Optimal production-inventory policy for closed-loop supply chain with remanufacturing under random demand and return

Abstract

This paper presents a centralized production-inventory model for an infinite planning horizon of a two-echelon closed-loop supply chain (CLSC) consisting a retailer, manufacturer, and remanufacturer. The demand at the retailer is satisfied through the new and remanufactured products received from the manufacturer and remanufacturer, respectively. The proposed model considers demand at the retailer and return at the remanufacturer as random. The manufacturer and remanufacturer produce the products at a finite rate, and they deliver to the retailer alternatively in multiple batches. An algorithm is developed to find the optimal-lot sizing and shipment policies of each entity of the CLSC by minimizing the expected joint total cost of the system. The results show that the CLSC is more profitable than the forward supply chain when the remanufacturing cost of the returned product is significantly low compared to the manufacturing cost of the new product, the demand variation at the retailer is low, and the fraction of demand returned is close to 1.

Appendices

Appendix 1: Derivation of the expected shortage of the returned product at the remanufacturer during each remanufacturing cycle

From Eq. (6), we have

$$B\left( {R_{r} } \right) = \int\limits_{0}^{{R_{r} - S_{r} }} {\left( {R_{r} - S_{r} - x_{r} } \right)f\left( {x_{r} } \right)dx_{r} }$$

(24)

The expression (24) can be rewritten as

$$\begin{aligned} B\left( {R_{r} } \right) & = \int\limits_{0}^{\infty } {\left( {R_{r} - S_{r} - x_{r} } \right)f\left( {x_{r} } \right)dx_{r} - } \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {R_{r} - S_{r} - x_{r} } \right)f\left( {x_{r} } \right)dx_{r} } \\ & = \left( {R_{r} - S_{r} } \right)\int\limits_{0}^{\infty } {f\left( {x_{r} } \right)dx_{r} - \int\limits_{0}^{\infty } {x_{r} f\left( {x_{r} } \right)dx_{r} + } } \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {x_{r} - \left( {R_{r} - S_{r} } \right)} \right)f\left( {x_{r} } \right)dx_{r} } \\ \end{aligned}$$

Since x_r ~ N(μrT, σ²rT), the above equation can be written as

$$B\left( {R_{r} } \right) = \left( {R_{r} - S_{r} - \mu rT} \right) + \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {x_{r} - \left( {R_{r} - S_{r} } \right)} \right)f\left( {x_{r} } \right)dx_{r} } .$$

(25)

Let $I = \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {x_{r} - \left( {R_{r} - S_{r} } \right)} \right)f\left( {x_{r} } \right)dx_{r} }$, which can be written as

$$\begin{aligned} & I = \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {x_{r} - \mu rT + \mu rT - \left( {R_{r} - S_{r} } \right)} \right)f\left( {x_{r} } \right)dx_{r} } ,\;{\text{therefore}} \\ & I = \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {x_{r} - \mu rT} \right)f\left( {x_{r} } \right)dx_{r} } + \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {\mu rT - \left( {R_{r} - S_{r} } \right)} \right)f\left( {x_{r} } \right)dx_{r} } \\ \end{aligned}$$

On substituting $f\left( {x_{r} } \right) = \frac{1}{{\sigma \sqrt {rT} \sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( {\frac{{x_{r} - \mu rT}}{{\sigma \sqrt {rT} }}} \right)^{2} }}$, we get

$$I = \int\limits_{{R_{r} - S_{r} }}^{\infty } {\left( {x_{r} - \mu rT} \right)\frac{1}{{\sigma \sqrt {rT} \sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( {\frac{{x_{r} - \mu rT}}{{\sigma \sqrt {rT} }}} \right)^{2} }} dx_{r} } + \left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right]\int\limits_{{R_{r} - S_{r} }}^{\infty } {\frac{1}{{\sigma \sqrt {rT} \sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( {\frac{{x_{r} - \mu rT}}{{\sigma \sqrt {rT} }}} \right)^{2} }} dx_{r} }$$

Let $\frac{{x_{r} - \mu rT}}{{\sigma \sqrt {rT} }} = y$, therefore $dx_{r} = \sigma \sqrt {rT} dy$

Hence, $I = \sigma \sqrt {rT} \int\limits_{{\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}}}^{\infty } {y\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}y^{2} }} dy} + \left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right]\int\limits_{{\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}}}^{\infty } {\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}y^{2} }} dy}$.

