Electric Multiple Unit Spare Parts Vendor-Managed Inventory Contract Mechanism Design

Abstract

As electric multiple unit (EMU) operations and maintenance demands have expanded, spare parts supply chain management has become increasingly crucial. This study emphasizes the supply challenges of EMU spare parts, including inadequate minimum inventory levels and prolonged response times. Redesigning the OEM–railway bureau vendor-managed inventory (VMI) model contract incentive and penalty system is the key goal. Connecting the spare parts supply system with its characteristics yields a game theory model. This study analyzes and compares the equilibrium strategies and profits of supply chain members under different mechanisms for managing critical spare parts. The findings demonstrate that mechanism contracts can enhance supply chain performance in a Pareto-improving manner. An in-depth analysis of downtime loss costs, procurement challenges, and order losses reveals their effects on supply chain coordination and profit allocation, providing railway bureaus and OEMs with a theoretical framework for supply chain decision-making. This study offers theoretical justification and a framework for decision-making on cooperation between OEMs and railroad bureaus in the management of spare parts supply chains, particularly for extensive EMU operations.

1. Introduction

China’s EMU inventory is considerable and is steadily entering the overhaul period, leading to an increasingly prominent demand for vehicle spare parts replacement. By the end of 2024, the operating length of high-speed trains in our country will exceed 48,000 km, with a nationwide EMU fleet of 4806 standard sets. A maintenance system for EMU is presently utilized, predominantly reliant on mileage cycles, augmented by time cycles. Since the commencement of the EMU in 2007, the initial fleet of vehicles has been operational for nearly 15 years. According to temporal cycles, cars introduced in later years will progressively transition into advanced maintenance phases, resulting in a substantial need for spare components. The EMU spare parts inventory must cover both emergency and normal operational needs, while also considering considerations such as capital occupation, unknown demand fluctuations, and concerns originating from model discontinuance and phase-out, like excess inventory and scrapping risk. It is essential to maintain equilibrium between transportation production and capital utilization, as the supply assurance for EMU replacement parts encounters considerable difficulties.

To tackle significant challenges, including the extensive inventory of EMU spare parts and inadequate emergency supply assurance capabilities at the railway bureau, GT Group has entered into a strategic cooperation agreement with ZC Company. Under this arrangement, the OEM under ZC Company will develop and maintain the parts center and parts sub-center for train components, serving as warehouses for VMI model spare parts reserves. This structure will enable the centralized procurement, overall storage, and coordinated distribution of major EMU spare parts by the OEM. The OEM and the railway bureau will engage in negotiations to formulate a framework agreement for the “inventory catalog”, appointing the OEM as the exclusive supplier. The VMI model presents several benefits compared to the conventional self-managed inventory system of the railway bureau; however, challenges persist, including a low fulfillment rate for the “minimum inventory level” of certain emergency critical spare parts and extended response times for non-emergency critical spare parts. Under the VMI approach, the OEM provides EMU spare parts to the railway bureau in accordance with the established “inventory catalog”. Two supply methods exist: emergency critical spare parts are quantitatively held at the railway bureau parts sub-center, whereas non-emergency critical spare parts are centrally stored at the parts center and distributed as required. When reserves are inadequate, supply is facilitated via market procurement. At present, the disparity in cost distribution between the OEM, which incurs inventory management expenses and additional risks, and the railway bureau results in inadequate incentives to sustain supply reserves. The OEM typically reserves high-value, fast-turnover obligatory replacement components for emergency vital spare parts with a “minimum inventory level” to mitigate risk, leading to a reserve structure that fails to satisfy the railway bureau’s requirements. The real reserve for the CR400BF model (manufactured by CRRC Changchun Railway Vehicles Co., Ltd., Changchun, China and CRRC Tangshan Co., Ltd., Tangshan, China) of a certain railway agency attained just 82.95% of the objective. For non-emergency vital spare parts without a “minimum inventory level”, some parts confront procurement problems, extended cycles, and high comprehensive procurement costs. Of the 15 pantograph camera demand plans filed in October 2023, just one was fulfilled by January 2024, impacting maintenance operations. In summary, in the actual implementation of the VMI model, the OEM needs to take on more duties such as inventory management and supply channel development, leading to increasing operational costs. Nonetheless, the revenue share has not proportionately risen, directly influencing its propensity to collaborate. This underscores the significance of judiciously establishing the OEM incentive mechanism. Therefore, based on the two types of supply methods for the two categories of spare parts, establishing a reasonable reward and penalty mechanism to encourage the OEM and the railway bureau to actively cooperate in implementing the VMI model is of significant practical importance to improve the demand satisfaction rate of EMU spare parts for the railway bureau. This paper’s primary contributions are as follows:

• Existing research on the EMU spare parts mainly focuses on demand forecasting, inventory optimization, and maintenance strategy and only considers decision optimization for individual entities in the supply chain, with insufficient attention to cooperation between upstream and downstream enterprises in the supply chain. This paper studies the design of the contract mechanism for EMU spare parts under the VMI model, aiming to improve the overall efficiency and responsiveness of the supply chain, enriching the research on spare parts coordination.
• The coordination role of different contracts under uncertain demand environments in various supply methods is explored in depth, enriching the perspective of scenario description in the design of contractmechanisms under the VMI model. The model takes into account factors such as the downtime loss cost of spare parts and the difficulty of spare parts procurement, thereby expanding the research perspective of the VMI model.

2. Literature Review

Existing research on EMU spare parts primarily focuses on spare parts classification, demand forecasting, maintenance strategy, and location strategy optimization. Bacchetti and Saccani [1] examined the gap between theory and practice in spare parts management and proposed directions for bridging this gap. Teixeira et al. [2] researched the inventory management of spare parts from the perspectives of maintenance and logistics. He Yong et al. [3] developed a phased preventive maintenance model based on the EMU maintenance system. Zhang et al. [4] examined the integrated optimization of location and inventory for the distribution of EMU maintenance parts. Li et al. [5] investigated the vehicle scheduling problem during disruptions in the high-speed railway network. Research on upstream and downstream collaboration in the equipment supply chain has mainly focused on after-sales maintenance services. Rahim et al. [6] studied the design of multi-resource service supply chain contracts between frontline service supply providers and emergency service supply providers under conditions of information asymmetry. Qin et al. [7] explored the design of equipment after-sales service contracts considering interruption insurance, finding that performance-based contracts outperform resource-based contracts in terms of system performance due to inherent distorted incentives. Kim et al. [8] analyzed the pros and cons of three types of contracts: fixed payment, cost-sharing, and performance-based (PBC). Some scholars have considered the main characteristics of spare parts, such as the diversity of parts, high uncertainty, and the critical impact of immediate replacement needs, incorporating flexible response time in after-sales contracts. Lamghari et al. [9] introduced a flexible time contract in spare parts service contracts, where the service provider pays a penalty for each new downtime if the number of times downtime exceeds the threshold surpasses the agreed upon number. A limited number of studies have considered contracts related to the spare parts supply chain reserve supply. Zhang et al. [10] examined the design of a performance-based contract for sharing inventory cost between suppliers and operators, further exploring how the inventory cost-sharing ratio affects operator profits and contract parameters. Shi Zhenyang [11] investigated decision-making issues within a service spare parts supply chain composed of product manufacturers, spare parts suppliers, and aftermarket manufacturing service providers, in the context of service spare parts supply disruption following product discontinuation, and explored a coordination mechanism based on a buyback contract. Currently, there is a lack of research on the coordination of the EMU spare parts supply chain, thus studying the contract mechanism of the EMU spare parts supply chain is of significant practical importance.

