# Numeric Evaluation of Game-Theoretic Collaboration Modes in Supplier Development
## 4 Model Description
### Derivation of the Optimizer's Cost Function  
Here, we consider a linear price distribution curve \(p(d)=a-b\cdot d\), where \(d\) represents the production quantity and \(p\) represents the sale price, cf., e.g., [7; 33; 34; 35], and where \(a>0\) and \(b>0\) signify the prohibitive price and the price elasticity of the production. In the present study, prohibitive price is the maximum price of the manufacturer's products that consumer can afford and the price elasticity is the range of product's price that the supplier offers to the manufacturer.  
Note that this system shows only one decision variable, which is given by the sales of the manufacturer. The supplier cannot take a decision with regards the production quantity and just produces the components to meet the market demand \(d\). Due to revenue maximization, the production quantity \(d\) which adopted by \(M\) is determined by the zero of the first derivative of the profit given by \(d(p(d)-c_{\text{M}}-c_{\text{SC}})\), where \(c_{\text{M}}\) and \(c_{\text{SC}}\) denote the unit production costs of \(M\) and the supply costs per unit of \(S\), cf. [7; 8]. Hence, we obtain  
\[p(d)-c_{\text{M}}-c_{\text{SC}}-bd=0\]  
and thereby the optimal production quantity and optimal sale price  
\[d^{*}=\frac{a-c_{\text{M}}-c_{\text{SC}}}{2b},\quad p(d^{*})=\frac{a+c_{\text {M}}+c_{\text{SC}}}{2}\]  
In order to include fixed revenues per unit \(r\) of the supplier \(S\) similar to [7; 8; 33; 34; 35], we define \(c_{\text{SC}}:=r+c_{\text{S}}\), where \(c_{\text{S}}\) represents the unit production costs of the supplier \(S\). Now, we maximize the overall profit \(J\), which is the summation of manufacturer profit \(J^{M}\) and the supplier profit \(J^{S}\)  
\[J(c_{\text{S}})=J^{M}(c_{\text{S}})+J^{S}(c_{\text{S}})=\frac{(a-c_{\text{M}} -c_{\text{SC}})^{2}}{4b}+\frac{a-c_{\text{M}}(r+c_{\text{S}})}{2b}r.\]  
Greater experience or development in human resource capabilities, production process or machines enable the supplier to improve its capability. Learning causes less effort and time, and in turn also fewer production costs per unit. To formulate the problem more realistic, we take the supplier learning rate into account. We suppose that the manufacturer wants to increase the overall revenue of the supply chain by decreasing the unit production costs \(c_{\text{S}}\) of the supplier via supplier development projects. To this end, we introduce a time-dependent function \(x:\mathbb{R}\rightarrow\mathbb{R}\)  
\[c_{\text{S}}=c_{0}x(t)^{m},\text{ with }\dot{x}(t)=u(t),\qquad x(0)=x_{0}=1\]  
where \(c_{0}>0\) represents the base unit production cost of the supplier \(S\), and \(m<0\) characterizes its learning rate. Last, \(x\) donates the number of supplier development projects, which can be changed by the manufacturer \(M\) via \(u\in[0,\omega]\). The bound \(\omega>0\) to represents a resources availability limitation, which may be due to time, manpower, or budget. A variety of former studies have proposed similar models of cost reduction through learning, cf., e.g., [33; 34; 35; 36]. To account for the respective development projects, we integrate supplier development costs \(c_{\text{SD}}u(t)\) into the profit function and obtain  
\[J_{T}(u;x_{0})=\int\limits_{0}^{T}\frac{(a-c_{\text{M}}-c_{0}x(t)^{m})^{2}-r^ {2}}{4b}-c_{\text{SD}}u(t)dt, \tag{1}\]  
where \(c_{\text{SD}}\geq 0\) represents the costs of a supplier development action. To have a more realistic setting, we do not consider problem parameters to be deterministic but some to be stochastic. While parameterssuch as the learning rate \(m\) are definitely stochastic and time-varying, parameters like the contract period \(T\) are most likely deterministic and fixed.