Let $I_{\text{I}} = \sigma \sqrt {rT} \int\limits_{{\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}}}^{\infty } {y\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}y^{2} }} dy}$ and $I_{\text{II}} = \left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right]\int\limits_{{\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}}}^{\infty } {\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}y^{2} }} dy}$.

Consider the expression $I_{\text{I}} = \sigma \sqrt {rT} \int\limits_{{\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}}}^{\infty } {y\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}y^{2} }} dy}$ and let ${{y^{2} } \mathord{\left/ {\vphantom {{y^{2} } 2}} \right. \kern-0pt} 2} = t$, therefore $ydy = dt$.

Hence, $I_{\text{I}} = \sigma \sqrt {rT} \int\limits_{{\frac{1}{2}\left( {\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}} \right)^{2} }}^{\infty } {\frac{1}{{\sqrt {2\pi } }}e^{ - t} dt} ,$ that is

$$I_{\text{I}} = \sigma \sqrt {rT} \frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( {\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}} \right)^{2} }} .$$

Using $S_{r} = k_{r} \sigma \sqrt {rT}$, and $R_{r} = \mu rT - k_{r} \sigma \sqrt {rT}$ from Eq. (4), we have

$$\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }} = - 2k_{r} .$$

Let z_r = 2k_r, and so

$$\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }} = - z_{r}$$

(26)

Hence, $I_{\text{I}} = \sigma \sqrt {rT} \frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}\left( { - z_{r} } \right)^{2} }}$ that is

$$I_{\text{I}} = \sigma \sqrt {rT} \phi ( - z_{r} ) = \sigma \sqrt {rT} \phi (z_{r} ),\quad {\text{as}}\;\phi ( - z_{r} ) = \phi (z_{r} ),$$

where $\phi ( \cdot )$ is pdf of the standard normal distribution.

Next,

$$I_{\text{II}} = \left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right]\int\limits_{{\frac{{R_{r} - S_{r} - \mu rT}}{{\sigma \sqrt {rT} }}}}^{\infty } {\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}y^{2} }} dy} = \left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right]\int\limits_{{ - z_{r} }}^{\infty } {\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{1}{2}y^{2} }} dy} .$$

Therefore,

$$I_{\text{II}} = \left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right]\left[ {1 - \Phi ( - z_{r} )} \right] = \left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right]\Phi (z_{r} ),$$

where $\Phi ( \cdot )$ is cdf of the standard normal distribution.

Now, from Eq. (30), $\left[ {\mu rT - \left( {R_{r} - S_{r} } \right)} \right] = z_{r} \sigma \sqrt {rT}$, and so

$$I_{\text{II}} = z_{r} \sigma \sqrt {rT} \Phi (z_{r} ).$$

Therefore, from Eq. (29), $B(R_{r} ) = \left( {R_{r} - S_{r} - \mu rT} \right) + \sigma \sqrt {rT} \left[ {\phi (z_{r} ) + z_{r} \Phi (z_{r} )} \right]$ and on substituting $\left( {R_{r} - S_{r} - \mu rT} \right) = - z_{r} \sigma \sqrt {rT}$ from Eq. (30), we get

$$B(R_{r} ) = \sigma \sqrt {rT} \left[ {\phi (z_{r} ) - z_{r} + z_{r} \Phi (z_{r} )} \right].$$

(27)

Appendix 2: The first and second partial derivatives of EJTC(·)

$$\frac{\partial EJTC\left( \cdot \right)}{{\partial k_{m} }} = \frac{{\sigma \sqrt {L_{m} } }}{T}\left[ {\frac{{h_{1} }}{\mu }\left( {\left( {1 - \alpha r} \right)\mu T + \frac{{\alpha z_{r} \sigma \sqrt {rT} }}{2}} \right) - m\pi \left( {1 - \Phi \left( {k_{m} } \right)} \right)} \right],$$

(28)

$$\frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{m}^{2} }} = \frac{{m\pi \sigma \sqrt {L_{m} } \phi \left( {k_{m} } \right)}}{T} > 0,$$

