Abstract
To maintain the competence, an increasing number of original equipment manufacturers (OEMs) outsource their production manufacturing to contract manufacturers (CMs). While the outsourcing strategy benefits the manufacturers greatly, it also brings huge risk derived from uncertain environment, which may directly affect the supply chain members’ competitive profits and then their leadership preferences in competition. To address this problem, this paper considers a model in which an OEM outsources its production partly to a competitive CM (CCM), who also sells her own products; moreover, they hold different risk attitudes toward the uncertain demand, characterized by the confidence level in the framework of uncertainty theory. Based on the framework, to explore the OEM’s and the CCM’s leadership preferences, we derive and compare the simultaneous game, the OEM-as-leader game and the CCM-as-leader game. We present an interesting insight that, with a high wholesale price, the leadership position is more attractive for a relatively risk-aversion OEM and a relatively risk-loving CCM, which demonstrates contrary effects of both one’s risk attitudes. Furthermore, we also find that both the OEM and the CCM would like to play the leadership when both the wholesale price and the outsourcing rate to the CCM is relatively low. However, in the case of a relatively high outsourcing rate and wholesale price, both parties would like to compromise to move last rather than sticking on moving first.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grant No. 71471126 and Hubei Province Natural Science Foundation of Key Project under Grant No. 2015CFA144.
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Appendix
Appendix
Proof of Theorem 1
Because the function \(\pi _{1}(q_{1},q_{2};x,y)\) is strictly increasing with x and is strictly decreasing with y, the chance constraint \(M\{\pi _{1}(q_{1},q_{2};\alpha _{1})\geqslant \pi _{01}\}\geqslant \alpha _{1}\) is equivalent to
Since \(\pi _{m1}(q_{1},q_{2};\alpha _{1})=max\{\pi _{01}\}\), the OEM’s maximum profit \(\pi _{m1}(\cdot ,\cdot )\) under its acceptable confidence level \(\alpha _{i}\) satisfies
Similarly, the CCM’s maximum profit \(\pi _{m2}(\cdot ,\cdot )\) under her acceptable confidence level \(\alpha _{2}\) is
By Eqs. (11) and (12), we obtain Eqs. (5) and (6). The proof of the theorem is completed. \(\square \)
Proof of Proposition 1
From Theorem 1, Eqs. (5) and (6) can be rewritten as
and
From Eqs. (13) and (14), the best response functions are:
Solving these two equations, the conclusion in Theorem 2 can be obtained. \(\square \)
Proof of Theorem 2
The proof of this theorem is similar to that of Theorem 1. \(\square \)
Proof of Proposition 2
Model (7) can be turned into the following one,
It can be shown that the optimal production quantity of the CCM is
Substituting \(q^{f}_{2}(q_{1})\) into the OEM’s profit function and maximizing the objective function yields the optimal production quantity:
Moreover, the corresponding optimal decision for the CCM is
Substituting \(q^{l}_{1}\) and \(q^{f}_{2}\) into the OME’s and CCM’s profit functions yields profits of two participants, and the conclusion in Theorem 2 can be obtained. The proof of the theorem is completed. \(\square \)
Proof of Theorem 3
The proof of Theorem 3 is similar to that of Theorem 1. \(\square \)
Proof of Proposition 3
The proof of Proposition 3 is similar to that of Proposition 2. \(\square \)
Proof of Proposition 4
When \(w<\min \{w^{O}, w^{C}\}\) and \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\), both the OEM and CCM exist in the market in all three basic games. At first, we compare \(w^{O}\) and \(w^{C}\), shown as follows.
