It is supposed that the decision-making process is structured such that
M determines the quantity supplied to the market obeying the paradigm of profit maximization. Note that we do not distinguish market demand from the production quantity of the manufacturer because the market price is endogenous to the quantity sold. Moreover, the supplier produces the components to satisfy the demand
d and thus does not decide on the production quantity. Because the manufacturer’s goal is profit maximization, the production quantity
d chosen by
M is determined by differentiating
$$\begin{aligned} d \cdot ( p(d) - c_M - c_{SC} ) \end{aligned}$$
(1)
with respect to
d and setting the resulting expression equal to zero, i.e.,
$$\begin{aligned} p(d) - c_M - c_{SC} - bd \mathop {=}\limits ^{!} 0, \end{aligned}$$
(2)
which yields the optimum production quantity
\(d^\star = \frac{a - c_M - c_{SC}}{2b}\) and the optimal sale price
\(p(d^\star ) = \frac{a + c_M + c_{SC}}{2}\). Here,
\(c_M\) and
\(c_{SC}\) denote the manufacturer’s unit production costs and the supply costs per unit charged by
S, respectively. We further assume that the supplier wants to earn a fixed profit margin
r. Thus, the supply costs
\(c_{SC}\) consist of the supplier’s fixed profit margin
r and the supplier’s unit production costs
\(c_S\), i.e.,
\(c_{SC} = r + c_S\). This assumption is not completely new: Honda Motor Company, e.g., first learns extensively about a suppliers cost structure and then specifies a target price that combines both the suppliers unit production cost and a percent margin [
29]. Similar approaches to specify the supply costs have been proposed by [
4,
21,
27]. Summing up, the supply chain profit is given by
$$\begin{aligned} J = J^M + J^S = \frac{(a-c_M-c_{SC})^2}{4b} + \frac{a-c_M-(r+c_S)}{2b}r = \frac{(a-c_M-c_{S})^2-r^2}{4b}. \end{aligned}$$
It is supposed that the manufacturer wants to decrease the supplier’s unit production costs
\(c_S\) by conducting supplier development projects to increase the market share if that increases the overall profit of the supply chain. To this end, the sustainable effect of supplier development on the supplier’s unit production costs
\(c_S\) is modelled by
\(c_S(x) = c_0 x^m\),
1 where
\(c_0 > 0\) denotes the supplier’s unit production cost at the outset,
\(m < 0\) characterizes the supplier’s learning rate, and
x defines the cumulative number of realized supplier development projects. The latter is modelled as a time-dependent function
\(x: [0,T] \rightarrow \mathbb {R}_{\ge 0}\) governed by the ordinary differential equation
$$\begin{aligned} \dot{x}(t) := \frac{\mathrm {d}}{\mathrm {d}t} x(t) = u(t),\qquad x(0) = x_0 = 1, \end{aligned}$$
(3)
with
\(u \in \mathcal {L}^\infty (\mathbb {R}_{\ge 0},[0,\omega ])\). Here,
u(
t) describes the number of supplier development projects at time
t; with capacity bound
\(\omega > 0\) to reflect limited availability of resources in terms of time, manpower, or budget. Similar models of cost reduction through learning have been proposed by [
4,
11,
20,
27,
56].
The costs of supplier development are integrated into the proposed model by a penalization term
\(c_{SD} u(t)\),
\(c_{SD} \ge 0\). Overall, this yields the supply chain’s profit function
\(J^{SC}: u\;\mapsto\;\mathbb{R}\)
$$\begin{aligned} J_{T}(u;x_0):= \int _0^T \frac{(a-c_M-c_0 x(t)^m)^2-r^2}{4b} - c_{SD} u(t)~\mathrm {d}t \end{aligned}$$
(4)
for a given time interval [0,
T], which must be maximized subject to the control constraints
\(0 \le u(t) \le \omega\),
\(t \in [0,T)\), and the system dynamics (
3). The contract period
T is of particular interest since investments into the cost structure of the supply chain require their amortization during the runtime of the contractual agreement. A summary of the parameters is given in Table
1.
Table 1
List of parameter
T
| Contract period | 60 |
a
| Prohibitive price | 200 |
b
| Price elasticity | 0.01 |
\(c_M\)
| Variable cost per unit (M) | 70 |
\(c_0\)
| Variable cost per unit (S) | 100 |
r
| Fixed profit margin (S) | 15 |
\(c_{SD}\)
| Supplier development cost per unit | 100,000 |
\(\omega\)
| Resource availability | 1 |
m
| Learning rate |
\(-0.1\)
|