(29)

$$\frac{\partial EJTC\left( \cdot \right)}{{\partial k_{l} }} = \frac{{\sigma \sqrt {L_{l} } }}{T}\left[ {\frac{{h_{1} }}{\mu }\left( {\alpha r\mu T - \frac{{\alpha z_{r} \sigma \sqrt {rT} }}{2}} \right) - l\pi \left( {1 - \Phi \left( {k_{l} } \right)} \right)} \right],$$

(30)

$$\frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{l}^{2} }} = \frac{{l\pi \sigma \sqrt {L_{l} } \phi \left( {k_{l} } \right)}}{T} > 0,$$

(31)

$$\frac{\partial EJTC\left( \cdot \right)}{{\partial z_{r} }} = \left\{ \begin{array}{l} \frac{{z_{r} \alpha^{2} \sigma^{2} rT}}{4\mu }\left\{ {h_{1} \left( {\frac{1}{m} + \frac{1}{l}} \right) + \frac{{h_{2} }}{m}\left[ {m\left( {1 - \frac{\mu }{P}} \right) - \left( {1 - \frac{2\mu }{P}} \right)} \right] + \frac{{h_{3} \mu }}{{\alpha P_{r} }} + \frac{{h_{4} }}{l}\left[ {l\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right]} \right\} \hfill \\ + \frac{{T\alpha \sigma \sqrt {rT} }}{2}\left\{ \begin{array}{l} h_{1} \left[ {\frac{{\left( {1 - \alpha r} \right)}}{m} + \frac{\alpha r}{l}} \right] + \frac{{h_{2} \left( {1 - \alpha r} \right)}}{m}\left[ {m\left( {1 - \frac{\mu }{P}} \right) - \left( {1 - \frac{2\mu }{P}} \right)} \right] \hfill \\ + h_{3} \left( {\frac{1}{2\alpha } - \frac{r\mu }{{P_{r} }}} \right) - \frac{{h_{4} \alpha r}}{l}\left[ {l\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right] \hfill \\ \end{array} \right\} \hfill \\ + \frac{{h_{1} \sigma \alpha \sigma \sqrt {rT} \left( {k_{m} \sqrt {L_{m} } - k_{l} \sqrt {L_{l} } } \right)}}{2\mu } - \alpha \sigma \sqrt {rT} \left[ {1 - \Phi \left( {z_{r} } \right)} \right]\pi_{r} \hfill \\ \end{array} \right\},$$

(32)

$$\frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial z_{r}^{2} }} = \frac{{\alpha^{2} \sigma^{2} rT}}{4\mu }\left\{ \begin{array}{l} h_{1} \left( {\frac{1}{m} + \frac{1}{l}} \right) + h_{2} \left[ {\left( {1 - \frac{\mu }{P}} \right) - \frac{1}{m}\left( {1 - \frac{2\mu }{P}} \right)} \right] \hfill \\ + \frac{{h_{3} \mu }}{{\alpha P_{r} }} + h_{4} \left[ {\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \frac{1}{l}\left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right] \hfill \\ \end{array} \right\} + \alpha \sigma \sqrt {rT} \phi \left( {z_{r} } \right)\pi_{r} > 0,$$

(33)

$$\frac{\partial EJTC\left( \cdot \right)}{\partial m} = \frac{1}{T}\left[ \begin{array}{l} A_{1} - \frac{{h_{1} }}{{2\mu m^{2} }}\left( {\left( {1 - \alpha r} \right)\mu T + {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right)^{2} \hfill \\ + \frac{{h_{2} }}{{2\mu m^{2} }}\left( {1 - \frac{2\mu }{P}} \right)\left( {\left( {1 - \alpha r} \right)\mu T + {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right)^{2} \hfill \\ + \sigma \sqrt {L_{m} } \left[ {\phi \left( {k_{m} } \right) - k_{m} + k_{m} \Phi \left( {k_{m} } \right)} \right]\pi \hfill \\ \end{array} \right],$$

(34)

$$\frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial m^{2} }} = \frac{1}{T}\left[ {\frac{1}{{\mu m^{3} }}\left( {\left( {1 - \alpha r} \right)\mu T + {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right)^{2} \left( {h_{1} - h_{2} + h_{2} \frac{2\mu }{P}} \right)} \right] > 0,\quad \left( {{\text{as}}\;h_{1} > h_{2} } \right),$$

