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Sustainable Supply Chain Model for Multi-stage Manufacturing with Partial Backlogging Under the Fuzzy Environment with the Effect of Learning in Screening Process

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Abstract

The primary concern of every business manager of supply chain system is to obtain economical sustainability. To achieve this goal they adopt different polices such as multi-stage manufacturing process, promotional strategies, learning effect, screening process etc. In the present study, a supply chain model consisting one retailer and one manufacturer is examined regarding the financial viability. An imperfect multi-stage manufacturing process is considered here with a probabilistic deteriorating item. The screening process under the effect of learning is performed in each stage of production and further the imperfect products are reworked in the same stage. Promotional efforts are initiated by the retailer to boost up demand. Shortages are allowed at the retailer end with partially backlogging. All the cost parameters are imprecise parameters due to the presence of uncertainty in the market. The presence of impreciseness in cost parameters is handled by applying the fuzzy set theory. To defuzzify the objective function of the system, the centroid method is used. Aim of this work is to minimize the average inventory cost so that order quantity and backorder quantity are optimal. Objective function in the developed model is nonlinear optimization problem which is solved with the help of calculus based classical optimization technique. Further, convexity of the objective function is explored with the help of graphs and Hessian matrix. Results indicate that average inventory cost decreases by 5% as the supply chain system shifted to three-stage manufacturing process to five-stage manufacturing process. Further, analysis shows that incorporating impreciseness in costs capture the real picture of business. Sustainability of the proposed model is explored with the help of numerical example and sensitive analysis. Form sensitivity analysis, positive impact of screening rate and promotional efforts are observed on the average cost of the system. Analysis also reflects that inventory cost of the system is high due to high backlogging rate.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Vardhaman College, Bijnor, U.P., India

    Dharmendra Yadav

  2. Department of Mathematics, Meerut College, Meerut, U.P., India

    Rachna Kumari

  3. Government Girls Degree College, Kharkhauda, Meerut, U.P., India

    Narendra Kumar

Authors
  1. Dharmendra Yadav
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  2. Rachna Kumari
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  3. Narendra Kumar
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Correspondence to Dharmendra Yadav.

