Joint optimization and coordination of fresh-product supply chains with quality-improvement effort and fresh-keeping effort

ABSTRACT

We consider a fresh-product supply chain (FPSC) in which the supplier and the e-tailer invest in quality-improvement effort and fresh-keeping effort, respectively. One centralized model and two decentralized models are developed to investigate the optimal effort and pricing decisions for double-effort (de), quality-improvement effort only (qe), fresh-keeping effort only (fe) and none effort (ne) strategies, respectively. We also conduct a comparative analysis to find out the superiority among four different strategies, to reveal the effect of different power structures, and to show the distortion of effort decisions. Our research reveals several insights. First, we find that the de strategy is a dominant strategy for both the supplier and the e-tailer. Second, although channel power structures have a significant impact on the effort decisions, it neither guarantees a greater effort nor more profit for the FPSC. Third, the optimal quality-improvement effort and fresh-keeping effort are always distorted in a decentralized setting, and the channel efficiency is lost the most under a de strategy. Finally, we propose the net-revenue and cost sharing contract and the two-part tariff contract to coordinate this decentralized FPSC. We further demonstrate several sensitivity effects based on computational studies, providing additional insights for better understanding the theoretical results.

KEYWORDS:

• Quality-improvement effort
• fresh-keeping effort
• fresh-product supply chain
• net-revenue and cost sharing contract
• two-part tariff contract

Disclosure statement

No potential conflict of interest was reported by the authors.

Proof of Proposition 1

Given $\mathrm{\Gamma }>0$, then, we obtain ${\pi }_{sc}^{de}-{\pi }_{sc}^{qe}=\frac{{\alpha }^2{\mathrm{\Omega }}^2\mathrm{\Theta }}{2\mathrm{\Gamma }(2\alpha -\mathrm{\Phi })}>0$, ${\pi }_{sc}^{de}-{\pi }_{sc}^{fe}=\frac{{\mathrm{\Omega }}^2\mathrm{\Phi }}{2\mathrm{\Gamma }(2\alpha -{\alpha }^2\mathrm{\Theta })}>0,$

${\pi }_{sc}^{qe}-{\pi }_{sc}^{ne}=\frac{{\mathrm{\Omega }}^2\mathrm{\Phi }}{2(2\alpha -\mathrm{\Phi })}>0$ and ${\pi }_{sc}^{fe}-{\pi }_{sc}^{ne}=\frac{{\alpha }^2{\mathrm{\Omega }}^2\mathrm{\Theta }}{2(2\alpha -{\alpha }^2\mathrm{\Theta })}>0$. End.

Proof of Proposition 2

Given $\mathrm{\Gamma }>0$, follows $e_1^{de\ast }-e_1^{qe\ast }=\frac{{\alpha }^2\mathrm{\Omega }\sqrt{\mathrm{\Phi }}\mathrm{\Theta }}{(2\alpha -\mathrm{\Phi })\mathrm{\Gamma }\sqrt{{\rho }_1}}>0$ and $e_2^{de\ast }-e_2^{fe\ast }=\frac{\alpha \mathrm{\Omega }\sqrt{\mathrm{\Theta }}\mathrm{\Phi }}{(2\alpha -{\alpha }^2\mathrm{\Theta })\mathrm{\Gamma }\sqrt{{\rho }_2}}>0$. End.

Proof of Theorem 2 and Theorem 3

Based on backwards induction, the proof is similar to the Proof of Theorem 1, thus, we omit it.

Proof of Proposition 3

(1) Given $\mathrm{\Gamma }>0$, then

${\pi }_e^{sd-fe}-{\pi }_e^{sd-ne}=\frac{\mathrm{\Theta }{\mathrm{\Omega }}^2}{16(2-\alpha \mathrm{\Theta })}>0$, ${\pi }_s^{sd-fe}-{\pi }_s^{sd-ne}=\frac{\mathrm{\Theta }{\mathrm{\Omega }}^2}{8(2-\alpha \mathrm{\Theta })}>0$ and ${\pi }_{sc}^{sd-fe}-{\pi }_{sc}^{sd-ne}>0$.

