Choosing reverse channels under collection responsibility sharing in a closed-loop supply chain with re-manufacturing

This paper considers a closed-loop supply chain with re-manufacturing consisting of retailers, manufacturers and third-party logistics service providers; all participating in the product recycling responsibilities. The effectiveness of methods that can be used to share responsibilities amongst these parties is quantified using different reverse channels. First, re-manufacturing models with three different reverse channels for retailer collection, manufacturer collection and third-party collection are developed using collection responsibility sharing. Next, by comparing these models with the case of no collection responsibility sharing, the effectiveness of responsibility sharing is analysed and quantified. The results for the three models support the following conclusions: (i) from the point of view of the retailer, third-party collection is always the worst choice; (ii) the choice between retailer collection and manufacturer collection depends on the cost parameter representing the resources required in performing the reverse collection tasks; (iii) from the point of view the manufacturer, when the value of the cost parameter is small, collection by manufacturer is the best choice; retailer collection will be best for high values of the cost parameter.

The authors would like to acknowledge the financial support of National Natural Science Foundation of China (71090402, 71101120 and 71101055), Fundamental Research Funds for the Central universities, SCUT, (No. 2011ZM0079), Educational Commission of Guangdong Province (wym11011) and the Guangdong Natural Science Foundation (S2011040002521).

Authors and Affiliations

1. School of Economics and Commerce, South China University of Technology, Guangzhou Higher Education Mega Center, Guangzhou, 510006, People’s Republic of China

   Yuan Shi

2. School of Economics and Management, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

   Jiajia Nie

3. Faculty of Electromechanical Engineering, Guangdong University of Technology, Guangzhou, 510006, People’s Republic of China

   Ting Qu

4. Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, People’s Republic of China

   Lap-Keung Chu & Domenic Sculli

Corresponding author

Correspondence to Jiajia Nie.

Appendix

Proof of Proposition 2

The differences of optimum selling prices \((p)\), collection rates \((\tau )\) and wholesale prices \((\omega )\) between Model R and Model RM are,

To analyse whether the formula (46) is positive or negative, it is only needed to analyse the expression of \(( 4k( {5\varpi -2\Delta } )- b\varpi ^{2}( {2\Delta +\varpi } ) )\). Three situations are identified: (1) when \(0<\varpi <{2\Delta }/5\), \(4k\left( {5\varpi -2\Delta } \right) - b\varpi ^{2}\left( {2\Delta +\varpi } \right) <0\), then \(\omega ^{RM}<\omega ^{R}\); (2) when \({2\Delta }/5<\varpi <{2\Delta }/3\), let \(k_1 ={b\varpi ^{2}\left( {2\Delta +\varpi } \right) }/{\left( {4\left( {5\varpi -2\Delta } \right) } \right) }\), which is larger than \(k_0 \), then, if \(k>k_1 \), \(\omega ^{RM}>\omega ^{R}\); and if \(k_0 <k<k_1 \), \(\omega ^{RM}<\omega ^{R}\); and (3) when \({2\Delta }/3<\varpi <\Delta \),\(k_1 <k_0 \), then \(\omega ^{RM}>\omega ^{R}\).

Proof of Proposition 3

The differences of the retailer’s and manufacturer’s profits between Model R and Model RM can be simplified as

where

When \(0<\varpi <{2\Delta }/3\), it can be shown that \(k>k_0 \), \(H_1 >0\),then \(\Pi _R^{RM} >\Pi _R^R \). When \({2\Delta }/3<\varpi <\Delta \), \(H_1 \) is quadratic function of \(k\) (a parabola as an inverted U-shape). It can be shown that the discriminant of the quadratic equation \((H_1 \left( k \right) =0)\) is positive. Therefore, two solutions for quadratic equation \(H_1 \left( k \right) =0\) exist, which are defined as \(k_2 \) and \(k_3 \). For \(k=k_0 \), since \(H_1 \left( k \right) <0\), both \(k_2 \) and \(k_3\) are greater than \(k_0\). Thus, when \({2\Delta }/3<\varpi <\Delta \), if \(k\in \left( {k_2 ,k_3 } \right) \),\(H_1 >0\),then \(\Pi _R^{RM} >\Pi _R^R \); and if \(k\in \left( {k_0 ,k_2 } \right) \cup \left( {k_3 ,\infty } \right) \), \(H_1 <0\), then \(\Pi _R^{RM} <\Pi _R^R \).

Proof of Proposition 5

The differences of optimum selling prices, collection rates and wholesale prices between Model M and Model MR are,

Proof of Proposition 6

The differences of optimum profits of retailer and manufacturer between Model M and Model MR are,

Then the relationships between profits increments and \(a,b,c_m ,k\) and \(\Delta \) can be concluded from Eqs. (53) and (54).

