Development of a Multi-Tier Collaborative Optimization Model for Semiconductor Supply Chain Management

With the rapid development of information technology and the advancement of globalization, the semiconductor industry, as an important support for modern technology, has gradually become an indispensable part of the global economy [1, 2]. The complexity and high dependency of the semiconductor industry chain make its supply chain management a highly challenging field [3-6]. Under the background of global competition and market demand fluctuations, how to effectively manage and optimize the supply links in the semiconductor supply chain [7, 8], especially the collaborative cooperation among multi-tier suppliers, has become a hot topic of concern in both academia and industry. In order to respond to the constantly changing market demand, improve supply chain efficiency, and reduce costs, the management of multi-tier supply links and collaborative decision optimization in the semiconductor supply chain is particularly important.

In the field of semiconductor supply chain management, researchers have proposed a variety of management and optimization strategies [9-11], aiming to improve the flexibility, response speed, and cost-effectiveness of the supply chain. However, existing research mainly focuses on the optimization of single suppliers or only considers the collaborative effects of partial segments in the supply chain, lacking a comprehensive analysis of the collaborative relationships among supply links in the entire multi-tier supply chain [12, 13]. Therefore, exploring a multi-tier supply link management and collaborative decision optimization model suitable for the semiconductor industry can provide more operational solutions for practical supply chain management, which has important academic significance and application value.

At present, research methods in supply chain management mostly focus on static optimization or local optimization, lacking effective response mechanisms for dynamic changes [14-17]. For example, Meshalkin and Rakitina [18] and Sawik [19] solve supply chain management problems through traditional linear programming or integer programming methods. Although these methods are theoretically effective, they often ignore the complex interaction relationships among supply chain members as well as the real-time and flexibility of supply chain response in practical operation. In addition, Mahmoudi and Fazlollahtabar [20] ignore the role of information sharing and coordination mechanisms, resulting in unsatisfactory practical application effects. Therefore, existing research methods have limitations such as excessive model simplification, insufficient dynamics, and poor practical applicability in solving the optimization problem of collaborative decision-making among multi-tier suppliers.

This paper mainly focuses on the management and collaborative optimization of multi-tier supply links in the semiconductor supply chain, and proposes an optimization model based on three-tier supply link management and collaborative decision-making. Firstly, this paper constructs a multi-tier supply chain management framework covering upstream raw material suppliers, midstream manufacturers, and downstream distributors, and optimizes the behavior of suppliers at all tiers through a collaborative decision-making model. Secondly, this paper uses advanced optimization algorithms to solve the model, aiming to improve decision-making efficiency, reduce costs, and enhance the overall response ability and stability of the supply chain in practical supply chain management. The research results not only provide theoretical basis and practical guidance for supply chain management in the semiconductor industry, but also offer new ideas and methods for supply chain optimization in related fields, with high research value and practical application prospects.

In the solution of the three-tier management and collaborative decision optimization model in the supply segment of the semiconductor supply chain, the fuzzy membership principle is first used to transform the multi-tier objectives into a single-objective programming problem for global optimization. Specifically, the objectives of customer satisfaction, manufacturer profit, and supplier profit involve certain fuzziness and uncertainty. Therefore, the fuzzy membership function is used to quantify the respective weights and influence degrees of these objectives. The introduction of fuzzy membership can effectively handle the conflicts and trade-offs between the objectives, so that each tier’s objectives can be solved through a unified single-objective function. In this model, the three-tier decision-makers interact and coordinate continuously, adjusting their decision parameters, and use the max-min method to obtain an acceptable equilibrium solution. The core of the equilibrium solution is to balance customer demand, manufacturer and supplier profit, while ensuring that each tier of decision-maker can find a mutually acceptable decision result in a changing market environment. In the second step of the optimization process, with customer satisfaction as the top-priority decision objective, an appropriate coordination mechanism is designed to meet the minimum requirements of manufacturers and suppliers, and to maximize customer satisfaction. For the semiconductor supply chain, the diversity of customer demand and the high requirements on product quality and delivery time make the optimization of customer satisfaction a key factor. By prioritizing customer satisfaction, the competitiveness and customer loyalty of semiconductor products in the market can be ensured. Meanwhile, during the optimization process, the objectives of manufacturer profit and supplier profit are also effectively considered. By setting a minimum target satisfaction for each tier, it ensures that each decision-maker can achieve overall optimization under the condition of meeting the minimum requirements. Finally, based on the Nash equilibrium principle, the model further explores whether there is room for improvement in each tier’s objectives by appropriately relaxing some decision variable constraints.