(35)

$$\frac{\partial EJTC\left( \cdot \right)}{\partial l} = \frac{1}{T}\left[ \begin{array}{l} A_{4} - \frac{{h_{1} }}{{2\mu l^{2} }}\left( {\alpha r\mu T - {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right)^{2} \hfill \\ + \frac{{h_{4} }}{{2\mu l^{2} }}\left( {1 - \frac{2\mu }{{P_{r} }}} \right)\left( {\alpha r\mu T - {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right)^{2} \hfill \\ + \pi \sigma \sqrt {L_{l} } \left[ {\phi \left( {k_{l} } \right) - k_{l} + k_{l} \Phi \left( {k_{l} } \right)} \right] \hfill \\ \end{array} \right],$$

(36)

$$\frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial l^{2} }} = \frac{1}{T}\left[ {\frac{1}{{\mu l^{3} }}\left( {\alpha r\mu T - {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right)^{2} \left( {h_{1} - h_{4} + h_{4} \frac{2\mu }{{P_{r} }}} \right)} \right] > 0,\quad \left( {{\text{as}}\;h_{1} > h_{4} } \right),$$

(37)

$$\frac{\partial EJTC\left( \cdot \right)}{\partial T} = \left\{ \begin{array}{l} \frac{\mu }{2}\left\{ \begin{array}{l} h_{1} \left( {\frac{{\left( {1 - \alpha r} \right)^{2} }}{m} + \frac{{\alpha^{2} r^{2} }}{l}} \right) + \left( {1 - \alpha r} \right)^{2} h_{2} \left[ {\left( {1 - \frac{\mu }{P}} \right) - \frac{1}{m}\left( {1 - \frac{2\mu }{P}} \right)} \right] + \hfill \\ rh_{3} \left( {1 + \frac{\alpha r\mu }{{P_{r} }}} \right) + \alpha^{2} r^{2} h_{4} \left[ {\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \frac{1}{l}\left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right] \hfill \\ \end{array} \right\} \hfill \\ + \frac{{z_{r} \alpha \sigma \sqrt {rT} }}{4T}\left\{ \begin{array}{l} h_{1} \left( {\frac{{\left( {1 - \alpha r} \right)}}{m} - \frac{\alpha r}{l}} \right) + \left( {1 - \alpha r} \right)h_{2} \left[ {\left( {1 - \frac{\mu }{P}} \right) - \frac{1}{m}\left( {1 - \frac{2\mu }{P}} \right)} \right] \hfill \\ + h_{3} \left( {\frac{1}{2\alpha } - \frac{r\mu }{{P_{r} }}} \right) - \alpha rh_{4} \left[ {\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \frac{1}{l}\left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right] \hfill \\ \end{array} \right\} \hfill \\ - \frac{{\alpha \sigma \sqrt {rT} }}{{2T^{2} }}\left[ {\frac{{h_{1} z_{r} \sigma }}{2\mu }\left( {k_{m} \sqrt {L_{m} } - k_{l} \sqrt {L_{l} } } \right) + \left( {\phi \left( {z_{r} } \right) - z_{r} + z_{r} \Phi \left( {z_{r} } \right)} \right)\pi_{r} } \right] \hfill \\ - \frac{1}{{T^{2} }}\left[ {\left( {mA_{1} + A_{2} + A_{3} + lA_{4} } \right) + \left[ \begin{array}{l} m\sigma \sqrt {L_{m} } \left( {\phi \left( {k_{m} } \right) - k_{m} + k_{m} \Phi \left( {k_{m} } \right)} \right) \hfill \\ + l\sigma \sqrt {L_{l} } \left( {\phi \left( {k_{l} } \right) - k_{l} + k_{l} \Phi \left( {k_{l} } \right)} \right) \hfill \\ \end{array} \right]\pi } \right] \hfill \\ \end{array} \right\}$$