Proof of Proposition 7

The result of case (1) is obvious and we will focus on the proofs for cases (2) to (4). Differentiating \(z_M^{3PRM} \) with respect to \(k,b,\varpi \) and \(\Delta \) respectively,

The results for cases (2) and (3) can be obtained from Eqs. (55) and (56); and since \(k>k_0 \), it can be obtained that the formula (57) is negative. Let

The result for Case (4) can be established.

Proof of Proposition 8

The differences of optimum selling prices, collection rates and wholesale prices between Model R and Model RM are,

To analyse if the formula (61) is positive or negative, three situations are needed to be discussed:

1. (1)

   when \(0<\varpi <{2\Delta }/5\), let

   $$\begin{aligned} k_4 =\frac{b\varpi \left( {2\Delta -\varpi } \right) \left( {2\Delta -5\varpi } \right) }{32\left( {2\Delta -3\varpi } \right) }>0. \end{aligned}$$

   (63)

   Since \(k>k_0 \), it can be shown that \(k_4 <k_0 \) when comparing \(k_0 \) with \(k_4 \). Thus formula (61) is positive when \(0<\varpi <{2\Delta }/5\); (2) when \({2\Delta }/5<\varpi <{2\Delta }/3\), both \(\left( {2\Delta -3\varpi } \right) \) and \(\left( {5\varpi -2\Delta } \right) \) are positive, then formula (61) is positive when \({2\Delta }/5<\varpi <{2\Delta }/3\); (3) when \({2\Delta }/3<\varpi <\Delta \), \(k_4 >0\) and let

   $$\begin{aligned} \frac{\Delta }{\varpi }=\sqrt{\frac{12\Delta -5\varpi }{31\varpi -18\Delta }}{\mathop {=}^{\Delta }} \eta . \end{aligned}$$

   (64)

   If \(\Delta /\varpi >\eta \), \(k_4 >k_0 \), formula (61) is positive when \(k_0 <k<k_4 \) and formula (61) is negative when \(k>k_4 \); and if \(\Delta /\varpi <\eta \), then \(k_4 <k_0 \), formula (61) is positive. To recap, the conclusion in (2) is obtained.

Proof of Proposition 9

The differences of optimum profits of the retailer and manufacturer between Model 3P and Model 3PRM are

where

Hence, the results are established.

Proof of Proposition 10

The differences of optimum selling price of the retailer between Model RM and Model 3PRM is,

To analyse if the formula (69) is negative or positive, Let

And then, \(k_0 \) minus \(k_5 \),

Then let \(H_6 \left( \varpi \right) {\mathop {=}^{\Delta }} 144\varpi \Delta ^{3}+8\varpi ^{2}\Delta ^{2}-\varpi ^{4}-16\Delta ^{4}\), it is obvious that \(H_6 \left( {\varpi =0} \right) <0\) and \(H_6 \left( {\varpi =\Delta } \right) >0\). Notice that \(H_6 \left( \varpi \right) \) is an increasing function of \(\varpi \). There exists a unique \(\varpi _1 \in \left( {0,\Delta } \right) \) that makes \(H_6 \left( {\varpi _1 } \right) =0\). Therefore, when \(\varpi \in \left( {\varpi _1 ,\Delta } \right) \) and \(k_0 >k_5 \), \(p^{3PRM}>p^{RM}\) when \(\varpi \in \left( {0,\varpi _1 } \right) \) and \(k_0 <k_5 \),if \(k_0 <k<k_5 \), then \(p^{3PRM}<p^{RM}\); and if \(k>k_5 \), then \(p^{3PRM}>p^{RM}\). Then the differences of optimum selling price of the retailer between Model 3PRM and Model MR is,

The difference of optimum selling price of the retailer between Model MR and Model RM is,

Let

It is obvious that when \(\varpi <{2\left( {3\sqrt{7}-7} \right) \Delta }/7{\mathop {=}^{\Delta }} \varpi _2 \), then \(k_6 >k_0 \); and if \(k>k_6 \), then \(p^{MR}>p^{RM}\); and if \(k<k_7 \), then \(p^{MR}<p^{RM}\). When \(\varpi >\varpi _2 \), then \(p^{MR}>p^{RM}\). Since \(H_6 \left( {\varpi _2 } \right) >0\), then \(\varpi _2 >\varpi _1\). Hence, we get the results in Proposition 10.

