Optimizing production and maintenance for the service-oriented manufacturing supply chain

Abstract

This work investigates a service-oriented manufacturing supply chain in which a manufacturer and an operator make decisions about equipment quality and maintenance service. Both the manufacturer and the operator have to make tradeoffs between equipment quality and maintenance service to maximize their own profit, which can lead to supply chain conflict. Decision models under decentralized decisions are formulated first for the manufacturer and the operator to make their respective independent optimal decisions, and a decision model under centralized decisions is formulated to obtain optimal decisions for the supply chain. The results show that channel coordination is not achievable and an agreement cannot be reached with decentralized decisions. To address this issue, two, i.e., a cost-sharing and a performance-based, strategies are introduced to coordinate the supply chain. The results reveal that the manufacturer and the operator are motivated to find the optimal decisions to maximize the profit of the supply chain when the subsidy rate or the penalty rate is equal to the profit margin of the operator. The models and the coordination strategies are extended to the situation considering the learning behavior of the manufacturer. The results show that the learning behavior impacts the profit of the supply chain with coordination and the preferences of the coordination strategy in the supply chain.

Appendix

1.1 A1: Proof of Lemma 1

Taking the first and the second partial derivatives of $\varPi_{m}$ with respect to $T_{i}$ gives $\frac{{\partial \varPi_{m} }}{{\partial T_{i} }} = P - \frac{{PT_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } - \frac{{C_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } + \xi = 0$ and $\frac{{\partial^{2} \varPi_{m} }}{{\partial T_{i}^{2} }} = - \frac{{\left( {PT_{r} + C_{r} } \right)\lambda_{0}^{'} \left( {T_{i} } \right)}}{\tau } < 0$, respectively. Thus, $\varPi_{m}$ is concave in $T_{i}$. The first-order condition $\partial \varPi_{m} /\partial T_{i} = 0$ is satisfied when $\lambda_{0} \left( {T_{i} } \right) =$$[(P + \xi )\tau ]/(PT_{r} + C_{r} )$. Since $\lambda_{0} (T_{i} )$ strictly increases as $T_{i}$ increases, $T_{1m}^{*} = T_{2m}^{*} = \cdots = T_{Nm}^{*} =$$Y/N - T_{p}$ can be obtained. Plugging $T_{1m}^{*} = T_{2m}^{*} = \cdots = T_{Nm}^{*} = \frac{Y}{N} - T_{p}$ into (6), $\varPi_{m} =$$P\left[ {Y - NT_{p} - NT_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right] - C\left( \tau \right) - N\left[ {C_{p} + C_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right] + W$ can be obtained.

Also $\frac{{\partial^{2} \varPi_{m} }}{{\partial N^{2} }} = \frac{1}{\tau }\left( {PT_{r} + C_{r} } \right)g^{'} \left( N \right) < 0$ and $\frac{{\partial^{2} \varPi_{m} }}{{\partial \tau^{2} }} = \frac{{ - 2N\left( {PT_{r} + C_{r} } \right)}}{{\tau^{3} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - C''\left( \tau \right) < 0$ exist. From the first-order conditions $\partial \varPi_{m} /\partial N = \frac{1}{\tau }\left( {PT_{r} + C_{r} } \right)g\left( N \right) - \left( {PT_{p} + C_{p} } \right) = 0$ and $\partial \varPi_{m} /\partial \tau = \frac{{NPT_{r} }}{{\tau^{2} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - C^{'} \left( \tau \right) + \frac{{NC_{r} }}{{\tau^{2} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } = 0$, the decisions of the manufacturer $(\tau_{m}^{*} ,N_{m}^{*} )$ can be obtained.

1.2 A2: Proof of Lemma 2

Similar to the proof of Lemma 1, the Lagrangian function is applied to the profit function of the operator, and the first-order condition is satisfied when $\lambda_{0} (T_{1o}^{*} ) = \lambda_{0} (T_{2o}^{*} ) = \cdots = \lambda_{0} (T_{No}^{*} )$. Hence, $T_{1o}^{*} = T_{2o}^{*} = \cdots = T_{No}^{*} = \frac{Y}{N} - T_{p}$ can be obtained. Plugging $T_{1o}^{*} = T_{2o}^{*} = \ldots = T_{No}^{*} = \frac{Y}{N} - T_{p}$ into the profit function of the operator, $\varPi_{o} = \left( {R - P} \right)\left[ {Y - NT_{p} - NT_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right] - W$ can be obtained.