Under the VMI model, suppliers are responsible for determining the inventory levels of retailers, necessitating a reconsideration of the supply chain contract. Numerous studies have been conducted on supply chain contracts under the VMI model, such as the risk-sharing contract [12], which features risk-sharing and unilateral compensation characteristics, and the option contract [13], widely applied in industries such as fashion, electronics, and fast-moving consumer goods. The subsidy contract is common in the mobile phone industry [14]. Ma Chao et al. [15], considering the replenishment strategy adopted by suppliers, constructed risk diversification, option, and subsidy contracts under a two-level VMI supply chain and explored the impact of suppliers’ loss aversion. D Wang et al. [16] explored how coordination of the VMI supply chain can be achieved through commitment–penalty contracts under conditions of bilateral asymmetric information. Zhu et al. [17] constructed a two-level VMI supply chain model with a revenue-sharing contract and theoretically derived the optimal revenue-sharing ratio and optimal replenishment quantity. Liu et al. [18] examined three different types of contracts—risk diversification and cost-sharing contracts, option and cost-sharing contracts—within a VMI supply chain consisting of a supply with risk aversion and a risk-neutral retailer. They also considered the coordination of the supply chain with a replenishment strategy. Qu Jiali and Hu Benyong [19] introduced a revenue-sharing contract under VMI, achieving a Pareto improvement in the performance of the supply chain amid supply and demand uncertainty. Cai et al. [20] facilitated the coordination of the VMI supply chain under output uncertainty through an option contract. Fan Chen et al. [21], building on the concept of risk-sharing, integrated the replenishment strategy with contract design under random demand, thereby providing the VMI supply chain with a certain level of robustness against parameters affecting its operational efficiency. Yin and Ma [22] investigated scenarios where manufacturers face output uncertainty and retailers face demand uncertainty, introducing unit reward and one-time reward contracts to achieve a Pareto improvement in the supply chain. However, there is limited research on designing contracts tailored to different supply scenarios. This paper will closely relate to practical situations, considering the design of a contract mechanism under quantity commitment and time response supply methods.

Currently, research on quantity commitment primarily investigates the impact of retailer quantity commitment on the output of the supply. Focusing on how changes in supply prices affect quantity commitment, under the framework of the minimum quantity commitment contract, Gong Xiting [23] develops a dynamic pricing strategy by coordinating inventory decisions and price discounts, thereby jointly optimizing the enterprise’s procurement decisions. The research results identified the optimal ordering decisions of retailers under different discounts. Cai et al. [24] studied the commitment order contract under the uncertainty of the VMI model supply and demand. They explored the optimal input quantity for the supply at various wholesale prices and examined how the retailer’s commitment order quantity affects the supplier’s input decisions. Han et al. [25] investigated the decision-making process for determining the minimum commitment quantity and long-term ordering strategy at the start of the planning period, in scenarios where the retailer commits to an order quantity with a supply offering price discounts. This paper fully considers the uniqueness of spare parts in EMU depots, focusing on the reserve decision of the parts center, which is directly constrained by the railway bureau’s “minimum inventory level” requirement. It emphasizes the impact of spare parts downtime loss cost on the decision-making and contract parameter settings for both the railway bureau and the parts center. Furthermore, in the study of response time, most of the literature focuses on delivery time decision research. Qiu Ruozhen et al. [26] examined the decision-making problem in a dual-channel supply chain under demand sensitive to price and delivery time. Research on supply chaincontracts mostly revolves around deterministic demand. Modak et al. [27] constructed a linear demand model influenced by price and delivery time, using a quantity discount contract for supply chain coordination. This paper will expand the research on supply chain contracts by considering response time under uncertain demand.

This paper extends existing research by focusing on the design of an EMU spare parts supply chain VMI contract mechanism, considering two supply modes: spare parts characteristics and the OEM. Through the mathematical modeling method, it establishes two mechanism models under quantity commitment: residual subsidy and cost sharing under time response. It analyzes the optimal parameter values for maximizing the benefits of the supply chain within the mechanism, facilitating decision-making for the railway bureau and OEM. This aims to improve the satisfaction rate of railway bureau EMU maintenance spare parts while avoiding excessive capital occupation by OEM in spare parts, optimizing the inventory structure at the parts center, and achieving maximum benefits for the EMU spare parts VMI supply chain.

3. Problem Description and Model Assumptions

3.1. Problem Description

There are two channels for the supply of EMU spare parts, as shown in Figure 1. The primary channel is the OEM channel, which utilizes the VMI model and establishes a parts center within the railway bureau’s EMU maintenance depots. These centers supply EMU spare parts to both the EMU maintenance base and the EMU depot. The secondary channel involves market diffusion, where the railway bureau procures EMU spare parts from various manufacturers through open bidding, subsequently maintaining and managing the inventory of these EMU spare parts.

Currently, the reserve of EMU parts is maintained through self-managed reserves, parts center-managed consignment stock, and third-party consignment stock. The existing inventory catalog encompasses a centralized inventory catalog for the EMU parts center, consignment stock catalogs, and emergency inventory catalogs, among others. Within the catalog, a clear distinction is made between “emergency critical spare parts” and “non-emergency critical spare parts”. Non-emergency vital spare parts are centrally stored at the parts center and procured through market acquisition. When a failure need occurs for non-emergency critical spare parts listed in the “inventory catalog” at the parts center, the entire network’s inventory is swiftly distributed or supplied through market procurement within a certain reaction time, as shown in Figure 2. This category of spare components aims to motivate the OEM to expedite response times by introducing a cost-sharing structure that encourages quicker delivery to satisfy the maintenance requirements of the railway bureau EMU. Emergency vital spare parts are provided by the on-site inventory of the parts sub-center. The classifications of EMU spare parts inventory at the parts sub-center are dictated by the inventory catalog, principally housing emergency vital spare parts to fulfill the “planned requisition” requirements of the railway bureau maintenance facility, as seen in Figure 3. This category of spare part seeks to motivate the OEM to enhance the fulfillment rate of the “minimum inventory level” by presenting a residual-subsidy mechanism contingent upon quantity commitments to encourage the OEM to augment stock levels.

For emergency critical spare parts, the OEM must complete the reserve plan in advance under demand uncertainty to meet the planned requisition needs of the railway bureau. The sequence of events is shown in Figure 4. Before actual demand $D$ arises, the railway bureau issues a “minimum inventory level” requirement $S_{\alpha }$, which serves only as a reference for the OEM’s reserve decision and is not mandatory. The OEM actually reserves $Q$ at the parts sub-center. After actual demand $D$ occurs, the railway bureau requisitions according to the plan. If the actual demand exceeds the reserve amount of the OEM at the parts sub-center, each unmet demand unit will result in a unit downtime loss $o$ for the railway bureau. If the actual demand is less than the reserve amount of the OEM at the parts sub-center, the OEM gains the unit residual value $v$ of the remaining portion, and the railway bureau pays according to the actual demand $D$ that occurred.

For non-emergency critical spare parts, a two-tier supply chain is considered, consisting of one OEM and one railway bureau. Due to the uncertainty of spare parts demand, the OEM generally does not stock these parts in advance or only maintains a small, centralized stock at the parts center. As illustrated in Figure 5, when the railway bureau generates demand, the OEM ensures timely supply by centrally stocking the required quantity of spare parts $q$ across the entire railway. In the event of a sudden failure at the EMU depot, a specific quantity $D$ of spare parts is requested from the OEM, which accepts the order and allocates from the entire railway inventory $q$. Any shortfall in inventory is addressed $D-q-\theta t$ through market procurement. Referring to the research of Cachon [28], the OEM determines the response time $t$, represents the time interval between the railway bureau’s spare part request and the actual delivery of spare parts to the EMU depot by the OEM.

3.2. Model Assumptions

The notation used in this paper is shown in

Table 1. Model parameters and meanings.

Parameter	Parameter Description
$f\left(x\right)$	Railway bureau spare parts demand probability density function
$F\left(x\right)$	Railway bureau spare parts demand cumulative distribution function
$D$	Maximum railway bureau spare parts demand, with a value of $u$
$S_{\alpha }$	Railway bureau spare parts “minimum inventory level” requirement
$Q$	OEM spare parts inventory at parts sub-center
$c$	OEM production or procurement unit cost for spare parts
$p$	OEM selling price of spare parts
$h$	OEM unit inventory holding cost at parts sub-center
$v$	Unit spare parts salvage value
$b$	Railway bureau unit subsidy for OEM remaining spare parts
$o$	Unit downtime loss incurred by railway bureau due to unmet demand
${\pi }_m$	OEM expected profit
${\pi }_r$	Railway bureau expected profit
$t$	OEM response time
$\mu$	Difficulty of spare parts procurement
$q$	OEM advance reserve quantity, i.e., centralized reserve quantity at parts center
$c_s$	OEM response-effort cost
$\theta$	Spare parts order loss coefficient
$r_0,$$r$	Response-effort cost coefficient
$\phi$	Cost-sharing coefficient

Note: Variables with subscript “$c$ ” indicate decisions under the centralized mode, subscript “$m$ ” indicates OEM decisions, subscript “$r$ ” indicates railway bureau decisions, superscript “$h$ ” indicates decisions under the contract mode, superscript “*” indicates optimal decision, overline “¯” indicates upper bound, underline “_” indicates lower bound, and symbols ${\left(\cdot \right)}^{+}=max\{\cdot ,0\}$ and $E\left(\cdot \right)$ are defined as expected values.

.