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Appendices

Appendix 1

$$\begin{aligned} \frac{{\partial TIC_{SC}^{j*} }}{\partial Q} & = \frac{{ j^{b} }}{{\mathop \sum \nolimits_{i = 1}^{n} \frac{{Q\left( {1 + \mu } \right)}}{{P^{i,j} }}\left( {\left( {1 + \frac{{P^{i,j} }}{{M^{i,j} }}} \right) + \left( {\frac{{a^{i,j} + b^{i,j} }}{2}} \right)\left( {1 - \frac{{P^{i,j} }}{{M^{i,j} }}} \right)} \right)}} \\ & \quad \left( {\mathop \sum \limits_{i = 1}^{n} \left( {C_{pi} + \frac{1}{3}\left( {\Delta_{4} - \Delta_{3} } \right)} \right)\left( {1 + \frac{{a^{i,j} + b^{i,j} }}{2}} \right)} \right. \\ & \quad + \mathop \sum \limits_{i = 1}^{n} \left( {I_{pi} + \frac{1}{3}\left( {\Delta_{6} - \Delta_{5} } \right)} \right)\left( {1 + \frac{{a^{i,j} + b^{i,j} }}{2}} \right) \\ & \quad + Q\mathop \sum \limits_{i = 1}^{n} \frac{{\left( {H_{pi} + \frac{1}{3}\left( {\Delta_{8} - \Delta_{7} } \right)} \right)}}{{P^{i,j} }} \\ & \quad \left. {\left( {1 + \frac{{a^{i,j} + b^{i,j} }}{2} - \left( {\frac{{a^{i,j} + b^{i,j} }}{2}} \right)^{2} + \left( {1 - \frac{{a^{i,j} + b^{i,j} }}{2}} \right)\frac{{2P^{i,j} }}{{M^{i,j} }}} \right)} \right) \\ & \quad + X_{1} \left( {C_{T} + \frac{1}{3}\left( {\Delta_{12} - \Delta_{11} } \right)} \right) \\ & \quad - \frac{{ j^{b} }}{{\mathop \sum \nolimits_{i = 1}^{n} \frac{{Q^{2} \left( {1 + \mu } \right)}}{{P^{i,j} }}\left( {\left( {1 + \frac{{P^{i,j} }}{{M^{i,j} }}} \right) + \left( {\frac{{a^{i,j} + b^{i,j} }}{2}} \right)\left( {1 - \frac{{P^{i,j} }}{{M^{i,j} }}} \right)} \right)}} \\ & \quad \left( {\mathop \sum \limits_{i = 1}^{n} \left( {S_{i} + \frac{1}{3}\left( {\Delta_{2} - \Delta_{1} } \right)} \right)} \right. + \mathop \sum \limits_{i = 1}^{n} \left( {C_{pi} + \frac{1}{3}\left( {\Delta_{4} - \Delta_{3} } \right)} \right)Q\left( {1 + \frac{{a^{i,j} + b^{i,j} }}{2}} \right) \\ & \quad + \mathop \sum \limits_{i = 1}^{n} \left( {I_{pi} + \frac{1}{3}\left( {\Delta_{6} - \Delta_{5} } \right)} \right)Q\left( {1 + \frac{{a^{i,j} + b^{i,j} }}{2}} \right) \\ & \quad + \frac{{Q^{2} }}{2}\mathop \sum \limits_{i = 1}^{n} \frac{{\left( {H_{pi} + \frac{1}{3}\left( {\Delta_{8} - \Delta_{7} } \right)} \right)}}{{P^{i,1} }} \\ & \quad \left( {1 + \frac{{a^{i,j} + b^{i,j} }}{2} - \left( {\frac{{a^{i,j} + b^{i,j} }}{2}} \right)^{2} + \left( {1 - \frac{{a^{i,j} + b^{i,j} }}{2}} \right)\frac{{2P^{i,j} }}{{M^{i,j} }}} \right) \\ & \quad + \left. {\left( {\left( {F + \frac{1}{3}\left( {\Delta_{10} - \Delta_{9} } \right)} \right) + QX_{1} \left( {C_{T} + \frac{1}{3}\left( {\Delta_{12} - \Delta_{11} } \right)} \right)} \right)} \right) \\ & \quad + \frac{{j^{b} }}{{\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}}} \\ & \quad \left[ {\left( {P_{C} + \frac{1}{3}\left( {\Delta_{16} - \Delta_{15} } \right)} \right) + \left( {S_{c} + \frac{1}{3}\left( {\Delta_{18} - \Delta_{17} } \right)} \right)} \right. \\ & \quad \left. { + \left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{{\left( {1 - p} \right)\left( {\left( {1 - p} \right)Q - B} \right)}}{D\left( \varepsilon \right)} + \frac{2pQ}{x}} \right)} \right] \\ & \quad - \frac{{\frac{{\left( {1 - p} \right)}}{D\left( \varepsilon \right)}j^{b} }}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\left( \varepsilon \right)} \right)^{2} }} \\ & \quad \left[ {\left( {K + \frac{1}{3}\left( {\Delta_{14} - \Delta_{13} } \right) + Q\left( {P_{C} + \frac{1}{3}\left( {\Delta_{16} - \Delta_{15} } \right)} \right) + \left( {S_{c} + \frac{1}{3}\left( {\Delta_{18} - \Delta_{17} } \right)} \right)Q} \right)} \right. \\ & \quad + \left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right) \times \left( {\frac{{\left( {\left( {1 - p} \right)Q - B} \right)^{2} }}{2D\left( \varepsilon \right)} + \frac{{pQ^{2} }}{x}} \right) + \frac{1}{2}\left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right) \\ & \quad \frac{{B^{2} }}{\delta D\left( \varepsilon \right)}\left. { + \left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)B} \right] \\ \end{aligned}$$
$$\begin{aligned} & \frac{{\partial TIC_{SC}^{j*} }}{\partial B} = \frac{{j^{b} }}{{\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}}}\left[ {\left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( { - \frac{{\left( {1 - p} \right)Q - B}}{D\left( \varepsilon \right)}} \right)} \right. \\ & \quad - \left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{B}{\delta D\left( \varepsilon \right)} + \left. {\left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)} \right] \\ & \quad - \frac{{\left( {\frac{1}{\delta } - 1} \right)\frac{1}{D\left( \varepsilon \right)}j^{b} }}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}} \right)^{2} }}\left[ {\left( {K + \frac{1}{3}\left( {\Delta_{14} - \Delta_{13} } \right)} \right) + Q\left( {P_{C} + \frac{1}{3}\left( {\Delta_{16} - \Delta_{15} } \right)} \right)} \right. \\ & \quad + \left( {S_{c} + \frac{1}{3}\left( {\Delta_{18} - \Delta_{17} } \right)} \right)Q + \left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{{\left( {\left( {1 - p} \right)Q - B} \right)^{2} }}{2D\left( \varepsilon \right)} + \frac{{pQ^{2} }}{x}} \right) \\ & \quad + \frac{1}{2}\left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{{B^{2} }}{\delta D\left( \varepsilon \right)} + \left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)B \\ \end{aligned}$$