3.1 Equilibrium satisfaction solution

In the solution of the three-tier management and collaborative decision optimization model in the supply segment of the semiconductor supply chain, the specific idea of obtaining equilibrium satisfaction based on the max-min method starts from independently solving the objective function of each tier. Each decision-maker—customer, manufacturer, and supplier—performs independent optimization under constraints, considering only their own objective, and solves for their respective ideal solutions. Customer satisfaction, manufacturer profit, and supplier profit are quantified by their respective membership functions, and η^D1, η^D2, and η^D3 represent the satisfaction values of each tier decision-maker. In the semiconductor supply chain, customers have high requirements on product quality, delivery time, and technical support, so customer satisfaction as the primary objective needs to be given priority. Manufacturers and suppliers focus respectively on their own production costs, profit maximization, and supply capability, so their objective functions focus on profit optimization. After each tier independently solves the problem, a set of objective values and satisfaction values is obtained, which serve as the basis for the next step of decision-making.

(1) Interactive Negotiation Between the First Two Tiers

In the model, customer satisfaction is expressed through the objective membership function ω(D₁), and the satisfaction with respect to decision variables (such as product pricing, delivery time, etc.) is also quantified by the decision variable membership function ω(a₁). The first-tier decision-maker performs optimization under its own objective and constraints, derives an ideal solution, and calculates the corresponding satisfaction value η^D1. This satisfaction value and the objective membership function will serve as constraints for the second-tier decision-maker during optimization, ensuring that the second-tier decisions can be adjusted and optimized within the range of customer satisfaction. To obtain an effective solution, the max-min method should be applied. Let:

$\eta =MIN\left\{ \omega \left( {{D}_{1}} \right),\omega \left( {{D}_{2}} \right),\omega \left( {{a}_{1}} \right) \right\}$     (5)

In the second-tier decision optimization, the manufacturer and supplier need to consider the constraints provided by the first-tier decision-maker, including the customer satisfaction objective membership function ω(D₁) and the decision variable membership function ω(a₁). The second-tier decision-makers will maximize their own profits while ensuring that their decisions meet the customer satisfaction requirements provided by the first tier. To obtain an effective solution, the max-min method is used to transform the second-tier problem into a single-tier programming model. During the solution process, it is necessary to maximize the overall satisfaction η of all decision-makers’ objectives and decision variables.

$\begin{align}  & MAX\ \eta  \\ & s.t.\omega \left( {{D}_{1}} \right)\ge \eta \ \quad \omega \left( {{D}_{2}} \right)\ge \eta \quad \omega \left( {{a}_{1}} \right)\ge \eta \quad  \\ & {{h}_{2}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0\quad {{a}_{1}},{{a}_{2}},{{a}_{3}}\ge 0 \\\end{align}$     (6)

In practical applications of the semiconductor supply chain, due to the uncertainty of customer demand, conflicts of interest between manufacturers and suppliers, and the complexity of products, this interactive and collaborative optimization process is very important. After obtaining the result of the second-tier decision, the solution result will be fed back to the first-tier decision-maker for evaluation. If the first-tier decision-maker is satisfied with the result, the optimization process ends and the result is passed to the third-tier decision-maker. If the first tier is not satisfied, the decision-maker needs to adjust the objective membership function or satisfaction level, and pass the adjusted membership function back to the second-tier decision-maker. Upon receiving the feedback from the first tier, the second-tier decision-makers also need to adjust their own membership functions and satisfaction levels according to the satisfaction level of the first tier and re-optimize until both the first and second-tier decision-makers can accept the final result.

