)}{Q}_{sj}{x}_{jkl}$$

(20)

$$G_s^{out\left(fix\right)}={\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{\displaystyle \sum }_{v=1}^V{g}_v^{out\left(fix\right)}{y}_{klv}$$

(21)

$$G_s^{out\left(var\right)}={\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}{\displaystyle \sum }_{v=1}^V{g}_{klv}^{out\left(var\right)}{Q}_{sklv}$$

(22)

$$T{G}_s=G_s^{in\left(fix\right)}+G_s^{in\left(var\right)}+G_s^{out\left(fix\right)}+G_s^{out\left(var\right)}$$

(23)

3.5.3. Optimization Model

To summarize, the optimization model of PCSS involved in this paper can be formulated as shown in Equations (24)–(34). The first objective is to maximize the total profit of the product family as shown in Equation (24). The second objective is to minimize the total GHG emissions of the product family as shown in Equation (25).

Equations (26)–(28) define the value of the binary decision variable x_jkl and enable only one module instance to be selected from each basic module for product variant j, and all configured product variants are different from each other.

Equations (29)–(31) define the value of the binary decision variable y_klv and restricts any supplier from being selected for the unselected module instance, while only one supplier is selected for a selected module instance.

Equations (32)–(33) define the value of the binary decision variable γ_klvt and make only one level of the piecewise fortified levels effective for module instance M_klv.

$$\mathrm{Max}P\left(x_{jkl}, y_{klv },{\gamma }_{vt } \right)={\sum }_{s=1}^{S}T{P}_s{\beta }_s$$

(24)

$$\mathrm{Min}TG\left(x_{jkl}, y_{klv },{\gamma }_{vt }\right)={\sum }_{s=1}^{S}T{G}_s{\beta }_s$$

(25)

subject to

$$x_{jkl}\in \left\{0,1\right\}$$

(26)

$${\displaystyle \sum }_{l=1}^{L_k}{x}_{jkl}=1$$

(27)

$${\displaystyle \sum }_{k=1}^K{\displaystyle \sum }_{l=1}^{L_k}\left|x_{jkl}-x_{j^{\prime }kl}\right|>0, j\ne j^{\prime }$$

(28)

$$y_{klv}\in \left\{0,1\right\}$$

(29)

$$y_{klv} \le {\displaystyle \sum }_{j=1}^J{x}_{jkl}$$

(30)

$$x_{jkl}\le {\displaystyle \sum }_{v=1}^V{y}_{klv}\le 1$$

(31)

$${\gamma }_{klvt }\in \left\{0,1\right\}$$

(32)

$${\displaystyle \sum }_{t=1}^{T_v}{\gamma }_{klvt }=1$$

(33)

$$\mathrm{Equations}\left(1\right)–\left(23\right)$$

(34)

3.6. Risk-Averse Model

In the risk-averse co-decision model of PCSS under disruption risks, the decision-makers need to decide on the confidence level ζ to control the risk of low-profit/high-cost due to supply disruption. It is assumed that the decision-makers are only willing to accept a PCSS scheme where the total probability of scenarios with the total profit less than the Value-at-Risk is not greater than (1 − ζ)%. The greater the confidence level ζ, the more risk aversive are decision-makers and the smaller the percentage of low-profit/high-cost. Moreover, because the Conditional Value-at-Risk has been a plausible coherent risk measure in representing the risk preference of a decision-maker, as well as mitigating losses under the risk of uncertainty [47], the Conditional Value-at-Risk is proposed to accommodate the issue. Based on the paper of Rockafellar and Uryasev [48], the risk-averse model of co-decision for PCSS, which aims to reduce the risk of low-profit/high-cost while minimizing the expected carbon emissions, is formulated as shown in Equations (35)–(38).