Proof of Proposition 11

The differences of optimum collection rates of the retailer between Model 3PRM and Model RM is,

When \(\varpi >{2\Delta }/5\), formula (75) is positive, which means \(\tau ^{3PRM}>\tau ^{RM}\); and when \(\varpi <{2\Delta }/5\), let

Also, \(k_7 >0\), and then \(k_0 \) minus \(k_7 \),

\(H_7 \left( \varpi \right) \) is an increasing function of \(\varpi \) within interval \(\left( {0,{2\Delta }/5} \right) \), which is also continuous with \(H_7 \left( {\varpi =0} \right) <0\) and \(H_7 \left( {\varpi ={2\Delta }/5} \right) <0\). Then there exists only one solution \(\varpi _3 \in \left( {0,{2\Delta }/5} \right) \) to make \(H_7 \left( {\varpi =\varpi _3 } \right) =0\), where \(\varpi _3 \) is the unique solution of equation \(H_7 \left( \varpi \right) =0\). Hence, when \(\varpi \in \left( {0,\varpi _3 } \right) \), \(k_0 <k_7 \), if \(k>k_7 \), then formula (75) will be positive, which means \(\tau ^{3PRM}>\tau ^{RM}\); and if \(k_0 <k<k_7 \), then formula (75) will be negative, which means \(\tau ^{3PRM}<\tau ^{RM}\). When \(\varpi \in \left( {\varpi _3 ,{2\Delta }/5} \right) \), \(k_0 >k_7 \), then formula (75) will be positive, which means \(\tau ^{3PRM}>\tau ^{RM}\).

Also, the differences of optimum collection rates of the retailer between Model 3PRM and Model MR is,

Finally, the differences of optimum collection rates of the retailer between Model MR and Model RM is,

Hence, we get the results in Proposition 11 from above analysis.

Proof of Proposition 12

The differences of optimum wholesale price between Model 3PRM and Model RM is,

It is obvious that when \(\varpi <{2\Delta }/{11}\), formula (80) will be negative, which means \(\omega ^{3PRM}<\omega ^{RM}\); and when \(\varpi >{2\Delta }/{11}\), let

then \(k_8 >0\),and when \(k=k_8 \), the formula (80) is equal to zero. And since \(k_8 <k_0 \),then formula (80) is negative, which means \(\omega ^{3PRM}<\omega ^{RM}\).

And the differences of optimum wholesale price between Model 3PRM and Model MR is,

Finally, the differences of optimum wholesale price between Model MR and Model MR is

Hence, we get the results in Proposition 12.

Proof of Proposition 13

The difference of the optimum profits of the retailer between Model 3PRM and Model RM is,

where \(H_8 =b\left( {2\Delta +\varpi } \right) \left( {4\Delta ^{2}-16\Delta \varpi +3\varpi ^{2}} \right) \). For \(k>k_0 \), it is easy to prove that formula (84) is negative, which means \(\Pi _R^{3PRM} <\Pi _R^{RM} \). And the difference of the optimum profits of the retailer between Model 3PRM and Model MR is,

Finally, the difference of the optimum profits of the retailer between Model MR and Model RM is,

Let

Since \(H_9 \left( k \right) \) is quadratic function of \(k\) with Parabola’s opening side down. When \(k=k_0 \), \(H_9 \left( k \right) <0\); and when \(k\rightarrow \infty \), \(H_9 \left( k \right) >0\). Thus there must exist a unique \(k_9 \in \left( {k_0 ,\infty } \right) \) to make \(H_9 \left( k \right) =0\). So, when \(k>k_9 \), the formula (86) will be positive and \(\Pi _R^{MR} >\Pi _R^{RM} \); and when \(k_0 <k<k_9 \), formula (86) will be negative and \(\Pi _R^{MR} <\Pi _R^{RM} \). Hence, we get the results in Proposition 13.

Proof of Proposition 14

The difference of the optimum profits of the manufacturer between Model 3PRM and Model RM is,

Let

When \(k>k_{10} \), the formula (88) will be negative, which means \(\Pi _M^{3PRM} \!<\!\Pi _M^{RM} \); and when \(k\!\!<\!\!k_{10} \), the formula

(88) will be positive, which means \(\Pi _M^{3PRM} >\Pi _M^{RM} \). The difference of the optimum profits of the manufacturer between Model 3PRM and Model MR is,

Finally, the difference of the optimum profits of the manufacturer between Model MR and Model RM is,

Let

It is obvious that \(k_{11} >k_{10} \). When \(k>k_{11} \), the formula (91) will be negative and \(\Pi _M^{MR} <\Pi _M^{RM} \) and when \(k<k_{11} \), the formula (91) will be positive and \(\Pi _M^{MR} >\Pi _M^{RM} \). Hence, we get the results in Proposition 14.

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