Blockchain-Based Long-Term Capacity Planning for Semiconductor Supply Chain Manufacturers

Abstract

The long-term production capacity planning of semiconductor supply chain manufacturers has a series of characteristics, such as large capital investment, fast technology upgrading, long lead-time of manufacturing equipment, and unstable market environment, which leads to the uncertainty of demand. In addition, blockchain technology is widely used to build consortium chains for information sharing across upstream and downstream enterprises, thus having the potential ability to help finance semiconductor manufacturers. This paper combines two uncertainty-oriented methods (stochastic programming and robust optimization) to examine the conversion of capacity between different product types and construct a two-stage mathematical planning model maximizing the net profit of manufacturers. Through the introduction of blockchain technology and information sharing among enterprises, we improve the effectiveness of our model to realize the optimal allocation of long-term capacity planning. Finally, we reformulate the model into a tractable MINLP, construct numerical examples to verify the solvability, and carry out sensitivity analysis.

1. Introduction

As critical national strategic resources, semiconductors have emerged as a key field of international tugs-of-war under the impact of deglobalization, with the stability of semiconductor supply chains constantly disrupted by trade policies. In September 2021, the Department of Commerce of the United States (US) launched a Request for Information (RFI) on the semiconductor supply chain that asked major manufacturers worldwide to share their inventories, production capacity, the procurement of raw materials, sales, and customer information within 45 days, in order to cope with the global chip crisis. The RFI targeted all stakeholders involved in the semiconductor supply chain, including designers and manufacturers, suppliers of materials and equipment, intermediaries, as well as terminal users. It has exacerbated the uncertainties and risks borne by the semiconductor industry. In September 2022, the US officially enacted the CHIPS and Science Act, announcing the investment of more than 50 billion US dollars in manufacturing and R&D of the semiconductor industry. Meanwhile, the Act sets forth targeted restrictive provisions, providing that chip manufacturers receiving financial assistance shall refrain from conducting the material expansion of semiconductor manufacturing capacity in China or any other foreign country of concern during the following ten-year period. The enactment of the Act will affect the optimized structure, security, and stability of the global industrial chain and supply chain of semiconductors.

On the one hand, semiconductor manufacturers have to face shocks from the international market. On the other hand, the supply chain of small and medium-sized enterprises also confronts financial challenges [1]. Capacity planning is especially important for the semiconductor industry due to the highly uncertain and capital-intensive nature of semiconductor manufacturing; while in urgent need, these features make it fairly difficult for semiconductor manufacturers to conduct proper long-term capacity planning.

Capacity planning provides a method to determine the total production capacity generated by a mix of such capital-intensive resources as equipment, tools, facilities, and the overall labor force size, thereby extending robust support to manufacturers regarding their long-term competition strategies and policies. Such planning plays a vital role in nurturing and maintaining the competitive edges of semiconductor manufacturers. It is true that many other industries require similar capital-intensive capacity planning. Still, only very few industries feature a short product life cycle, constant technological innovation, and long delivery lead time of manufacturing machines, all situations faced by the semiconductor industry [2]. Given that these characteristics bring about many uncertainties for the capacity planning of manufacturers, semiconductor manufacturers must deal with the huge discrepancies between the actual capacity planning and their targets, which causes the waste of resources or lack of production capacity. Therefore, manufacturers find it imperative to work out a feasible approach to capacity planning [3,4,5].

The semiconductor industry faces significant challenges when it comes to managing capacity planning due to the uncertainty of demand. Forecasting demand accurately is a complex and challenging task, particularly in a market that is subject to rapid technological change and frequent fluctuations. To this end, methods for optimization under uncertainty are utilized to tackle demand uncertainty. Two of these methods are stochastic programming and robust optimization. Stochastic programming requires that the distribution model of uncertain parameters be determined first, and then the expectation or chance constraints are introduced, which can be transformed into a deterministic problem. Robust optimization describes the uncertainty of parameters in the form of uncertainty sets, and the optimization objective is to make the worst case the best.

In addition, the emergence of blockchain technology also presents new opportunities for the semiconductor industry. Blockchain technology, as a decentralized basic architecture and a paradigm of distributed computing, leverages an encrypted chain block structure to verify and store data. In addition, it adopts consensus algorithms among distributed nodes to generate and update data and employs automation scripts for programming and data manipulation. The application of distributed ledgers, consensus mechanisms, and smart contracts renders a blockchain technology that is decentralized and tamper-proof with traceable data, which in turn creates a trustworthy data-sharing environment for on-chain manufacturers and safeguards the security and reliability of information sharing [6]. At present, blockchains can be classified as public, private, and consortium blockchains by degree of openness. A public blockchain is one shared by a group. Anyone who joins the blockchain could gain access to all the information on the chain, which makes it an open system. A private blockchain is only accessible to specific users. One can only join the private blockchain upon system authentication. A consortium blockchain is best suited when there is a need for both types of blockchains, namely public and private blockchains. The consortium designates the pre-selected nodes, which determine the generation of every block. In the meantime, other nodes only participate in transactions and are excluded from consensus building. It is a better solution for data sharing between manufacturers involved in supply chains. As a result, financial institutions can use blockchain to monitor the activities of manufacturers based on transparent data shared on the platform [7,8]. The semiconductor industry is characterized by a high level of capital intensity, which often requires substantial financing to support its growth and development. Leveraging blockchain technology can offer an innovative solution to address these challenges by enabling the sharing of critical data and adjusting capital costs.

In this study, we propose a blockchain technology-based capacity planning decision-making model featuring a two-stage production launch. Considering the financial support of blockchain for semiconductor production, the blockchain technology implementation effort is introduced, and the stochastic programming and robust optimization are combined to depict the capacity planning decision models of the first and second stages, respectively, with an aim at semiconductor enterprises making long-term capacity planning decisions.

We summarize the main contributions of this paper as follows. (1) To our knowledge, we are the first to study the application of blockchain technology to the production of semiconductors. (2) We propose a two-stage production decision model that can be used in long-term capacity planning. We reformulate it into a tractable form and verify its solvability. (3) To handle the demand uncertainty, we devise an innovative approach that incorporates stochastic programming and robust optimization into our two-stage model.

The remainder of this paper is organized as follows. We position this study in the related literature in Section 2. Section 3 describes the problem and presents our mathematical model. In Section 4, we convert the problem into a tractable formulation. Numerical studies are provided in Section 5. Finally, we conclude this paper in Section 6.

2. Literature Review

This study is related to three streams of literature: capacity planning, optimization under uncertainty, and blockchain technology.

Capacity planning can be classified as long-term, medium-term, and short-term on the basis of duration [9]. At present, the prevailing approaches to capacity planning in the semiconductor industry include the spreadsheet approach [9], the formulation approach [10], linear programming (LP) [11], and stochastic programming [12]. Among them, the spreadsheet approach is the most popular, applicable to calculations in simple production environments, such as the quantity of equipment. In cases where the processing period is defined as the key point of work indicators, the formulation approach applies. However, a large number of experiments should be conducted so that the spreadsheets and formulation technology can yield sound solutions. LP can be used to construct objective functions and related constraints, which allows for calculating the quantity of equipment required and optimizing the product mix for maximum profits. Bermon et al. [11] proposed an LP model for capacity planning. This LP-based optimization system became the decision support system for capacity planning of the largest semiconductor production line of IBM in 1996. Therefore, LP is one of the effective solutions to that problem based on the comparison of various capacity-planning approaches. Although LP is easily accessible, it only functions when demand forecasts are known, or in the case of a highly accurate demand forecast. Unfortunately, in the semiconductor industry, demand can be highly uncertain, which poses a great challenge to demand forecasts. Such uncertainties trigger more acute problems in long-term capacity planning. Existing literature shows that scholarly attention remains few on the long-term capacity planning of semiconductor manufacturers. Decision-making models for medium-term capacity planning are not directly applicable to long-term problems, because larger uncertainties exist over a longer time frame. We contribute to this stream of research by proposing a tractable two-stage production decision model that can be used in long-term capacity planning. Numerical examples demonstrate the solvability and effectiveness of our model.

