2.1. Yield Rate Modeling in Disassembly

Existing research on yield rate modeling in disassembly processes can be broadly classified into two approaches: deterministic and stochastic modeling. Deterministic yield models assume fixed yield rates for all take-back products. For instance, Kwak and Kim (Reference Kwak and Kim2017) used a transition matrix with predefined, constant yield rates for production planning. While simplifying modeling, this approach demands simultaneous yield rate inputs, making it time-consuming and often unrealistic. Similarly, Huang et al. (Reference Huang, Yi, Shi and Guo2018) classified used products into three quality grades—high, medium, and low—using modal intervals. While this method introduces quality uncertainty, yield rates within each grade remain deterministic, failing to capture real-world variability.

Disassembly involves multiple stakeholders—OEMs, third-party remanufacturers, and logistics providers—who influence core recovery. Priyono et al. (Reference Priyono, Ijomah and Bititci2016) outlined key disassembly stages: inspection, sorting, selective disassembly, and cleaning, all of which impact yield variability. Effective stakeholder coordination enhances disassembly efficiency, improving material recovery while reducing economic and environmental costs.

Stochastic models address disassembly uncertainty by representing take-back core quality probabilistically. Yang et al. (Reference Yang, Wang and Ji2015) modeled core quality distributions, considering remanufacturing time. Other studies explored exponential and Weibull distributions (Kumar, Reference Ke, Su and Wang2014; Ke et al., Reference Kumar2009). Despite capturing yield uncertainty, these methods overlook interdependencies across disassembly levels. Further exploration of uncertainty propagation could provide a more realistic representation.

2.2. Diffusion Process

Diffusion models, first introduced by Sohl-Dickstein et al. (Reference Sohl-Dickstein, Weiss, Maheswaranathan and Ganguli2015), are deep generative frameworks originally designed for image generation but now widely used in constrained optimization (Reference Ho, Jain and AbbeelHo et al., 2020, Reference Mazé and AhmedMazé and Ahmed, 2022). The diffusion process, denoted as q, represents the forward step, where noise is added iteratively via a Markov chain to transform data into a noise-like distribution. A variance schedule β_t controls the noise at each time step t, as formulated in Equation 1:

(1) $$q(x_t |x_{t - 1} = {\cal{N}}(x_t ;\sqrt {\beta _t } x_{t - 1} ,(1 - \beta _t {\mathbb{I}}))$$

where ${\cal{N}}$ is a Gaussian distribution, x_t is the data state at time t, and ${\mathbb{I}}$ is the identity matrix. This process gradually deconstructs the data structure into pure noise.

The diffusion process effectively models yield rate variability and sequential dependencies in disassembly. Yield rates are stochastic, influenced by take-back quality, efficiency, and subpart conditions. By introducing controlled noise at each step, this Markov chain-based approach aligns with the hierarchical nature of disassembly, where each level’s yield rate depends on the previous one. The variance schedule further allows stakeholders to adjust variability for specific products, making diffusion modeling a flexible tool for remanufacturing.

2.3. Sustainable Design

Sustainable design balances economic, environmental, and social factors by integrating resourceefficient practices into production. In remanufacturing, studies have optimized demand, supply, core acquisition, and production to support OEMs, retailers, and policymakers (Kwak and Kim, Reference Kwak and Kim2017, Kwak, Reference Kwak2018, Sun and Li, Reference Sun and Li2023).

