Maintenance optimisation and coordination with fairness concerns for the service-oriented manufacturing supply chain

ABSTRACT

We consider a service-oriented supply chain consisting of a manufacturer and an operator both with fairness concerns. The manufacturer provides equipment maintenance by choosing an optimal maintenance service level to maximise its profit, which may reduce the profit of the operator and the efficiency of the supply chain. We initially establish optimisation models with fairness concerns, and then a novel incentive strategy (LB strategy) is proposed to achieve the supply chain coordination. The LB strategy is compared with the existing cost subsidy (CS) strategy. Also, our analyses reveal the impact of fairness concern on the service-oriented manufacturing supply chain.

KEYWORDS:

• Service-oriented manufacturing
• maintenance optimisation
• coordination strategy
• fairness concern
• supply chain

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (71971052, 71671033, 71971048), the Fundamental Research Funds for the Central Universities (N2006006, N160601001), the Fund for Innovative Research Groups of the National Natural Science Foundation of China (71621061), the Major Internatonal Joint Research Project of the National Natural Science Foundation of China (71520107004), the Project of Promoting Talents in Liaoning Province (XLYC1807252, XLYC1907015) and the 111 Project (B16009).

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

A2.Proof of Proposition 6

For analytical simplicity and notational convenience, let $E=(1+\lambda )C_r-\lambda R{T}_r$ and $F=[(1+\lambda )A-\lambda (R-A)]T_r+(1+\lambda )C_r$. The discussions in Sections 4.1 and 5.1 indicate $E>0$ and $F>0$. Taking the derivative of $\mathrm{\Delta }g({T_{M_{UR}}}^{\ast })$ with respect to $\lambda$ gives $\frac{\mathrm{\partial }\mathrm{\Delta }g({T_{M_{UR}}}^{\ast })}{\mathrm{\partial }\lambda }=\frac{(T_r{C}_p-T_p{C}_r)\{(F^2-E^2)R+A{E}^2\}}{E^2{F}^2}$. Clearly, $F-E=(1+2\lambda )A{T}_r>0$, $F+E>0$ and thus $F^2-E^2>0$. Therefore, when $C_p/C_r=$$T_p/T_r$, the first-order condition $\mathrm{\partial }\mathrm{\Delta }g({T_{M_{UR}}}^{\ast })/\mathrm{\partial }\lambda =0$ is satisfied. Furthermore, when $C_p/C_r>T_p/T_r$, $\mathrm{\partial }\mathrm{\Delta }g({T_{M_{UR}}}^{\ast })/\mathrm{\partial }\lambda >0$ exists; and when $C_p/C_r<T_p/T_r$, $\mathrm{\partial }\mathrm{\Delta }g({T_{M_{UR}}}^{\ast })/\mathrm{\partial }\lambda <0$ exists.

A3.Proof of Proposition 7

For analytical simplicity and notational convenience, let $\theta =(R{T}_p+C_p)/(R{T}_r+C_r)$. According to the analysis and results in Section 4.3, $[\lambda R{T}_p-(1+\lambda )C_p^{\prime }]/[\lambda R{T}_r-(1+\lambda )C_r]=(R{T}_p+C_p)/(R{T}_r+C_r)=$ $\theta$, and the subsidy of the operator to the manufacturer for each preventive maintenance by using the CS strategy is $\mathrm{\Delta }{C}_p=(C_p-\theta C_r)-\frac{\lambda \cdot R}{1+\lambda }(T_p-\theta T_r)$. For the LB strategy, the limit on the number of breakdowns $N$ and the award amount for each unrealised breakdown $C_n$ satisfy $\frac{\left(1+\lambda \right)C_p-\lambda R{T}_p-C_{n}N}{\left(1+\lambda \right)C_r-\lambda R{T}_r+C_n}=\frac{R{T}_p+C_p}{R{T}_r+C_r}$. Thus, $C_n[N-M(T_{M{O}_U}^{\ast }){]}^{+}=[(C_p-\theta C_r)-\frac{\lambda R}{1+\lambda }(T_p-\theta T_r)]$$\frac{[N-M(T_{M{O}_U}^{\ast })](1+\lambda )}{N+\theta }$. Let $\mathrm{\Delta }{C}_p=C_n[N-M(T_{M{O}_U}^{\ast }){]}^{+}$, then ${\lambda }_1=\frac{\theta +M(T_{M{O}_U}^{\ast })}{{[N-M(T_{M{O}_U}^{\ast })]}^{+}}$. When $\lambda \le {\lambda }_1$, the coordination cost satisfies $C_n[N-M(T_{M{O}_U}^{\ast }){]}^{+}\le \mathrm{\Delta }{C}_p$. When $\lambda >{\lambda }_1$, the coordination cost satisfies $C_n[N-M(T_{M{O}_U}^{\ast }){]}^{+}>\mathrm{\Delta }{C}_p$. Thus, there exists a threshold for the fairness concern parameter of the service-oriented manufacturer, below which the operator is better off with the LB strategy, and above which the operator is better off with the CS strategy. The results in the case of $C_p/C_r<T_p/T_r$ under the UOS mode can be compared in similar ways.