The following assumptions are made in this paper:

• Both the OEM and the railway bureau are fully rational decision-makers with complete information symmetry. That is, both parties are aware of the probability density and cumulative distribution functions of spare parts demand, which is similar to general equipment supply chain studies [29].
• Spare parts repair is not considered. Damaged parts must be replaced. Spare parts reserved at the OEM but unused by the railway bureau during the stock period are transferred or scrapped, retaining some residual value $v$.
• Spare parts shortages may lead to maintenance delays or failures. A unit shutdown loss cost $o$ is introduced to measure the adverse effects of such situations.
• $v<c+h<p$ ensures the OEM always gains profit; $p<o$ is the condition for the railway bureau to purchase spare parts, $0<b<c+h-v$; otherwise, the primary manufacturer would infinitely stockpile parts.
• Different parts have varying procurement difficulties. Let $\mu \in \left[1,1.5\right]$ represent the procurement difficulty. When $\mu =1$, the difficulty is normal. Higher difficulty requires greater response-effort costs.
• Market procurement involves response-effort costs, such as supplier collaboration and channel development. Response-effort cost $c_s$ is defined as the increase in the marginal cost of accelerating response time. Following the research of Yang, X. et al. [30], the following quadratic form is used to represent response-effort cost: $c_s={\left(\mu r_0-rt\right)}^2$, where $r_0$ is the maximum response-effort cost and $r$ is the cost saved by delaying a unit’s response time $\mu r_0/r>t$. Larger response-effort costs enable meeting more spare parts demand.
• In real-world railway operations, long response times for non-emergency critical parts may delay EMU maintenance and cause shutdown losses. Following Wen et al. [31], we introduce $\theta$ as the spare parts order loss coefficient. For each unit time delay in response, the OEM loses $\theta$ orders, leading to $\theta$ spare parts-related shutdown losses for the EMU depot.

4. Spare Parts Supply Chain VMI Contract Mechanism Design

This section establishes supply chain optimization models. Proofs for propositions are consolidated in Appendix A.

4.1. Reward–Punishment Mechanism for Emergency Critical Spare Parts: A Quantity Commitment-Based Design

4.1.1. Without Reward and Penalty Mechanism

In a centralized supply chain, the goal is to maximize the profit of the supply chain. At this point, the expected profit function of the supply chain can be expressed as

$${\pi }_c=vE{\left(Q_c-D\right)}^{+}-\left(c+h\right)Q_c-oE{\left(D-Q_c\right)}^{+}$$

(1)

By taking the first and second derivatives of Equation (1) with respect to $Q_c$, obtain

$${\displaystyle \frac{\partial {\pi }_c}{\partial Q_c}}=o-c-h+\left(v-o\right)F\left(Q_c\right)$$

(2)

$${\displaystyle \frac{{\partial }^2{\pi }_c}{\partial Q_c^2}}=\left(v-o\right)f\left(Q_c\right)$$

(3)

Since $v<o$, the second-order partial derivative ${\displaystyle \frac{{\partial }^2{\pi }_c}{\partial Q_c^2}}<0$. Thus, the supply chain’s expected total profit function is concave in $Q_c$. When ${\displaystyle \frac{\partial {\pi }_c}{\partial Q_c}}=0$, there is a unique optimal spare parts reserve quantity $Q_c^{\ast }$ for the original equipment manufacturer, resulting in

$$Q_c^{\ast }=F^{-1}\left({\displaystyle \frac{o-c-h}{o-\nu }}\right)$$

(4)

The optimal expected total profit of the supply chain is obtained as

$$\mathrm{a} {\pi }_c^{\ast }=vE{\left(Q_c^{\ast }-D\right)}^{+}-\left(c+h\right)Q_c^{\ast }-oE{\left(D-Q_c^{\ast }\right)}^{+}=1,$$

(5)

In decentralized decisions, the railway bureau department specifies a minimum reserve requirement $S_{\alpha }$, and then the OEM reserves $Q$ units. When the OEM cannot ensure compensation for extra reserves, it will seldom consider the railway bureau department’s requirement in production decisions. Instead, the OEM reserves units according to the demand $D$. The OEM’s expected profit can be expressed as follows:

$$\mathrm{a} {\pi }_m=-\left(c+h\right)Q+pEmin\left\{Q,D\right\}+vE{\left(Q-D\right)}^{+}=1,$$

(6)

In Equation (6), the first part is the procurement and holding cost, the second part is the actual spare parts sales revenue to the railway bureau, and the third part is the salvage value when spare parts inventory remains. Simplifying Equation (6) yields

$${\pi }_m=pQ-(c+h)Q+(p-v){\displaystyle {\int }_0^{Q}D}f(D)dD+(v-p)Q{\displaystyle {\int }_0^{Q}f}(D)dD$$

(7)

The expected profit of the railway bureau can be expressed as follows:

$${\pi }_r=-pEmin\left\{Q,D\right\}-oE{\left(D-Q\right)}^{+}$$

(8)

In Equation (8), the first part is the actual spare parts cost to the railway bureau, and the second part is the downtime loss caused by the spare parts shortage to the railway bureau.

Through Equation (8), the first-order partial derivative and second-order partial derivative of the OEM’s reserve quantity can be obtained as shown in Equations (9) and (10):

$${\displaystyle \frac{\partial {\pi }_m}{\partial Q}}=-c-h+p+(-p+v)F(Q)$$

(9)

$${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial Q^2}}=(v-p)f\left(Q\right)$$

(10)

Since $p>v$, the second-order derivative ${\displaystyle \frac{d^2{\pi }_m}{d{Q}^2}}<0$. Thus, under decentralize decisions, the OEM’s profit function is concave in $Q$. Let ${\displaystyle \frac{\partial {\pi }_m}{\partial Q}}=0$ and solve for the unique optimal reserve quantity $Q^{\ast }$:

$$Q^{\ast }=F^{-1}\left({\displaystyle \frac{p-c-h}{p-v}}\right)$$

(11)

The optimal expected profit of the OEM, railway bureau, and supply chain are respectively,

$${\pi }_m^{\ast }=-\left(c+h\right)Q^{\ast }+pEmin\left\{Q^{\ast },D\right\}+vE{\left(Q^{\ast }-D\right)}^{+}$$

(12)

$${\pi }_r^{\ast }=-pEmin\left\{Q^{\ast },D\right\}-oE{\left(D-Q^{\ast }\right)}^{+}$$

(13)

$${\pi }^{\ast }=-\left(c+h\right)Q^{\ast }+vE{(Q^{\ast }-D)}^{+}-oE{(D-Q^{\ast })}^{+}$$

(14)

 Proposition 1. 

$Q^{\ast }<Q_c^{\ast }$, ${\pi }^{\ast }<{\pi }_c^{\ast }$.

Proposition 1 indicates that centralized decision-making yields greater total profit for the supply chain and an appropriate inventory level of spare parts compared to decentralized decision-making. Quantity commitment constitutes a form of a relational contract that depends on mutual trust between supply and demand parties and is motivated by self-enforcement to guarantee contract fulfillment. In the decentralized decision-making approach, the OEM mitigates demand uncertainty by lowering inventory levels, therefore optimizing its profits. The OEM and railway bureau, as two autonomous decision-making institutions, experience a conflict of interest during their cooperation. As a result, the double marginalization effect prohibits decentralized decisions from maximizing overall cooperative gains, implying that supply chain coordination cannot be realized. Therefore, to resolve the conflicts of interest between the OEM and railway bureau in their cooperation, enhance the final cooperation outcome, and achieve a win–win collaboration, this paper further explores the coordination problem of the EMU spare parts supply chain through contract design based on the aforementioned research.

4.1.2. Residual-Subsidy Mechanism Under the Quantity Commitment

Under a residual-subsidy agreement, the sequence of events is as follows: As shown in Figure 6, before actual demand occurs, the railway bureau department determines the “minimum reserve quantity” requirement $S_{\alpha }$ and the residual-subsidy coefficient $b$. The OEM promises to reserve at least $S_{\alpha }$ units of spare parts, i.e., $Q\ge S_{\alpha }$. After actual demand occurs, if the OEM’s reserve quantity is less than the actual demand, the railway bureau department bears the corresponding downtime loss. Conversely, when the OEM’s reserve $S_{\alpha }$ exceeds the actual demand $Q$. In this scenario, the OEM receives the salvage value of the remaining spare parts, along with a surplus subsidy for the quantity $S_{\alpha }-Q$, representing the portion by which the actual demand falls short of the “minimum inventory level”. The railway bureau then makes a payment based on the actual demand and provides a surplus subsidy for the $S_{\alpha }-Q$ portion, where actual demand is less than the “minimum inventory level”, at a rate of $b$ per unit.