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and

$$\frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial T^{2} }} = \left\{ \begin{array}{l} - \frac{{z_{r} \alpha \sigma \sqrt {rT} }}{{8T^{2} }}\left[ \begin{array}{l} h_{1} \left( {\frac{{\left( {1 - \alpha r} \right)}}{m} - \frac{\alpha r}{l}} \right) + \left( {1 - \alpha r} \right)h_{2} \left[ {\left( {1 - \frac{\mu }{P}} \right) - \frac{1}{m}\left( {1 - \frac{2\mu }{P}} \right)} \right] \hfill \\ + h_{3} \left( {\frac{1}{2\alpha } - \frac{r\mu }{{P_{r} }}} \right) - \alpha rh_{4} \left[ {\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \frac{1}{l}\left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right] \hfill \\ \end{array} \right] \hfill \\ + \frac{{3\alpha \sigma \sqrt {rT} }}{{4T^{3} }}\left[ {\frac{{h_{1} z_{r} \sigma }}{2\mu }\left( {k_{m} \sqrt {L_{m} } - k_{l} \sqrt {L_{l} } } \right) + \left( {\phi \left( {z_{r} } \right) - z_{r} + z_{r} \Phi \left( {z_{r} } \right)} \right)\pi_{r} } \right] \hfill \\ + \frac{2}{{T^{3} }}\left[ {\left( {mA_{1} + A_{2} + A_{3} + lA_{4} } \right) + \left[ \begin{array}{l} m\sigma \sqrt {L_{m} } \left( {\phi \left( {k_{m} } \right) - k_{m} + k_{m} \Phi \left( {k_{m} } \right)} \right) \hfill \\ + l\sigma \sqrt {L_{l} } \left( {\phi \left( {k_{l} } \right) - k_{l} + k_{l} \Phi \left( {k_{l} } \right)} \right) \hfill \\ \end{array} \right]\pi } \right] \hfill \\ \end{array} \right\}.$$

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Appendix 3: Test of convexity of the proposed model

To show the convexity of cost function in Eq. (11), all principal minors of the Hessian matrix of the cost function must be positive, i.e. $H_{1} > 0$,$H_{2} > 0$, $H_{3} > 0$, $H_{4} > 0$, $H_{5} > 0$, $H_{6} > 0$.

Here,

$$\begin{aligned} & H_{1} = \left| {\lambda_{11} } \right|,\quad H_{2} = \left| {\begin{array}{*{20}l} {\lambda_{11} } \hfill & {\quad \lambda_{12} } \hfill \\ {\lambda_{21} } \hfill & {\quad \lambda_{22} } \hfill \\ \end{array} } \right|,\quad H_{3} = \left| {\begin{array}{*{20}l} {\lambda_{11} } \hfill & {\quad \lambda_{12} } \hfill & {\quad \lambda_{13} } \hfill \\ {\lambda_{21} } \hfill & {\quad \lambda_{22} } \hfill & {\quad \lambda_{23} } \hfill \\ {\lambda_{31} } \hfill & {\quad \lambda_{32} } \hfill & {\quad \lambda_{33} } \hfill \\ \end{array} } \right|,\quad H_{4} = \left| {\begin{array}{*{20}l} {\lambda_{11} } \hfill & {\quad \lambda_{12} } \hfill & {\quad \lambda_{13} } \hfill & {\quad \lambda_{14} } \hfill \\ {\lambda_{21} } \hfill & {\quad \lambda_{22} } \hfill & {\quad \lambda_{23} } \hfill & {\quad \lambda_{24} } \hfill \\ {\lambda_{31} } \hfill & {\quad \lambda_{32} } \hfill & {\quad \lambda_{33} } \hfill & {\quad \lambda_{34} } \hfill \\ {\lambda_{41} } \hfill & {\quad \lambda_{42} } \hfill & {\quad \lambda_{43} } \hfill & {\quad \lambda_{44} } \hfill \\ \end{array} } \right| \\ & H_{5} = \left| {\begin{array}{*{20}l} {\lambda_{11} } \hfill & {\quad \lambda_{12} } \hfill & {\quad \lambda_{13} } \hfill & {\quad \lambda_{14} } \hfill & {\quad \lambda_{15} } \hfill \\ {\lambda_{21} } \hfill & {\quad \lambda_{22} } \hfill & {\quad \lambda_{23} } \hfill & {\quad \lambda_{24} } \hfill & {\quad \lambda_{25} } \hfill \\ {\lambda_{31} } \hfill & {\quad \lambda_{32} } \hfill & {\quad \lambda_{33} } \hfill & {\quad \lambda_{34} } \hfill & {\quad \lambda_{35} } \hfill \\ {\lambda_{41} } \hfill & {\quad \lambda_{42} } \hfill & {\quad \lambda_{43} } \hfill & {\quad \lambda_{44} } \hfill & {\quad \lambda_{45} } \hfill \\ {\lambda_{51} } \hfill & {\quad \lambda_{52} } \hfill & {\quad \lambda_{53} } \hfill & {\quad \lambda_{54} } \hfill & {\quad \lambda_{55} } \hfill \\ \end{array} } \right|\quad {\text{and}}\quad H_{6} = \left| {\begin{array}{*{20}l} {\lambda_{11} } \hfill & {\quad \lambda_{12} } \hfill & {\quad \lambda_{13} } \hfill & {\quad \lambda_{14} } \hfill & {\quad \lambda_{15} } \hfill & {\quad \lambda_{16} } \hfill \\ {\lambda_{21} } \hfill & {\quad \lambda_{22} } \hfill & {\quad \lambda_{23} } \hfill & {\quad \lambda_{24} } \hfill & {\quad \lambda_{25} } \hfill & {\quad \lambda_{26} } \hfill \\ {\lambda_{31} } \hfill & {\quad \lambda_{32} } \hfill & {\quad \lambda_{33} } \hfill & {\quad \lambda_{34} } \hfill & {\quad \lambda_{35} } \hfill & {\quad \lambda_{36} } \hfill \\ {\lambda_{41} } \hfill & {\quad \lambda_{42} } \hfill & {\quad \lambda_{43} } \hfill & {\quad \lambda_{44} } \hfill & {\quad \lambda_{45} } \hfill & {\quad \lambda_{46} } \hfill \\ {\lambda_{51} } \hfill & {\quad \lambda_{52} } \hfill & {\quad \lambda_{53} } \hfill & {\quad \lambda_{54} } \hfill & {\quad \lambda_{55} } \hfill & {\quad \lambda_{56} } \hfill \\ {\lambda_{61} } \hfill & {\quad \lambda_{62} } \hfill & {\quad \lambda_{63} } \hfill & {\quad \lambda_{64} } \hfill & {\quad \lambda_{65} } \hfill & {\quad \lambda_{66} } \hfill \\ \end{array} } \right|, \\ \end{aligned}$$

