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An inventory management for global supply chain through reworking of defective items having positive inventory level under multi-trade-credit-period

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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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References

  • Alkahtani, M., et al. (2020). An agricultural products supply chain management to optimize resources and carbon emission considering variable production rate: Case of nonperishable corps. Processes, 8(11), 1505.

    Article  Google Scholar 

  • Alkahtani, M., et al. (2021a). E-Agricultural supply chain management coupled with blockchain effect and cooperative strategies. Sustainability, 13(2), 816.

    Article  Google Scholar 

  • Alkahtani, M., et al. (2021b). A covid-19 supply chain management strategy based on variable production under uncertain environment conditions. International Journal of Environmental Research and Public Health, 18(4), 1662.

    Article  Google Scholar 

  • Bonney, M., & Jaber, M. Y. (2011). Environmentally responsible inventory models: Non-classical models for a non-classical era. International Journal of Production Economics, 133(1), 43–53.

    Article  Google Scholar 

  • Bouchery, Y. (2012). Supply chain optimization with sustainability criteria: A focus on inventory models.

  • Cárdenas-Barrón, L. E. (2008). Optimal manufacturing batch size with rework in a single-stage production system–a simple derivation. Computers & Industrial Engineering, 55(4), 758–765.

    Article  Google Scholar 

  • Cárdenas-Barrón, L. E. (2011). The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra. Applied Mathematical Modelling, 35(5), 2394–2407.

    Article  Google Scholar 

  • Chiu, S. W. (2007). Optimal replenishment policy for imperfect quality EMQ model with rework and backlogging. Applied Stochastic Models in Business and Industry, 23(2), 165–178.

    Article  Google Scholar 

  • Chiu, S. W., Ting, C.-K., & Chiu, Y.-S.P. (2007). Optimal production lot sizing with rework, scrap rate, and service level constraint. Mathematical and Computer Modelling, 46(3–4), 535–549.

    Article  Google Scholar 

  • Chu, P., Chung, K.-J., & Lan, S.-P. (1998). Economic order quantity of deteriorating items under permissible delay in payments. Computers & Operations Research, 25(10), 817–824.

    Article  Google Scholar 

  • Eroglu, A., & Ozdemir, G. (2007). An economic order quantity model with defective items and shortages. International Journal of Production Economics, 106(2), 544–549.

    Article  Google Scholar 

  • Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36(4), 335–338.

    Article  Google Scholar 

  • Hariga, M., As’ad, R., & Shamayleh, A. (2017). Integrated economic and environmental models for a multi stage cold supply chain under carbon tax regulation. Journal of Cleaner Production, 166, 1357–1371.

    Article  Google Scholar 

  • Huang, Y. F. (2003). Optimal retailer’s ordering policies in the EOQ model under trade credit financing. Journal of the Operational Research Society, 54(9), 1011–1015.

    Article  Google Scholar 

  • Huang, Z., & Li, S. X. (2001). Co-op advertising models in manufacturer–retailer supply chains: A game theory approach. European Journal of Operational Research, 135(3), 527–544.

    Article  Google Scholar 

  • Jaber, M. Y., Zanoni, S., & Zavanella, L. E. (2014). Economic order quantity models for imperfect items with buy and repair options. International Journal of Production Economics, 155, 126–131.

    Article  Google Scholar 

  • Jamal, A. M. M., Sarker, B. R., & Wang, S. (1997). An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. Journal of the Operational Research Society, 48(8), 826–833.

    Article  Google Scholar 

  • Jinsong, H., et al. (2010). Fuzzy economic order quantity model with imperfect quality and service level. In 2010 Chinese control and decision conference.

  • Kang, C. W., et al. (2019). A single-stage manufacturing model with imperfect items, inspections, rework, and planned backorders. Mathematics, 7(5), 446.

    Article  Google Scholar 

  • Lashgari, M., Taleizadeh, A. A., & Sana, S. S. (2016). An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity. Journal of Industrial & Management Optimization, 12(3), 1091.

    Article  Google Scholar 

  • Liao, J.-J., Huang, K.-N., & Chung, K.-J. (2012). Lot-sizing decisions for deteriorating items with two warehouses under an order-size-dependent trade credit. International Journal of Production Economics, 137(1), 102–115.

    Article  Google Scholar 

  • Mashud, A. H. M., Uddin, M. S., & Sana, S. S. (2019). A two-level trade-credit approach to an integrated price-sensitive inventory model with shortages. International Journal of Applied and Computational Mathematics, 5(4), 1–28.