Stochastic programming and robust optimization are the two main methods to deal with model uncertainty. Due to the uncertainties of demand, stochastic programming is widely applied to the capacity planning research of the semiconductor industry. Having considered the uncertainties brought by capacity and demand fluctuations, Robert et al. [13] presented a multi-stage stochastic programming model for capacity planning and adopted a scenario-based method to show uncertainties to minimize the gap between product demands and the actual capacity. Catay [14] developed a multi-period mixed-integer programming model for semiconductor manufacturing capacity planning based on the variability of demand and capacity, multi-stage production, and capacity utilization. In Catay’s study, two new scalable distributed parallel optimization algorithms were developed to help solve problems faster to provide a large number of scenarios and enormous first-stage integer variables necessary to describe uncertainties in applications. Zhao et al. [15] designed a novel decision rule-based method for the multiple facility capacity expansion problem. Barahona and Bermon [16] used the concept of tool groups to link product demand to production equipment demand, enabling direct decision-making on the quantity of equipment. Given the technological shift between products, a single-period decision-making model for the quantity of equipment was constructed to minimize capacity shortage and the number of tool groups. Chien et al. [17] pointed out that demand fulfillment and capacity utilization are directly associated with customer satisfaction, market growth, and the profitability of the company in the semiconductor industry. Therefore, a two-stage stochastic programming (2-SSP) model was constructed to facilitate the decision-making on the production plans of wafers, thereby ensuring the fulfillment rate of orders in the case of uncertain demand. Karabuk et al. [18] identified two unique characteristics of the capacity planning of semiconductor manufacturers: (1) wafer demands and manufacturing capacity are both main sources of uncertainty, and (2) capacity planning must consider the distinct viewpoints of marketing and manufacturing. With these characteristics taken into account, multi-stage stochastic programming with demand and capacity uncertainties was then constructed. At the same time, to reconcile the marketing and manufacturing perspectives, a decomposition of the planning problem resembling decentralized decision-making should be taken into consideration.

In addition to stochastic programming, robust optimization is also considered an effective tool for optimizing uncertainties. Assuming mean-covariance information about the demand distribution, Zhang et al. [19] applied distributionally robust models to determine production periods and production quantities in face of uncertain demand. To handle uncertainties and enhance the robustness of capacity planning of wafer plants, Chen et al. [20] developed a mathematical model of robust production capacity planning. Meanwhile, to solve this problem, the team also proposed a heuristic algorithm and conducted numerical experiments to evaluate its robustness. Zheng et al. [21] applied the minimax regret strategy to newly established product lines and the capacity conversion between different products, whereas considering the capacity of different product types, the capacity conversion also enhances the robustness of capacity planning and reduces capital input. Apart from the above-mentioned prevailing approaches, there are studies on the capacity planning and scheduling of semiconductor manufacturers in respect of investment planning of equipment [22], resource portfolio planning [23], and game models of production [24].

Considering the advantages and disadvantages of stochastic programming and robust optimization, in our two-stage model, we adopt stochastic programming in the first stage to handle uncertainty and variability in demand, and robust optimization in the second stage to ensure a high-quality solution that can withstand worst-case scenarios. This innovative approach has broad practical applications in semiconductor manufacturing.

Research about applying blockchain technology to production practice is still a growing trend. Based on the smart contract and consensus mechanism, Tian [25] brought forward a blockchain-based manufacturing capacity-sharing model to achieve efficient capacity utilization based on smart contracts and consensus mechanisms. Ren [26] incorporated blockchains into the Internet systems of energy and accordingly developed a half-centralized two-stage robust optimization model. The study made use of blockchain technology to acquire historical data and established data-driven uncertainty sets to solve regulation schemes and exclude some extreme scenarios, thus making the model less conservative. Besides, the verification function of consensus mechanisms of blockchains was fully leveraged to get rid of the information tampered with by malicious nodes and enhance the fault tolerance of the system. Leng et al. [27] leveraged the real-time network synchronization of blockchains to establish an intelligent production safety protection framework, thereby boosting the robustness and reliability of manufacturing. To the best of our knowledge, studies on the application of blockchains to the production of semiconductors remain missing. Taking the advantage of consortium blockchain that is able to share the demand information of upstream and downstream enterprises, and hence provide financial support for semiconductor production, we incorporate blockchain technology implementation efforts into our model to better utilize this new technology.

3. Problem Descriptions and Modeling

3.1. Problem Descriptions

Forecasts on production capacity demand and decisions made thereby are of critical importance in long-term capacity planning. Demand forecasts lay the foundation of capacity planning. However, demand faces a great many uncertainties, which becomes an even more prominent problem in long-term capacity planning. Currently, there are two decision-making approaches to handling uncertainties, namely stochastic programming and robust optimization. Stochastic programming represents the uncertainty of the decision-making process with a number of scenarios by fitting the probability distribution functions with uncertain parameters through historical data or by means of scenario generation. Such a method converts optimization problems under uncertainty into deterministic optimization problems under certain probability. In the decision-making on the long-term capacity planning of semiconductor manufacturers, it is not easy to estimate the probability distribution of uncertain parameters, which is why the majority of capacity decision-making models based on stochastic programming often adopt the scenario generation method [20].

The scenario-generated single-stage stochastic programming decision-making model with minimum objectives is presented as follows:

$$\underset{x\in X}{min}{E}_{s\in S}\left[g(x,s)\right]$$

In the single-stage stochastic programming model, x represents a decision variable; X represents the feasible set of the decision variable x; S represents a set of possible scenarios. In the decision-making model, when a decision is made on the decision variable x, no other adjustment is rendered, so it is unable to reflect the actual decision-making layer. For this reason, two-stage stochastic programming models are often adopted to reflect the multiple layers of the decision-making process. The mathematical form of a two-stage stochastic programming model is presented as follows:

$$\begin{array}{c}\underset{x\in X}{min}{E}_{s\in S}\left[g\left(y_s\right)\right]\\ F\left(x,s,y_s\right)\le d\forall s\in S\end{array}$$

A two-stage stochastic programming decision-making model comprises two layers of decision-making. The first layer is defined as “upper-level decision-making”, which is made first and affects the subsequent decision-making process and the results thereof. For example, in the above two-stage stochastic programming model, the decision $x\in X$ represents the upper-level decision-making. The second layer is “lower-level decision-making”, namely a decision based on the result x of the upper-level decision-making and the actual scenario s. For instance, in the above two-stage stochastic programming model, $y_s$ represents the lower-level decision-making and satisfies $F\left(x,s,y_s\right)\le d$. Suppose that a manufacturer seeks to make a decision on production, in this case, it has to decide on the site planning and selection for the production line before gaining a clear picture of the demand scenario. This constitutes upper-level decision-making. After that, the manufacturer needs to adjust the production plan in line with the actual product demand, and the decision on actual production is regarded as the lower-level decision-making.