where $H_{6} = \left| H \right|$; $\left| \bullet \right|$ is the Hessian determinant and λ_ij (i, j = 1, 2, 3, 4, 5, 6) are the second order partial derivatives of the cost function given by Eq. (11).

The second order partial derivatives of the cost function (11) are given below:

$$\lambda_{12} = \lambda_{21} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{m} \partial k_{l} }} = 0,\quad \lambda_{13} = \lambda_{31} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{m} \partial z_{r} }} = \frac{{\sigma \sqrt {L_{m} } }}{2T}\frac{{h_{1} }}{\mu }\frac{{\alpha \sigma \sqrt {rT} }}{2},$$

$$\lambda_{14} = \lambda_{41} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{m} \partial m}} = - \frac{{\sigma \sqrt {L_{m} } \pi \left( {1 - \Phi \left( {k_{m} } \right)} \right)}}{T},\quad \lambda_{15} = \lambda_{51} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{m} \partial l}} = 0,$$

$$\lambda_{16} = \lambda_{61} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{m} \partial T}} = \frac{{\sigma \sqrt {L_{m} } }}{{T^{2} }}\left[ { - \frac{{h_{1} }}{\mu }\frac{{\alpha z_{r} \sigma \sqrt {rT} }}{4} + m\pi \left( {1 - \Phi \left( {k_{m} } \right)} \right)} \right],$$

$$\lambda_{23} = \lambda_{32} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{l} \partial z_{r} }} = - \frac{{\sigma \sqrt {L_{l} } }}{T}\frac{{h_{1} }}{\mu }\frac{{\alpha \sigma \sqrt {rT} }}{2},\quad \lambda_{24} = \lambda_{42} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{l} \partial m}} = 0,$$

$$\lambda_{25} = \lambda_{52} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{l} \partial l}} = - \frac{{\sigma \sqrt {L_{l} } \pi \left( {1 - \Phi \left( {k_{l} } \right)} \right)}}{T},$$