Next the CS and the LB coordination strategies under the ROS mode are compared. Without loss of generality, the case of $C_p/C_r>T_p/T_r$ is taken as an example. The payoffs of the operator using the CS and the LB coordination strategies are shown in (37) and (38), respectively, as follows.

(37) ${\pi }_{O_R}(T_{M{O}_R}^{\ast })=RU(T_{M{O}_R}^{\ast })-AU(T_{M{O}_R}^{\ast })-\frac{\mathrm{\Delta }{C}_p}{T_{M{O}_U}^{\ast }+T_P}$(37)

(38) ${\pi }_{O_R}(T_{M{O}_R}^{\ast })=RU(T_{M{O}_R}^{\ast })-AU(T_{M{O}_R}^{\ast })-\frac{C_n{[N-M(T_{M{O}_R}^{\ast })]}^{+}}{T_{M{O}_R}^{\ast }+T_p}$(38)

Similarly, only $\mathrm{\Delta }{C}_p$ and $C_n[N-M(T_{M{O}_R}^{\ast }){]}^{+}$ need to be compared. According to the analysis and results in Section 5.3,

$\begin{array}{rl}& {C^{\prime }}_p=\frac{(R{T}_p+C_p)\{[(1+\lambda )A-\lambda (R-A)]T_r+(1+\lambda )C_r\}-[(1+\lambda )A-\lambda (R-A)]T_p(R{T}_r+C_r)}{(1+\lambda )(R{T}_r+C_r)}\\ & =C_r\theta +\frac{[(1+\lambda )A-\lambda (R-A)](T_r\theta -T_p)}{1+\lambda }\end{array}$

The subsidy of the operator to the manufacturer for each preventive maintenance is $\mathrm{\Delta }{C}_p=C_p-C_r\theta -\frac{[(1+\lambda )A-\lambda (R-A)](T_r\theta -T_p)}{1+\lambda }$. For the LB strategy, the limit on the number of breakdowns $N$ and the award amount for each unrealised breakdown $C_n$ satisfy $\frac{[(1+\lambda )A-\lambda (R-A)]T_p+(1+\lambda )C_p-C_{n}N}{[(1+\lambda )A-\lambda (R-A)]T_r+(1+\lambda )C_r+C_n}=\frac{R{T}_p+C_p}{R{T}_r+C_r}$.

Thus, $C_n[N-M(T_{M{O}_R}^{\ast }){]}^{+}=$$\frac{{[N-M(T_{M{O}_R}^{\ast })]}^{+}(1+\lambda )}{N+\theta }\left\{C_p-C_r\theta -\frac{[(1+\lambda )A-\lambda (R-A)](T_r\theta -T_p)}{1+\lambda }\right\}$. With $\mathrm{\Delta }{C}_p=$ $C_n[N-M(T_{M{O}_R}^{\ast }){]}^{+}$, ${\lambda }_2=\frac{\theta +M(T_{M{O}_R}^{\ast })}{{[N-M(T_{M{O}_R}^{\ast })]}^{+}}$. When $\lambda \le {\lambda }_2$, the coordination cost satisfies $C_n[N-M(T_{M{O}_R}^{\ast }){]}^{+}\le \mathrm{\Delta }{C}_p$. When $\lambda >{\lambda }_2$, the coordination cost satisfies $C_n[N-M(T_{M{O}_R}^{\ast }){]}^{+}>\mathrm{\Delta }{C}_p$. Thus, there exists a threshold for the fairness concern parameter of the service-oriented manufacturer, below which the operator is better off with the LB strategy, and above which the operator is better off with the CS strategy. The results for $C_p/C_r<T_p/T_r$ under the ROS mode can be compared in similar ways. As a result, Proposition 7 is proved.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [71971052, 71671033, 71971048]; Fund for Innovative Research Groups of the National Natural Science Foundation of China [71621061]; Major International Joint Research Project of the National Natural Science Foundation of China [71520107004]; Project of Promoting Talents in Liaoning Province [XLYC1807252, XLYC1907015]; Fundamental Research Funds for the Central Universities [N2006006, N160601001]; 111 Project [B16009].