The widespread use of stochastic programming in the decision-making research on semiconductor capacity planning brings about two acute problems. First, it is difficult to acquire the occurrence probability of a scenario. Second, with increasing uncertainties, the number of scenarios represented is likely to grow exponentially. In the meantime, however, long-term capacity planning features a long timeframe and outstanding uncertainties. Unlike stochastic programming, robust optimization represents the uncertain parameters in an optimization problem as an uncertainty set with boundaries or represents uncertainties as a set of possible extreme scenarios. In this way, when a decision is made under the most unfavorable circumstances, it can ensure the feasibility and robustness of the optimal solution under any scenario.

The scenario-generated single-stage robust optimization decision-making model is presented as follows:

$$\underset{x\in X}{min}\underset{s\in S}{max}f(x,s)$$

Similar to the single-stage stochastic programming model, in a single-stage robust optimization model, $s\in S$ represents uncertainties. When a decision is made on the decision variable x with no further adjustment, the model selects the most unfavorable scenario in accordance with the decision results of the decision variable x. Likewise, in the actual decision-making process, once this kind of uncertainty is fixed, especially once the scenario is known, the decision-maker will adopt the corresponding adjustment method to mitigate or cushion the influences resulting from the fixed scenario. This constitutes an adaptive robust optimization model, also known as a two-stage robust optimization model. The mathematical form is as follows:

$$\underset{x\in X}{min}\underset{s\in S}{max}\underset{y\in Y(x,s)}{min}f(x,s,y)$$

In the adaptive robust optimization model, the first layer represents the decision on planning, which is made to minimize the objective function before the uncertain scenario is determined. The decision variable of the layer is represented as x, which satisfies $x\in X$. In contrast, the second layer represents the determination of the “uncertain scenario” based on the results of the planning decision, giving rise to the maximum objective function. The third layer represents the decision on operation. Once the scenario is determined, the decision on operation is used to mitigate the impacts associated with the most unfavorable scenario, making this layer is the minimum objective function. It is worth noting that, given that the operation decision depends on the results of the planning decision and the most unfavorable scenario, it satisfies $y\in Y(x,s)$.

Robust optimization can effectively represent uncertainties without having to deal with the hard-to-obtain probability distribution. Nevertheless, due to the conservative nature of decision-making, manufacturers must increase investment to ensure the fulfillment of orders or cut financial input out of concerns over excess capacity, which may lead to a severe capacity shortage.

The consortium blockchain is a kind of distributed ledger technology. It allows multiple organizations or individuals to maintain historical data in a shared database. These organizations or individuals are often “participants” of the consortium blockchain who are involved in database maintenance by installing nodes of the consortium blockchain. A consortium blockchain provides a trustworthy data-sharing platform that facilitates information transmission in the course of semiconductor production. For example, manufacturers can create a decentralized and transparent network that tracks the information of upstream and downstream manufacturers, helping to improve coordination and increase profits. Thus, semiconductor manufacturers could benefit from high blockchain implementation efforts.

The present study takes into account both the merits and demerits of stochastic programming and robust optimization, assuming that the upstream and downstream manufacturers of the semiconductor industry could form a consortium blockchain to lower the uncertainties arising out of production and manufacturing. Two methods are adopted to make use of the application capacity of blockchains to build up a decision-making model for long-term capacity planning. In the meantime, since the decision-making process in actual production features “multiple stages and multiple layers”, the present study hereby develops a capacity decision-making model based on two-stage stochastic programming models and robust optimization models. We make several remarks on our analysis and assumptions.

(1) Semiconductor manufacturers are capital-intensive businesses. The industry witnesses rapid and frequent technological updates whereas equipment manufacturing faces long delivery lead times. Meanwhile, long-term capacity planning often features long time frames. Given these characteristics, the present study hereby proposes stage-by-stage investment and construction, namely progressive investment, which is expected to help manufacturers better respond to demand uncertainties. To be more specific, a two-stage production launch approach is explored. A certain amount of capacity is put into use at the first stage. After a period of time, another decision on capacity will be made at the decision point in the second stage.

(2) On the basis of stage-by-stage investment and construction, a model combining stochastic programming and robust optimization is developed. In this case, it is important to highlight two important points. The decisions made in the first stage will affect those made in the second stage. As to demand forecasts, the forecast accuracy of the first stage is higher than that of the second stage because long-term demand forecasts are easier than short-term demand forecasts. For example, the market demand in China’s semiconductor industry takes on a trend of fast growth in the long run. However, it can be more challenging to deliver an accurate forecast of the short-term market demand due to the influences of various policies, domestic and international environments, and fluctuations of upstream and downstream orders. Hence, as it is easier to obtain the occurrence probability of a scenario in the first stage, stochastic programming is applied. In the second stage, adjustments and other decisions are made on the basis of the capacity decision in the previous stage. Robust optimization is used for the modeling of decision-making in the second stage to ensure the robustness of the decision.

(3) The demand of semiconductor manufacturers is generated by orders rather than by highly market-based free transactions, which means sufficient order fulfillment is of crucial importance. Semiconductor manufacturers face very stringent customer requirements. The demands are unpredictable and are lost if the manufacturer does not have enough capacity during a period of high demand [22]. The marginal cost of unmet demand is significantly higher than that of idle capacity in the semiconductor industry. Semiconductor manufacturers often err on the side of having overcapacity and keeping equipment idle to ensure customer goodwill and loyalty [28]. However, the intensive efforts to lower unmet demand in the first stage will likely lead to overcapacity and losses. For this reason, it is essential to consider the cost of overcapacity while pondering over the objectives of their decisions.

(4) When a decision on capacity is made, the manufacturer must adjust its capacity in line with the actual demand instead of keeping it unchanged. Facing the actual demand, it may have to deal with the overcapacity or capacity shortage of different products. Capacity conversion is launched in this case to coordinate capacity between products. Capacity conversion means that the capacity of a product type can be converted to that of some designated product types.

(5) The consortium blockchain provides a trustworthy data-sharing platform that keeps track of the information of upstream and downstream manufacturers. From the perspective of supply chain finance, as a result, financial institutes are able to monitor the operations of manufacturers based on the transparent data shared in the blockchain. With greater blockchain implementation efforts, semiconductor manufacturers could derive more financial support and improve their own financial situation.

(6) Product inventories are not considered in the model.