$$\lambda_{26} = \lambda_{62} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{l} \partial T}} = \frac{{\sigma \sqrt {L_{l} } }}{{T^{2} }}\left[ { - \frac{{h_{1} }}{\mu }\frac{{\alpha z_{r} \sigma \sqrt {rT} }}{4} + l\pi \left( {1 - \Phi \left( {k_{l} } \right)} \right)} \right],$$

$$\lambda_{34} = \lambda_{43} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial z_{r} \partial m}} = - \frac{{\left[ {h_{1} - \left( {1 - \frac{2\mu }{P}} \right)h_{2} } \right]\left[ {\frac{{z_{r} \alpha^{2} \sigma^{2} rT}}{2\mu } + \left( {1 - \alpha r} \right)T\alpha \sigma \sqrt {rT} } \right]}}{{2m^{2} }},$$

$$\lambda_{35} = \lambda_{53} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial z_{r} \partial l}} = - \frac{{\left[ {h_{1} - \left( {1 - \frac{2\mu }{{P_{r} }}} \right)h_{4} } \right]\left( {\frac{{z_{r} \alpha^{2} \sigma^{2} rT}}{2\mu } + \alpha rT\alpha \sigma \sqrt {rT} } \right)}}{{2l^{2} }},$$

$$\lambda_{36} = \lambda_{63} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial z_{r} \partial T}} = \left\{ \begin{array}{l} \frac{{z_{r} \alpha^{2} \sigma^{2} r}}{4\mu }\left\{ \begin{array}{l} h_{1} \left( {\frac{1}{m} + \frac{1}{l}} \right) + \frac{{h_{2} }}{m}\left[ {m\left( {1 - \frac{\mu }{P}} \right) - \left( {1 - \frac{2\mu }{P}} \right)} \right] \hfill \\ + \frac{{h_{3} \mu }}{{\alpha P_{r} }} + \frac{{h_{4} }}{l}\left[ {l\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right] \hfill \\ \end{array} \right\} \hfill \\ \quad + \frac{{3\alpha \sigma \sqrt {rT} }}{4}\left\{ \begin{array}{l} h_{1} \left[ {\frac{{\left( {1 - \alpha r} \right)}}{m} + \frac{\alpha r}{l}} \right] + \frac{{h_{2} \left( {1 - \alpha r} \right)}}{m}\left[ {m\left( {1 - \frac{\mu }{P}} \right) - \left( {1 - \frac{2\mu }{P}} \right)} \right] \hfill \\ + h_{3} \left( {\frac{1}{2\alpha } - \frac{r\mu }{{P_{r} }}} \right) - \frac{{h_{4} \alpha r}}{l}\left[ {l\left( {1 - \frac{\mu }{{P_{r} }}} \right) - \left( {1 - \frac{2\mu }{{P_{r} }}} \right)} \right] \hfill \\ \end{array} \right\} \hfill \\ \quad + \frac{1}{2T}\left[ {\frac{{h_{1} \sigma \alpha \sigma \sqrt {rT} \left( {k_{m} \sqrt {L_{m} } - k_{l} \sqrt {L_{l} } } \right)}}{2\mu } - \alpha \sigma \sqrt {rT} \left[ {1 - \Phi \left( {z_{r} } \right)} \right]\pi_{r} } \right] \hfill \\ \end{array} \right\},$$

$$\lambda_{45} = \lambda_{54} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{\partial m\partial l} = 0,$$

$$\begin{aligned} & \lambda_{46} = \lambda_{64} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{\partial m\partial T} = - \left\{ \begin{array}{l} \frac{1}{{T^{2} }}\left( {A_{1} + \pi \sigma \sqrt {L_{m} } \left[ {\phi \left( {k_{m} } \right) - k_{m} + k_{m} \Phi \left( {k_{m} } \right)} \right]} \right) \hfill \\ \quad + \frac{{\left( {1 - \alpha r} \right)\left[ {\left( {1 - \alpha r} \right)\mu T - {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right]\left[ {h_{1} - \left( {1 - \frac{2\mu }{P}} \right)h_{2} } \right]}}{{2m^{2} T}} \hfill \\ \end{array} \right\} \\ \quad {\text{and}}\;\lambda_{56} = \lambda_{65} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{\partial l\partial T} = - \left\{ \begin{array}{l} \frac{1}{{T^{2} }}\left( {A_{4} + \pi \sigma \sqrt {L_{l} } \left[ {\phi \left( {k_{l} } \right) - k_{l} + k_{l} \Phi \left( {k_{l} } \right)} \right]} \right) \hfill \\ \quad + \frac{\alpha r}{{2l^{2} T}}\left( {\alpha r\mu T - {{\alpha z_{r} \sigma \sqrt {rT} } \mathord{\left/ {\vphantom {{\alpha z_{r} \sigma \sqrt {rT} } 2}} \right. \kern-0pt} 2}} \right)\left[ {h_{1} - \left( {1 - \frac{2\mu }{{P_{r} }}} \right)h_{4} } \right] \hfill \\ \end{array} \right\}. \\ \end{aligned}$$