Under a residual-subsidy agreement, the OEM’s expected profit function is

$${\pi }_m^h=pE\min \left\{Q,D\right\}-Q\left(c+h\right)+bE{(S_{\alpha }-D)}^{+}+vE{\left(Q-D\right)}^{+}$$

(15)

In Equation (15), the first term represents the revenue from the spare parts requisitioned according to actual demand, the second term indicates the production and holding costs, the third term is the remaining subsidy provided by the railway bureau, and the fourth term denotes the residual value of unutilized spare parts.

The expected profit function of the railway bureau is

$${\pi }_r^h=-pEmin\left\{Q,D\right\}-bE{\left(S_{\alpha }-D\right)}^{+}-oE{\left(D-Q\right)}^{+}$$

(16)

Equation (16) yields the first- and second-order partial derivatives of ${\pi }_m^h$ with respect to $Q$:

$${\displaystyle \frac{\partial {\pi }_m^h}{\partial Q}}=p-c-h-\left(p-v\right)F\left(Q\right)$$

(17)

$${\displaystyle \frac{{\partial }^2{\pi }_m^h}{\partial Q^2}}=-\left(p-v\right)f(Q)$$

(18)

Since $p>v$, the second-order partial derivative ${\displaystyle \frac{{\partial }^2{\pi }_m^h}{\partial Q^2}}<0$. Thus, under decentralized decisions, the OEM’s profit function is concave in $Q$. Let ${\displaystyle \frac{\partial {\pi }_m^h}{\partial Q}}=0$ and solve for $Q$:

$$Q=Q^{\ast }=F^{-1}\left({\displaystyle \frac{p-c-h}{p-v}}\right)$$

(19)

Compared to the OEM’s expected profit function under decentralized decisions without a reward–punishment mechanism, the additional terms in the contract mechanism’s profit function are related to the railway bureau department’s “minimum reserve requirement” and actual demand, not the OEM’s actual reserves. Thus, the OEM’s optimal reserve quantity under the contract mechanism, aimed at maximizing its own profits, is the same as under decentralized decisions.

Per the contract $Q\ge S_{\alpha }$, the OEM’s optimal reserve quantity is

$$Q^{h\ast }=max\left\{S_{\alpha },F^{-1}\left({\displaystyle \frac{p-c-h}{p-v}}\right)\right\}$$

(20)

Thus, the optimal expected profits for the OEM and the railway bureau department are

$${\pi }_m^{h^{\ast }}=pEmin\left\{Q^{h\ast },D\right\}-Q^{h\ast }\left(c+h\right)+vE{(Q^{h\ast }-D)}^{+}+bE{(S_{\alpha }-D)}^{+}$$

(21)

Then, the railway bureau department’s decision-making problem is discussed. Its optimal expected profit is

$${\pi }_r^{h\ast }=-pEmin\left\{Q^{h\ast },D\right\}-bE{\left(S_{\alpha }-D\right)}^{+}-oE{\left(D-Q^{h\ast }\right)}^{+}$$

(22)

In Equation (22): The first term is the cost of actually used spare parts. The second term is the residual subsidy for unclaimed spare parts. The third term is the downtime loss from stockouts.

Using backward induction, the optimal “minimum reserve requirement” for the railway bureau department is discussed.

 Proposition 2. 

Railway bureau “Minimum Inventory Level”.

(1)

When $S_{\alpha }\le Q^{\ast }$, ${\pi }_r^h$ is a concave function and a monotonically decreasing function with respect to $S_{\alpha }$, $S_{\alpha }^{\ast }=0$;

(2)

When $S_{\alpha }>Q^{\ast }$, $o>o_1$, ${\pi }_r^h$ is a concave function with respect to $S_{\alpha }$, $S_{\alpha }^{\ast }=F^{-1}\left({\displaystyle \frac{o-p}{o-p+b}}\right)$.

Proposition 2 reveals the relevant properties of the railway bureau’s optimal expected profit. When the “minimum inventory level” set by the railway bureau is lower than the optimal inventory level $Q^{\ast }$ determined by OEM under decentralized decision-making, the reserve decision of OEM is not influenced by the “minimum inventory level” requirement but is instead based on its own profit maximization principle. In this scenario, if the railway bureau issues a “minimum inventory level” and the OEM reserves are not utilized, the railway bureau needs to provide a certain surplus subsidy to the OEM. Therefore, in such a case, not issuing a “minimum inventory level” requirement is the best choice for the railway bureau to achieve its optimal expected profit. When the “minimum inventory level” set by the railway bureau exceeds the optimal inventory level $Q^{\ast }$ under OEM decentralized decision-making, the railway bureau faces a trade-off between surplus subsidies and downtime loss cost: the higher the “minimum inventory level” requirement, the more surplus subsidy the railway bureau has to pay, but it can also effectively reduce the occurrence of spare parts shortages.

Furthermore, when the railway bureau “minimum inventory level” requirement is greater than the optimal inventory level $Q^{\ast }$ under OEM decentralized decision-making, the spare parts downtime loss exceeds a certain threshold $o_1$. This implies that only when the spare parts downtime loss is sufficiently large will the optimal “minimum inventory level” required by the railway bureau be higher than the optimal inventory level of OEM. At this point, the railway bureau is motivated to issue a “minimum inventory level” requirement and provide surplus subsidies; conversely, if this condition is not met, not issuing a “minimum inventory level” requirement remains the optimal decision.

 Proposition 3. 

Compared to no contract mechanisms, when$o>o_1$, setting residual subsidy$b$within a certain parameter range can effectively increase the OEM’s spare parts reserves, boosting the railway bureau’s demand satisfaction rate.

(1) 

When $0\le b\le \underset{¯}{b}$,  the OEM rejects the contract;

(2) 

When $\underset{¯}{b}< b\le \overline{b}$,  the railway bureau’s optimal “minimum reserve quantity” is $S_{\alpha }^{\ast }=F^{-1}\left({\displaystyle \frac{o-p}{o-p+b}}\right)$, and the OEM’s reserve quantity is $Q^{h\ast }=F^{-1}\left({\displaystyle \frac{o-p}{o-p+b}}\right)$, with ${\displaystyle \frac{\partial S_{\alpha }^{\ast }}{\partial b}}<0$ and ${\displaystyle \frac{\partial Q^{h\ast }}{\partial b}}<0$;

(3) 

When $b>\overline{b}$,  the railway bureau’s optimal “minimum reserve quantity” is $S_a^{\ast }=0$, and the OEM’s reserve quantity is $Q^{h\ast }=F^{-1}\left({\displaystyle \frac{p-c-h}{p-v}}\right)$, aligning with decentralized decisions.

Proposition 3 indicates that certain conditions must be met by the downtime loss cost and the remaining subsidy parameters to ensure that the reserve level of the OEM under the contract mechanism is higher than without it and to guarantee profitability for both the railway bureau and the OEM. Only under these conditions is supply chain coordination possible. According to Proposition 2, the railway bureau needs to impose a non-zero “minimum inventory level” requirement only when the unit downtime loss exceeds $o_1$. In this scenario, it is possible for the reserve level of the OEM under the contract mechanism to match the reserve level under centralized decision-making. Furthermore, as the remaining subsidy coefficient $b$ increases, the railway bureau’s expected profit correspondingly decreases. Therefore, there must be a threshold value $\overline{b}$, beyond which the railway bureau’s expected profit will drop to the same level as when no “minimum inventory level” requirement is imposed. Thus, when $b>\overline{b}$, the railway bureau’s optimal decision is not to impose a “minimum inventory level” requirement. On the other hand, as the remaining subsidy coefficient $b$ decreases, the OEM’s expected profit will also decrease. Consequently, there exists a threshold value $\underset{¯}{b}$, below which the OEM’s profit is less than zero. Only when the remaining subsidy exceeds this lower limit $\underset{¯}{b}$ can the OEM be ensured to participate in the contract; otherwise, the OEM’s optimal inventory level is unaffected by the railway bureau, and in this case, the OEM’s optimal inventory level is consistent with that under decentralized decision-making.

Finally, when the downtime loss cost exceeds both $o_1$ and the remaining subsidy parameter $b\in (\underset{¯}{b},\overline{b}]$, since the OEM’s optimal inventory level depends on the railway bureau’s “minimum inventory level” requirement, the “minimum inventory level” set by the railway bureau will decrease as the remaining subsidy parameter increases. This also means that the OEM’s optimal inventory level will similarly decrease as the remaining subsidy increases. A reduction in reserve levels increases the risk of stockouts for the railway bureau, which in turn leads to greater downtime losses. Therefore, during contract negotiations, both parties in the supply chain should set subsidy parameters judiciously to maximize the supply chain’s profits.