Beyond economic and operational optimization, sustainable design enhances disassemblability and remanufacturing efficiency. Ijomah et al. (Reference Ijomah, McMahon, Hammond and Newman2007) proposed design-for-remanufacturing (DfR) guidelines to improve component recovery and reduce costs. Ijomah (Reference Ijomah2009) addressed decision-making challenges in remanufacturing, advocating structured design approaches to enhance take-back core quality. Dong et al. (Reference Dong, Zhang, Tong and Dong2006) introduced a hierarchical disassembly planning approach using a hierarchical attributed liaison graph (HALG) to improve remanufacturing efficiency while reducing computational complexity. These studies highlight sustainable design’s role in shaping remanufacturing strategies and take-back core quality. Additionally, Life Cycle Assessment (LCA) is widely used to assess remanufacturing’s sustainability benefits (Zheng et al., Reference Zheng, Li, Wang, Shi, Xu and Yang2019, Zhang et al., Reference Zhang, Zhang, Zhang, Jiang, Liu and Cai2020), and this study derives its environmental impact parameters from LCA calculations.

These studies underscore sustainable design’s impact on remanufacturing and take-back core quality. Among them, Kwak and Kim (Reference Kwak and Kim2017) developed a mixed-integer nonlinear programming (MINLP) model integrating pricing, production, and environmental considerations for new and remanufactured products. The study assumes deterministic yield rates for the disassembly process, represented in the form of a transition matrix, as illustrated in Figure 1.

Figure 1. Disassembly structure of example product ABC and its corresponding transition matrix (Reference Kwak and KimKwak and Kim, 2017)

In Figure 1, the letters in parentheses represent the component condition: ‘(R)’ denotes ‘remanufactured’, ‘W’ denotes ‘working’ and ‘N’ denotes ‘not-working’. Operations 1 to 4 represent disassembly operations. For example, in operation 1, an EoL product ABC is taken as input, producing 70% working AB, 30% not-working AB, 90% working component C and 10% not-working component C. These proportion of reusable parts (i.e. 0.7 and 0.9) are referred to as yield rates.

3.1. Yield Dependency Modeling in Disassembly

Previous studies (Lambert, Reference Lambert2002; Kwak et al., Reference Kwak, Hong and Cho2009; Kang et al., Reference Kang, Kwak, Cho and Hong2010; Kwak and Kim, Reference Kwak and Kim2017) have commonly used transition matrices (Figure 1) to represent the relationship between product design and remanufacturing operations. Remanufacturing typically involves three stages: disassembly, reconditioning, and reassembly. This study focuses on the disassembly stage, as the quality of EoL products is inherently uncertain when collected by third-party takeback companies. It is assumed that a third-party company gathers used products from customers, and the OEM repurchases these products from the third party, with the buyback price determined in this study.

Given an EOL product, let t = {0,1,2, ..., T} represent the disassembly levels, and let $\{ y_t \} _{t = 0}^T $ denote the sequence of random variable corresponding to the yield rates at each disassembly level. For simplicity, it is assumed that all parts within a given disassembly level share the same yield rate. while this assumption can be relaxed to follow distinct yield rates for each part, it is adopted in this study for demonstration purposes.

At t = 0, y ₀ represents the yield rate of the initial disassembly level and is modeled as a truncated normal distribution, $y_0 \sim {\cal{N}}(\mu _0 ,\sigma _0^2 )$ , truncated to the interval [0,1]. A third-party takeback company assesses the EOL product and provides the OME with a rough quality score ranging from 0 to 1, which serves as μ ₀. The variance $\sigma _0^2 $ is determined by the OEM’s production planning experts based on domain knowledge and historical data.

For subsequent disasembly levels (t ≥ 0), the OEM defines a sequence of variance schedule, $\{ \beta _t \} _{t = 1}^T $ , to control the variabilioty of yield rates at each level. At disassembly level t, the yield rate y_t is modeled as a conditional truncated normal distribution, dependent on the previous yield rate y _t−1, as expressed in Equation 2.

(2) $$q(y_t |y_{t - 1} ) = {\cal{N}}(y_t ;\sqrt {\beta _t } x_{t - 1} ,(1 - \beta _t ))$$

The use of the variance schedule $\{ \beta _t \} _{t = 1}^T $ provides OEM experts with the flexibility to adapt the model to specific disassembly cases. This formulation, inspired by diffusion processes, assumes that the yield rate at each level depends only on the yield rate at the preceding level, consistent with the properties of a Markov process. This assumption is reasonable, as the yield rate of a subassembly or part naturally depends on the quality observed at the upper disassembly level (Reference Darghouth and Abdel-AalDarghouth and Abdel-Aal, 2021).