The remaining expressions of the second order partial derivatives, i.e. $\lambda_{11} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{m}^{2} }}$, $\lambda_{22} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial k_{l}^{2} }}$,$\lambda_{33} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial z_{r}^{2} }}$, $\lambda_{44} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial m^{2} }}$, $\lambda_{55} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial l^{2} }}$, and $\lambda_{66} = \frac{{\partial^{2} EJTC\left( \cdot \right)}}{{\partial T^{2} }}$ are mentioned in Appendix 2.

Appendix 4: Derivation of the FSC model

The expression of the EJTC per unit time of the FSC corresponding to the CLSC under study is obtained from Eq. (11) by setting the parameters associated with the returned and remanufactured products as zero (i.e. A₃, h₃, A₄, h₄, r, L_l, π_r= 0). Thus, the expression for the EJTC per unit time of the FSC is given by

$$EJTC\left( {k_{m} ,m,T} \right) = \left\{ \begin{array}{l} \frac{{mA_{1} + A_{2} }}{T} + \left( {\frac{\mu T}{2m} + k_{m} \sigma \sqrt {L_{m} } } \right)h_{1} + \frac{\mu T}{2m}\left[ {\left( {m - 1} \right) - \left( {m - 2} \right)\frac{\mu }{P}} \right]h_{2} \hfill \\ \quad + \frac{m}{T}\sigma \sqrt {L_{m} } \left[ {\phi \left( {k_{m} } \right) - k_{m} + k_{m} \Phi \left( {k_{m} } \right)} \right]\pi \hfill \\ \end{array} \right\}.$$

(40)

To minimize the EJTC in Eq. (40), we take the first partial derivatives of the EJTC in Eq. (40) with respect to each decision variable while keeping other decision variables fixed, and setting them to zero, we obtain

$$\frac{{\partial EJTC\left( {k_{m} ,m,T} \right)}}{{\partial k_{m} }} = h_{1} \sigma \sqrt {L_{m} } - \frac{m}{T}\sigma \sqrt {L_{m} } \left[ {1 - \Phi \left( {k_{m} } \right)} \right]\pi ,$$

(41)

$$\frac{{\partial EJTC\left( {k_{m} ,m,T} \right)}}{\partial m} = \frac{{A_{1} }}{T} - \frac{{\mu Th_{1} }}{{2m^{2} }} + \frac{\mu T}{{2m^{2} }}\left( {1 - \frac{2\mu }{P}} \right)h_{2} + \frac{1}{T}\sigma \sqrt {L_{m} } \left[ {\phi \left( {k_{m} } \right) - k_{m} + k_{m} \Phi \left( {k_{m} } \right)} \right]\pi ,$$

(42)

and

$$\frac{{\partial EJTC\left( {k_{m} ,m,T} \right)}}{\partial T} = \left\{ \begin{array}{l} - \frac{{\left( {mA_{1} + A_{2} } \right)}}{{T^{2} }} + \frac{{\mu h_{1} }}{2m} + \frac{\mu }{2m}\left[ {\left( {m - 1} \right) - \left( {m - 2} \right)\frac{\mu }{P}} \right]h_{2} \hfill \\ \quad - \frac{m}{{T^{2} }}\sigma \sqrt {L_{m} } \left[ {\phi \left( {k_{m} } \right) - k_{m} + k_{m} \Phi \left( {k_{m} } \right)} \right]\pi \hfill \\ \end{array} \right\}.$$

(43)

The optimal value of the decision variables k_m, m (integer) and T, and the corresponding EJTC of the FSC are obtained from Eqs. (41), (42), (43), and (40) by using the iterative method similar to the CLSC model as explained in Sect. 4.