Taking the first and the second partial derivatives of $\varPi_{o}$ with respect to $N$ gives $\frac{{\partial \varPi_{o} }}{\partial N} =$$- \left( {R - P} \right)T_{p} { + }\frac{1}{\tau }\left( {R - P} \right)T_{r} g\left( N \right)$ and $\frac{{\partial^{2} \varPi_{o} }}{{\partial N^{2} }} = \frac{1}{\tau }\left( {R - P} \right)T_{r} g^{\prime}\left( N \right) < 0$, respectively, and taking the first and the second partial derivatives of $\varPi_{o}$ with respect to $\tau$ gives $\frac{{\partial \varPi_{o} }}{\partial \tau } = \frac{{N\left( {R - P} \right)T_{r} }}{{\tau^{2} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega }$ and $\frac{{\partial^{2} \varPi_{o} }}{{\partial \tau^{2} }} = \frac{{ - 2N\left( {R - P} \right)T_{r} }}{{\tau^{3} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } < 0$, respectively. Apparently, $\varPi_{o}$ is a concave function of $N$ and $\tau$. From the first-order conditions $\partial \varPi_{o} /\partial N = 0$ and $\partial \varPi_{o} /\partial \tau = 0$, the optimal solutions of the operator can be obtained.

1.3 A3: Proof of Lemma 3

The Lagrangian function is applied to the profit function under centralized decisions

$$ \;\varPi_{c} = R\sum\limits_{i = 1}^{N} {\left[ {T_{i} - T_{r} M\left( {\tau ,T_{i} } \right)} \right]} - C\left( \tau \right) - \sum\limits_{i = 1}^{N} {\left[ {C_{p} + C_{r} M\left( {\tau ,T_{i} } \right)} \right]} + \xi \left[ {NT_{p} + \sum\limits_{i = 1}^{N} {T_{i} } - Y} \right] . $$

(37)

Taking the first and the second partial derivatives of $\varPi_{c}$ with respect to $T_{i}$ gives $\frac{{\partial \varPi_{c} }}{{\partial T_{i} }} = R - \frac{{RT_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } - \frac{{C_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } + \xi \;$ and $\frac{{\partial^{2} \varPi_{c} }}{{\partial T_{i}^{2} }} = - \frac{{\left( {RT_{r} + C_{r} } \right) \cdot \lambda^{\prime}_{0} \left( {T_{i} } \right)}}{\tau } < 0$, respectively. Therefore, $\varPi_{c}$ is concave in $T_{i}$. The first-order condition $\partial \varPi_{c} /\partial T_{i} = 0$ is satisfied when $\lambda_{0} \left( {T_{i} } \right) =$$[(R + \xi )\tau ]/(RT_{r} + C_{r} )$. Since $\lambda_{0} (T_{i} )$ strictly increases as $T_{i}$ increases, $T_{1c}^{*} = T_{2c}^{*} = \cdots = T_{Nc}^{*} = \frac{Y}{N} - T_{p}$ can be obtained. Plugging $T_{1c}^{*} = T_{2c}^{*} = \cdots = T_{Nc}^{*} = \frac{Y}{N} - T_{p}$ into (10), $\varPi_{c} = R\left[ {Y - NT_{p} - NT_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right] - N\left[ {C_{p} + C_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right]$ is obtained. Taking the first and the second partial derivatives of $\varPi_{c}$ with respect to $N$ gives $\frac{{\partial \varPi_{c} }}{\partial N} = - \left( {RT_{p} + C_{p} } \right) + \frac{1}{\tau }\left( {RT_{r} + C_{r} } \right)g\left( N \right)$ and $\frac{{\partial^{2} \varPi_{c} }}{{\partial N^{2} }} =$$\frac{1}{\tau }\left( {RT_{r} + C_{r} } \right)g^{\prime}\left( N \right) < 0$, respectively, and taking the first and the second partial derivatives of $\varPi_{c}$ with respect to $\tau$ gives $\frac{{\partial \varPi_{c} }}{\partial \tau } = \frac{{NRT_{r} }}{{\tau^{2} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - C^{\prime}\left( \tau \right) + \left( {\frac{Y}{N} - T_{p} } \right)^{\omega } \frac{{NC_{r} }}{{\tau^{2} }}$ and $\frac{{\partial^{2} \varPi_{c} }}{{\partial \tau^{2} }} = \frac{{ - 2N\left( {RT_{r} + C_{r} } \right)}}{{\tau^{3} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - C^{\prime\prime}\left( \tau \right) < 0$, respectively. Apparently, $\varPi_{c}$ is a concave function of $N$ and $\tau$. From the first-order conditions $\partial \varPi_{c} /\partial N = 0$ and $\partial \varPi_{c} /\partial \tau = 0$, the optimal solution of centralized decisions can be obtained.