Given $\mathrm{\Gamma }>0$, then

${\pi }_s^{sd-de}-{\pi }_s^{sd-fe}=\frac{\mathrm{\Phi }{\mathrm{\Omega }}^2}{4\alpha (2\mathrm{\Gamma }+\mathrm{\Phi })(2-\alpha \mathrm{\Theta })}>0$, ${\pi }_e^{sd-de}-{\pi }_e^{sd-fe}=\frac{\mathrm{\Phi }{\mathrm{\Omega }}^2[2(2\alpha -{\alpha }^2\mathrm{\Theta })+(2\mathrm{\Gamma }+\mathrm{\Phi })]}{8{(2\mathrm{\Gamma }+\mathrm{\Phi })}^2(2\alpha -{\alpha }^2\mathrm{\Theta })}>0\mathrm{a}\mathrm{n}\mathrm{d}{\pi }_{sc}^{sd-de}-{\pi }_{sc}^{sd-fe}>0$.

Thus, we have ${\pi }_e^{sd-de}>{\pi }_e^{sd-fe}>{\pi }_e^{sd-ne}$, ${\pi }_s^{sd-de}>{\pi }_s^{sd-fe}>{\pi }_s^{sd-ne}$ and ${\pi }_{sc}^{sd-de}>{\pi }_{sc}^{sd-fe}>{\pi }_{sc}^{sd-ne}$.

The proof of (2), (3) and (4) is similar to (1), thus, we omit it. End.

Proof of Proposition 4

The proof is similar to the Proof of Proposition 2, thus, we omit it.

Proof of Proposition 5

(1) For $e_1^{sd-de\ast }-e_1^{ed-de\ast }=\frac{\mathrm{\Omega }\sqrt{\mathrm{\Phi }}({\alpha }^2\mathrm{\Theta }-\mathrm{\Phi })}{(2\mathrm{\Gamma }+\mathrm{\Phi })(2\mathrm{\Gamma }+{\alpha }^2\mathrm{\Theta })\sqrt{{\rho }_1}}$ and $e_2^{sd-de\ast }-e_2^{ed-de\ast }=\frac{\alpha \mathrm{\Omega }\sqrt{\mathrm{\Theta }}({\alpha }^2\mathrm{\Theta }-\mathrm{\Phi })}{(2\mathrm{\Gamma }+\mathrm{\Phi })(2\mathrm{\Gamma }+{\alpha }^2\mathrm{\Theta })\sqrt{{\rho }_2}}$.

Therefore, when $\alpha >\sqrt{\frac{\mathrm{\Phi }}{\mathrm{\Theta }}}$, $e_1^{sd-de\ast }>e_1^{ed-de\ast }$ and $e_2^{sd-de\ast }>e_2^{ed-de\ast }$; otherwise, $e_1^{sd-de\ast }\le e_1^{ed-de\ast }$ and $e_2^{sd-de\ast }\le e_2^{ed-de\ast }$.

The proof of (2) is similar to (1), thus, we omit it. End.

Proof of Proposition 6

(1) $e_1^{sd-qe\ast }-e_1^{qe\ast }=\frac{-2\alpha \mathrm{\Omega }\sqrt{\mathrm{\Phi }}}{(2\alpha -\mathrm{\Theta })(4\alpha -\mathrm{\Phi })\sqrt{{\rho }_1}}<0$ and $e_1^{ed-qe\ast }-e_1^{qe\ast }=\frac{-\mathrm{\Omega }\sqrt{\mathrm{\Phi }}}{2(2\alpha -\mathrm{\Theta })\sqrt{{\rho }_1}}<0$.

The proof of (2) and (3) is similar to (1), thus, we omit it. End.

Proof of Proposition 7

(1) For ${\psi }_{\hspace{0.17em}}^{sd-ne}=\frac{{\pi }_{sc}^{sd-ne}}{{\pi }_{sc}^{ne}}=\frac{3}{4}$, ${\psi }_{\hspace{0.17em}}^{sd-qe}=\frac{{\pi }_{sc}^{sd-qe}}{{\pi }_{sc}^{qe}}=1-\frac{4{\alpha }^2}{16{\alpha }^2+\mathrm{\Phi }(\mathrm{\Phi }-8\alpha )}<\frac{3}{4}$, ${\psi }_{\hspace{0.17em}}^{sd-fe}=\frac{{\pi }_{sc}^{sd-fe}}{{\pi }_{sc}^{fe}}=\frac{3}{4}$ and ${\psi }_{\hspace{0.17em}}^{sd-de}=\frac{{\pi }_{sc}^{sd-de}}{{\pi }_{sc}^{de}}=1-\frac{{(2\alpha -{\alpha }^2\mathrm{\Theta })}^2}{{(2\mathrm{\Gamma }+\mathrm{\Phi })}^2}<\frac{3}{4}$.