3.2. Definitions of Relevant Parameters, Sets, and Variables

• Sets

I: Sets of product types;

$i\in I$: Product type i

$j\in J\left(i\right)$: Set of product type conversions allowed for by the capacity of product type i;

$k\in K\left(i\right)$: Set of product types whose capacity can be converted into that of the product type i;

$T_1$: Set of the time when decisions on actual production are made at the first stage;

$t_1\in T_1$: Time when decisions on actual production are made at the first stage;

$T_2$: Set of the time when decisions on actual production are made at the second stage;

$t_2\in T_2$: Time when decisions on actual production are made at the second stage;

$s_1\in S_1$: First-stage scenarios;

$S_2$: Set of second-stage scenarios;

$s_2\in S_2$: Second-stage scenarios;

• Parameters

${\alpha }_i$: Coefficient of excess production of product type i;

$e_i$: Existing capacity of product type i;

${\gamma }_i$: Maximum capacity conversion coefficient possible for product type i;

${\beta }_i$: Maximum capacity reduction coefficient possible for product type i;

${\delta }_{k,i}$: Efficiency coefficient of capacity converted from product type k to product type i;

$p_i$: Investment cost in capacity per unit of product type i at the first stage;

$q_i$: Investment cost in capacity per unit of product type i at the second stage;

$g_i$: Net profit per unit of product type i;

$h_i$: Shortage cost per unit of product type i;

$f_i$: Overcapacity cost per unit of product type i;

$b_1$: Upper limit of the investment cost at the first stage;

$b_2$: Upper limit of the investment cost at the second stage;

$P_s$: Probability distribution of the first stage;

$d_{i,t_1}^{s_1}$: Demand for product i at $t_1$, the actual decision-making point at the first stage under Scenario $s_1$;

$d_{i,t_2}^{s_2}$: Demand for product i at $t_2$, the actual decision-making point at the first stage under Scenario $s_2$;

${\omega }_1$: Weight coefficient of target value at the first stage;

${\omega }_2$: Weight coefficient of target value at the second stage;

c: blockchain technology implementation cost

$\varphi$: blockchain technology effectiveness parameter

• Decision Variables

$x_i$: Initial capacity building invested for product type i at the first stage;

$\rho$: blockchain technology implementation effort;

• Auxiliary Decision Variables

$o_{i,t_1}^{s_1}$: Capacity of product type i available for production at $t_1$, the actual decision-making point at the first stage under scenario $s_1$;

$c_{i,t_1}^{s_1}$: Remaining capacity of product type i after capacity conversion at $t_1$, the actual decision-making point at the first stage under scenario $s_1$;

$m_{i,j,t_1}^{s_1}$: Capacity of product type i converted into that of the product type j at $t_1$, the actual decision-making point at the first stage under scenario $s_1$;

$r_{i,t_1}^{s_1}$: Actual capacity of product type i after capacity conversion and adjustments at $t_1$, the actual decision-making point at the first stage under scenario $s_1$;

$u_{i,t_1}^{s_1}$: Insufficient capacity of product type i at $t_1$, the actual decision-making point at the first stage under scenario $s_1$;

$v_{i,t_1}^{s_1}$: Excess capacity of product type i at $t_1$, the actual decision-making point at the first stage under scenario $s_1$;

$y_i^{s_2}$: Capacity expansion of product type i at the initial period of the second stage under scenario $s_2$;

$w_i^{s_2}$: Capacity adjustments of product type i after capacity reduction at the first stage during the initial period of the second stage under scenario $s_2$;

$o_{i,t_2}^{s_2}$: Capacity of product type i available for manufacturing at $t_2$, the actual decision-making point at the second stage (after capacity adjustments) under scenario $s_2$;

$c_{i,t_2}^{s_2}$: Remaining capacity of product type i after capacity conversion at $t_2$, the actual decision-making point at the second stage under scenario $s_2$;

$m_{i,j,t_2}^{s_2}$: Capacity of product type i converted into that of the product type j at $t_2$, the actual decision-making point at the second stage under scenario $s_2$;

$r_{i,t_2}^{s_2}$: Actual capacity of product type i after capacity conversion and adjustments at $t_2$, the actual decision-making point at the second stage under scenario $s_2$;

$u_{i,t_2}^{s_2}$: Insufficient capacity of product type i at $t_2$, the actual decision-making point at the second stage under scenario $s_2$;

$v_{i,t_2}^{s_2}$: Excess capacity of product type i at $t_2$, the actual decision-making point at the second stage under scenario $s_2$;

3.3. Mathematical Models

3.3.1. Decision-Making at the First Stage

• Objective Function

$$\underset{x,\rho }{max}\left\{-\sum _{i=1}^I{p}_i\times x_i+\sum _{s_1\in S_1}{P}_s\times \sum _{t_1=1}^{T_1}\left(\sum _{i=1}^I{g}_i\times r_{i,t_1}^{s_1}-h_i\times u_{i,t_1}^{s_1}\right)\right\}-c{\rho }^2$$

(1)

In Objective Function (1), the expected net profit at the first stage is maximized, which is calculated by deducting the cost of capacity building, the lost sales penalty cost, and the cost of blockchain construction investment from the revenue.

2.

Decision-Making on Capacity

$$\sum _{i=1}^I{p}_i\times x_i\le b_1$$

(2)

For Decision-Making on Capacity (2), at the first stage, the constraint on decision-making levied by the investment budget of capacity building, namely Constraint (2), should be taken into consideration, representing that the investment in various product types does not exceed the total investment budget of the first stage.

3.

Decision-Making on Actual Production

$$\begin{array}{c}{x}_i+e_i\le o_{i,t_1}^{s_1}\le {\alpha }_i\times x_i+e_i\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1\end{array}$$

(3)

$$\begin{array}{c}{o}_{i,t_1}^{s_1}=c_{i,t_1}^{s_1}+\sum _{j\in J\left(i\right)}{m}_{i,j,t_1}^{s_1}\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1\end{array}$$

(4)

$$\begin{array}{c}\sum _{j\in J\left(i\right)}{m}_{i,j,t_1}^{s_1}\le {\gamma }_i\times \left(x_i+e_i\right)\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1\end{array}$$

(5)

$$\begin{array}{c}{r}_{i,t_1}^{s_1}=c_{i,t_1}^{s_1}+\sum _{k\in K\left(i\right)}{\delta }_{k,i}\times m_{k,i,t_1}^{s_1}\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1\end{array}$$

(6)

$$\begin{array}{c}{r}_{i,t_1}^{s_1}+u_{i,t_1}^{s_1}-v_{i,t_1}^{s_1}=d_{i,t_1}^{s_1}\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1\end{array}$$

(7)

$$\begin{array}{c}0\le u_{i,t_1}^{s_1}\le d_{i,t_1}^{s_1}\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1\end{array}$$

(8)

$$\begin{array}{c}0\le v_{i,t_1}^{s_1}\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1\end{array}$$

(9)

The capacities of different types of products will be adjusted according to the changes in actual demands in the actual decision-making process in production. Constraint (3) represents the actual available capacity of products, which allows excess production with new or expanded capacity and does not allow existing capacity to do so. Meanwhile, an upper limit exists for such excess production, which is determined by the excess production coefficient ${\alpha }_i$. Constraint (4) represents the allocation of the available capacity, more specifically, the capacity that is readily available for direct use in actual production or converted into the capacity for manufacturing exotic products. Therefore, the actual available capacity equals the direct production capacity (remaining capacity) plus the total converted capacity. In the meantime, there is also an upper limit for capacity conversion, namely Constraint (5), which is the total capacity converted from a single type of product must not exceed the upper limit given by the maximum capacity conversion coefficient ${\gamma }_i$ under the rated capacity. After the actual capacity is adjusted, the actual capacities of different types of products equal the sum of the remaining capacity plus the capacity converted into this product type, which comes as Constraint (6). Constraint (7) is the equilibrium equation of capacity, and the actual production plus the stockout with the overcapacity deducted equals the actual demand. Constraint (8) refers to the restriction on capacity shortage, which will not be less than zero, but will not exceed the actual demand for the product, either. In comparison, Constraint (9) represents the restriction on overcapacity, which cannot be negative.