In summary, effective cooperation between both parties can be promoted only by keeping the remaining subsidy parameters within a reasonable range $b\in (\underset{¯}{b},\overline{b}]$. This approach incentivizes the OEM to proactively increase the reserve levels of EMU spare parts, thereby improving the demand satisfaction rate of the railway bureau’s maintenance.

 Proposition 4. 

Indicates that, compared to decentralized decision-making without a quantity commitment, the introduction of a residual-subsidy contract creates a scenario where $o\ge \max \left\{o_1,o_2\right\}$. When $b\in (\underset{¯}{b},\overline{b}]$, the railway bureau always benefits, and only when the subsidy parameter $b$ falls within a certain range will the OEM’s optimal expected profit increase, achieving a Pareto improvement. Specifically,

$$b\in \left({\displaystyle \frac{-pEmin\left\{Q^{h\ast },D\right\}+Q^{h\ast }\left(c+h\right)-vE{(Q^{h\ast }-D)}^{+}+\mathrm{pEmin}\left\{Q^{\ast },D\right\}-Q^{\ast }\left(\mathrm{c}+\mathrm{h}\right)+\mathrm{vE}{(Q^{\ast }-D)}^{+}}{E{(Q^{h\ast }-D)}^{+}}},\overline{b}\right]$$

Proposition 4 demonstrates that when the downtime loss cost and the residual-subsidy parameter are within a certain range, the residual-subsidy contract under the quantity commitment not only enhances the overall profit of the supply chain but also results in profits for both the OEM and the railway bureau that exceed those without a contract mechanism, achieving a Pareto improvement. When the downtime loss cost of spare parts is significant and the residual-subsidy parameter is $b\in (\underset{¯}{b},\overline{b}]$, the OEM earns less profit by maintaining an inventory level $Q^{h\ast }$ greater than the decentralized decision-making optimal inventory level $Q^{\ast }$. However, due to the increased demand satisfaction rate of spare parts, the downtime loss cost decreases, and the railway bureau’s profit increases with the OEM’s inventory level. Therefore, it is necessary to compensate for the loss caused by the OEM’s inventory $Q^{h\ast }$ through the residual subsidy. Additionally, for the OEM, only when the residual-subsidy parameter exceeds a certain threshold can it benefit from the contract. In summary, there exists an interval of $b$ that can achieve a Pareto improvement in the supply chain.

4.2. Reward–Punishment Mechanism for Non-Emergency Critical Spare Parts: A Response Time-Based Design

4.2.1. No Reward–Punishment Mechanism

Under centralized decision-making, the objective is to maximize the supply chain’s profit. The supply chain’s expected profit function can be expressed as

$${\pi }_c=-\left(c+h\right)q+\nu E{\left(q-D\right)}^{+}-cE{\left(D-q-\theta t_c\right)}^{+}-{\left(\mu r_0-r{t}_c\right)}^2-oE\left(\theta t_c\right)$$

(23)

Taking the first- and second-order partial derivatives of Equation (23) with respect to $t_c$,

$${\displaystyle \frac{\partial {\pi }_c}{\partial t_c}}=c\theta \left(1-F\left(q+\theta t_c\right)\right)+2r\left(\mu r_0-r{t}_c\right)-o\theta$$

(24)

$${\displaystyle \frac{{\partial }^2{\pi }_c}{\partial t_c^2}}=-c{\theta }^{2}f\left(q+\theta t_c\right)-2r^2$$

(25)

Since ${\displaystyle \frac{{\partial }^2{\pi }_c}{\partial t_c^2}}<0$, the supply chain’s total profit function is concave in $t_c$. Let ${\displaystyle \frac{\partial {\pi }_c}{\partial t_c}}=0$ and solve for the unique optimal response time $t_c^{\ast }$. The optimal $t_c^{\ast }$ satisfies

$$t_c^{\ast }={\displaystyle \frac{c\theta \left(1-F\left(q+\theta t_c^{\ast }\right)\right)+2r\mu r_0-o\theta }{2r^2}}$$

(26)

To ensure $t_c^{\ast }>0$, it requires $r_0>{\displaystyle \frac{o\theta -c\theta \left(1-F\left(q+\theta t_c^{\ast }\right)\right)}{2r\mu }}$.

The optimal profit for the supply chain under centralized decision-making is

$${\pi }_{\mathrm{c}}^{\ast }=-\left(c+h\right)q+\nu E{\left(q-D\right)}^{+}-cE{\left(D-q-\theta t_c^{\ast }\right)}^{+}-{\left(\mu r_0-r{t}_c^{\ast }\right)}^2-oE\left(\theta t_c^{\ast }\right)$$

(27)

Under decentralized decisions, the OEM aims to maximize its own profit. The OEM’s expected profit function can be expressed as

$${\pi }_m=pEmin\{D,q\}-\left(c+h\right)q+vE{\left(q-D\right)}^{+}+\left(p-c\right)E{\left(D-q-\theta t\right)}^{+}-{\left(\mu r_0-rt\right)}^2$$

(28)

In Equation (28), the first part represents the benefit from advance spare parts inventory, the second part covers the procurement and holding costs of advance inventory, the third part is the salvage value when the spare parts inventory remains, the fourth part represents the benefit from market procurement, and the fifth part is the response-effort cost. At this point, the expected profit function of the railway bureau can be expressed as

$${\pi }_r=-pEmin\{D,q\}-pE{\left(D-q-\theta t\right)}^{+}-oE\left(\theta t\right)$$

(29)

In Equation (29), the first and second parts represent the spare parts procurement cost of the railway bureau, while the second term represents the downtime loss caused by unmet spare parts demand.

 Proposition 5. 

If $p<p^{\prime }$, then the profit function of the OEM under decentralized decision-making is a concave function with respect to response time, and there exists an optimal response time that satisfies

$$-\left(p-c\right)\theta \left(1-F\left(q+\theta t^{\ast }\right)\right)+2r\left(\mu r_0-r{t}^{\ast }\right)=0$$

Proposition 5 indicates that when the profit from spare parts satisfies specific criteria, a singular response time exists that optimizes the predicted profit of the OEM. If the OEM’s final decision does not align with the best reaction time, the OEM’s maximum profit will invariably be inferior to the profit achieved by selecting this optimal response time. Therefore, if the OEM can ultimately select this response time as the ideal response time, the OEM will have the ability to optimize the predicted profit response.

In conclusion, the current ideal earnings for the OEM and the railway bureau are as follows:

$${\pi }_{\mathrm{m}}^{\ast }=pEmin\{D,q\}-\left(c+h\right)q+vE{\left(q-D\right)}^{+}+\left(p-c\right)E{\left(D-q-\theta t^{\ast }\right)}^{+}-{\left(\mu r_0-r{t}^{\ast }\right)}^2$$

(30)

$${\pi }_{\mathrm{r}}^{\ast }=-pEmin\{D,q\}-pE{\left(D-q-\theta t^{\ast }\right)}^{+}-oE\left(\theta t^{\ast }\right)$$

(31)

 Proposition 6. 

$t_c^{\ast }<t^{\ast }$, ${\pi }_c^{\ast }>{\pi }^{\ast }$.

Proposition 6 indicates that under centralized decision-making, the overall profit of the supply chain and the optimal response time for spare parts are both superior to those under decentralized decision-making. Under decentralized decision-making, the OEM often confronts a trade-off between reaction time and effort cost and prefers to pick a comparatively longer response time to meet the railway bureau’s demand because accelerating the response time would lead to an increase in the effort cost invested. For example, to quicken the response time, the OEM may need to boost inventory levels, invest in technology, create channels, or commit additional people and material resources. These measures will increase operating costs, resulting in a longer response time, which exacerbates the downtime losses of the railway bureau and impacts the overall profit of the supply chain. Therefore, synchronizing the supply chain is vital.

 Proposition 7. 

When $p<p^{\prime }$:

(1) 

Under both decentralized decision-making and centralized decision-making, as the procurement difficulty coefficient increases, the optimal response time will increase, and the total profit of the supply chain will decrease;

(2) 

Under centralized decision-making, as the spare parts order loss coefficient increases, the optimal response time decreases, and the total profit of the supply chain decreases; under decentralized decision-making, as the spare parts order loss coefficient increases, the optimal response time first decreases and then increases, and the total profit of the supply chain decreases.