1.4 A4: Proof of Proposition 1

From $g\left( {N_{c}^{*} } \right) = \frac{{RT_{p} + C_{p} }}{{RT_{r} + C_{r} }}\tau_{c}^{*}$, $g\left( {N_{c}^{*} } \right)$ obviously increases as $\tau_{c}^{*}$ increases. Given $g\left( {N_{c}^{*} } \right) = \frac{Y}{{N_{c}^{*} }}\lambda_{0} \left( {\frac{Y}{{N_{c}^{*} }} - T_{p} } \right) - M_{0} \left( {\frac{Y}{{N_{c}^{*} }} - T_{p} } \right)$, taking the first derivative of $g\left( {N_{c}^{*} } \right)$ with respect to $N_{c}^{*}$ gives $g^{\prime}\left( {N_{c}^{*} } \right) = - \frac{{Y^{2} }}{{N_{c}^{*3} }}\lambda^{\prime}_{0} \left( {\frac{Y}{{N_{c}^{*} }} - T_{p} } \right) < 0$. Obviously, $g\left( {N_{c}^{*} } \right)$ decreases as $N_{c}^{*}$ increases, and thus, $N_{c}^{*}$ decreases as $\tau_{c}^{*}$ increases.

1.5 A5: Proof of Proposition 2

Similar to the proofs of Lemmas 1 and 2, the Lagrangian function is applied to the profit function (13). Taking the first and the second partial derivatives of $\varPi_{m}$ with respect to $T_{i}$ gives $\frac{{\partial \varPi_{m} }}{{\partial T_{i} }} = P - \frac{{PT_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } - \frac{{\left( {1 - \alpha } \right)C_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } + \xi \;$ and $\frac{{\partial^{2} \varPi_{m} }}{{\partial T_{i}^{2} }} = - \frac{{\left[ {PT_{r} + \left( {1 - \alpha } \right)C_{r} } \right]\lambda_{0}^{'} \left( {T_{i} } \right)}}{\tau } < 0$, respectively. Therefore, $\varPi_{m}$ is concave in $T_{i}$. The first-order condition $\partial \varPi_{m} /\partial T_{i} = 0$ is satisfied when $\lambda_{0} \left( {T_{i} } \right) = \left[ {\left( {P + \xi } \right)\tau } \right]/\left[ {PT_{r} + \left( {1 - \alpha } \right)C_{r} } \right]$. Since $\lambda_{0} (T_{i} )$ strictly increases as $T_{i}$ increases,$T_{1csm}^{*} =$$T_{2csm}^{*} = \cdots = T_{Ncsm}^{*} = \frac{Y}{N} - T_{p}$ can be obtained. Plugging $T_{1csm}^{*} = T_{2csm}^{*} = \cdots = T_{Ncsm}^{*} = \frac{Y}{N} - T_{p}$ into (13), $\varPi_{m} = P\left[ {Y - NT_{p} - NT_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right] - \left( {1 - \alpha } \right)\left[ {C\left( \tau \right) + NC_{p} + C_{r} \sum\nolimits_{i = 1}^{N} {M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} } \right]{ + }W$ can be obtained. Taking the first and the second partial derivatives of $\varPi_{m}$ with respect to $N$ gives $\frac{{\partial \varPi_{m} }}{\partial N}\;\; = - \left[ {PT_{p} + \left( {1 - \alpha } \right)C_{p} } \right] + \frac{1}{\tau }\left[ {PT_{r} + \left( {1 - \alpha } \right)C_{r} } \right]g\left( N \right)$ and $\frac{{\partial^{2} \varPi_{m} }}{{\partial N^{2} }} = \frac{1}{\tau }\left[ {PT_{p} + \left( {1 - \alpha } \right)C_{p} } \right]g^{'} \left( N \right)$$= - \frac{{Y^{2} }}{{\tau N^{3} }}\left[ {PT_{p} + \left( {1 - \alpha } \right)C_{p} } \right]\lambda_{0}^{'} \left( {\frac{Y}{N} - T_{p} } \right) < 0$, respectively, and taking the first and the second partial derivatives of $\varPi_{m}$ with respect to $\tau$ gives $\frac{{\partial \varPi_{m} }}{\partial \tau } = \frac{{NPT_{r} }}{{\tau^{2} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - \left( {1 - \alpha } \right)C^{'} \left( \tau \right) + \frac{{NC_{r} \left( {1 - \alpha } \right)}}{{\tau^{2} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega }$ and $\frac{{\partial^{2} \varPi_{c} }}{{\partial \tau^{2} }} = \frac{{ - 2N\left[ {PT_{r} + \left( {1 - \alpha } \right)C_{r} } \right]}}{{\tau^{3} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - \left( {1 - \alpha } \right)C''\left( \tau \right) < 0$, respectively. Apparently, $\varPi_{m}$ is a concave function of $N$ and $\tau$. From the first-order conditions $\partial \varPi_{m} /\partial N = 0$ and $\partial \varPi_{m} /\partial \tau = 0$, the decisions of the manufacturer $\left( {N_{csm}^{*} ,\tau_{csm}^{*} } \right)$ can be obtained by solving the system of equations $g\left( {N_{csm}^{*} } \right) = \frac{{PT_{p} + \left( {1 - \alpha } \right)C_{p} }}{{PT_{r} + \left( {1 - \alpha } \right)C_{r} }} \cdot \tau_{csm}^{*}$ and $L\left( {\tau_{csm}^{*} } \right){ = }\frac{{N_{csm}^{*} \left[ {PT_{r} + \left( {1 - \alpha } \right)C_{r} } \right]M\left( {\frac{Y}{{N_{csm}^{*} }} - T_{p} } \right)}}{1 - \alpha }$.

