# A Coordinated Supply Contract for a Two-Echelon Supply Chain Considering Learning Effects
## 3 Basic Supply-Chain Models without Learning-Curve Effects  
We consider a two-echelon supply chain with one manufacturer and one retailer in which the manufacturer distributes its products through the retailer to the single-selling season market. The market demand is subject to high demand uncertainty and can be characterized by a stochastic variable \(X\) that follows a probability density function \(f(x)\) and a cumulative distribution function \(F(x)\). The product requires a long production lead time, and the order quantity must be determined well before the sale season. Any products unsold until the end of the selling season are salvaged at discounted prices. In our research, we have chosen model Equation (1) as the learning curve, with \(b=-log_{2}r\), where r represents the learning rate. Then, the learning curve model is as follows:  
\[c_{Q}=c_{1}Q^{-b}=c_{1}Q^{log_{2}r} \tag{2}\]  
where \(Q\) is the production quantity and \(c_{Q}\) is the cumulative average per-unit production cost for the \(Q\) units produced. This model is relatively easy to comprehend and interpret. The model facilitates intuitive and convenient data fitting and analysis. The supply contract scheme in our study is based on a Stackelberg game framework, with the manufacturer acting as the leader, determining the supply contract and offering it to the retailer, and the retailer as the follower, deciding on order quantity based on the offered contract. The manufacturer anticipates the retailer's decisions and tries to coordinate the supply chain by setting the contract parameters, including wholesale price and revenue-sharing fractions, so it is optimal for the retailer to determine the order quantities that are parallel to the decision in the centralized system. The notations in Nomenclature are used throughout this article. It is assumed that the sales price is greater than the production cost, and the excess inventory is sold at a lower price than the production cost, i.e., \(p>c>s\). To begin with, we discuss contract models for a supply chain with static production costs (without learning effects). These models are used as benchmarks for our model, which will be presented in the upcoming section. In a two-tier supply chain composed of a manufacturer and a retailer, the optimal production quantity for the overall supply chain in the face of uncertain demand can be obtained by treating these two tiers as a single organization and making decisions accordingly [41]. If the supply chain produces \(Q\) products and \(x\) is a realized demand, the sales quantity will be min (\(Q\), \(x\)), and the excess inventory will be max (\(Q-x\), \(0\)). It is straightforward to have \(S(Q)=E[min(Q,x)]=Q-\int_{0}^{Q}F(x)dx\) and \(I(Q)=\int_{0}^{Q}F(x)dx=Q-S(Q)\). Then, the expected profit of the supply chain is as follows:  
\[\pi_{Nc}=pS(Q)-cQ+sI(Q)=(p-c)Q-(p-s)\int_{0}^{Q}F(x)dx \tag{3}\]  
The first term is sales revenue, the second term is production cost, and the final term is salvage value from unsold products. The first and second derivatives of \(\pi_{Nc}\) with respect to \(Q\) are as follows:  
\[\frac{d\pi_{Nc}}{dQ}=(p-c)-(p-s)F(Q) \tag{4}\]  
\[\frac{d^{2}\pi_{Nc}}{dQ^{2}}=-(p-s)f(Q)<0 \tag{5}\]  
We can determine that \(\pi_{Nc}\) is a concave function in \(Q\) because its second derivative is negative. The optimal production quantity giving the maximum expected profit is obtained by the first order condition. Let \(Q_{Nc}^{*}\) denote the optimal production supply quantity that maximizes the overall profit of the supply chain, then the following relationship holds. \[Q_{Nc}^{*}=F^{-1}\bigg{(}\frac{p-c}{p-s}\bigg{)} \tag{6}\]  
From the system's perspective, it is optimal to produce \(Q_{Nc}^{*}\) to maximize the whole supply-chain profit. When each member of the supply chain makes decisions independently, it is natural for them to make decisions to maximize their individual benefits. A commonly employed strategy in decentralized decision-making systems is wholesale contracts.