Proposition 7 elucidates the correlation among ideal response time, overall supply chain profit, procurement difficulty coefficient, and spare parts order loss coefficient in both centralized and decentralized decision-making contexts. When the procurement difficulty coefficient rises, the response cost for the OEM will directly increase. The cost pressure hampers the OEM’s ability to sustain its initial response speed, resulting in an elongation of response time as the procurement difficulty coefficient escalates. Meanwhile, the total profit of the supply chain would fall due to the growth in response-effort cost and the increase in response time.

In the centralized decision-making model, an increase in the replacement parts order loss coefficient will greatly worsen the OEM’s order loss and also raise the downtime losses of the railway bureau. To minimize these losses, centralized decision-making tends to shorten the reaction time, hence the optimal response time is inversely related to the order loss coefficient. It is important to acknowledge that while the response time diminishes as the order attrition coefficient increases, the variation is rather minimal. In contrast, the overall profit of the supply chain is more susceptible to the order attrition coefficient, declining as the order attrition coefficient grows. In the decentralized decision-making paradigm, the OEM’s principal worry with the attrition coefficient of spare part orders is the aggravation of its own order loss. When the attrition coefficient of spare part orders is quite low, the OEM will choose to accelerate the response time to reduce order loss. However, when the attrition coefficient of spare part orders climbs to a certain level, further lowering the response time would not only lead to a considerable loss in orders but also impose large response costs on the OEM. In this case, the OEM is more motivated to increase the reaction time to avoid losses. Therefore, as the attrition coefficient of spare part orders grows, the OEM’s optimal response time first decreases and subsequently increases. Similarly, because the negative impact of the order attrition coefficient on the total profit of the supply chain is greater than the positive impact through its effect on response time, the total profit of the supply chain under decentralized decision-making will ultimately decrease as the order attrition coefficient increases.

In summary, a rise in either the procurement difficulty coefficient or the attrition coefficient of spare part orders will lead to a reduction in the total profit of the supply chain.

4.2.2. Residual-Subsidy Mechanism Under Response Time

To enable the supply chain to promptly respond to the railway bureau’s spare parts demand, a response-effort cost-sharing contract is introduced between the OEM and the railway bureau. The contract details are as follows: During spare parts delivery, the railway bureau shares a proportion $\phi$ of the response-effort cost $\phi {\left(\mu r_0-rt\right)}^2$, while the OEM pays the remaining response-effort cost $\left(1-\phi \right){\left(\mu r_0-rt\right)}^2$.

The profit functions for the OEM and the railway bureau are

$${\pi }_m^h=pEmin\{D,q\}-\left(c+h\right)q+vE{\left(q-D\right)}^{+}+\left(p-c\right)E{\left(D-q-\theta t\right)}^{+}-\left(1-\phi \right){\left(\mu r_0-rt\right)}^2$$

(32)

$${\pi }_r^h=-pEmin\{D,q\}-pE{\left(D-q-\theta t\right)}^{+}-oE\left(\theta t\right)-\phi {\left(\mu r_0-rt\right)}^2$$

(33)

Taking the first- and second-order partial derivatives of Equation (33) with respect to $t$,

$${\displaystyle \frac{\partial {\pi }_m^h}{\partial t}}=-\left(p-c\right)\theta \left(1-F\left(q+\theta t\right)\right)+2r\left(1-\phi \right)\left(\mu r_0-rt\right)$$

(34)

$${\displaystyle \frac{{\partial }^2{\pi }_m^h}{\partial t^2}}=\left(p-c\right){\theta }^{2}f\left(q+\theta t\right)-2\left(1-\phi \right)r^2$$

(35)

Similarly to the proof of Proposition 5, when $\left(p-c\right){\theta }^{2}f\left(q+\theta t\right)-2\left(1-\phi \right)r^2<0$, the OEM’s profit function is concave in $t$. Let ${\displaystyle \frac{\partial {\pi }_m^h}{\partial t}}$ = 0 and solve for the unique optimal response time $t^{h\ast }$:

$$t^{h\ast }={\displaystyle \frac{2\left(1-\phi \right)r\mu r_0-\left(p-c\right)\theta \left(1-F\left(q+\theta t^{h\ast }\right)\right)}{2\left(1-\phi \right)r^2}}$$

(36)

At this point, the optimal expected profits for the OEM, railway bureau, and supply chain are

$$\begin{array}{l}{\pi }_m^{h\ast }=pEmin\{D,q\}-\left(c+h\right)q+vE{\left(q-D\right)}^{+}\\ +\left(p-c\right)E{\left(D-q-\theta t^{h\ast }\right)}^{+}-\left(1-\phi \right){\left(\mu r_0-r{t}^{h\ast }\right)}^2\end{array}$$

(37)

$${\pi }_r^{h\ast }=-pEmin\{D,q\}-pE{\left(D-q-\theta t^{h\ast }\right)}^{+}-oE\left(\theta t^{h\ast }\right)-\phi {\left(\mu r_0-r{t}^{h\ast }\right)}^2$$

(38)

 Proposition 8. 

When $p<p^{″}$, compared to decentralized decision-making without a cost-sharing contract, the cost-sharing contract can accelerate the response time, increase the demand satisfaction rate, and the profit of the OEM will rise with an increase in the cost-sharing coefficient, while the profit of the railway bureau will initially increase and then decrease as the cost-sharing coefficient increases.

Proposition 8 indicates that implementing a cost-sharing contract encourages the OEM to focus more on speeding up the reaction time, providing a prompt provision of spare parts when needed by the railway bureau, thereby sustaining the stability and dependability of EMU operations. The cost-sharing contract establishes an incentive mechanism. For the OEM, while accelerating the reaction time entails certain operating expenses, the cost-sharing agreement offers financial reimbursement from the railway bureau, and enhancing the demand satisfaction rate of the railway bureau can lead to increased revenues. Additionally, as the cost-sharing coefficient rises, the response cost for the OEM falls, leading to an increase in the OEM’s predicted profit. For the railway bureau, profits after cost-sharing will first climb and then decline, demanding a reasonable range for the cost-sharing coefficient to accomplish a dual profit increase for both the OEM and the railway bureau relative to the absence of a contract mechanism.

 Proposition 9. 

After introducing a cost-sharing contract, when $p<p^{″}$, only if the cost-sharing ratio $\phi$ satisfies Equation (39) will the optimal expected profit of both the railway bureau and the OEM increase simultaneously, achieving a Pareto improvement. Where

$$\phi \in \left(1-{\displaystyle \frac{\left(p-c\right)\left[A-B\right]-D}{C}},{\displaystyle \frac{p\left[A-B\right]+o\left[E\left(\theta t^{\ast }\right)-E\left(\theta t^{h\ast }\right)\right]}{C}}\right)$$

(39)

Proposition 9 indicates that the cost-sharing contract is advantageous for both the OEM and the railway bureau only when the proportion of response effort cost sharing is within a specific range. If the cost-sharing ratio is below a certain threshold, the cost incurred by the OEM to improve response time exceeds the cost shared by the railway bureau, resulting in the OEM bearing a larger share of the response-effort cost and lacking motivation to improve response time, which would harm the interests of the supply chain. If the cost-sharing ratio surpasses a specific threshold, the advantages to the railway bureau are insufficient to balance the shared expenses, then the railway bureau will not engage in cost-sharing with the OEM. In summary, when the response-effort cost-sharing ratio falls within a given range, the performance of the supply chain can achieve a Pareto improvement. The OEM and the railway bureau can change the cost-sharing coefficient to redistribute costs, thereby boosting the overall efficiency of the supply chain.

5. Case Analysis

The analysis above examined the conditions and the impact of the relevant parameters for achieving Pareto improvement in supply chain performance under the quantity commitment with the residual-subsidy mechanism and the cost-sharing mechanism under response time. Based on previous conclusions, both the residual-subsidy mechanism under quantity commitment and under response time can achieve a win–win situation for the railway bureau and the OEM in terms of expected profit under certain conditions, and their effectiveness is related to relevant parameters. The spare parts demand distribution is estimated through fitting methods applied to historical procurement records and inventory data. Parameters are set in accordance with survey results and then numerical examples are analyzed.