3.3.2. Investment Decision-Making at the Second Stage

• Objective Function

$$\underset{x}{max}\underset{s_2\in S_2}{min}\underset{Y}{max}\left\{-\sum _{i=1}^I{q}_i\times y_i^{s_2}+\sum _{t_2=1}^{T_2}\left(\sum _{i=1}^I{g}_i\times r_{i,t_2}^{s_2}-h_i\times u_{i,t_2}^{s_2}-f_i\times v_{i,t_2}^{s_2}\right)\right\}$$

(10)

In Objective Function (10), the net profit under the most unfavorable scenario at the second stage is maximized, which is calculated by deducting the cost of capacity adjustment, the lost sales penalty cost, and the overcapacity penalty cost from the revenue. It should be noted that the decision-making of the second stage depends on the capacity planning of the first stage because the second stage focuses on capacity adjustments after the decision-making on capacity at the first stage, and its adjustment decision is also affected by the most unfavorable demand scenario after the decision on capacity is made at the first stage. Therefore, in the structure of Objective Function (10), the first layer remains the capacity planning decision-making at the first stage, and its decision-making is still represented by Constraint (2), which is designed to maximize the net profit. The second layer determines the most unfavorable scenario, where $S_2$ serves as the uncertainty set constructed by different demand scenarios, minimizing the net profit. Furthermore, the third layer represents capacity adjustment and decision-making on actual production under an explicit demand scenario in the second stage, maximizing the net profit. It should be noted that, in the expression of the above mathematical models, we still regard the first and the second stage as independent decision-making and the rest as a multi-stage decision-making process.

2.

Capacity Adjustment

$$\begin{array}{c}\sum _{i=1}^I{q}_i\times y_i^{s_2}\le b_2\forall s_2\in S_2\end{array}$$

(11)

$$\begin{array}{c}{\beta }_i\times \left(x_i+e_i\right)\le w_i^{s_2}\le x_i+e_i\forall i\in I,\forall s_2\in S_2\end{array}$$

(12)

The capacity can be adjusted in advance during capacity adjustment decision-making in the second stage. If the demand is predicted to grow, the capacity may be expanded continuously with restrictions on the budget placed, which is Constraint (11). If the future demand decreases sharply due to technological changes and updates or market uncertainties, the capacity will be cut accordingly, which is Constraint (12). However, since the capacity cannot be completely cut or cut on a large scale, restrictions are imposed on maximum capacity reduction, which is determined by the maximum capacity reduction coefficient ${\beta }_i$.

3.

Decision-Making on Actual Production

$$\begin{array}{c}{y}_i^{s_2}+w_i^{s_2}\le o_{i,t_2}^{s_2}\le \alpha \times y_i^{s_2}+w_i^{s_2}\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2\end{array}$$

(13)

$$\begin{array}{c}{o}_{i,t_2}^{s_2}=c_{i,t_2}^{s_2}+\sum _{j\in J\left(i\right)}{m}_{i,j,t_2}^{s_2}\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2\end{array}$$

(14)

$$\begin{array}{c}\sum _{j\in J\left(i\right)}{m}_{i,j,t_2}^{s_2}\le {\gamma }_i\times \left(y_i^{s_2}+w_i^{s_2}\right)\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2\end{array}$$

(15)

$$\begin{array}{c}{r}_{i,t_2}^{s_2}=c_{i,t_2}^{s_2}+\sum _{k\in K\left(i\right)}{\delta }_{k,i}\times m_{k,i,t_2}^{s_2}\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2\end{array}$$

(16)

$$\begin{array}{c}{r}_{i,t_2}^{s_2}+u_{i,t_2}^{s_2}-v_{i,t_2}^{s_2}=d_{i,t_2}^{s_2}\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2\end{array}$$

(17)

$$\begin{array}{c}0\le u_{i,t_2}^{s_2}\le d_{i,t_2}^{s_2}\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2\end{array}$$

(18)

$$\begin{array}{c}0\le v_{i,t_2}^{s_2}\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2\end{array}$$

(19)

The constraints of decision-making on actual production at the second stage have the same representations as that at the first stage, so detailed explanations are omitted.

3.3.3. Multi-Stage Decision-Making Models

The outermost decision-making at both the first and the second stages is the first decision made on capacity planning, namely the establishment and expansion of capacity and the input of the blockchain at the first stage. Such a decision will affect the decision on actual production made in the first stage and the decision on capacity adjustment and actual production in the second stage, with both maximizing the net profit of these two stages. Therefore, the decision on capacity planning made at the first stage can be represented as Objective Function (20):

$$\begin{array}{c}\hfill \underset{x}{max}\left\{-\sum _{i=1}^I{p}_i\times x_i+\sum _{s_1\in S_1}{P}_s\times \sum _{t_1=1}^{T_1}(\sum _{i=1}^I{g}_i\times r_{i,t_1}^{s_1}-h_i\times u_{i,t_1}^{s_1})-c{\rho }^2,\right.\\ \hfill \left.\underset{s_2\in S_2}{min}\underset{Y}{max}\left[-\sum _{i=1}^I{q}_i\times y_i^{s_2}+\sum _{t_2=1}^{T_2}(\sum _{i=1}^I{g}_i\times r_{i,t_2}^{s_2}-h_i\times u_{i,t_2}^{s_2}-f_i\times v_{i,t_2}^{s_2})\right]\right\}\end{array}$$

(20)

In Objective Function (20), the outer-layer decision, also the first-stage decision on capacity planning, will affect the net profit of the two stages. To be specific, excess capacity input at the first stage may lead to overcapacity at the second stage, or inadequate capacity input at the first stage may cause capacity shortage at the second stage. Therefore, we utilize a linear weighting method. Let ${\omega }_1$ and ${\omega }_2$ represent the importance of the net profits of different stages, and thereby we can integrate multiple objectives into a single one. Since investing in consortium blockchains may help track information of upstream and downstream manufacturers, larger investments in the blockchain at the first stage will help manufacturers better coordinate with other enterprises, get financed, and alter the profit structure in the second stage. Therefore, when ${\omega }_1=1,{\omega }_2=e^{-\varphi \rho }$, Objective Function (21) is presented as follows:

$$\begin{array}{cc}\hfill & \underset{x,\rho }{max}\left\{-\sum _{i=1}^I{p}_i\times x_i+\sum _{s_1\in S_1}{P}_s\times \sum _{t_1=1}^{T_1}(\sum _{i=1}^I{g}_i\times r_{i,t_1}^{s_1}-h_i\times u_{i,t_1}^{s_1})-c{\rho }^2\right.\hfill \\ \hfill & \left.+e^{-\varphi \rho }\times \underset{s_2\in S_2}{min}\underset{Y}{max}\left[-\sum _{i=1}^I{q}_i\times y_i^{s_2}+\sum _{t_2=1}^{T_2}(\sum _{i=1}^I{g}_i\times r_{i,t_2}^{s_2}-h_i\times u_{i,t_2}^{s_2}-f_i\times v_{i,t_2}^{s_2})\right]\right\}\hfill \end{array}$$

(21)

Objective Function (21) and Constraints (2)–(9) and (11)–(19) constitute a multi-stage decision-making model, namely the long-term capacity planning model.

4. Solving the Model

In this model, two-stage stochastic programming and robust optimization are combined, leading to a multi-layered decision-making structure with various directions in the second decision-making stage. Therefore, the model cannot be solved directly. At the same time, capacity conversion may lead to a cycle of capacity conversion in solving the problem, which means the decision-making results cannot reflect the actual situation of overcapacity. These are two main difficulties to solve the model. To address the difficulties, we introduce auxiliary variables and stratify the model, transforming the original model into a solvable planning model.