Appendix 2

$$\begin{aligned} \frac{{\partial^{2} TIC_{SC}^{j*} }}{{\partial Q^{2} }} & = \frac{{ j^{b} }}{{\mathop \sum \nolimits_{i = 1}^{n} \frac{{Q\left( {1 + \mu } \right)}}{{P^{i,1} }}\left( {\left( {1 + \frac{{P^{i,1} }}{{M^{i,1} }}} \right) + \theta^{i,1} \left( {1 - \frac{{P^{i,1} }}{{M^{i,1} }}} \right)} \right)}} \\ & \quad \left( {\mathop \sum \limits_{i = 1}^{n} \frac{{\left( {H_{pi} + \frac{1}{3}\left( {\Delta_{8} - \Delta_{7} } \right)} \right)}}{{P^{i,1} }}} \right.\left. { \times \left( {1 + \frac{{a^{i,1} + b^{i,1} }}{2} - \left( {\frac{{a^{i,1} + b^{i,1} }}{2}} \right)^{2} + \left( {1 - \frac{{a^{i,1} + b^{i,1} }}{2}} \right)\frac{{2P^{i,1} }}{{M^{i,1} }}} \right)} \right) \\ & \quad + \frac{{2 j^{b} }}{{\mathop \sum \nolimits_{i = 1}^{n} \frac{{Q^{2} \left( {1 + \mu } \right)}}{{P^{i,1} }}\left( {\left( {1 + \frac{{P^{i,1} }}{{M^{i,1} }}} \right) + \theta^{i,1} \left( {1 - \frac{{P^{i,1} }}{{M^{i,1} }}} \right)} \right)}} \\ & \quad \left( {\mathop \sum \limits_{i = 1}^{n} \left( {C_{pi} + \frac{1}{3}\left( {\Delta_{4} - \Delta_{3} } \right)} \right)\left( {1 + \frac{{a^{i,1} + b^{i,1} }}{2}} \right) + \mathop \sum \limits_{i = 1}^{n} \left( {I_{pi} + \frac{1}{3}\left( {\Delta_{6} - \Delta_{5} } \right)} \right)\left( {1 + \frac{{a^{i,1} + b^{i,1} }}{2}} \right)} \right. \\ & \quad Q\mathop \sum \limits_{i = 1}^{n} \frac{{\left( {H_{pi} + \frac{1}{3}\left( {\Delta_{8} - \Delta_{7} } \right)} \right)}}{{P^{i,1} }} \\ & \quad + \left. {\left( {1 + \frac{{a^{i,1} + b^{i,1} }}{2} - \left( {\frac{{a^{i,1} + b^{i,1} }}{2}} \right)^{2} + \left( {1 - \frac{{a^{i,1} + b^{i,1} }}{2}} \right)\frac{{2P^{i,1} }}{{M^{i,1} }}} \right) + X_{1} \left( {C_{T} + \frac{1}{3}\left( {\Delta_{12} - \Delta_{11} } \right)} \right)} \right) \\ & \quad + \frac{{2 j^{b} }}{{\mathop \sum \nolimits_{i = 1}^{n} \frac{{Q^{3} \left( {1 + \mu } \right)}}{{P^{i,1} }}\left( {\left( {1 + \frac{{P^{i,1} }}{{M^{i,1} }}} \right) + \theta^{i,1} \left( {1 - \frac{{P^{i,1} }}{{M^{i,1} }}} \right)} \right)}} \\ & \quad (\mathop \sum \limits_{i = 1}^{n} \left( {S_{i} + \frac{1}{3}\left( {\Delta_{2} - \Delta_{1} } \right)} \right) + \mathop \sum \limits_{i = 1}^{n} \left( {C_{pi} + \frac{1}{3}\left( {\Delta_{4} - \Delta_{3} } \right)} \right)Q\left( {1 + \frac{{a^{i,1} + b^{i,1} }}{2}} \right) \\ & \quad + \mathop \sum \limits_{i = 1}^{n} \left( {I_{pi} + \frac{1}{3}\left( {\Delta_{6} - \Delta_{5} } \right)} \right)Q\left( {1 + \frac{{a^{i,1} + b^{i,1} }}{2}} \right) \\ & \quad + \frac{{Q^{2} }}{2}\mathop \sum \limits_{i = 1}^{n} \frac{{\left( {H_{pi} + \frac{1}{3}\left( {\Delta_{8} - \Delta_{7} } \right)} \right)}}{{P^{i,1} }}\left( {1 + \frac{{a^{i,1} + b^{i,1} }}{2} - \left( {\frac{{a^{i,1} + b^{i,1} }}{2}} \right)^{2} + \left( {1 - \frac{{a^{i,1} + b^{i,1} }}{2}} \right)\frac{{2P^{i,1} }}{{M^{i,1} }}} \right) \\ & \quad + \left( {\left( {F + \frac{1}{3}\left( {\Delta_{10} - \Delta_{9} } \right)} \right) + QX_{1} \left( {C_{T} + \frac{1}{3}\left( {\Delta_{12} - \Delta_{11} } \right)} \right)} \right)) \\ \end{aligned}$$