3.2. Model Formulation

Since yield rates at each disassembly level are stochastic, the mixed-integer programming model from Kwak and Kim (Reference Kwak and Kim2017) is revised to maximize expected profit:

(3) $$\max {\mathbb{E}}_Y [{\rm{Profit}}(P_n ,Z_n ,P_r ,Z_r ,M_i ,N_i ,P_k ,X_k ,O_j )],$$

where ${\mathbb{E_Y}}$ represents the expectation over stochastic yield rates $\{ y_t \} _{t = 0}^T $ , and Profit(·) follows Equation 4 from Kwak and Kim (Reference Kwak and Kim2017). The decision variables are:

• P_n , Z_n : Selling price and production volume of new products.

• P_r , Z_r : Selling price and production volume of remanufactured products.

• P_k , X_k : Buyback price and return volume of end-of-life product k (\forallk \in K).

• M_i , N_i , O_i : Recycling volume, new part production (\foralli \in I), and operation frequency (\forallj \in J).

The profit function for a single-period production horizon is:

(4) $$\matrix{ \hfill {{\rm{Profit}}(P_n ,Z_n ,P_r ,Z_r ,M_i ,N_i ,P_k ,X_k ,O_j ) = (P_n - C_n )Z_n + P_r Z_r \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \cr \hfill { - (\mathop \sum \limits_{i \in I} c_i^M M_i + \mathop \sum \limits_{k \in K} P_k X_k + \mathop \sum \limits_{j \in J} c_j O_j + \mathop \sum \limits_{i \in I} c_i^N N_i + c_d Z_r )} \cr }$$

The model maintains constraints on production capacities, disassembly flow balances, and green profit thresholds, consistent with Kwak and Kim (Reference Kwak and Kim2017). For full constraint details, refer to their work. The numerical solution method is described in Section 3.3.

Environmental impact is assessed through carbon savings, measured as the reduction in CO₂ emissions from remanufacturing versus new production. This metric, widely used in sustainability assessments and regulations, was chosen for its standardization. Other factors (e.g., resource depletion, water usage, material recovery) are important but beyond this study’s scope (Reference Zhang, Zhang, Zhang, Jiang, Liu and CaiZhang et al., 2020).

3.3. Numerical Method

The model formulation described in Section 3.2 introduces significant complexity in solving the problem. Even the deterministic version is a nonlinear mixed-integer optimization problem, which requires considerably more computational resources compared to linear programming. Incorporating yield dependency modeling in the disassembly step further complicates the problem, as all decision variables depend on random variables, and the interdependencies among these variables make the problem even less tractable. To address this challenge, this study employs a numerical approach called Scenario Generation, as outlined in Algorithm 3.3.

In Algorithm 3.3, Monte Carlo simulation is employed to generate an initial set of N scenarios (Reference MooneyMooney, 1997), with N set to 100,000 to comprehensively capture yield rate variability. For each scenario, yield rates $\{ y_t \} _{t = 0}^T $ are sampled based on predefined parameters: the initial yield mean (μ ₀), initial standard deviation (σ₀), and variance schedule $(\{ \beta _t \} _{t = 1}^T )$ . To balance computational efficiency and accuracy, k-means clustering (Reference Sinaga and YangSinaga and Yang, 2020) is subsequently applied to select K representative scenarios, from which the empirical expected profit is calculated.