4.1. Introduction of 0–1 Auxiliary Decision Variables

In the original model, the sets $j\in J\left(i\right),k\in K\left(i\right)$ cannot be expressed directly, and the defined variables $m_{i,j,t_1}^{s_1},m_{i,j,t_2}^{s_2}$ cannot be directly used to construct constraints as a method to prevent capacity conversion cycles. The 0–1 variables are introduced to facilitate the mathematical expression of $j\in J\left(i\right),k\in K\left(i\right)$ as $z_{i,j,t}^s=0$ or 1. More specifically, $z_{i,j,t}^s=0$ means that the capacity of the product type i is not allowed to convert into that of the product type j, whereas, if the result is 1, the above capacity conversion is allowed. By using the properties of 0–1 variables, the linear constraints are constructed as follows:

$$m_{i,j,t}^s\le M\times z_{i,j,t}^s\forall i,j\in I,\forall t\in T,\forall s\in S$$

(22)

In Constraint (22), M represents a positive number that is big enough, which represents the restriction on capacity conversion. If $z_{i,j,t}^s=0$, the capacity of the product type i cannot be converted into that of the product type j. In the meantime, in order to prevent the cycle of capacity conversion, the “forbidden circle method” is hereby adopted to construct the constraint to prevent the cycle of capacity conversion; that is, to prevent $z_{i,j,t}^s$ in the cycle of all product types from being 1. The specific constraint is presented as follows:

$$z_{i,i,t}^s=0\forall i\in I,\forall t\in T,\forall s\in S$$

(23)

Constraint (23) represents the prevention of conversion with the same product type;

$$z_{i,j,t}^s+z_{j,i,t}^s\le 1\forall n_m\in I,\forall t\in T,\forall s\in S$$

(24)

Constraint (24) indicates that any two product types can make one-way capacity conversion only;

$$\begin{array}{c}{z}_{n_1,n_2,t}^s+z_{n_2,n_3,t}^s+z_{n_3,n_1,t}^s\le 2\forall n_m\in I,\forall t\in T,\forall s\in S\end{array}$$

(25)

$$\begin{array}{c}{z}_{n_1,n_2,t}^s+z_{n_2,n_3,t}^s+z_{n_3,n_4,t}^s+z_{n_4,n_1,t}^s\le 3\forall n_m\in I,\forall t\in T,\forall s\in S\\ \cdots \end{array}$$

(26)

$$\begin{array}{c}{z}_{n_1,n_2,t}^s+z_{n_2,n_3,t}^s+\dots \dots +z_{N-1,N,t}^s+z_{N,1,t}^s\le N-1\forall n_m\in I,\forall t\in T,\forall s\in S\end{array}$$

(27)

Constraints (25)–(27) represent that the capacity conversion cycle is prohibited among three and more product types, where N is the cardinal number of Set I. Thus, the representational constraints on $j\in J\left(i\right)$ and $k\in K\left(i\right)$ can be constructed to prevent the emergence of capacity conversion cycles, and all the constraints introduced hereby are linear constraints.

Introduction of Auxiliary Variables of Objective Functions: $\lambda ,\eta$

The decision-making at the second stage is conducted under the most unfavorable demand scenario, so auxiliary decision variables are hereby used to express such decision-making in mathematical form. Next, we transform the multi-layered decision-making structure into a single-layered one.

The original objective function of the model is presented as follows:

$$max\{min[max\left(\mathrm{profit}\right)\left]\right\}$$

Under the structure, the main decision made at the outermost layer is the investment decision made in the first stage. For the actual production decision under the first-stage demand, this decision is a single-stage maximization mathematical model instead of a multi-layered structure. For the second stage, both capacity adjustment decisions and actual production decisions depend on the results of the capacity investment decisions of the first stage, so the subsequent deep structure cannot be eliminated. However, if the results of the investment decision made at the first stage are known, the outermost max can be eliminated, and the decision of the second stage can be converted into $minmax$(profit), reducing the structural layers of the decision-making model, and then directly solve the $minmax$(profit) with the model properties.

First-stage decision-making model: in this model, considering merely the actual production decision made at the first stage, set the capacity input decision $x\in X$ as a known variable. The model structure is presented as follows:

$$\begin{array}{cc}\hfill f_1\left(x\right)=& max\left(\lambda \right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \lambda =-p_i\times x_i+\sum _{s_1\in S_1}{P}_s\times \sum _{t_1=1}^{T_1}\left(\sum _{i=1}^I{g}_i\times r_{i,t_1}^{s_1}-h_i\times u_{i,t_1}^{s_1}\right)-c{\rho }^2\hfill \\ & \left(3\right)\text{–}\left(9\right)\hfill \end{array}$$

The first-stage decision-making model, $f_1\left(x\right)$, is a planning model that can be solved directly.

In the second-stage decision-making model, regard the result of the capacity investment decision made at the first stage present as a known variable, and represent it by x. In the second-stage $minmax$(profit) decision-making structure, the outer layer is to minimize the net profit, and its decision is made under the most unfavorable demand scenario, which means, when x is known, the most unfavorable demand scenario is chosen to minimize the inner max(profit). Based on the above analysis, the auxiliary decision variable $\eta$ is hereby introduced with a constraint form presented as follows:

$$\eta \le -\sum _{i=1}^I{q}_i\times y_i^{s_2}+\sum _{t_2=1}^{T_2}\left(\sum _{i=1}^I{g}_i\times r_{i,t_2}^{s_2}-h_i\times u_{i,t_2}^{s_2}-f_i\times v_{i,t_2}^{s_2}\right)\forall s_2\in S_2$$

Specifically, $\eta$ is less than or equal to the decision made under all demand scenarios. This constraint can represent the minimized decision of the outer layer. After introducing $\eta$, the maximized $\eta$ can represent the $minmax$(profit) decision-making structure. Therefore, the second-stage decision-making model is presented as follows:

$$\begin{array}{cc}\hfill f_2\left(x\right)=& max\left(\eta \right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \eta \le -\sum _{i=1}^I{q}_i\times y_i^{s_2}+\sum _{t_2=1}^{T_2}\left(\sum _{i=1}^I{g}_i\times r_{i,t_2}^{s_2}-h_i\times u_{i,t_2}^{s_2}-f_i\times v_{i,t_2}^{s_2}\right)\forall s_2\in S_2\hfill \\ & \left(11\right)\text{–}\left(19\right)\hfill \end{array}$$

4.2. Solvable Planning Models

Based on the above decomposition and transformation of models, the long-term capacity planning model can be represented as:

$$max\left\{f_1\left(x\right)+e^{-\varphi \rho }\times f_2\left(x\right)\right\}$$