$$\begin{aligned} & \frac{1}{{\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}}}\left[ {\left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{{\left( {1 - p} \right)^{2} }}{D} + \frac{2p}{x}} \right)} \right] \\ & \quad - \frac{{2\frac{{\left( {1 - p} \right)}}{D\left( \varepsilon \right)}}}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}} \right)^{2} }}\left[ {\left( {P_{C} + \frac{1}{3}\left( {\Delta_{16} - \Delta_{15} } \right)} \right) + \left( {S_{c} + \frac{1}{3}\left( {\Delta_{18} - \Delta_{17} } \right)} \right)} \right. \\ & \quad \left. { + \left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{{\left( {1 - p} \right)\left( {\left( {1 - p} \right)Q - B} \right)}}{D\left( \varepsilon \right)} + \frac{2pQ}{x}} \right)} \right] \\ & \quad + \frac{{2\left( {\frac{{\left( {1 - p} \right)}}{D\left( \varepsilon \right)}} \right)^{2} }}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}} \right)^{3} }} \left[ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} \left( {K + \frac{1}{3}\left( {\Delta_{14} - \Delta_{13} } \right)} \right) + Q\left( {P_{C} + \frac{1}{3}\left( {\Delta_{16} - \Delta_{15} } \right)} \right)} \right. \\ & \quad + \left( {S_{c} + \frac{1}{3}\left( {\Delta_{18} - \Delta_{17} } \right)} \right)Q + \left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{{\left( {\left( {1 - p} \right)Q - B} \right)^{2} }}{2D\left( \varepsilon \right)} + \frac{{pQ^{2} }}{x}} \right) \\ & \quad + \frac{1}{2}\left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{{B^{2} }}{\delta D\left( \varepsilon \right)} + \left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)B\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right] \\ \end{aligned}$$
$$\begin{aligned} \frac{{\partial^{2} TIC_{SC}^{j*} }}{{\partial B^{2} }} & = \frac{1}{{\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}}} \\ & \quad \left[ {\left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{1}{D}} \right) + \left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{1}{\delta D\left( \varepsilon \right)}} \right] \\ & \quad - \frac{{2\left( {\frac{1}{\delta } - 1} \right)\frac{1}{D}}}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}} \right)^{2} }} \\ & \quad \left[ {\left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( { - \frac{{\left( {1 - p} \right)Q - B}}{D\left( \varepsilon \right)}} \right)} \right. \\ & \quad \left. { + \frac{1}{2}\left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{{B^{2} }}{\delta D\left( \varepsilon \right)} + \left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)B} \right] \\ & \quad + \frac{{\left( {\left( {\frac{1}{\delta } - 1} \right)\frac{1}{D\left( \varepsilon \right)}} \right)^{2} }}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}} \right)^{2} }} \\ & \quad \left[ {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} \left( {K + \frac{1}{3}\left( {\Delta_{14} - \Delta_{13} } \right)} \right) + Q\left( {P_{C} + \frac{1}{3}\left( {\Delta_{16} - \Delta_{15} } \right)} \right)} \right. \\ & \quad + \left( {S_{c} + \frac{1}{3}\left( {\Delta_{18} - \Delta_{17} } \right)} \right)Q + \left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{{\left( {\left( {1 - p} \right)Q - B} \right)^{2} }}{2D\left( \varepsilon \right)} + \frac{{pQ^{2} }}{x}} \right) \\ & \quad + \frac{1}{2}\left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{{B^{2} }}{\delta D\left( \varepsilon \right)} + \left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)B\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right] \\ \end{aligned}$$