For ${\psi }_{\hspace{0.17em}}^{sd-de}-{\psi }_{\hspace{0.17em}}^{sd-qe}=\frac{-{\alpha }^2\mathrm{\Theta }\mathrm{\Phi }\left(\frac{2\alpha }{4\alpha -\mathrm{\Phi }}+\frac{2\alpha -{\alpha }^2\mathrm{\Theta }}{2\mathrm{\Gamma }+\mathrm{\Phi }}\right)}{(4\alpha -\mathrm{\Phi })(2\mathrm{\Gamma }+\mathrm{\Phi })}<0$, then ${\psi }_{\hspace{0.17em}}^{sd-de}<{\psi }_{\hspace{0.17em}}^{sd-qe}$. Thus, we have

${\psi }_{\hspace{0.17em}}^{sd-de}<{\psi }_{\hspace{0.17em}}^{sd-qe}<{\psi }_{\hspace{0.17em}}^{sd-fe}={\psi }_{\hspace{0.17em}}^{sd-ne}=\frac{3}{4}.$

The proof of (2) and (3) is similar to (1), thus, we omit it. End.

Proof of Proposition 8

Given $w^{snrcs}$, $\gamma$, $\lambda$ and $e_1^{snrcs}=e_1^{de\ast }$, we can quickly obtain

$e_2^{snrcs}=\frac{[\gamma (\delta +\beta {\theta }_0-\alpha r{\eta }_0+\frac{\mathrm{\Phi }\mathrm{\Omega }}{\mathrm{\Gamma }})-\alpha w]\sqrt{\mathrm{\Theta }}}{(2\lambda -\gamma \alpha \mathrm{\Theta })\sqrt{{\rho }_2}}\mathrm{a}\mathrm{n}\mathrm{d}{p}^{snrcs}=\left[\frac{{\varphi }_1(2-\alpha \mathrm{\Theta })\mathrm{\Omega }}{\mathrm{\Gamma }}+{\varphi }_{1}c-w\right]\frac{{\varphi }_2-{\varphi }_1\alpha \mathrm{\Theta }}{(2{\varphi }_2-{\varphi }_1\alpha \mathrm{\Theta }){\varphi }_1}+\frac{{\varphi }_{1}r{\eta }_0+w}{{\varphi }_1}$.

Let $e_2^{snrcs}=e_2^{de\ast }$ and $p^{snrcs}=p^{de\ast }$, we can obtain:

when $\lambda =\gamma$, then $w^{snrcs}=\gamma c$; when $\lambda =\frac{\alpha \mathrm{\Theta }\gamma }{2}$, then $w^{snrcs}=\frac{\gamma (2-\alpha \mathrm{\Theta })\mathrm{\Omega }}{\mathrm{\Gamma }}+\gamma c$. End.

Proof of Proposition 9

Given $e_1^{snrcs}=e_1^{de\ast }$, $p^{snrcs}=p^{de\ast }$, $e_2^{snrcs}=e_2^{de\ast }$, $w^{snrcs}=\gamma c$, $\lambda =\gamma$, and let

${\pi }_e^{snrcs}(p_{\hspace{0.17em}}^{snrcs},e_2^{snrcs})=\frac{\alpha \gamma (2-\alpha \mathrm{\Theta }){\mathrm{\Omega }}^2}{2{\mathrm{\Gamma }}^2}>{\pi }_e^{sd-d\mathrm{e}}$ and

${\pi }_s^{snrcs}(w^{snrcs},\gamma ,\lambda ,e_1^{snrcs})=\frac{[\alpha (1-\gamma )(2-\alpha \mathrm{\Theta })-\mathrm{\Phi }]{\mathrm{\Omega }}^2}{2{\mathrm{\Gamma }}^2}>{\pi }_s^{sd-d\mathrm{e}}$, then $\frac{{\mathrm{\Gamma }}^2}{{(2\mathrm{\Gamma }+\mathrm{\Phi })}^2}<\gamma <\frac{\mathrm{\Gamma }}{2\mathrm{\Gamma }+\mathrm{\Phi }}$. End.

Proof of Proposition 10

The proof is similar to the proof of Proposition 8.

Proof of Proposition 11

The proof is similar to the proof of Proposition 9.