The first objective is to maximize the potential worst-case profit below VaR by maximizing CVaR as shown in Equation (35). The second objective is to minimize the expected total GHG emissions (TG) as shown in Equation (36). By measuring CVaR, the magnitude of the tail profit is considered to achieve a more accurate estimate of the risks. The tail profit $T{P}_s^{\prime }$ for scenario s is defined as the non-negative amount of profit less than VaR in scenario s as shown in Equation (37).

$$\mathrm{Max}CVaR=VaR-{\left(1-\zeta \right)}^{-1}{\sum }_{s=1}^{S}T{P}_s^{\prime }{\beta }_s$$

(35)

$$\mathrm{Min}TG\left(x_{jkl}, y_{klv },{\gamma }_{vt }\right)={\sum }_{s=1}^{S}T{G}_s{\beta }_s$$

(36)

subject to

$$T{P}_s^{\prime }=\max \left\{0,VaR-T{P}_s\right\}$$

(37)

Equations (1) ~ (23); Equations (26) ~ (33)

(38)

4. Algorithm Design

Since the non-dominated sorting genetic algorithm with the elite strategy (NSGA-II) has been proven to have good performance in multi-objective optimization problems, it is adopted to solve the established bi-objective model in this paper.

The main process of the NSGA-II has been illustrated in the literature and can be briefly described as follows. (1) The initial population is randomly generated by designing a chromosome coding strategy. (2) The offspring population is attained by the genetic operators (i.e., selection, crossover and mutation). (3) The intermediate population is generated by merging the offspring population and the parent population, and ranked based on their non-dominated levels, crowding distances of the individuals. (4) The parent population is filled by the intermediate population ordered by ranks from high to low.

In the above process, chromosome coding is an important foundation of NSGA-II, which aims to express a series of problems as a series of strings [21]. Figure 1 shows an example of the chromosome structure. It can be found that a chromosome structure is composed of three sections: product variant section, supplier selection section and fortified supplier section. For the product variant section, a product variant includes three basic modules, and the number of module instance for the basic modules are 3, 2, 2, respectively. The product variant can be configurated by selecting one module instance from each basic module. The gene fragment information for product variant 1 is compiled as: variant 1 = {M₁₃, M₂₁, M₃₂}, and the value 9 in price 1 represents that the 9th discrete price is selected for product variant 1.

To construct the fixed-length chromosome, suppliers were selected for each module instance in the supplier selection section. The selected supplier for each module instance was also given a fortified level as shown in the fortified supplier section. For the above two gene fragments, therefore, the gene length is equal to 7, i.e., the total number of module instances. In the supplier selection section, the selected supplier number for module instance 1 to 7 are {5, 0, 3, 1, 0, 0, 3}, where 0 means no supplier is selected. For the fortified supplier selection, the decoding is exemplified as: the supplier 1 for module instance M₂₁ are fortified to level 2, and the supplier 3 for module instance M₁₃ are fortified to level 1.

In addition, the roulette selection, the single-point crossover operator and the random mutation method are adopted for selection, crossover and mutation, respectively. Since the above genetic operators and the non-dominated sorting strategy have been widely used, please refer to the literature of Deb et al. [49] for more details.

5. Case Study

5.1. Case Description

In this section, a case illustration of an electronic dictionary’s product family architecture is conducted to evaluate the performance of the proposed bi-objective optimization model for low-carbon product family configuration and resilient supplier selection under disruption risks. The experiments are also intended to compare the risk-neutral solutions and risk-averse solutions under two different ranges of disruption probability. This case accounts for 12 suppliers and 6 basic modules, including product case module (M₁), storage module (M₂), voice module (M₃), key module (M₄), control module (M₅) and display module (M₆). The number of the module instance for the above modules are 4, 4, 3, 3, 3 and 3, respectively. Through market research and expert analysis, it was finally decided to develop two types of product variants, namely J = 2.

As shown in

Table 1. The related parameters in proposed model.

Parameter	Value	Parameter	Value
$d_i,u_{ikl},{\lambda }_{ij}^e,{\lambda }_{ij}^c,P_j,c_{kl}^{in\left(var\right)}$	From literature [8]	$c_j^{sho}$	$0.3\ast P_j$
$c_{vkl}^{pur},g_{kl}^{in\left(var\right)},g_{vkl}^{out\left(var\right)},distance$	From literature [8]	$g_J^{in\left(fix\right)}$	1500
$c_J^{in\left(fix\right)}$	400,000	$g_v^{out\left(fix\right)}$	$0.066\ast distance$
$c_{vkl}^{out\left(fix\right)}$	$5000\ast c_{vkl}^{pur}$	${\theta }_v$	$\mathrm{U}\left[0.15;0.20\right]$

, the data used in this paper are generated from or based on the literature [6]. Then, it was assumed that all module instances from the same supplier have the same fortified cost percentage and supply volume percentage in the same fortified level. The information about the fortified cost percentage and supply percentage under disruption are shown in

Table 2. The information about the fortified cost percentage and supply volume percentage under disruption.