Similarly, $T_{1cso}^{*} = T_{2cso}^{*} = \cdots = T_{Ncso}^{*} = \frac{Y}{N} - T_{p}$ can be obtained. Constraint (18) can be expressed as $P\sum\nolimits_{i = 1}^{N} {\left[ {T_{i} - T_{r} M\left( {\tau ,T_{i} } \right)} \right]} - \left( {1 - \alpha } \right)\left[ {C\left( \tau \right) + NC_{p} + C_{r} \sum\nolimits_{i = 1}^{N} {M\left( {\tau ,T_{i} } \right)} } \right] + W \ge \pi$. Therefore, when the profit of the manufacturer is not less than its reservation profit, the profit of the operator will be maximized. Therefore, $W + P\sum\nolimits_{i = 1}^{N} {\left[ {T_{i} - T_{r} M\left( {T_{i} ,\tau } \right)} \right]} = \pi + \left( {1 - \alpha } \right)\left[ {C\left( \tau \right) + NC_{p} + C_{r} \sum\nolimits_{i = 1}^{N} {M\left( {\tau ,T_{i} } \right)} } \right]$ can be obtained, meaning that the operator is most profitable. After plugging the above equation and $T_{1cso}^{*} = T_{2cso}^{*} = \cdots = T_{Ncso}^{*} = \frac{Y}{N} - T_{p}$ into (16), the following can be obtained$\varPi_{o} =$$R\left[ {Y - NT_{p} - NT_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right]$$- C\left( \tau \right) - N\left[ {C_{p} + C_{r} M\left( {\tau ,\frac{Y}{N} - T_{p} } \right)} \right] - \pi$.