    Article  Google Scholar 

  • Mohanty, D. J., Kumar, R. S., & Goswami, A. (2018). Vendor-buyer integrated production-inventory system for imperfect quality item under trade credit finance and variable setup cost. RAIRO-Operations Research, 52(4), 1277–1293.

    Article  Google Scholar 

  • Omair, M., et al. (2019). The quantitative analysis of workers’ stress due to working environment in the production system of the automobile part manufacturing industry. Mathematics, 7(7), 627.

    Article  Google Scholar 

  • Omair, M., Sarkar, B., & Cárdenas-Barrón, L. E. (2017). Minimum quantity lubrication and carbon footprint: A step towards sustainability. Sustainability, 9(5), 714.

    Article  Google Scholar 

  • Pal, B., Sana, S. S., & Chaudhuri, K. (2014). Three stage trade credit policy in a three-layer supply chain—A production-inventory model. International Journal of Systems Science, 45(9), 1844–1868.

    Article  Google Scholar 

  • Pal, B., Sana, S. S., & Chaudhuri, K. (2016). Two-echelon competitive integrated supply chain model with price and credit period dependent demand. International Journal of Systems Science, 47(5), 995–1007.

    Article  Google Scholar 

  • Papachristos, S., & Konstantaras, I. (2006). Economic ordering quantity models for items with imperfect quality. International Journal of Production Economics, 100(1), 148–154.

    Article  Google Scholar 

  • Porteus, E. L. (1986). Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research, 34(1), 137–144.

    Article  Google Scholar 

  • Rosenblatt, M. J., & Lee, H. L. (1986). Economic production cycles with imperfect production processes. IIE Transactions, 18(1), 48–55.

    Article  Google Scholar 

  • Salameh, M., & Jaber, M. (2000). Economic production quantity model for items with imperfect quality. International Journal of Production Economics, 64(1–3), 59–64.

    Article  Google Scholar 

  • Sana, S. S. (2015). An EOQ model for stochastic demand for limited capacity of own warehouse. Annals of Operations Research, 233(1), 383–399.

    Article  Google Scholar 

  • Sana, S. S., & Chaudhuri, K. (2008). A deterministic EOQ model with delays in payments and price-discount offers. European Journal of Operational Research, 184(2), 509–533.

    Article  Google Scholar 

  • Sarkar, B., et al. (2014). An economic production quantity model with random defective rate, rework process and backorders for a single stage production system. Journal of Manufacturing Systems, 33(3), 423–435.

    Article  Google Scholar 

  • Sarkar, B., Omair, M., & Choi, S.-B. (2018). A multi-objective optimization of energy, economic, and carbon emission in a production model under sustainable supply chain management. Applied Sciences, 8(10), 1744.

    Article  Google Scholar 

  • Shu, T., et al. (2017). Manufacturers’/remanufacturers’ inventory control strategies with cap-and-trade regulation. Journal of Cleaner Production, 159, 11–25.

    Article  Google Scholar 

  • Taleizadeh, A. A., et al. (2016a). Imperfect economic production quantity model with upstream trade credit periods linked to raw material order quantity and downstream trade credit periods. Applied Mathematical Modelling, 40(19–20), 8777–8793.

    Article  Google Scholar 

  • Taleizadeh, A. A., Khanbaglo, M. P. S., & Cárdenas-Barrón, L. E. (2016b). An EOQ inventory model with partial backordering and reparation of imperfect products. International Journal of Production Economics, 182, 418–434.

    Article  Google Scholar 

  • Taleizadeh, A. A., Moshtagh, M. S., & Moon, I. (2018). Pricing, product quality, and collection optimization in a decentralized closed-loop supply chain with different channel structures: Game theoretical approach. Journal of Cleaner Production, 189, 406–431.

    Article  Google Scholar 

  • Taleizadeh, A. A., Sarkar, B., & Hasani, M. (2019). Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering. Journal of Industrial & Management Optimization, 2019, 1684–1688.

    Google Scholar 

  • Tang, C. S., & Zhou, S. (2012). Research advances in environmentally and socially sustainable operations. European Journal of Operational Research, 223(3), 585–594.

    Article  Google Scholar 

  • Tayyab, M., Sarkar, B., & Yahya, B. (2019). Imperfect multi-stage lean manufacturing system with rework under fuzzy demand. Mathematics, 7(1), 13.