Scenario Generation Algorithm [1] Initial yield mean μ ₀, initial yield standard deviation σ₀, variance schedule $(\{ \beta _t \} _{t = 1}^T )$ , truncation bounds [0,1], number of initial scenarios N, number of reduced scenarios K Reduced set of scenarios {Y⁽¹⁾, Y⁽²⁾, . . . , Y^(K)} with estimated joint probabilities {P ⁽¹⁾, P ⁽²⁾, . . . , P ^(K)} Initialize an empty list of scenarios n = 1 to N Generate N initial scenarios Sample initial yield Y ₀ ∼ TruncatedNormal(μ ₀, σ ₀, [0,1]) t = 1 to T Generate yields for each level Set standard deviation σ_t = 1 − β_t Sample Y_t ∼ TruncatedNormal $(\sqrt {\beta _t } Y_{t - 1} ,\sigma _t ,[0.1])$ based on previous yield Y _t−1 Store yield sequence as a scenario

Use k-means clustering to reduce scenarios to K representative scenarios Calculate joint probabilities for each reduced scenario

Reduced scenarios with joint probabilities

To compute the empirical expected profit using the K representative scenarios, the joint probability distribution for each scenario k ∈ K is required. Given the Markov property in the diffusion process, the joint distribution for a scenario k can be expressed as:

(5) $$P^{(k)} = q^k (y_0 ,y_1 \ldots ,y_T ) = q(y_0 )\mathop \prod \limits_{t = 1}^T q(y_t |y_{t - 1} ),$$

where q(y ₀) is defined as the truncated normal distribution: q(y ₀) = TruncatedNormal(μ ₀, σ ₀, [0,1]). Algorithm 3.3 outputs a set of K representative scenarios along with their corresponding joint probabilities. These scenarios serve as realized data points to be input into the MINLP problem, transforming it into a deterministic version for each scenario k. The empirical expected profit is then computed using Equation 6:

(6) $${\mathbb{E}}_Y [{\rm{Profit}}( \cdot )] = \mathop \sum \limits_{k \in K} P^{(k)} {\rm{Profit}}^{(k)} ( \cdot ),$$

where P ^(k) is the joint probability of scenario k, and Profit^(k)(·) represents the profit evaluated for scenario k. This numerical approach balances computational efficiency with the need to account for stochastic yield rates in the optimization process.

4.1. Model Setup

This study evaluates two distinct product types—smartphones and recreational boat engines—to demonstrate the proposed stochastic methodology’s effectiveness in OEM decision-making. These products differ in production scale, market size, EoL availability, and remanufacturing processes, providing a comprehensive comparison.

Smartphones are mass-produced with high EoL availability and mature takeback systems. Their disassembly process is relatively simple, involving three levels and 23 operations. In contrast, recreational boat engines serve a niche market with fewer EoL products and a less developed remanufacturing business. Their disassembly process is more complex, assuming five levels and 122 operations (details withheld for confidentiality).

These contrasting characteristics make smartphones and boat engines ideal case studies for evaluating the methodology’s robustness across different remanufacturing environments. Table 1 summarizes model assumptions, with smartphone data from Kwak and Kim (Reference Kwak and Kim2017) and boat engine data based on Brunswick Corporation. Due to confidentiality, boat engine parameters are withheld, and results are normalized.

Table 1. Model Assumptions for Smartphone and Boat Engine

Given the two product types and their model assumptions, two scenarios are analyzed:

• Baseline Case: Yield rates are uniform across disassembly levels. For smartphones, EoL product type 1 has a deterministic yield of 1.0, while type 2 is 0.5. For boat engines, the yield for EoL product type 1 is 0.8.

• Stochastic Case: Yield rates vary across disassembly levels. Table 2 summarizes the yield dependency modeling parameters. For smartphones, only the yield rate of the poor EoL product type is stochastic, while the good type remains deterministic. Three sub-cases assess the methodology’s accuracy with 10, 100, and 1000 generated scenarios.