Taking the first and the second partial derivatives of $\varPi_{o}$ with respect to $N$ gives $\frac{{\partial \varPi_{o} }}{\partial N} = - \left[ {RT_{p} + C_{p} } \right] - \frac{1}{\tau }\left[ {RT_{r} + C_{r} } \right]g\left( N \right)$ and $\frac{{\partial^{2} \varPi_{o} }}{{\partial N^{2} }} = \frac{1}{\tau }\left( {RT_{r} + C_{r} } \right)g^{\prime}\left( N \right) < 0$, respectively, and taking the first and the second partial derivatives of $\varPi_{o}$ with respect to $\tau$ gives $\frac{{\partial \varPi_{o} }}{\partial \tau } = \frac{{NRT_{r} }}{{\tau^{2} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - C^{\prime}\left( \tau \right) + \left( {\frac{Y}{N} - T_{p} } \right)^{\omega } \;\;\frac{{NC_{r} }}{{\tau^{2} }}$ and $\frac{{\partial^{2} \varPi_{o} }}{{\partial \tau^{2} }} = \frac{{ - 2N\left( {RT_{r} + C_{r} } \right)}}{{\tau^{3} }}\left( {\frac{Y}{N} - T_{p} } \right)^{\omega } - C^{\prime\prime}\left( \tau \right) < 0$, respectively. Apparently, $\varPi_{o}$ is a concave function of $N$ and $\tau$. From the first-order conditions $\partial \varPi_{o} /\partial N = 0$ and $\partial \varPi_{o} /\partial \tau = 0$, the optimal solutions of the operator can be obtained by solving the system of equations $g\left( {N_{cso}^{*} } \right) = \frac{{RT_{p} + C_{p} }}{{RT_{r} + C_{r} }}\tau_{cso}^{*}$ and $L\left( {\tau_{cso}^{*} } \right){ = }N_{cso}^{*} \left( {RT_{r} + C_{r} } \right)\left( {\frac{Y}{{N_{cso}^{*} }} - T_{p} } \right)^{\omega }$ with $\left( {\tau_{csm}^{*} ,N_{csm}^{*} } \right) = \left( {\tau_{cso}^{*} ,N_{cso}^{*} } \right) = \left( {\tau_{c}^{*} ,N_{c}^{*} } \right)$, or equivalently, $\frac{{PT_{p} + \left( {1 - \alpha } \right)C_{p} }}{{PT_{r} + \left( {1 - \alpha } \right)C_{r} }} = \frac{{RT_{p} + C_{p} }}{{RT_{r} + C_{r} }}{ = }\frac{{RT_{p} + C_{p} }}{{RT_{r} + C_{r} }}$ and $\frac{{PT_{r} + \left( {1 - \alpha } \right)C_{r} }}{1 - \alpha } = RT_{r} + C_{r} = RT_{r} + C_{r}$, and the subsidy rate can be obtained after solving this system of equations. Plugging $\left( {T_{ic}^{*} ,\tau_{c}^{*} ,N_{c}^{*} } \right)$, $\left( {T_{im}^{*} ,\tau_{m}^{*} ,N_{m}^{*} } \right)$ and $\left( {T_{io}^{*} ,\tau_{o}^{*} ,N_{o}^{*} } \right)$ into (10), (2) and (4), respectively, the fixed payment $W$ can be obtained by solving this system of inequalities $\varPi_{m}^{*} \left( {T_{ic}^{*} ,\tau_{c}^{*} ,N_{c}^{*} } \right) >$$\varPi_{m}^{*} \left( {T_{im}^{*} ,\tau_{m}^{*} ,N_{m}^{*} } \right)$ and $\varPi_{o}^{*} \left( {T_{ic}^{*} ,\tau_{c}^{*} ,N_{c}^{*} } \right) > \varPi_{o}^{*} \left( {T_{io}^{*} ,\tau_{o}^{*} ,N_{o}^{*} } \right)$.

Propositions 3 and 7 can be proved in similar ways. Therefore, their proofs are omitted.

1.6 A6: Proof of Proposition 5

For analytical simplicity, the approximation $\sum\nolimits_{i = 1}^{N} {i^{ - a} = \int_{0}^{N} {i^{ - a} di} } = \frac{{N^{1 - a} }}{1 - a}$ is adopted. This approximation works very well if $0 < a \le 0.6$ and it also applicable if $0. 6< a \le 1$ (Keachie and Fontana 1966; Camm et al. 1987; Tarakci et al. 2009). Although this approximation has a tendency to slightly overstate the total cost, it doesn’t affect our main analysis.

Let $g_{l} \left( N \right) = \left( {\frac{Y}{{N^{1 - a} }} - \frac{{aT_{p} }}{1 - a}} \right)\lambda_{0} \left( {\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right) - N^{a} M_{0} \left( {\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right)$.