5.1. Effectiveness Analysis of the Residual-Subsidy Mechanism Under Quantity Commitment

The following section first uses numerical analysis to examine the impact of spare parts downtime loss and the railway bureau residual-subsidy coefficient on the optimal decision and optimal expected profit of supply chain members under quantity commitment in different modes: decentralized decision-making, centralized decision-making, and the introduction of the residual subsidy. Referring to spare parts data from a certain railway bureau, values are assigned to the relevant parameters in the model. Spare parts demand follows normal distribution $D~N(90,{28}^2)$, $c=850$, $p=1500$, $v=400$, $o=3500$, and $h=100$, where it is calculated that when the supply chain is coordinated, $b=431.37$. These parameters are substituted into the formulas for the three modes without the reward and penalty mechanism, centralized decision-making, decentralized decision-making, and contract mechanism. The results of parameter changes are shown in

Table 2. Decision variables under different decision-making modes.

Variable	Centralized Decision-Making	Decentralized Decision-Making	Residual-Subsidy Contract Decision
$Q$	115	90	115
${\pi }_m$	—	37,212.58	44,631.92
${\pi }_r$	—	−157,340.77	−152,702.10
${\pi }_c$	−108,070.18	−120,128.19	−108,070.18

.

According to the data in Table 2, under the centralized decision-making mode, the spare parts inventory of the OEM is significantly higher than the reserve under the decentralized decision-making mode. Meanwhile, the overall profit of the supply chain under the centralized decision-making mode also exceeds the profit under the decentralized decision-making mode. After the introduction of the “residual subsidy” contract mechanism, the reserve quantity of the OEM and the total profit of the supply chain align with the centralized decision-making model. At this point, the profit of the entire supply chain increases by 10%. Additionally, the profits of both the OEM and railway bureau improve compared to the decentralized decision-making model. This indicates that the “residual subsidy under quantity commitment” contract can effectively coordinate the parties in the supply chain, achieving a Pareto improvement.

5.1.1. The Impact of Downtime Loss on Supply Chain Members’ Optimal Decision and Optimal Expected Profit

Firstly, the effect of downtime loss cost on reserve levels is examined. According to Propositions 2 and 3, when the downtime loss of spare parts is substantial and the subsidy coefficient falls within a certain range, specifically when $o>o_1$ and $\underset{¯}{b}<b\le \overline{b}$, it becomes necessary for the railway bureau to require the OEM to maintain a “minimum inventory level” reserve. Under the contract mechanism, the optimal inventory level will increase as $o$ increases. Based on the parameter settings, it is determined that when $o>1931.37$, and the downtime loss due to spare parts shortage is less than CNY 1931.37, the optimal inventory level determined by the OEM under the decentralized decision-making model is greater than that under the contract mechanism. In this situation, the railway bureau does not impose a “minimum inventory level” requirement, and the OEM reserve level matches the optimal inventory level under decentralized decision-making, as shown in Figure 7. This indicates that, in this scenario, the railway bureau does not need to set reserve level requirements. If the railway bureau does impose such requirements, it would lead to additional surplus subsidy expenses and increased costs, assuming the same number of spare parts supplies.

Figure 8 shows that when $b>\overline{b}$, the optimal inventory level under both decentralized decision-making and the contract mechanism is consistent and unaffected by the downtime loss cost. This aligns with the conclusion from Proposition 3, as the surplus subsidy is excessively large, making the railway bureau’s optimal “minimum inventory level” requirement 0. The optimal inventory level under both decentralized decision-making and the contract mechanism is determined solely by maximizing their own interests and remains independent of the downtime loss cost.

Next, the impact of downtime loss costs on the profits of supply chain members is examined. When $b\in (\underset{¯}{b},\overline{b}]$, supply chain members cooperate based on the “minimum inventory level” requirement. As the downtime cost $o$ increases, the optimal inventory level $Q^{h\ast }$ will correspondingly increase. Although this somewhat raises the risk of OEM inventory surplus, the residual subsidy will also increase, thereby enhancing OEM’s expected profit. For the railway bureau, an increase in downtime cost $o$ means that the losses incurred during stockouts become more severe, necessitating a higher residual subsidy to incentivize the OEM to reserve more, which directly results in a decrease in the railway bureau’s expected profit, as shown in Figure 9. Furthermore, as the downtime loss $o$ continues to increase, the optimal inventory level gradually approaches the maximum value of the actual demand. In this scenario, under the contract mechanism, the railway bureau’s profit tends to stabilize, whereas under the decentralized decision-making model, the railway bureau’s profit shows a linear downward trend.

5.1.2. Impact of Contract Residual-Subsidy Parameters on Optimal Decision and Optimal Expected Profit of Supply Chain Members

First, the impact of residual-subsidy parameters on the optimal decision of supply chain members is discussed. Figure 10 illustrates the impact of residual-subsidy parameters on the railway bureau’s “minimum inventory level” requirement and the OEM’s reserve level. According to Proposition 3, in the case where $o=3500$, when $0<b\le 1.28$, since the subsidy cannot make the OEM reserve according to the railway bureau’s “minimum reserve level” requirement, the expected profit is greater than 0, and at this point, the supply chain parties do not reach a contract; when $1.28<b\le 603.93$, OEM’s reserve level equals the railway bureau’s “minimum inventory level”. For the railway bureau, the higher the residual subsidy, the greater the cost. Therefore, as the residual subsidy increases, the “minimum inventory level” requirement will decrease, and the OEM’s central optimal inventory level will also decrease accordingly; when $b>603.93$, due to the excessively high residual subsidy, the railway bureau’s optimal expected profit is less than the expected profit without the “minimum inventory level” requirement, at which point the railway bureau’s “minimum inventory level” requirement is 0, and the OEM’s optimal inventory level is equal to that under its decentralized decision-making. Next, the impact of the remaining subsidy parameter on the expected profit of supply chain members is discussed.

According to Proposition 3, when $o=3500$, the range for the subsidy parameter $b$ that allows for contract cooperation is CNY 1.28–603.93. When $b=431.37$, the total profit of the supply chain under the contract model is equivalent to the total profit under the centralized model. Additionally, each member of the supply chain earns more profit under the contract model than under the decentralized decision-making model.

As shown in Figure 11, as the remaining subsidy parameter $b$ increases, the profit of the OEM under the contract mechanism rises, while the profit of the railway bureau decreases. Since the OEM receives compensation for excess reserves, under the optimal inventory level of the supply chain, it is calculated through Proposition 4 that when $b\in \left[234.6,603.93\right]$, the remaining subsidy contract achieves Pareto improvement. This means that both the OEM and railway bureau gain more profit under the remaining subsidy contract than in the decentralized supply chain. It is evident that by adjusting the size of $b$, profit coordination among members of the supply chain can be achieved, thereby incentivizing the OEM to increase the spare parts inventory, which supports the healthy and sustainable development of the supply chain.

5.2. Effectiveness Analysis of Cost-Sharing Mechanism Under Response Time

Subsequently, numerical analysis is conducted to examine the impact of procurement difficulty, the order loss coefficient, and the cost-sharing coefficient on the optimal decision and optimal expected profit of supply chain members under response time in different modes, including decentralized decision-making, centralized decision-making, and the introduction of cost-sharing. The spare parts data from a certain railway bureau are used to assign values to the relevant parameters in the model. The spare parts demand follows normal distribution $D~N(85,{28}^2)$, and other related parameter assignments in the model are as follows: $c=650$, $p=1600$, $v=400$, $o=2200$, $h=100$, $\mu =1$, $\theta =7$, $r_0=325$, $r=30$, $q=30$. Upon calculation, when the supply chain is coordinated, $\phi =0.54$. By substituting these parameters into the formulas for the three modes—no reward and penalty mechanism, centralized decision-making, decentralized decision-making, and contract mechanism—the results of the parameter changes are shown in

Table 3. Decision variables under different decision-making modes.

Variable	Centralized Decision-Making	Decentralized Decision-Making	Cost-Sharing Contract Decision
$t$	4.3 days	9.6 days	4.3 days
${\pi }_m$	—	23,845.59	33,867.84
${\pi }_r$	—	−195,619.44	−178,947.43
${\pi }_c$	−145,079.59	−171,773.85	−145,079.59

.

According to the data in Table 3, the response time of the OEM is significantly shorter under the centralized decision-making model compared to the decentralized decision-making model. At the same time, the overall profit of the supply chain is higher under the centralized decision-making model than under the decentralized decision-making model. With the introduction of the “response time cost-sharing” contract mechanism, the optimal response time of the OEM and the total profit of the supply chain reached the same level as in the centralized decision-making model, and the total cost of the supply chain was reduced by 15.5%. This result indicates that the “cost-sharing” contract can effectively coordinate the members of the supply chain, thereby achieving Pareto improvement.