(2) Three-Tier Interactive Negotiation

When solving the three-tier management and collaborative decision optimization model in the supply segment of the semiconductor supply chain, the first step of obtaining equilibrium satisfaction based on the max-min method is to pass the objective membership functions and decision variable membership functions of the first-tier customer and the second-tier manufacturer and supplier to the third-tier decision-maker. The membership functions of the customer satisfaction and the profit objectives of the supplier and manufacturer become constraints at this stage and are input into the optimization model of the third tier. This means that the third-tier decision-makers not only need to maximize their own profit objectives, but also must consider the customer satisfaction and the profit requirements of the manufacturer. Similarly, let:

$\eta =MIN\left\{ \omega \left( {{D}_{1}} \right),\omega \left( {{D}_{2}} \right),\omega \left( {{D}_{3}} \right),\omega \left( {{a}_{1}} \right),\omega \left( {{a}_{2}} \right) \right\}$     (7)

Further, transform the single-tier programming model with the objective of maximizing the overall satisfaction of all decision-makers’ objectives and decision variables. In the semiconductor supply chain, the decision-makers include customers, manufacturers, and suppliers, and there are certain conflicts and coordination needs among the three parties’ objectives. The application of the max-min method means to seek an equilibrium solution by maximizing the overall satisfaction of customer satisfaction, manufacturer profit, and supplier profit. This new single-tier programming model integrates the objectives of the three parties and obtains a balance point by maximizing the satisfaction of all decision-makers, so as to achieve an optimal solution commonly accepted by the three parties. The new single-tier programming model expression is:

$\begin{align}  & MAX\ \eta  \\ & s.t.\omega \left( {{D}_{1}} \right),\omega \left( {{D}_{2}} \right),\omega \left( {{D}_{3}} \right), \\ & \omega \left( {{a}_{1}} \right),\omega \left( {{a}_{2}} \right)\ge \eta \quad {{h}_{3}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0 \\ & {{a}_{1}},{{a}_{2}},{{a}_{3}}\ge 0\quad 0\le \eta \le 1 \\\end{align}$     (8)

After obtaining the solution of the new single-tier programming model, the result needs to be fed back to the first and second-tier decision-makers for evaluation. If all three tiers of decision-makers are satisfied with the current solution, then the solution process ends and proceeds to the next step. If any tier of decision-makers is not satisfied with the result, corresponding adjustments need to be made. For example, if the customer is not satisfied with the current product quality or delivery time, the customer will adjust its satisfaction requirements, modify the objective membership function, and pass the adjusted result to the second-tier decision-maker. Similarly, if the supplier or manufacturer is dissatisfied with the result, they will also adjust their own objectives and decision variable membership functions and re-input them into the optimization model. Through such interactive adjustments, each tier maximizes its own objective while considering the objectives of other tiers, thus achieving final coordination and win-win results.

Finally, through the iterative solution of the max-min method, the equilibrium solution (a₁,a₂,a₃) and equilibrium satisfaction λ of the three-tier objective programming are obtained. In this process, the three decision-makers set their respective minimum expected satisfaction values (σ₁,σ₂,σ₃) based on the obtained equilibrium satisfaction λ and proceed to the next step of the solution process.

3.2 Priority optimization of the first-tier objective

In the three-tier management and collaborative decision optimization model of the semiconductor supply chain, the priority optimization of the first-tier objective is crucial to ensure that customer satisfaction is fully guaranteed. Due to the specific and diverse demands of customers for semiconductor products, encompassing aspects such as product quality, delivery time, and technical support, optimizing customer satisfaction should be the primary goal in supply chain decision-making. To prioritize the optimization of customer satisfaction, the supply chain coordination center will first ensure that customer demands for products are met, and then proceed with subsequent decisions. Specifically, the priority optimization of the first-tier objective is reflected in the decision-making process by adjusting the customer satisfaction membership function and setting priorities, so that the tier of customer satisfaction becomes a prerequisite for other decision objectives. This optimization method not only considers the basic needs of customers but also ensures that customer satisfaction, as a constraint, fully guides the decisions of the second tier, avoiding the sacrifice of basic customer needs while satisfying the profits of suppliers or manufacturers, thereby enhancing the overall efficiency and stability of the supply chain.

