# A Coordinated Supply Contract for a Two-Echelon Supply Chain Considering Learning Effects
## 1 Introduction  
Supply chain coordination is crucial for optimizing operational efficiency and enhancing overall performance, especially in today's rapidly changing business environment [1]. Effective supply chain coordination can achieve a balance of interests among the various participants in the supply chain and drive collaborative efforts [2], i.e., the total profit of the decentralized supply chain is equal to that achieved under a centralized system. Supply contracts serve as key tools widely applied to establish cooperative relationships among the parties involved in the supply chain and ensure the realization of mutual benefits. Several channel coordination schemes have been introduced, including revenue-sharing contracts [3, 4], cost-sharing contracts [5, 6], and option contracts [7, 8], etc. It is important to note that these supply contracts assume constant production costs throughout a production period. However, in real-world industries, the per-unit production cost is often reduced as production quantity increases or over time due to learning from repetitive activities [9, 10, 11]. This phenomenon is known as the learning curve (LC) effect. The LC effect has been widely observed in such industries as assembly tasks, aircraft manufacturing, apparel manufacturing, transportation, and semiconductor industries [1, 8, 10, 11, 12, 13, 14]. [PERSON] [9] is regarded as the first to address the operations management (OM) problems under LC effects. Since then, a substantial body of literature has emerged, incorporating the LC effects as a crucial concept applied to achieve cost efficiency and optimize operational efficiency in OM problems [12, 13]. The existing literature has addressed problems in pricing strategy [3, 15, 16, 17],lot size [18; 19; 20; 21], outsourcing decisions [22; 23; 24; 25], scheduling [26; 27; 28; 29; 30], and supplier selection strategy [31; 32], to name a few. Their findings demonstrate the effectiveness of implementing learning curve effects across various domains, showcasing improvements in system performance. However, few studies exist on coordinated supply-contract-design models in situations where LC effects are present. This study aims to fill the research gap by integrating the learning effects into the development of supply contracts. Specifically, we aim to provide a coordinated supply contract scheme for a two-stage supply chain with decreasing production costs due to the LC effects. The supply chain under consideration consists of two independent parties, a single manufacturer and a single retailer. It is assumed that the manufacturer acts as a Stackelberg game leader who takes the first move by designing and offering the contract. The contract is designed in a way that the retailer orders a quantity amount that aligns with the globally optimal solution. This paper contributes to the literature twofold: (1) We develop a coordinated contract scheme for the manufacturer under LC effects. The scheme includes a procedure to determine a globally optimal order quantity for the supply chain under LC effects. Then, the revenue-sharing contract is applied for channel coordination. Our findings reveal that the contract with LC effects can provide additional profits for manufacturers and retailers compared to the existing coordinated contracts with static production costs. (2) We address the problems related to bargaining powers among supply chain members. We present how the contract parameters are chosen for different bargaining-power structures. The relationships between bargaining-power structures and revenue-sharing ratios are identified in the coordinated contracts. This article is organized as follows. Section 2 reviews the related literature. In Section 3, we present basic supply-contract models for the centralized and decentralized system under static production costs, which are used as benchmarks. Section 4 presents the supply contract scheme that incorporates the LC effect in a supply-contract-design process.