Solving the decision model in (29)–(30) and taking the first and the second partial derivatives of $\varPi_{cl}$ with respect to $T_{i}$ gives $\frac{{\partial \varPi_{cl} }}{{\partial T_{i} }} = R - \frac{{RT_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } - \frac{{C_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } + \xi = 0$ and $\frac{{\partial^{2} \varPi_{cl} }}{{\partial T_{i}^{2} }} =$$- \frac{{\left( {RT_{r} + C_{r} } \right)\lambda_{0}^{'} \left( {T_{i} } \right)}}{\tau } < 0$, respectively. Therefore, $\varPi_{cl}$ is concave in $T_{i}$. The first-order condition $\partial \varPi_{cl} /\partial T_{i} = 0$ is satisfied when $\lambda_{0} \left( {T_{i} } \right) = [(R + \xi )\tau ]/(RT_{r} + C_{r} )$. Since $\lambda_{0} (T_{i} )$ strictly increases as $T_{i}$ increases, $T_{1cl}^{*} = T_{2cl}^{*} = \cdots = T_{Ncl}^{*} = \frac{Y}{{N_{cl}^{*} }} - \frac{{N_{Lc}^{{* - a_{cl} }} }}{{1 - a_{cl} }}T_{p}$ can be obtained. Similarly, from the first-order conditions $\partial \varPi_{cl} /\partial N = 0$ and $\partial \varPi_{cl} /\partial \tau = 0$, the optimal solutions of the other two variables $\left( {\tau_{cl}^{*} ,N_{cl}^{*} } \right)$ under centralized decisions can be obtained by solving the linear systems $g_{l} \left( {N_{cl}^{*} } \right) = \frac{{RT_{p} + C_{p} }}{{RT_{r} + C_{r} }}\tau_{cl}^{*}$ and $L\left( {\tau_{cl}^{*} } \right) = N_{cl}^{*} \left( {RT_{r} + C_{r} } \right)\left( {\frac{Y}{{N_{cl}^{*} }} - \frac{{T_{p} N_{cl}^{{* - a_{cl} }} }}{{1 - a_{cl} }}} \right)^{\omega }$.

Given $L\left( {\tau_{cl}^{*} } \right) = \tau_{cl}^{*2} C^{'} \left( {\tau_{cl}^{*} } \right) = N_{cl}^{*} \left( {RT_{r} + C_{r} } \right)\left( {\frac{Y}{{N_{cl}^{*} }} - \frac{{T_{p} N_{cl}^{{* - a_{cl} }} }}{{1 - a_{cl} }}} \right)^{\omega }$, the first partial derivatives of $L\left( {\tau_{cl}^{*} } \right)$ with respect to $\tau_{cl}^{*}$ and $a_{cl}$ are given by $\partial L\left( {\tau_{cl}^{*} } \right)/\partial \tau_{cl}^{*} = 2\tau_{cl} C^{'} \left( {\tau_{cl} } \right) \ge 0$ and $\partial L\left( {\tau_{cl}^{*} } \right)/\partial a_{cl} = \frac{{T_{p} N_{cl}^{{*1 - a_{cl} }} \left( {RT_{r} + C_{r} } \right)\left[ {\left( {1 - a_{cl} } \right)\ln N_{cl}^{*} - 1} \right]}}{{\left( {1 - a_{cl} } \right)^{2} }}\lambda_{0} \left( {\frac{Y}{{N_{cl}^{*} }} - \frac{{T_{p} N_{cl}^{{* - a_{cl} }} }}{{1 - a_{cl} }}} \right)$, respectively. Therefore $\partial L\left( {\tau_{cl}^{*} } \right)/\partial a_{cl} > 0$ and $\partial \tau_{cl}^{*} /\partial a_{cl} > 0$ can be obtained when $0 \le a_{cl} \le 1 - \frac{1}{{\ln N_{cl}^{*} }}$; and $\partial L\left( {\tau_{cl}^{*} } \right)/\partial a_{cl} < 0$ and $\partial \tau_{cl}^{*} /\partial a_{cl} < 0$ can be obtained when $1 - \frac{1}{{\ln N_{cl}^{*} }} < a_{cl} \le 1$.

Given $T_{icl}^{*} = \frac{Y}{{N_{cl}^{*} }} - \frac{{N_{cl}^{{* - a_{cl} }} }}{{1 - a_{cl} }}T_{p}$, the partial derivative of $T_{icl}^{*}$ with respect to $a_{cl}$ is given by $\partial T_{icl}^{*} /\partial a_{cl} = \frac{{N_{cl}^{{* - a_{cl} }} \left[ {1 - \left( {1 - a_{cl} } \right)\ln N_{cl}^{*} } \right]}}{{\left( {1 - a_{cl} } \right)^{2} }}$. Therefore $\partial T_{icl}^{*} /\partial a_{cl} > 0$ can be obtained when $0 \le a_{c} \le 1 - \frac{1}{{\ln N_{cl}^{*} }}$; and $\partial T_{icl}^{*} /\partial a_{cl} < 0$ can be obtained when $1 - \frac{1}{{\ln N_{cl}^{*} }} < a_{c} \le 1$. Similarly, $\partial N_{cl}^{*} /\partial a_{cl} > 0$ can be obtained.