(1) Maximizing the First-Tier Objective

One of the specific steps in the priority optimization of the first-tier objective is to consider customer satisfaction as the most important goal, maximizing customer satisfaction while ensuring that the objectives of manufacturers and suppliers are reasonably met. First, the first-tier decision-maker establishes a single-tier planning model, prioritizing the maximization of customer satisfaction under the premise of considering the minimum satisfaction of manufacturers and suppliers. This means that the first-tier decision-maker needs to adopt corresponding decision strategies, such as adjusting product quality and delivery time, to meet customer demands while ensuring that the minimum satisfaction levels of lower-tier decision-makers are not compromised. For the semiconductor supply chain, the diversity and high requirements of customer demands make optimization in aspects such as product quality and delivery time particularly important.

$\begin{align}  & MAX{{\sigma }_{1}} \\ & s.t.\omega \left( {{D}_{1}} \right)\ge {{\sigma }_{1}}\quad \omega \left( {{D}_{2}} \right)\ge {{\sigma }_{2}}\quad  \\ & \omega \left( {{D}_{3}} \right)\ge {{\sigma }_{3}}\quad \omega \left( {{D}_{1}} \right),\omega \left( {{D}_{2}} \right)\ge \eta  \\ & {{h}_{1}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0\quad {{a}_{1}},{{a}_{2}},{{a}_{3}}\ge 0 \\\end{align}$     (9)

In the actual operation of the supply chain, due to conflicts of interest among decision-makers at various tiers, the optimal solution of the first-tier decision-maker may not be achievable in practice. Therefore, to address the issue of infeasible solutions caused by conflicts of interest, the first-tier decision-maker needs to adjust the model according to the actual situation, relax some constraints or adjust the objective membership function to find a feasible solution. For example, the first-tier decision-maker may moderately relax certain requirements of customer satisfaction, allowing for some compromise to ensure that the operation of the entire supply chain is not overly restricted. In the semiconductor supply chain, the profit objectives of manufacturers and suppliers often conflict with customer satisfaction, especially in aspects such as product pricing and delivery time. Therefore, the first-tier decision-maker needs to flexibly adjust decisions to balance customer satisfaction with the overall efficiency of the supply chain.

(2) Optimization Using Adjustment Mechanism

When the model has no feasible solution, the first-tier decision-maker, in order to achieve its minimum satisfaction goal σ₁, will, when necessary, reduce the target satisfaction of the lower two tiers of decision-makers. The core of this adjustment mechanism lies in appropriately adjusting the target satisfaction values of the lower two tiers of decision-makers to ensure that the first-tier decision-maker can maximize the customer satisfaction objective while maintaining the acceptability of the final solution by the lower-tier decision-makers. In specific operations, the first-tier decision-maker will adjust the minimum satisfaction of manufacturers and suppliers to ensure that their satisfaction levels are not lower than the equilibrium satisfaction η obtained through the max-min method. In this process, manufacturers and suppliers in the semiconductor supply chain usually have strong profit demands. Therefore, adjusting their target satisfaction requires careful consideration to ensure that their profit expectations are not excessively compressed, avoiding intensified conflicts of interest. Next, to ensure the feasibility of the model and the acceptability of the lower-tier decision-makers, the first-tier decision-maker will minimize the reduction range of the lower-tier target satisfaction, i.e., minimize the value of β. The specific operation of this step is to further optimize the adjustment parameters under the constraints that the first-tier target satisfaction and the lower-tier target satisfaction are not lower than the equilibrium satisfaction, so that the reduction range is minimized. Manufacturers and suppliers in the semiconductor supply chain are highly sensitive to profits. Therefore, when the first-tier decision-maker adjusts the lower-tier satisfaction, it needs to accurately calculate the minimum adjustment amount to avoid excessive profit losses or supply chain instability. The model expression is as follows:

$\begin{align}  & MIN\beta  \\ & s.t.\omega \left( {{D}_{1}} \right)\ge {{\sigma }_{1}}\quad \omega \left( {{D}_{2}} \right)\ge {{\sigma }_{2}}-\beta \ge \eta \quad  \\ & \omega \left( {{D}_{3}} \right)\ge {{\sigma }_{3}}-\beta \ge \eta \quad \omega \left( {{a}_{1}} \right)\ge \eta  \\ & \omega \left( {{a}_{2}} \right)\ge \eta \quad {{h}_{1}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0\quad  \\ & {{h}_{2}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0\quad {{h}_{3}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0 \\ & {{a}_{1}},{{a}_{2}},{{a}_{3}}\ge 0\quad 0\le \beta <1 \\\end{align}$     (10)

Let Ξ be the ratio of the actual target satisfaction of the lower two tiers to the minimum expected satisfaction obtained in the first step. To fully optimize the solution obtained by the model, let:

$\Xi =\frac{MIN\left\{ \omega \left( {{D}_{u}} \right) \right\}}{\sigma },u=2,3$     (11)

$\Xi \in \left[ {{\Xi }^{\#}},{{\Xi }^{*}} \right]$     (12)

If Ξ>Ξ^*, it indicates that the first-tier decision-maker still needs to further improve its own target satisfaction. The set σ₁ can be increased by 1%, and the model is solved again until Ξ$\in$[Ξ^#, Ξ^*].

If Ξ<Ξ^* and ∆<∆^#, the minimum satisfaction value of the lower-tier targets should be increased, let:

${{\sigma }_{e}}=\sigma \cdot {{\Xi }^{\#}}$     (13)

Then further solve the model:

$\begin{align}  & MAX\eta  \\ & s.t.\omega \left( {{D}_{1}} \right),\omega \left( {{D}_{2}} \right),\omega \left( {{D}_{3}} \right),\omega \left( {{a}_{1}} \right),\omega \left( {{a}_{2}} \right)\ge \eta \quad  \\ & \omega \left( {{D}_{u}} \right)\ge {{\sigma }_{u}}\left( u\ne e \right)\quad \omega \left( {{D}_{e}} \right)\ge {{\sigma }_{e}} \\ & {{h}_{1}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0\quad {{h}_{2}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0\quad  \\ & {{h}_{3}}\left( {{a}_{1}},{{a}_{2}},{{a}_{3}} \right)\le 0\quad {{a}_{1}},{{a}_{2}},{{a}_{3}}\ge 0\text{  }0\le \eta \le 1 \\\end{align}$     (14)

If the above equation has a feasible solution, proceed to overall optimization. Otherwise, reduce the set σ₁ by 1% and solve the model again until Ξ$\in$[Ξ^#, Ξ^*]. Finally, the satisfaction of each tier's objectives and decision variables can be obtained: i*(D₁), i(D₂), i"(D₃), i*(a₁), i*(a₂).

3.3 Overall optimization of the three-tier objectives

After the interactive collaborative optimization, a solution satisfactory to all three tiers of decision-makers has been obtained. However, this solution does not fully consider the potential improvement space among the various objectives. Especially in the second step, due to the non-compensatory nature of the max-min operator while maximizing the satisfaction of the primary objective, there may still be room for further optimization of other objectives. Therefore, the first step of overall optimization is to seek further Pareto improvement on the premise that the satisfaction of each tier's objective is not lower than the result of the second step. Specifically, among customer satisfaction, manufacturer profit, and supplier profit, decision-makers need to balance customer demands and the profit objectives of supply chain members through a flexible coordination mechanism, avoiding damage to the interests of other objectives while pursuing one particular objective. In the semiconductor supply chain, customers have high requirements for product quality and delivery time, while manufacturers and suppliers have strong demands regarding price, delivery time, and profit. Therefore, the overall optimization process must emphasize coordination among the three, ensuring a mutually beneficial situation among the three tiers of decision-makers.

In actual operation, the optimization of the semiconductor supply chain not only involves the adjustment of objective satisfaction but also requires refined management of the interests of each tier of decision-makers. To achieve overall optimization, the key to the first step is to fine-tune the result obtained from the second step, ensuring that while meeting the minimum satisfaction of lower-tier decision-makers, the potential space for benefit improvement is explored and realized. Specifically, the profit objectives of manufacturers and suppliers may be improved by optimizing production scheduling, adjusting pricing strategies, or optimizing supply chain resource allocation. The improvement of customer satisfaction may be achieved by enhancing product quality, shortening delivery time, or providing better after-sales services.