The objective function of this decision is to maximize, with the outer layer maximizing the multi-stage net profit. The first-stage $f_1\left(x\right)$ and the second-stage $f_2\left(x\right)$ maximize the net profit of each stage, so the redundant maximized structure can be eliminated. It follows that the whole model can be transformed into a single maximized mathematical optimization model, whose mathematical form is presented as follows:

$$\begin{array}{cc}& max\{\lambda +e^{-\varphi \rho }\times \eta \}\hfill \\ \hfill s.t.& \lambda =-p_i\times x_i+\sum _{s_1\in S_1}{P}_s\times \sum _{t_1=1}^{T_1}(\sum _{i=1}^I{g}_i\times r_{i,t_1}^{s_1}-h_i\times u_{i,t_1}^{s_1})-c{\rho }^2\hfill \\ & z_{i,i,t_1}^{s_1}=0\forall i\in I,\forall t_1\in T_1,\forall s_1\in S_1.\hfill \\ & z_{i,j,t_1}^{s_1}+z_{j,i,t_1}^{s_1}\le 1\forall i,j\in I,\forall t_1\in T_1,\forall s_1\in S_1.\hfill \\ & z_{n_1,n_2,t_1}^{s_1}+z_{n_2,n_3,t_1}^{s_1}+z_{n_3,n_1,t_1}^{s_1}\le 2\forall n_m\in I,\forall t_1\in T_1,\forall s_1\in S_1.\hfill \\ & \cdots \cdots \hfill \\ & z_{n_1,n_2,t_1}^{s_1}+z_{n_2,n_3,t_1}^{s_1}+......+z_{N-1,N,t_1}^{s_1}+z_{N,1,t_1}^{s_1}\le N-1\forall n_m\in I,\forall t_1\in T_1,\forall s_1\in S_1.\hfill \\ & m_{i,j,t_1}^{s_1}\le M\times z_{i,j,t_1}^{s_1}\forall i,j\in I,\forall t_1\in T_1,\forall s_1\in S_1.\hfill \\ & \eta \le -\sum _{i=1}^I{q}_i\times y_i^{s_2}+\sum _{t_2=1}^{T_2}(\sum _{i=1}^I{g}_i\times r_{i,t_2}^{s_2}-h_i\times u_{i,t_2}^{s_2}-f_i\times v_{i,t_2}^{s_2})\forall s_2\in S_2\hfill \\ & z_{i,i,t_2}^{s_2}=0\forall i\in I,\forall t_2\in T_2,\forall s_2\in S_2.\hfill \\ & z_{i,j,t_2}^{s_2}+z_{j,i,t_2}^{s_2}\le 1\forall i,j\in I,\forall t_2\in T_2,\forall s_2\in S_2.\hfill \\ & z_{n_1,n_2,t_2}^{s_2}+z_{n_2,n_3,t_2}^{s_2}+z_{n_3,n_1,t_2}^{s_2}\le 2\forall n_m\in I,\forall t_2\in T_2,\forall s_2\in S_2.\hfill \\ & \cdots \cdots \hfill \\ & z_{n_1,n_2,t_2}^{s_2}+z_{n_2,n_3,t_2}^{s_2}+......+z_{N-1,N,t_2}^{s_2}+z_{N,1,t_2}^{s_2}\le N-1\forall n_m\in I,\forall t_2\in T_2,\forall s_2\in S_2.\hfill \\ & m_{i,j,t_2}^{s_2}\le M\times z_{i,j,t_2}^{s_2}\forall i,j\in I,\forall t_2\in T_2,\forall s_2\in S_2.\hfill \\ & \left(3\right)\text{–}\left(9\right),\left(11\right)\text{–}\left(19\right)\hfill \end{array}$$

The decision-making model is non-linear programming in which the integer variables are binary variables, and the number of integer variables depends on the number of demand scenarios at different stages, the time span, and the number of product types. This mixed integer nonlinear programming has a convex relaxation, and its solubility relies on the number of integer variables. For a moderate number of integer variables, the model can be directly solved via the programming software Python and Matlab to call the solver Gurobi or Cplex.

5. Numerical Studies

The first step of applying our algorithm to make long-term capacity planning is to estimate the specific parameters. For each product type i, by comparing the planned production quantity with the demand quantity in each scenario, we are able to calculate ${\alpha }_i$. In addition, those production parameters ${\gamma }_i,{\beta }_i,{\delta }_i$ are determined by the production line and can be directly derived from the factory data. $p_i,q_i,g_i,h_i,f_i$ are the cost parameters, and $b_1,b_2$ are the upper limit of the investment cost at the first stage and second stage, respectively.

Due to the lack of relevant data in the semiconductor industry, we can only construct a calculation example manually to verify the model’s solvability and conduct the sensitivity analysis. The manual calculation example assumes that the number of decision-making points of the two stages is six, and the capacity can be mutually converted among all product types. Other parameters are presented in

Table 1. List of parameters.

Parameter	Value	Remark
${\alpha }_i$	$1.2$	This value is assigned to all product types
${\gamma }_i$	$0.25$	This value is assigned to all product types
${\beta }_i$	$0.8$	This value is assigned to all product types
${\delta }_{k,i}$	$0.65$	Assigned to the conversion between any two product types
$p_i$	6	This value is assigned to all product types
$q_i$	6	This value is assigned to all product types
$g_i$	1	This value is assigned to all product types
$h_i$	2	This value is assigned to all product types
$f_i$	2	This value is assigned to all product types
$b_1$	75,000
$b_2$	21,000
c	3000
$\varphi$	0.04
${\omega }_1,{\omega }_2$	1
$e_1,e_2,e_3,e_4$	$1000,0,500,0$

. Since too many parameters are involved, part of them can be found in the Appendix A. This example is analyzed via MATLAB programming and calling the CPLEX to solve the problem. The operating environment is i7-9700KF CPU with 16 GB RAM. The solution of the basic case, as well as the running time, are reported in

Table 2. Running time of the basic case.

${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\sum }_{\mathit{i}\in \mathit{I}}{\mathit{x}}_{\mathit{i}}$	$\mathit{\lambda }$	$\mathit{\eta }$	$\mathit{\rho }$	Optimal Value	Time $\left(\mathit{s}\right)$
4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$56.48$

.

The sensitivity analysis is based on five key parameters in the above calculation example, namely the excess production coefficient (${\alpha }_i$), the maximum capacity reduction coefficient (${\beta }_i$), the maximum capacity conversion coefficient (${\gamma }_i$), the capacity conversion efficiency coefficient (${\delta }_{k,i}$), and blockchain effectiveness ($\varphi$).

The sensitivity analysis of the excess production coefficient and the maximum capacity reduction coefficient based on the basic calculation example is shown in

Table 3. Sensitivity analysis of excess production coefficient. Other parameters remain unchanged.

${\mathit{\alpha }}_{\mathit{i}}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\sum }_{\mathit{i}\in \mathit{I}}{\mathit{x}}_{\mathit{i}}$	$\mathit{\lambda }$	$\mathit{\eta }$	$\mathit{\rho }$	Optimal Value	Time $\left(\mathit{s}\right)$
$1.1$	3770	3280	3624	1826	12,500	11,382	76,506	1.920	87,888	$57.76$
$1.15$	4077	3137	3534	1752	12,500	17,522	77,564	1.801	95,086	$57.29$
$1.2$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$56.48$
$1.25$	4141	2996	3439	1924	12,500	27,140	78,959	1.498	106,099	$55.59$
$1.3$	4131	2996	3425	1848	12,500	31,130	79,466	1.190	110,596	$58.97$

and

Table 4. Sensitivity analysis of the coefficient of maximum capacity reduction. Other parameters remain unchanged.