1.7 A7: Proof of Proposition 6

By the Envelope Theorem (Milgrom and Segal 2002; Benjaafar et al. 2019), $\frac{{\partial \varPi_{cl}^{*} }}{\partial a}\left( a \right) = \frac{{\partial \varPi_{cl}^{*} }}{\partial a}\left( {T_{icl}^{*} ,N_{cl}^{*} ,\tau_{cl}^{*} ,a} \right)$ holds on any compact interval in (0, 1). Therefore,

$$ \begin{aligned} \frac{{\partial \varPi_{cl}^{*} }}{\partial a}\left( {T_{icl}^{*} ,N_{cl}^{*} ,\tau_{cl}^{*} ,a} \right) & = - C_{p} \frac{{ - \left( {1 - a} \right)N_{cl}^{*1 - a} \ln N_{cl}^{*} + N_{cl}^{*1 - a} }}{{\left( {1 - a} \right)^{2} }} - K^{\prime}\left( a \right) + \xi^{*} T_{p} \frac{{ - \left( {1 - a} \right)N_{cl}^{*1 - a} \ln N_{cl}^{*} + N_{cl}^{*1 - a} }}{{\left( {1 - a} \right)^{2} }} \\ & = \left( {\xi^{*} T_{p} - C_{p} } \right)\frac{{N_{cl}^{*1 - a} \left[ {1 - \left( {1 - a} \right)\ln N_{cl}^{*} } \right]}}{{\left( {1 - a} \right)^{2} }} - K^{\prime}\left( a \right) \\ \end{aligned} $$

is obtained. Since $\xi^{*} = \left( {RT_{r} + C_{r} } \right)\lambda \left( {\tau^{*} ,T_{i}^{*} } \right) - R = \left( {RT_{r} + C_{r} } \right)\frac{1}{\tau } \cdot \omega \cdot \left( {\frac{Y}{N} - \frac{{N^{ - a} }}{1 - a}T_{p} } \right)^{\omega - 1} - R < - \frac{{C_{p} }}{{T_{p} }}$, thus, $\partial \varPi_{cl}^{*} /\partial a \ge 0$ holds when $0 \le a_{cl} \le 1 - \frac{1}{{\ln N_{cl}^{*} }}$ and $K^{\prime}\left( {a_{cl} } \right) \le \hat{K^{\prime}}\left( {a_{cl} } \right)$. Furthermore, $\partial \varPi_{cl}^{*} /\partial a < 0$ holds when $1 - \frac{1}{{\ln N_{cl}^{*} }} < a_{cl} \le 1$ or when $0 \le a_{cl} \le 1 - \frac{1}{{\ln N_{cl}^{*} }}$ and $K^{\prime}\left( {a_{cl} } \right) > \hat{K^{\prime}}\left( {a_{cl} } \right)$.

1.8 A8: Proof of Lemma 4

Let $g_{l} \left( {N,a} \right) = \left( {\frac{Y}{{N^{1 - a} }} - \frac{{aT_{p} }}{1 - a}} \right)\lambda_{0} \left( {\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right) - N^{a} M_{0} \left( {\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right)$, $h_{cl} \left( {\tau ,N,a} \right) = N^{1 - a} \left[ {\left( {1 - a} \right)\ln N - 1} \right]\left[ {RT_{p} + C_{p} - \left( {RT_{r} + C_{r} } \right)T_{p} \lambda \left( {\tau ,\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right)} \right] - \left( {1 - a} \right)^{2} K^{'} \left( a \right)$, $h_{ml} \left( {\tau ,N,a} \right) = N^{1 - a} \left[ {\left( {1 - a} \right)\ln N - 1} \right]\left[ {PT_{p} + C_{p} - \left( {PT_{r} + C_{r} } \right)T_{p} \lambda \left( {\tau ,\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right)} \right] - \left( {1 - a} \right)^{2} K^{\prime}\left( a \right)$, $h_{ol} \left( {N,a} \right) = \left( {R - P} \right)T_{p} N^{1 - a} \left[ {\left( {1 - a} \right)\ln N - 1} \right]\left[ {1 - T_{r} \lambda \left( {\tau ,\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right)} \right]$ and $S_{l} \left( {\tau ,N,a} \right) = \frac{{N\left( {R - P} \right)T_{r} }}{{\tau^{ 2} }}\left( {\frac{Y}{N} - \frac{{T_{p} N^{ - a} }}{1 - a}} \right)^{\omega }$.