The first step in the overall optimization process is to use the principle of Pareto improvement. Based on the objective satisfaction obtained in the second step as the starting point of negotiation, a Nash cooperative solution form is used to design the objective function. Through Nash bargaining theory, the cooperative solution form can effectively promote the interest balance among decision-makers at each tier, thereby providing theoretical support for the overall optimization of the three-tier objectives. In the semiconductor supply chain, the objectives of customers, manufacturers, and suppliers usually conflict. Customers have strict requirements for product quality and delivery time, while manufacturers and suppliers have their own demands in terms of profit, production scheduling, and pricing. By using the Nash cooperative solution as the optimization objective function, three-tier decision-makers can find a balance of their respective interests in the process of seeking Pareto improvement, thus achieving the joint improvement of customer satisfaction, manufacturer profit, and supplier profit. The corresponding objective function is:

$\begin{align}  & \left[ \omega \left( {{D}_{1}} \right)-\omega *\left( {{D}_{1}} \right) \right]\left[ \omega \left( {{D}_{2}} \right)-\omega *\left( {{D}_{2}} \right) \right]  \left[ \omega \left( {{D}_{3}} \right)-\omega *\left( {{D}_{3}} \right) \right] \\\end{align}$     (15)

After the optimization of the previous two steps, the feasible domain of the third step may be greatly restricted, resulting in a smaller solution space. Therefore, after the second step has appropriately relaxed the objective satisfaction, the third step must further improve the rigidity of the constraint conditions to ensure that more possible optimal solutions can be explored. Specifically, at this stage, by appropriately relaxing the constraints on decision variables, more flexibility and adjustment space are allowed, thereby providing a broader solution space for optimization. In the actual operation of the semiconductor supply chain, this step may involve moderate adjustments to production planning, inventory management, or logistics scheduling. For example, allowing a certain degree of delivery time extension or appropriately lowering some quality standards, so that manufacturers and suppliers can obtain more profit space while ensuring product quality. By reducing the satisfaction of decision variables and introducing the adjustment parameter α, constraint conditions can be flexibly adjusted to obtain a more reasonable optimal solution.

$\omega \left( {{a}_{k}} \right)\ge \omega *\left( {{a}_{k}} \right)-\alpha ,k=1,2,3$     (16)

Finally, based on the relaxed constraints and the Nash cooperative solution, the decision optimization model of the third step is established. The objective of this optimization model is to explore solutions that can achieve greater benefits while maintaining a balance among customer satisfaction, manufacturer profit, and supplier profit. In the semiconductor supply chain, the optimization model of the third step not only requires the optimization of customer satisfaction and the profits of manufacturers and suppliers but also needs to consider the coordination efficiency of each link of the supply chain. For example, by optimizing raw material procurement, production processes, logistics configuration, etc., the operational efficiency of the entire supply chain can be further improved, thus providing a more acceptable and superior solution for the three tiers of decision-makers. The expression of the decision optimization model in the third step is as follows:

$\begin{aligned} & \operatorname{MAX}\left\{\begin{array}{l}{\left[\omega\left(D_1\right)-\omega^*\left(D_1\right)\right]\left[\omega\left(D_2\right)-\omega^*\left(D_2\right)\right]} \\ {\left[\omega\left(D_3\right)-\omega^*\left(D_3\right)\right]}\end{array}\right\} \\ & \text { s.t. } \omega\left(a_k\right) \geq \omega^*\left(a_k\right)-\alpha, k=1,2,3 \\ & \omega\left(D_u\right) \geq \omega^*\left(D_u\right), u=1,2,3 \\ & h_1\left(a_1, a_2, a_3\right) \leq 0 \quad h_2\left(a_1, a_2, a_3\right) \leq 0 \\ & h_3\left(a_1, a_2, a_3\right) \leq 0 \\ & a_1, a_2, a_3 \geq 0 \quad 0 \leq \eta \leq 1\end{aligned}$     (17)