Table 2. Model Parameters for Smartphone and Boat Engine

4.2. Optimization results

This section presents optimization results for both cases. All problems were solved using the Gurobi Python API under an academic license (Gurobi Optimization, LLC, 2024). Table 3 summarizes the baseline case, where yield rates are deterministic and uniform across disassembly operations. For smartphones, the optimal total profit is $2.16 million, with environmental savings of 73,767 kg CO ₂ equivalent. For boat engines, values are normalized (0 to 100), with an optimal total profit of $69 and environmental savings of 47.07 kg CO ₂ equivalent.

Table 3. Optimization results for the baseline case

Figures 2 and 3 illustrate the stochastic results obtained using the proposed methodology for the three sub-cases introduced in Section 4.1. The ‘x’ symbols represent the baseline values, as shown in Table 3.

Figure 2. Stochastic Distribution of Total Profit and Environmental Savings Across Different Numbers of Scenarios for Smartphones

Figure 3. Stochastic Distribution of Total Profit and Environmental Savings Across Different Numbers of Scenarios for Boat Engines

For smartphones, in the 1000-scenario sub-case, 43% of scenarios achieve higher total profit and greater environmental savings than the baseline. Similarly, for boat engines, 33% of scenarios exceed the baseline in both metrics. The total profit distributions under 1000 scenarios follow a normal distribution at a 5% significance level, validated by the Anderson-Darling test.

Moreover, for smartphones, the expected total profit is $2.157 million (10 scenarios), $2.159 million (100 scenarios), and $2.157 million (1000 scenarios). For boat engines, it is $68.24 (10 scenarios), $67.88 (100 scenarios), and $67.86 (1000 scenarios). The consistency across sub-cases highlights the effectiveness of using fewer scenarios with the k-means clustering algorithm, which efficiently captures possible outcomes while maintaining accuracy.

4.3. Discussions

The optimized results in Section 4.2 highlight the benefits of incorporating yield uncertainty into the proposed methodology. While the baseline results provide reasonable estimates of total profit and environmental savings (as ‘x’ markers are centrally located within the joint distributions), they lack insight into the range of possible operational outcomes. By adopting the stochastic methodology, the model becomes more robust to real-world variations in disassembly, reducing the risk of suboptimal remanufacturing planning.

The stochastic approach gives OEMs critical insights into outcome variability for both new and remanufactured products, which the deterministic baseline does not capture. For smartphones, total profit varies minimally (±2%) due to a stable market and standardized processes, whereas environmental savings fluctuate more significantly (from 10% higher to 3% lower than baseline). These results suggest that OEMs could leverage stable profits and environmental benefits to integrate carbon-saving metrics into long-term planning, such as regulatory compliance, eco-conscious marketing, or carbon credit strategies (Reference Wang, Zhou, Ma and WangWang et al., 2024).

In contrast, boat engines exhibit greater variability, with total profit ranging from +10% to -22% and environmental savings from +85% to -75% relative to baseline. This results from higher disassembly complexity, greater remanufacturing costs, and sensitivity to initial yield rates. When a take-back engine’s overall yield rate falls below 0.5, the model may opt against remanufacturing, amplifying environmental savings variability. These findings emphasize the need for high-quality take-back cores, reinforcing the roles of DfRem and AI-driven condition assessment in reducing variability and enhancing remanufacturing viability (Reference Ijomah, McMahon, Hammond and NewmanIjomah et al., 2007). For high-value goods like boat engines, adaptive disassembly strategies that account for yield uncertainty can help OEMs optimize cost-benefit trade-offs in remanufacturing (Reference Priyono, Ijomah and BititciPriyono et al., 2016).

These insights align with the products’ intrinsic differences. Smartphone remanufacturing ensures stable profitability with moderate carbon savings, making it suitable for sustainability-driven policies. Conversely, boat engine remanufacturing has a higher environmental impact but requires an adaptive approach due to yield uncertainty. As carbon savings become increasingly monetized, accounting for variability in disassembly outcomes is essential when formulating remanufacturing strategies.