• (i) Taking the first and the second partial derivatives of $\varPi_{ml}$ with respect to $T_{i}$ gives $\frac{{\partial \varPi_{ml} }}{{\partial T_{i} }} = P - \frac{{PT_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } - \frac{{C_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } + \xi = 0$ and $\frac{{\partial^{2} \varPi_{m} }}{{\partial T_{i}^{2} }} = - \frac{{\left( {PT_{r} + C_{r} } \right)\lambda_{0}^{'} \left( {T_{i} } \right)}}{\tau } < 0$, respectively. Therefore, $\varPi_{ml}$ is concave in $T_{i}$. The first-order condition $\partial \varPi_{ml} /\partial T_{i} = 0$ is satisfied when $\lambda_{0} \left( {T_{i} } \right) = [(P + \xi )\tau ]/(PT_{r} + C_{r} )$. Since $\lambda_{0} (T_{i} )$ strictly increases as $T_{i}$ increases, $T_{1ml}^{*} = T_{2ml}^{*} = \cdots = T_{Nml}^{*} = \frac{Y}{{N_{ml}^{*} }} - \frac{{N_{ml}^{{* - a_{ml}^{*} }} }}{{1 - a_{ml}^{*} }}T_{p}$ can be obtained. Similarly, from the first-order conditions $\partial \varPi_{ml} /\partial N = 0$, $\partial \varPi_{ml} /\partial \tau = 0$ and $\partial \varPi_{ml} /\partial a = 0$, the optimal solutions of the manufacturer can be obtained.

• (ii) These results can be proved in similar ways and, therefore, the proofs are omitted.

• (iii) From (35), $\mathop {\varPi_{cl} }\limits_{{T_{i} ,N,\tau ,a}} = R\sum\nolimits_{i = 1}^{N} {\left[ {T_{i} - M\left( {\tau ,T_{i} } \right)T_{r} } \right]} - C\left( \tau \right) - \left( {\sum\nolimits_{i = 1}^{N} {C_{p} } i^{ - a} + C_{r} \sum\nolimits_{i = 1}^{N} {M\left( {\tau ,T_{i} } \right)} } \right) - K\left( a \right) + \xi \left[ {\sum\nolimits_{i = 1}^{N} {T_{p} i^{ - a} + } \sum\nolimits_{i = 1}^{N} {T_{i} } - Y} \right]$. Taking the first and the second partial derivatives of $\varPi_{cl}$ with respect to $T_{i}$ gives $\frac{{\partial \varPi_{cl} }}{{\partial T_{i} }} = R - \frac{{RT_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } - \frac{{C_{r} \lambda_{0} \left( {T_{i} } \right)}}{\tau } + \xi$ and $\frac{{\partial^{2} \varPi_{cl} }}{{\partial T_{i}^{2} }} = - \frac{{\left( {RT_{r} + C_{r} } \right)\lambda^{\prime}_{0} \left( {T_{i} } \right)}}{\tau } < 0$. Apparently, $\varPi_{cl}$ is a concave function of $T_{i}$. The first-order condition $\partial \varPi_{cl} /\partial T_{i} = 0$ is satisfied when $\lambda_{0} \left( {T_{i} } \right) = [(R + \xi )\tau ]/(RT_{r} + C_{r} )$. Since $\lambda_{0} (T_{i} )$ strictly increases as $T_{i}$ increases, $T_{1cl} = T_{2cl} = \cdots = T_{Ncl} = \frac{Y}{N} - \frac{{N^{ - a} }}{1 - a}T_{p}$ is obtained. After plugging $T_{1cl} = T_{2cl} = \cdots = T_{Ncl} = \frac{Y}{N} - \frac{{N^{ - a} }}{1 - a}T_{p}$ into the profit function, from the first-order conditions $\partial \varPi_{cl} /\partial N = 0$, $\partial \varPi_{cl} /\partial \tau = 0$ and $\partial \varPi_{cl} /\partial a = 0$, the optimal solutions of the centralized decisions with learning behavior can be obtained.