${\mathit{\beta }}_{\mathit{i}}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\sum }_{\mathit{i}\in \mathit{I}}{\mathit{x}}_{\mathit{i}}$	$\mathit{\lambda }$	$\mathit{\eta }$	$\mathit{\rho }$	Optimal Value	Time $\left(\mathit{s}\right)$
$0.7$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$58.93$
$0.75$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$57.80$
$0.8$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$56.48$
$0.85$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$57.88$
$0.9$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$58.62$

. In Table 3, as the excess production coefficient continuously increases, the optimal value increases accordingly—the positive correlation results from two aspects. On the one hand, the increase in the excess production coefficient expands the actual input of the maximum capacity to offset the capacity shortage of the first stage; on the other hand, there is no possibility of excess production in the second stage by using the existing capacity of the first stage. Hence, when the excess production coefficient is large enough, the capacity reserved for the second stage is relatively smaller. However, capacity may also be adjusted in the second stage, such as the building of new capacity, which will also lead to a reduction in overcapacity. Meanwhile, thanks to excess production, demand can be met even without the input of excess capacity. Therefore, there will be less overcapacity with the maximum profit of the second stage expanded. In Table 4, the change of the maximum capacity reduction coefficient does not change the capacity input and optimal value at all because the maximum capacity reduction coefficient merely influences the decision-making of the second stage, which is made in the most unfavorable circumstances. Under such circumstances, demand grows or exceeds the first-stage capacity as a whole, meaning there will be no capacity reduction. On the contrary, new capacity will be built to make up for the insufficiency, which is why a change in the maximum capacity reduction coefficient will not change the optimal value and the optimal solution.

In the sensitivity analysis on the maximum capacity conversion coefficient, it is assumed that capacity conversion among different product types can increase the net profit of manufacturers, as shown in

Table 5. Sensitivity analysis of the coefficient of maximum capacity conversion. Other parameters remain unchanged.

${\mathit{\gamma }}_{\mathit{i}}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\sum }_{\mathit{i}\in \mathit{I}}{\mathit{x}}_{\mathit{i}}$	$\mathit{\lambda }$	$\mathit{\eta }$	$\mathit{\rho }$	Optimal Value	Time $\left(\mathit{s}\right)$
0	4176	3078	3244	2002	12,500	19,243	71,864	1.870	91,107	$54.65$
$0.15$	4073	3237	3277	1913	12,500	21,950	77,572	1.779	99,522	$55.85$
$0.2$	4187	3016	3188	1979	12,500	22,234	78,209	1.639	100,443	$57.79$
$0.25$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$56.48$
$0.3$	4186	3009	3610	1695	12,500	22,842	78,567	1.420	101,409	$57.17$
$0.35$	4015	3009	3798	1678	12,500	22,931	78,571	1.309	101,502	$55.34$

. Capacity conversion can help manufacturers adjust their capacity according to the actual demand and convert the capacity from product types with excess capacity to those with insufficient capacity. In doing so, they do not need to handle capacity shortages with excess capacity. Additionally, they could address overcapacity via capacity adjustment. At the same time, the expansion of capacity conversion will increase the net profit as well. Both Table 5 and

Table 6. Sensitivity analysis of efficiency coefficient of capacity conversion. Other parameters remain unchanged.

${\mathit{\delta }}_{\mathit{i},\mathit{k}}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\sum }_{\mathit{i}\in \mathit{I}}{\mathit{x}}_{\mathit{i}}$	$\mathit{\lambda }$	$\mathit{\eta }$	$\mathit{\rho }$	Optimal Value	Time $\left(\mathit{s}\right)$
$0.55$	4186	3016	3336	1962	12,500	21,633	78,725	1.855	100,358	$57.47$
$0.6$	4188	3016	3389	1907	12,500	22,122	78,577	1.742	100,699	$57.29$
$0.65$	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$56.48$
$0.7$	4186	3009	3533	1772	12,500	22,982	78,247	1.442	101,229	$56.59$
$0.75$	4186	3009	3600	1705	12,500	23,248	78,066	1.201	101,314	$56.40$

show that the expansion of maximum capacity conversion and the improvement of capacity conversion efficiency can bring about an increase in the net profit. The former enables an enlarged space for capacity adjustment and thereby deals with the capacity shortage and excess capacity, whereas the latter can further make up for the insufficient capacity in actual production, thereby reducing the capacity shortage.

The sensitivity analysis on blockchain implementation effectiveness is shown in

Table 7. Sensitivity analysis on blockchain effectiveness ($\varphi$).

$\mathit{\varphi }$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\sum }_{\mathit{i}\in \mathit{I}}{\mathit{x}}_{\mathit{i}}$	$\mathit{\lambda }$	$\mathit{\eta }$	$\mathit{\rho }$	Optimal Value	Time $\left(\mathit{s}\right)$
0.03	4178	3079	3248	2010	12,515	19,342	72,445	1.429	91,787	$55.98$
0.04	4188	3016	3463	1833	12,500	22,611	78,402	1.628	101,013	$56.48$
0.05	4065	3087	3598	1997	12,747	23,871	79,082	1.828	102,953	$57.87$
0.06	4089	3198	3601	2001	12,889	24,019	79,981	2.091	104,811	$60.12$

, which reflects the effectiveness of blockchain applications. The higher the effectiveness, the better the manufacturers could share information in the upstream and downstream markets, and the more financial support they could acquire in the second stage. Table 7 also shows that higher blockchain implementation effectiveness can lead to a higher net profit.

In addition, in order to verify the solvability of the mathematical model in large-scale calculation examples, four product types are considered. A manual calculation example is constructed with the following conditions: the number of scenarios at the first stage is four with the same probability of occurrence; the number of scenarios at the second stage is eight; the number of decision-making points of both stages is eight.

Table 8. Results of large-scale cases.

# Product Types	# Time Spans	# Scenarios of the First Stage	# Scenarios of the Second Stage	Running Time $\left(\mathit{s}\right)$
4	6	4	5	$56.48$
4	8	4	5	$76.16$
5	6	4	5	$804.20$
5	8	4	5	$1460.71$
6	6	4	5	>3600
6	8	4	5	>3600

shows the results of large-scale calculation examples.

As demonstrated in the running solving time of the large-scale calculation example, the increase in the number of product types and the number of time spans prolongs the solving time, and the increase in the number of product types imposes a significantly stronger effect on the increase in the solving time than the increase in the number of time spans. This is because the increase in the number of product types will not only increase the number of 0–1 variables in the model but also lead to a factorial growth of constraints in order to ensure acyclic capacity conversion. However, the 0–1 variables lead to greater difficulties, resulting in a significant increase in the solving time. In practice, semiconductor manufacturers produce a limited number of wafer product types, and the capacity conversion between different types is not always feasible. Therefore, when applying the decision-making model, the solving time can be shortened.

6. Conclusions

In the case of uncertain demand, this paper describes and models the long-term capacity planning of semiconductor enterprises, and proposes a two-stage mathematical planning model based on the combination of stochastic planning and robust optimization. Step by step, starting from putting capacity into production, we consider the capacity input by stage to reduce the overcapacity caused by the rapid decline in demand to reduce the risk borne on manufacturers in the long-term capacity building. At the same time, given the capacity conversion between different product types, sensitivity analysis is conducted on the manual calculation example to show that the existence of capacity conversion, the expansion of conversion, and the improvement of conversion efficiency can reduce capacity shortage and excess capacity, and thereby improve the maximum net profit of the manufacturers. Finally, since the consortium blockchain, as a data-sharing platform, can link the demand data in upstream markets to that of downstream markets, financial institutes are able to monitor the operations of manufacturers based on the transparent data shared in the blockchain, and thus semiconductor manufacturers get more financial support and hence possess a better profit structure.

Although we have considered the correlation between different stages (reflected in the connection between the objective function and capacity), we have not taken the correlation between demand into consideration (that is to say, the demand of different stages is regarded as independent from each other). In practice, such a correlation does exist. At the same time, the actual capacity conversion is a very complicated process. The description of capacity conversion in this paper needs to be more complex to fully reflect the actual situation. Therefore, more actual production conditions are required to enrich the description and build a more reasonable model.