Abstract
Supply chain management is facing serious challenges in the form of imperfections, which cause quality and environmental concerns. The supplier’s manufacturing system may not produce all perfect items and some chances of receiving a lot may include a proportion of imperfect items. It is a time-consuming, negative impact on the environment, and costly activity if the buyer instantly exchanges these defective items with the supplier. These defective products are still economically valuable and can be reworkable. It is more feasible to repair or rework the products at a local repair/service store for saving cost, environment, and time. The repaired products are expected to come back to the buyer when the inventory level is positive. In addition, global purchasing brings superfluous and continuing paybacks, where there are various scenarios when the sellers and buyers are working at a long distance and they are dealing in various businesses by importing and exporting products. Therefore, the supplier also offers a sustainable method of payment to the buyer known as a multi-trade credit period. An inventory model is developed to reduce the on-hand stock, save the environment, and benefit for interim financing. The objective is to optimize the profit of the supply chain by incorporating product reparation policy with the integration of multi-trade-credit policy and shortages simultaneously. A non-derivative approach is utilized to optimize the supply chain mathematical model by deciding the ordering lot size, cycle time, and proportion of backorder demand. The model also used numerical data of the firm to support decision-makers for transforming the proposed supply chain model into real practices. The proposed model has been checked for the sensitivity of various significant supply chain management parameters.
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Appendices
Appendix A
The formulation for F* and T* for Case2:
The profit function for Case2 in Eq. (13) for Case 1 can be formulated as:
The profit equation is simplified by adding + 1 and −1 to the Q. Therefore, TP can be given as.
By rearranging and soling the parameters, it is obtained as.
After putting the value of \(C_{z} = (P + l - C_{u} )\), the updated TP can be written as.
As \(D(P - C_{u} ) - C_{z} D(1 - \gamma ) - C_{u} I_{c1} DM\) terms are constant and can be arranged in the form of algebraic approach. Therefore, the Y(T, F) is:
The simplified and composed form of Y(T, F) can be given as.
(See “Appendix D” for all values).
We can rearrange Eq. (A6) as
where \(\lambda (F) = J_{2} - J_{4} F + J_{5} F^{2} = J_{2} - 2J_{2} F + J_{5} F^{2}\) and \(\alpha F = J_{3} F\).
The optimal value of cycle time T can be obtained as
After substituting T*, the total profit function can be given as.
It is observed that the T* is depending on F, where the optimal value of F is formulated as.
Substituting the values of J4, J3, and J5, it is obtained as.
From Eq. (A9)
Putting optimum value of F in Eq. (A12).
Substituting the values of J1, J2, J3, J4, and J5, it is obtained.
Appendix B
Optimal values of F* and T* for Case 3.
The profit function for Case 3 in Eq. (14) for Case 1 can be formulated as:
By putting \(c_{z} = (P + l - c_{u} )\) than the total profit function becomes
The profit function can be further simplified as
As \((P - C_{u} )D - C_{z} D(1 - \gamma ) - C_{u} I_{c2} DN + C_{u} I_{c1} DN - C_{u} I_{c1} DM\) terms are constant. Therefore, the Y (T, F) is:
The compact form of Y (T, F) can be expressed as:
(See “Appendix E” for all values).
We can re-write Eq. (B6) as
where \(\lambda (F) = J_{2} - J_{4} F + J_{5} F^{2} = J_{2} - 2J_{2} F + J_{5} F^{2}\) and \(\alpha F = J_{3} F\).
The total cost equation reaches at least value with respect to T when
The minimum value for the total cost by substituting T* in the cost equation is
After substituting J4, J3, and J5 in Eq. (B10)
From Eq. (B8)
By putting the value of F, then it is obtained.
Substituting J1, J2, J3, J4, and J5, then it is obtained.
Appendix C
Appendix D
Appendix E
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Ahmed, W., Jalees, M., Omair, M. et al. An inventory management for global supply chain through reworking of defective items having positive inventory level under multi-trade-credit-period. Ann Oper Res 315, 1–28 (2022). https://doi.org/10.1007/s10479-022-04646-y
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DOI: https://doi.org/10.1007/s10479-022-04646-y