Proof of Proposition 12

Given $w^{stpt}$, $F^{stpt}$, $e_1^{stpt}=e_1^{de\ast }$, we can quickly obtain

$e_2^{stpt}=\frac{(\delta +\beta {\theta }_0-\alpha w^{stpt}-\alpha r{\eta }_0+\frac{\mathrm{\Phi }\mathrm{\Omega }}{\mathrm{\Gamma }})\sqrt{\mathrm{\Theta }}}{(2-\alpha \mathrm{\Theta })\sqrt{{\rho }_2}}$

$\mathrm{a}\mathrm{n}\mathrm{d}{p}^{stpt}=\frac{(1-\alpha \mathrm{\Theta })(\delta +\beta {\theta }_0-\alpha w^{stpt}-\alpha r{\eta }_0+\frac{\mathrm{\Phi }\mathrm{\Omega }}{\mathrm{\Gamma }})}{2\alpha -{\alpha }^2\mathrm{\Theta }}+w^{stpt}+r{\eta }_0.$

Let $p^{stpt}=p_{\hspace{0.17em}}^{de\ast }$and $e_2^{stpt}=e_2^{de\ast }$, then $w^{stpt}=c$. End.

Proof of Proposition 13

The proof is similar to the Proof of Proposition 12.

Proof of Proposition 14

Given $e_1^{stpt}=e_1^{de\ast }$, $p^{stpt}=p^{de\ast }$, $e_2^{stpt}=e_2^{de\ast }$ and $w^{stpt}=c$, and let

${\pi }_e^{stpt}(p^{stpt},e_2^{stpt})=\frac{(2\alpha -{\alpha }^2\mathrm{\Theta }){\mathrm{\Omega }}^2}{2{\mathrm{\Gamma }}^2}-F^{stpt}>{\pi }_e^{sd-d\mathrm{e}}$and ${\pi }_s^{stpt}(w^{stpt},e_1^{stpt},F^{stpt})=-\frac{\mathrm{\Phi }{\mathrm{\Omega }}^2}{2{\mathrm{\Gamma }}^2}+F^{stpt}>{\pi }_s^{sd-d\mathrm{e}}$, then $\frac{{(\mathrm{\Gamma }+\mathrm{\Phi })}^2{\mathrm{\Omega }}^2}{2(2\mathrm{\Gamma }+\mathrm{\Phi }){\mathrm{\Gamma }}^2}<F^{stpt}<\frac{(3\mathrm{\Gamma }+\mathrm{\Phi }){(\mathrm{\Gamma }+\mathrm{\Phi })}^2{\mathrm{\Omega }}^2}{2{\mathrm{\Gamma }}^2{(2\mathrm{\Gamma }+\mathrm{\Phi })}^2}$. End.

Proof of proposition 15

The proof is similar to the Proof of Proposition 14.

Additional information

Funding

This paper was partly supported by the Humanities and Social Sciences foundation of Ministry of Education of China [Grant no. 19YJC630044,19YJC630035]; Natural Science Foundation of Zhjiang Province [Grant no. LY18G010016]; Zhoushan Science and Technology Project [Grant no. 2017C41018]; Foundation of Zhejiang Educational Committee [Grant no. Y201840337]; the Open Foundation from Marine Sciences in the Most Important Subjects of Zhejiang [Grant no. 20130212]; the Open Foundation from Fishery Sciences in the First-Class Subjects of Zhejiang [Grant no. 20160023]; the Startup Foundation of Zhejiang Ocean University [Grant no. 11085090318,11085090418] and Fundamental Research Funds for Zhejiang Provincial Universities and Research Institutes [Grant no. 2019J00055].

Notes on contributors

Bojun Gu

Bojun Gu is an associate professor in the School of Economics and Management, Zhejiang Ocean University, China. He received his BS degree and MS degree in 2003 and 2006 at Lanzhou University. He received his PhD degree in 2016 at Beijing Institute of Technology, majored in Management Science and Engineering. His research topics include quality management, fresh-product supply chain and application of game theory in operational research.

Yufang Fu

Yufang Fu is a lecturer in the School of Economics and Management, Zhejiang Ocean University, China. She received her PhD degree from Department of Management Science, Xiamen University in 2017. Her current research interests mainly include quality management, E-commerce operational management and application of game theory in operational research.

Jun Ye

Jun Ye is currently a lecturer in the School of Economics and Management, Zhejiang Ocean University, China. She received her BS degree in 2009 at Dalian Maritime University and obtained her MSC degree in 2011 at National University of Singapore majored in Transportation System & Management. Her research topics include quality management, fresh-product supply chain and international shipping management.