Supplier	The Fortified Cost Percentage${\mathit{\alpha }}_{\mathit{k}\mathit{l}\mathit{v}\mathit{t}}$ [The Supply Volume Percentage ${\mathit{\delta }}_{\mathit{k}\mathit{l}\mathit{v}\mathit{t}}]$					Disruption Probability
	Level 0	Level 1	Level 2	Level 3	Level 4
1	40 [0]	50 [12]	60 [24]	70 [36]	/	0.17
2	50 [0]	60 [16]	70 [32]	/	/	0.20
3	50 [0]	70 [28]	/	/	/	0.16
4	30 [0]	40 [7.2]	50 [14.4]	60 [21.6]	70 [28.8]	0.17
5	40 [0]	50 [10.4]	60 [20.8]	70 [31.2]	80 [41.6]	0.16
6	50 [0]	60 [15.2]	70 [30.4]	80 [45.6]	/	0.19
7	40 [0]	60 [16]	80 [32]	/	/	0.18
8	30 [0]	50 [14.4]	70 [28.8]	90 [43.2]	/	0.18
9	50 [0]	70 [21.6]	90 [43.2]	/	/	0.20
10	40 [0]	50 [13.6]	60 [27.2]	80 [40.8]	/	0.15
11	30 [0]	60 [20]	90 [40]	/	/	0.16
12	30 [0]	45 [11.2]	60 [22.4]	75 [33.6]	90 [44.8]	0.20

. The disruption probability θ_v are drawn from uniform distribution [0.15, 0.20], i.e., drawn from U[0.15; 0.20], as shown in last column of Table 2.

5.2. Results and Analysis

The NSGA-II algorithm described in Section 4 was programmed in the MATLAB 2021a platform. The relevant control parameters of the NSGA-II were set as follows: the population size is 1000; the number of iterations is 500; the crossover rate and the mutation ratio are 0.90 and 0.40, respectively.

5.2.1. The Convergence of NSGA-II

To assess the convergence of NSGA-II, the non-dominated solutions from the 50th to the 500th generation in the steps of 50 generations were extracted. The experiment results are shown in Figure 2. It can be seen from Figure 2 that the non-dominated solutions of the algorithm gradually get closer to the Pareto front with the increase in the iteration number and the non-dominated solutions approximately reach the Pareto front solutions when the iterations are approximately 250. Moreover, it also can be found that there is an approximate linear correlation between the expected total profit (TP) and the expected total GHG emissions (TG), and the maximum TP is 5.351 × 10³ $ with TG 1.496 × 10⁵ kg.

The managerial insight from the experiment is as follows: the NSGA-II has good convergence and uniformity, and the obtained solutions can gradually move to the Pareto front solutions of the optimization problem.

To explore the influence of the value of confidence levels ζ on the solutions, the confidence level ζ is set at five levels of 0.01, 0.25, 0.50, 0.75 and 0.99, which means that the objective function maximizes the lowest 99%, 75%, 50%, 25% and 1% of all scenario outcomes, i.e., the expected worst-case profit. The Pareto front solution under different confidence levels ζ are shown in Figure 3. It can be seen that there is an approximate linear correlation between CVaR and TG. As the confidence level increases, CVaR gradually decreases, which means that the expected worst-case profit for the lowest (1 − ζ)% of all scenarios decreases as the confidence level increases. Moreover, by comparing the Pareto front solution of the risk-neutral model and risk-averse model, it can be found that the risk-neutral solution is similar to the risk-averse solution when the confidence level is low enough (e.g., when ζ = 0.01).

Therefore, the managerial insight from the experiment is that choosing an appropriate confidence level and obtaining at least 1.3 × 10⁵ kg of the GHG Emission quota will help maximize profits while reducing risk losses.

5.2.2. Influence of Disruption Probability on Results

With two different ranges of disruption probability, the risk-neutral solutions obtained when TP was maximum are additionally illustrated in

Table 3. Risk-neutral solutions obtained when TP is maximum.