Then, the sign of \(w^{C}-w^{O}\) depends on that of \([2\gamma \Phi ^{-1}(1-\alpha _{1})-(1+\gamma \Psi ^{-1}(\alpha _{1}))\Phi ^{-1}(1-\alpha _{2})]\). This is because \(w^{C}-w^{O}<0\) if \(k< \frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) and \(w^{C}-w^{O}>0\) if \(k> \frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\). Thus, we still need to look at whether \(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) belongs to the range of k by comparison between \(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) and \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}\) and that between \(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) and \(\frac{4-\Psi ^{-1}(\alpha _{1})}{2}\). We show that
We can find that \(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }>\frac{4-\Psi ^{-1}(\alpha _{1})}{2}\) if \(\gamma <\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\) and \(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }<\frac{4-\Psi ^{-1}(\alpha _{1})}{2}\) if \(\gamma >\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\). In summary, both the OEM and CCM exist in the market in all three basic games. And the range of the wholesale price in which all three games exist is:
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(i) If \(0\leqslant \gamma <\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\), \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\), then \((0, w^{C})\).
-
(ii) If \(\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\leqslant \gamma <1\),
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If \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\), then \((0, w^{C})\),
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If \(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\), then \((0, w^{O})\).
-
Secondly, we compare the performances of the OEM and the CCM under the three basic games yields in Proposition 4. First, we have
Then, the sign of \(\pi _{1}^{l}-\pi _{1}^{f}\) depends on that of
Define \(w_{A}={\frac{\Phi ^{-1}(1-\alpha _{1})-2\,\Phi ^{-1}(1-\alpha _{2})}{\gamma \Psi ^{-1}(\alpha _{1})-4\,\gamma +1}}\). Then, according to the above-mentioned results and \(\frac{1}{4-\Psi ^{-1}(\alpha _{1})}<\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\), we analyze separately in \(\left[ 0, \frac{1}{4-\Psi ^{-1}(\alpha _{1})}\right) \), \(\left( \frac{1}{4-\Psi ^{-1}(\alpha _{1})}, \frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\right) \), \(\left[ \frac{1}{4-2\Psi ^{-1}(\alpha _{1})}, 1\right) \)
Case 1 When \(\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\leqslant \gamma <1\), we have
Thus, \(\pi _{1}^{f}>\pi _{1}^{s}\) if \(w>w_{A}\) and \(\pi _{1}^{f}<\pi _{1}^{s}\) if \(w<w_{A}\). Furthermore, we show that
It can be verified that the sign of the above two equations depends on \((1+\gamma \Psi ^{-1}(\alpha _{1}))\Phi ^{-1}(1-\alpha _{2})-2\gamma \Phi ^{-1}(1-\alpha _{1})\). Then, we have \(w_{A}<w^{O}<w^{C}\) if \(k>\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) and \(w^{C}<w^{O}<w_{A}\) if \(k<\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\). Hence, we have two subcases to consider:
-
a)
\(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) and \(0<w<w^{C}\), then \(\pi _{1}^{f}-\pi _{1}^{s}<0\).
-
b)
\(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\), then \(\pi _{1}^{f}-\pi _{1}^{s}<0\) if \(0<w<w^{A}\) while \(\pi _{1}^{f}-\pi _{1}^{s}>0\) if \(w^{A}<w<w^{O}\). Additionally, \(\pi _{1}^{f}-\pi _{1}^{s}=0\) if \(w=w_{A}\).
Case 2 When \(\frac{1}{4-\Psi ^{-1}(\alpha _{1})}<\gamma <\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\), we have \(\pi _{1}^{f}>\pi _{1}^{s}\) if \(w>w_{A}\) and \(\pi _{1}^{f}<\pi _{1}^{s}\) if \(w<w_{A}\). In addition, \(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }>\frac{4-\Psi ^{-1}(\alpha _{1})}{2}\), thus we obtain \(w^{C}<w^{O}<w^{A}\). In summary, we have \(\pi _{1}^{f}-\pi _{1}^{s}<0\) if \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\) and \(0<w<w^{C}\).
Case 3: When \(0\leqslant \gamma <\frac{1}{4-\Psi ^{-1}(\alpha _{1})}\), similar to the second case, we obtain \(\pi _{1}^{f}-\pi _{1}^{s}>0\) if \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\) and \(0<w<w^{C}\).
The proof of the theorem is completed. \(\square \)
Proof of Proposition 5
The proof of this theorem is similar to that of Theorem 1.