5.2.1. The Impact of Procurement Difficulty Coefficient and Order Loss Coefficient on the Optimal Decision and Optimal Expected Profit of Supply Chain Members

First, the impact of the procurement difficulty coefficient and the order loss coefficient on the ideal response time is explored. According to Figure 12, under the given spare parts downtime loss cost, in the centralized decision-making model, the optimal response time of the OEM is positively correlated with the procurement difficulty coefficient, meaning that as the procurement difficulty coefficient increases, the optimal response time gradually extends. Conversely, it is negatively connected with the order loss coefficient, indicating that as the order loss coefficient increases, the optimal response time rapidly shortens. In the decentralized decision-making model, the ideal reaction time of the OEM likewise extends with the growth of the procurement difficulty coefficient. Notably, as the order loss coefficient grows, the optimal response time of the OEM first decreases and subsequently increases. When the order loss coefficient is minimal, the OEM can reduce order loss by expediting the response time. However, when the order loss coefficient is substantial, incurring more response-effort costs would only cause the costs to grow sharply, and at this point, the OEM is unwilling to further accelerate the response time, which fits with the conclusion of Proposition 7.

Simultaneously, a comparison of the two decision-making models reveals that the optimal response time under centralized decision-making is faster than the response time under decentralized decision-making. Moreover, for spare parts with a high order loss coefficient and significant procurement complexity, centralized decision-making is more successful in reducing reaction time compared to decentralized decision-making. This implies that when spare parts confront significant order loss risk and substantial procurement difficulty, developing a coordination mechanism for optimization management becomes particularly critical.

Next, the influence of the procurement difficulty coefficient and the order loss coefficient on profit is addressed. As demonstrated in Figure 13, under both decentralized and centralized decision-making, the overall profit of the entire supply chain rapidly drops as the procurement difficulty coefficient and the order loss coefficient increase. Notably, as seen in Figure 12, in both centralized decision-making and decentralized decision-making models, an increase in the spare parts order loss coefficient leads to a faster response time, but the magnitude of this shift is rather restricted. In contrast, the overall profit of the supply chain is more sensitive to the order loss coefficient. Because the negative impact of the order loss coefficient on the total profit of the supply chain surpasses its positive impact by affecting the response time, as the order loss coefficient increases, the profit of the supply chain continues to drop. However, under a centralized system, the total profit of the supply chain is higher than that under decentralized decision-making, showing a considerable necessity for constructing a contract mechanism.

5.2.2. The Impact of Contract Cost-Sharing Parameters on Optimal Expected Profit

As shown in Figure 14, as the downtime loss cost $o$ increases, the cost-sharing ratio in the contract coordination process shows an upward trend. For the railway bureau, higher downtime costs for spare parts mean greater losses due to the inability to supply spare parts promptly. Therefore, the railway bureau urgently expects the OEM to respond quickly and complete delivery. In this situation, the railway bureau must share a higher proportion of the response cost to effectively incentivize the OEM to take measures to accelerate the response time, thereby reducing its own downtime losses caused by the untimely supply of spare parts.

As shown in Figure 15, with the increase in the cost-sharing coefficient $\phi$, the OEM consistently benefits from the introduction of the cost-sharing contract. However, for the railway bureau, it only benefits from the introduction of the cost-sharing contract when $0<\phi <0.64$ and only achieves a dual increase in profits for both the OEM and the railway bureau when $0<\phi <0.64$. It is evident that the railway bureau can adjust the cost-sharing coefficient $\phi$ to coordinate the profits of each member of the supply chain. This incentivizes the OEM to accelerate the response time of spare parts, resulting in improved profits for all members compared to the decentralized decision-making model. This is conducive to the healthy and sustainable development of the supply chain.

6. Conclusions

The inventory of EMUs in China is considerable and is progressively entering a phase of refurbishment, resulting in a heightened need for replacement vehicle spare parts. Currently, the supply of EMU spare parts mostly rests on the strategic cooperation agreement between GT Group and ZC Company, with the OEM adopting the VMI model to supply spare parts to the railway bureau. While this approach has greatly secured the timely supply of most replacement parts, it still highlights several serious concerns that need correction, such as the poor fulfillment rate of specific spare parts and frequent delays in supply periods. An in-depth investigation of these difficulties reveals that the absence of an effective reward and penalty mechanism is a core element. Under the VMI model, the sensible allocation of costs is closely tied to the success of collaboration among supply chain participants. This underscores the need for a fair incentive system for the OEM. In light of this, this article focuses on the two-tier supply chain encompassing the railway bureau and OEM, considering two supply methods: OEM on-site inventory and time response. It studies and analyzes the corresponding reward and penalty system, seeking to provide theoretical support and practical guidance for boosting EMU spare parts supply efficiency, optimizing supply chain management, and strengthening the cooperation between the railway bureau and OEM.

The main research findings of this paper are as follows:

• Consider the circumstance where emergency critical spare parts are supplied by the OEM at the parts sub-center under on-site inventory supply conditions. A residual-subsidy system under quantity commitment is proposed. Through the building and study of both centralized decision-making and decentralized decision-making models, it is found that the optimal decision attained under the centralized decision-making model is superior to the decisions made under the decentralized decision-making model. The primary research findings are as follows:
  • The decision of whether the railway bureau provides a “minimum inventory level” requirement and executes the residual-subsidy plan hinges on comparing the downtime loss to a certain threshold. If the downtime loss exceeds this threshold, the railway bureau is motivated to issue the requirement and implement the subsidy; conversely, if the downtime loss is below this threshold, it is optimal to maintain the decision not to issue the “minimum inventory level” requirement;
  • When the downtime loss of spare parts is significant, setting a residual subsidy within a certain parameter range is necessary to increase the inventory levels and reduce the risk of stockouts, indicating the potential for supply chain coordination. Case studies show that after the residual-subsidy contract mechanism was introduced, the OEM’s spare parts inventory increased from 90 under the decentralized decision-making mode to 115, and the total profit of the supply chain also increased accordingly, achieving a Pareto improvement for the supply chain. Within this range, as the residual subsidy increases, the OEM’s inventory level reduces, and the expected profit increases, while the railway bureau’s expected profit decreases. Additionally, the higher the downtime loss cost, the more pronounced the coordination effect produced by the contract mechanism.
• In the setting where non-emergency critical spare parts are centrally stored by the OEM at the parts center and supplied through market procurement, a cost-sharing mechanism based on response time has been designed. Initially, by creating and studying two models, centralized decision-making and decentralized decision-making, it is discovered that the optimal decision achieved under the centralized decision-making model is superior to that under the decentralized decision-making model. The principal research conclusions are as follows:
  • In both decentralized decision-making and centralized decision-making, the optimal response time is directly proportional to the procurement difficulty coefficient, while the total profit of the supply chain is inversely proportional to both the procurement difficulty coefficient and the order loss coefficient;
  • Compared to decentralized decision-making without a contract, the cost-sharing contract can accelerate the response time, increase the demand satisfaction rate, and enhance the profit of the OEM as the cost-sharing coefficient increases. Meanwhile, the cost for the railway bureau initially decreases and then increases with the rise in the cost-sharing coefficient;
  • Upon introducing the contract mechanism, it is observed that the OEM consistently benefits, while the railway bureau needs to meet certain conditions of the cost-sharing parameters to achieve cost reduction. Consequently, by modifying the pertinent parameters in the contract, the distribution of costs among members can be efficiently managed to achieve Pareto improvement. Case studies reveal that after implementing the response time-based cost-sharing mechanism, the OEM’s reaction time decreased from 9.6 days under the decentralized decision-making model to 4.3 days. Furthermore, the total cost of the supply chain was reduced by 15.5%, and the total profit approached the level of the centralized decision-making model. Additionally, the cost-sharing coefficient necessary for the supply chain to attain a coordinated condition grows with the rise in the downtime loss cost of spare parts. For spare parts with high downtime loss costs, the railway bureau must take higher response cost-sharing to assure timely supply.

These findings provide practical guidance for the OEM and railway bureau in contract design. The model clarifies cost–benefit trade-offs in inventory commitments and response time targets, supporting rational choices of subsidies, cost-sharing terms, and contract parameters under budget and reliability constraints. Both parties can use it to identify win–win solutions and design incentive-aligned contracts.

Future studies can expand on, but are not limited to, the following aspects. (1) Further consider the transshipment behavior amongst OEMs to explore more complicated supply chain configurations. (2) Broaden the range of EMU spare parts and examine discrete Poisson distribution or negative binomial distribution. (3) Imported parts may face supply interruptions. In the future, contract design can be considered for circumstances where supply interruptions occur. (4) Further considering supply disruption or delays caused by unexpected events, the contract design should be extended to scenarios that include uncertainty and risk mitigation.