Item		Module									Price ($)	Demand (×102)	TP (×103)	TG (×102)	0.99-VaR (×103)	0.99-CVaR (×103)
		M1		M2		M3	M4		M5	M6
θv ∈ U[0.15; 0.20]
Product configuration	Variant 1	M11		M21		M32	M42		M51	M61	74	366	5351	1496	−1438	−1438
	Variant 2		M14		M22	M32	M42		M51	M61	77	107
Supplier		8	8	2	2	8	2		8	2
(Fortified level)		(2)	(2)	(2)	(2)	(2)	(2)		(2)	(2)
θv ∈ U[0.01; 0.05]
Product configuration	Variant 1	M11		M22		M32	M42		M51	M62	74	263	9071	1566	−16182	−16182
	Variant 2	M11		M22		M32		M41	M51	M62	74	234
Supplier		8		11		8	11	11	8	8
(Fortified level)		(0)		(0)		(0)	(0)	(0)	(0)	(0)

. Comparing the corresponding TP for two different ranges of disruption probability demonstrates that for a higher disruption probability (for all suppliers with θ_v ∈ U [0.15; 0.20]), a smaller TP is achieved. In contrast, by comparing the corresponding 0.99-CVaR of the optimal risk-neutral solution under two different ranges of disruption probability, it was found that a smaller 0.99-CVaR is obtained for a lower disruption probability (for all suppliers with θ_v ∈ U[0.01; 0.05]), and this means that the expected worst-case profit for the lowest 1% of all scenarios is lower for a lower disruption probability.

Furthermore, as shown in Table 3, it was found that all selected suppliers with θ_v ∈ U [0.15; 0.20] were fortified to a certain level, while all selected suppliers with θ_v ∈ U [0.01; 0.05] were not fortified. These results indicate that low-probability risks are easily ignored in the risk-neutral model, so that serious losses are caused when risks occur. At the same time, the results also show that the fortified supplier is an effective resilient strategy in mitigating risk losses and increasing the potential worst-case profit.

In summary, some managerial insights can be summarized as follows. (i) Low-probability risks are easily ignored in the risk-neutral model; (ii) although the disruption probability may be extremely low, it is necessary to adopt appropriate strategies to mitigate the potential worst-case losses.

With different disruption probabilities, the risk-averse solutions at the maximum CVaR are additionally illustrated in

Table 4. Risk-averse solutions obtained when TP is maximum.

Item		Module										Price ($)	Demand (×102)	TP (×103)	TG (×102)	0.99-VaR (×103)	0.99-CVaR (×103)
		M1		M2	M3	M4		M5		M6
θv ∈ U[0.15; 0.20]
Product configuration	Variant 1	M14		M22	M31	M42		M51		M62		77	192	4438	1338	2191	2191
	Variant 2	M14		M22	M31		M41		M52	M62		77	226
Supplier		8		11	8	11	9	8	9	8
(Fortified level)		(3)		(2)	(3)	(2)	(2)	(3)	(2)	(3)
θv ∈ U[0.01; 0.05]
Product configuration	Variant 1	M12		M22	M31	M41		M52		M61		77	190	4608	1152	2140	2140
	Variant 2		M11	M22	M31	M41		M52			M62	77	162
Supplier		8	8	12	8	9		9		12	8
(Fortified level)		(3)	(3)	(4)	(3)	(2)		(2)		(4)	(3)

. Firstly, by comparing the risk-neutral solutions and the risk-averse solutions, respectively, shown in Table 3 and Table 4, it was found that for risk-neutral solutions, a higher TP was achieved. In contrast, for the risk-averse solutions, the 0.99-CVaR was higher, and the selected suppliers were fortified to a higher level. The results prove that the risk-averse model has higher reliability in uncertain environments than the risk-neutral model and it helps to mitigate the potential worst-case losses.

Moreover, comparing the corresponding risk-averse solutions for different ranges of disruption probability in Table 4, it can be noted that, although TG and TP for a lower disruption probability (for all suppliers with θ_v ∈ U[0.01; 0.05]) are better than that for higher disruption probability, 0.99-VaR and 0.99-CVaR are almost the same for different ranges of disruption probability. The above results demonstrate that the potential worst-case profit hardly changed as disruption probability θ_v varied when the confidence level ζ was large enough.

In conclusion, the managerial insight inspired by the above results are as follows. i) The risk-averse model helped to control the potential worst-case losses. ii) When the decision-makers were extremely risk-averse (e.g., when ζ = 0.99), the potential worst-case outcomes hardly change as disruption probability θ_v varied.