Here
We can see that the sign of \(\pi _{2}^{f}-\pi _{2}^{s}\) is the same as that of (17)
Define \(w_{B}=\frac{\left( \Psi ^{-1}(\alpha _{1})^{2}-16\Psi ^{-1}(\alpha _{1}\right) +32)\Phi ^{-1}(1-\alpha _{2})+\left( 6\Psi ^{-1}(\alpha _{1})-16\right) \Phi ^{-1}(1-\alpha _{1})}{8\gamma (2-\Psi ^{-1}(\alpha _{1}))(4-\Psi ^{-1}(\alpha _{1}))-16+6\Psi ^{-1}(\alpha _{1})}\).
And note that \(w_{A}<w_{B}<w^{O}<w^{C}\) if \(k>\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) and \(w^{C}<w^{O}<w_{B}<w_{A}\) if \(k<\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\), and note that \(\frac{16-6\Psi ^{-1}(\alpha _{1})}{8(2-\Psi ^{-1}(\alpha _{1}))(4-\Psi ^{-1}(\alpha _{1}))}<\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\). So we have three cases to consider:
-
(i) \(0\leqslant \gamma <\frac{8-3\Psi ^{-1}(\alpha _{1})}{4(2-\Psi ^{-1}(\alpha _{1}))(4-\Psi ^{-1}(\alpha _{1}))}\), \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\) and \(0<w<w^{C}\), then \(\pi _{2}^{f}-\pi _{2}^{s}>0\).
-
(ii) \(\frac{8-3\Psi ^{-1}(\alpha _{1})}{4(2-\Psi ^{-1}(\alpha _{1}))(4-\Psi ^{-1}(\alpha _{1}))}<\gamma <\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\), \(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\) and \(0<w<w^{C}\), then \(\pi _{2}^{f}-\pi _{2}^{s}<0\).
-
(iii) \(\frac{1}{4-2\Psi ^{-1}(\alpha _{1})}\leqslant \gamma <1\), we have two subcases
-
a)
\(\frac{2\Psi ^{-1}(\alpha _{1})}{4-\Psi ^{-1}(\alpha _{1})}<k\leqslant \frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }\) and \(0<w<w^{C}\), then \(\pi _{2}^{f}-\pi _{2}^{s}<0\).
-
b)
\(\frac{1+\gamma \Psi ^{-1}(\alpha _{1})}{2\gamma }<k\leqslant \frac{4-\Psi ^{-1}(\alpha _{1})}{2}\), \(\pi _{1}^{f}-\pi _{1}^{s}<0\) if \(0<w<w^{B}\) while \(\pi _{2}^{f}-\pi _{2}^{s}>0\) if \(w^{B}<w<w^{O}\). Additionally, \(\pi _{1}^{f}-\pi _{1}^{s}=0\) if \(w=w^{B}\). The proof of the theorem is completed.
-
a)
\(\square \)
Proof of Proposition 6
According to the results of above three games, we can obtain all the basic games exist when \(\gamma _{1}<\gamma <\gamma _{2}\). Here,
From Eqs. (16) and (17), we denote \(\gamma _{A}=\frac{2\Phi ^{-1}(1-\alpha _{2})-\Phi ^{-1}(1-\alpha _{1})+w}{w(4-\Psi ^{-1}(\alpha _{1})}\), and
Similar to the proofs of Propositions 5 and 6, we have \(\pi _{1}^{f}-\pi _{1}^{s}>0\) if \(\gamma >\gamma _{A}\) and \(\pi _{2}^{f}-\pi _{2}^{s}>0\) if \(\gamma >\gamma _{B}\). Because \(\gamma _1<\gamma _{A}<\gamma _{B}<\gamma _2\), the conclusion in Proposition 6 can be obtained. \(\square \)
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Chen, H., Yan, Y., Liu, Z. et al. Effect of risk attitude on outsourcing leadership preferences with demand uncertainty. Soft Comput 22, 5263–5278 (2018). https://doi.org/10.1007/s00500-017-2977-9
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DOI: https://doi.org/10.1007/s00500-017-2977-9