and

$$\begin{aligned} \frac{{\partial^{2} TIC_{SC}^{j*} }}{\partial Q\partial B} & = \frac{1}{{\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}}} \\ & \quad \left[ {\left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( { - \frac{{\left( {1 - p} \right)}}{D\left( \varepsilon \right)}} \right)} \right] \\ & \quad - \frac{{2\frac{{\left( {1 - p} \right)}}{D\left( \varepsilon \right)}}}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}} \right)^{2} }} \\ & \quad \left[ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right.\left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( { - \frac{{\left( {1 - p} \right)Q - B}}{D}} \right) \\ & \quad + \left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{B}{\delta D\left( \varepsilon \right)} + \left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right] \\ & \quad + \frac{{2\left( {\left( {\frac{1}{\delta } - 1} \right)\frac{1}{D\left( \varepsilon \right)}} \right)^{2} }}{{\left( {\frac{{\left( {1 - p} \right)Q}}{D\left( \varepsilon \right)} + \left( {\frac{1}{\delta } - 1} \right)\frac{B}{D\left( \varepsilon \right)}} \right)^{3} }} \\ & \quad \left[ {\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} \left( {K + \frac{1}{3}\left( {\Delta_{14} - \Delta_{13} } \right)} \right) + Q\left( {P_{C} + \frac{1}{3}\left( {\Delta_{16} - \Delta_{15} } \right)} \right)} \right. \\ & \quad + \left( {S_{c} + \frac{1}{3}\left( {\Delta_{18} - \Delta_{17} } \right)} \right)Q \\ & \quad + \left( {H_{c} + \frac{1}{3}\left( {\Delta_{20} - \Delta_{19} } \right)} \right)\left( {\frac{{\left( {\left( {1 - p} \right)Q - B} \right)^{2} }}{2D\left( \varepsilon \right)} + \frac{{pQ^{2} }}{x}} \right) \\ & \quad + \frac{1}{2}\left( {Sh_{c} + \frac{1}{3}\left( {\Delta_{22} - \Delta_{21} } \right)} \right)\frac{{B^{2} }}{\delta D\left( \varepsilon \right)} \\ & \quad + \left( {LS_{c} + \frac{1}{3}\left( {\Delta_{24} - \Delta_{23} } \right)} \right)\left( {\frac{1 - \delta }{\delta }} \right)B\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right] \\ \end{aligned}$$

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Yadav, D., Kumari, R. & Kumar, N. Sustainable Supply Chain Model for Multi-stage Manufacturing with Partial Backlogging Under the Fuzzy Environment with the Effect of Learning in Screening Process. Int. J. Appl. Comput. Math 7, 40 (2021). https://doi.org/10.1007/s40819-021-00951-5

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