In addition, taking Table 3 and Table 4 together, although the selected module instances are fortified to different levels, it can be found that the worst-case supply volume percentage (i.e., supply volume percentage when the failure occurs) is the same for the selected module instances in each scheme, as shown in Figure 4. For example, for two schemes as shown in Table 3, the worst-case supply volume percentage of all selected module instances are 70% and 30%, respectively; for two schemes as shown in Table 4, the worst-case supply volume percentage of all selected module instances are both 90%. The reason for the above results is that the quantity of product variants is determined by the smallest delivered quantity among all selected module instances (i.e., the delivery bottleneck) in co-decision for PCSS, which is different from the supply chain design for the single component/module.

Therefore, the managerial insight derived from the above results is that in co-decision for PCSS, the worst-case supply volume percentage of all selected module instances should be as close as possible to reduce the holding volume for surplus module instances.

5.2.3. Influence of Confidence Level on the Risk-Averse Solutions

To further explore how the confidence level affects the risk-averse solutions, the values of related indicators obtained by the risk-averse model when obtaining the maximum CVaR are listed in

Table 5. The related indicators obtained by risk-averse model when obtaining the maximum CVaR.

Item	Confidence Level ζ
	0.01	0.25	0.50	0.75	0.99
θv ∈ U[0.15; 0.20]
Price [P1; P2]	[74; 77]	[77; 77]	[77; 77]	[77; 77]	[77; 77]
Market share [d1; d2] (×103)	[366; 107]	[185; 220]	[159; 247]	[196; 226]	192; 226
TP (×103)	5351	4329	4406	4438	4438
TG (×102)	1496	1268	1286	1338	1338
VaR (×103)	8579	3645	3368	2577	2191
CVaR (×103)	5351	3463	3257	2537	2191
θv ∈ U[0.01; 0.15]
Price [P1; P2]	[74; 74]	[77; 77]	[77; 77]	[77; 77]	[77; 77]
Market share [d1; d2] (×103)	[175; 319]	[176; 225]	[258; 147]	[127; 262]	[190; 162]
TP (×103)	9051	7062	5216	4912	4608
TG (×102)	1558	1258	1293	1242	1152
VaR (×103)	9923	5975	3317	2673	2140
CVaR (×103)	9051	5450	3303	2673	2140

under different confidence levels and different ranges of disruption probability. The above indicators plotted as the confidence level ζ varied are also shown in Figure 5 and Figure 6. Taking Figure 5 and Figure 6 together, the curves of TP drop quickly at first and then stabilize as the confidence level ζ increases. The turning points are, respectively, at ζ = 0.25 and ζ = 0.50 for two different ranges of disruption probability. The above results show that when the confidence level is greater than the turning point, the potential worst-case losses further decrease as the confidence level increase, while TP does not significantly decay. Similarly, it can be found from Table 5 that when the confidence level is equal to 0.25, the price of product variants for two different ranges of disruption probability both rise from 74 USD to 77 USD, and then remain constant.

For two different ranges of disruption probability, in addition, the trend of TG curves is the same as that of the total market share curves as shown in Figure 5 and Figure 6. Here, it is worth noting that the above curves each have two turning points. The first turning points are both at ζ = 0.25, and the second turning points are at ζ = 0.75 and ζ = 0.50, respectively. For any of the above curves, when the confidence level is less than the first turning point, the curve drops quickly with the increase in the confidence level; when the confidence level is between the two turning points, the curve rises slightly as the confidence level increases; when the confidence level is greater than the second turning point, the curve again drops with the increase in the confidence level.

To illustrate the above phenomenon, the costs and GHG emissions, which are obtained by the risk-neutral model and the risk-averse model when obtaining the maximum TP and maximum CVaR, respectively, are shown in Figure 7 when θ_v ∈ U[0.01; 0.05]. Firstly, it can be found that the fixed GHG emission of the intra-manufacturer and the purchase cost are mainly contributed to TG and the expective total costs (TC), respectively. Moreover, it can be observed that the fluctuations of the fixed GHG emission of the intra-manufacturer, the shortage cost, the fortified cost and the purchase cost are most obvious as the confidence level varied. When ζ < 0.25, the shortage cost and the purchase cost are significantly reduced while the fortified cost gradually grows with the increase of the confidence level. When 0.25 < ζ < 0.50, the fortified cost continues to increase to the peak point while the shortage cost continues to decrease to the valley point, and the purchase cost rebounds slightly. When ζ > 0.75, the purchase cost drops again, and the fortified cost drops slightly while the shortage cost is basically unchanged.

The comparison of Figure 6 and Figure 7 demonstrates that the trend of TC, TG and the total market share is consistent as the confidence level varies. This is mainly because the total market share directly affects the fixed GHG emission of the intra-manufacturer and the purchase cost. Therefore, the indicators that are significantly affected by the confidence level can be summarized into the following three items: the total market share, the shortage cost and the fortified cost. Taking the case of θ_v ∈ U[0.01; 0.05] as an example, the above phenomenon can be properly explained by the following conjectures. (i) When ζ < 0.25, in order to mitigate the potential worst-case losses (i.e., the shortage cost in the potential worst-case scenarios), Product variant prices are rising while the total market share is reduced after a trade-off between price and market share. (ii) When 0.25 < ζ < 0.50, since all selected suppliers are fortified to a higher level, which make more contributions to mitigate the potential worst-case losses, the total market share can be appropriately expanded. (iii) When ζ > 0.75, it means a higher level of risk aversion. Since suppliers have been fortified to the highest level, however, only by giving up part of the market share again can the potential worst-case losses be mitigated.

To sum up, the following managerial insights can be summarized. (i) Appropriate controlling market share is also an effective strategy to mitigate the potential worst-case losses and improve the potential worst-case profit. (ii) When θ_v ∈ U[0.15; 0.20], it is recommended to set the confidence level to 0.50, 0.99 or any value between 0 and 0.25. (iii) When θ_v ∈ U[0.01; 0.05], it is recommended to set the confidence level to 0.75, 0.99 or any value between 0 and 0.25.

6. Conclusions

With the deep integration of the global economy, the stability of the supply chain is facing the challenge of more uncertain events, such as global pandemics, political unrest and natural disasters. Although many scholars have conducted extensive research on improving the resilience of the supply chain, the research on integrating product family configuration and supplier selection under disruption risks is limited. In this paper, therefore, an improved stochastic bi-objective mixed integer programming model was proposed to support co-decision for low-carbon product family configuration and resilient supplier selection under disruption risks. In order to improve the supply chain performance, the fortified supplier, one of the resilience strategies, was adapted to mitigate the potential worst-case losses. Then, two popular financial engineering percentile measures of risk, Value-at-Risk and Conditional Value-at-Risk, were applied to control the potential worst-case losses caused by supply disruptions.

Moreover, through studying the impact of disruption probability and confidence level on optimization results, several important managerial insights were summarized. For example: (i) The fortifying supplier is still an effective strategy to improve the resiliency of the supply chain in co-decision of PCSS. (ii) Appropriate controlling of the market share is an effective strategy to mitigate the potential worst-case losses and improve the potential worst-case profit. (iii) The effects of low-probability risk are more likely to be ignored in the risk-neutral solution, and it is necessary to adopt a risk-averse solution to reduce potential worst-case losses. (iv) Different from the supply chain design of a single component/module, the supply volume of each component/module should be balanced to reduce the potential worst-case losses in co-decision of PCSS with disruption considered.

A limitation of the proposed method is that only one resilience strategy is adapted to mitigate the potential worst-case losses. Future work may consider more risk mitigation strategies such as dual/multiple sourcing and back-up supplier and explore the use scenarios and optimal combinations of each risk mitigation strategy. GHG emissions in this paper manly involve the CO₂ emission of manufacturers in the process of R&D, assembly and procurement, and the measures to reduce GHG emissions are mainly to select module instances with low carbon emissions. In the future, it is possible to construct the carbon neutral model for suppliers, manufacturer, customers and recycling plants in the whole life cycle of products, and further study the optimal PCSS scheme to make a trade-off among suppliers, manufacturer, customers and government. In addition, as the number of disruption scenarios in the proposed model grows exponentially with the number of suppliers, the solution process is cumbersome and computationally expensive. In previous studies, although the FCM (Fuzzy C mean) and SBR (simultaneous backward reduction) have been used to reduce the number of disruption scenarios in the risk-neutral model, it is not suitable for the risk-averse model. To improve the efficiency of solutions, it is necessary to study scenario reduction techniques and algorithms suitable for risk-averse co-decision of PCSS in the future.