    Article  Google Scholar 

  • Teng, J.-T., Chang, C.-T., & Chern, M.-S. (2012). Vendor–buyer inventory models with trade credit financing under both non-cooperative and integrated environments. International Journal of Systems Science, 43(11), 2050–2061.

    Article  Google Scholar 

  • Tiwari, S., Daryanto, Y., & Wee, H. M. (2018). Sustainable inventory management with deteriorating and imperfect quality items considering carbon emission. Journal of Cleaner Production, 192, 281–292.

    Article  Google Scholar 

  • Tsao, Y.-C., Chen, T.-H., & Zhang, Q.-H. (2013). Effects of maintenance policy on an imperfect production system under trade credit. International Journal of Production Research, 51(5), 1549–1562.

    Article  Google Scholar 

  • Williamson, O. E. (2008). Outsourcing: Transaction cost economics and supply chain management. Journal of Supply Chain Management, 44(2), 5–16.

    Article  Google Scholar 

  • Yang, S. A., & Birge, J. R. (2013). How inventory is (should be) financed: Trade credit in supply chains with demand uncertainty and costs of financial distress. SSRN 1734682.

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Author information

Authors and Affiliations

  1. Department of Operations and Supply Chain, NUST Business School, National University of Sciences and Technology, H-12, Islamabad, 44000, Pakistan

    Waqas Ahmed & Muhammad Imran

  2. Department of Management Sciences, Bahria University, E-8, Islamabad, Pakistan

    Muhammad Jalees

  3. Department of Industrial Engineering, Jalozai Campus, University of Engineering & Technology, Peshawar, 25000, Pakistan

    Muhammad Omair

  4. Institute of Business Administration, University of the Punjab, Lahore, Pakistan

    Zainab Mukhtar

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  1. Waqas Ahmed
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  2. Muhammad Jalees
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  3. Muhammad Omair
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  4. Zainab Mukhtar
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  5. Muhammad Imran
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Correspondence to Muhammad Omair.

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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:

$$\begin{aligned} TP(F,T) &= P(FD + \gamma (1 - F)D) \\ &- \left[ \begin{array}{ll} & \frac{O}{T} + C_{u} (FD + \gamma (1 - F)D) + C_{s} FD \\ & + \beta FD(1 + m)\left[ {\frac{{s_{r} + 2A}}{\beta FTD} + c_{lm} + 2c_{t} + (h_{s} + h_{sc} )\left( {\frac{\beta FTD}{R} + t_{T} } \right)} \right] \\ & + (h + h_{c} )\left[ {\frac{{(1 - \beta )^{2} F^{2} TD}}{2} + \frac{{\beta T(FD)^{2} }}{x}} \right] \\ & + (h_{r} + h_{rc} )\left[ {\beta F^{2} TD - \beta FD\left( {\frac{FTD}{x} + \frac{\beta FTD}{R} + t_{T} } \right) - \frac{{\beta^{2} F^{2} TD}}{2}} \right] \\ & + \pi \frac{{\gamma (1 - F)^{2} TD}}{2} + l(1 - \gamma )(1 - F)D + (u + g)wFD \\ & + PI_{e} \frac{{(DM)^{2} }}{2TD} - C_{u} I_{c1} \frac{{(TD - DM)^{2} }}{2TD} \\ \end{array} \right]\end{aligned} $$
(A1)

The profit equation is simplified by adding + 1 and −1 to the Q. Therefore, TP can be given as.

$$\begin{aligned} TP(F,T) &= PD(1 - (1 - F)(1 - \gamma )) \\ &- \left[ \begin{array}{ll} & \frac{O}{T} + C_{u} D(1 - (1 - F)(1 - \gamma ) + C_{s} FD \\ & + \beta FD(1 + m)\left[ {\frac{{s_{r} + 2A}}{\beta FTD} + c_{lm} + 2c_{t} + (h_{s} + h_{sc} )\left( {\frac{\beta FTD}{R} + t_{T} } \right)} \right] \\ & + (h + h_{c} )\left[ {\frac{{(1 - \beta )^{2} F^{2} TD}}{2} + \frac{{\beta T(FD)^{2} }}{x}} \right] \\ & + (h_{r} + h_{rc} )\left[ {\beta F^{2} TD - \beta FD\left( {\frac{FTD}{x} + \frac{\beta FTD}{R} + t_{T} } \right) - \frac{{\beta^{2} F^{2} TD}}{2}} \right] \\ & + \pi \frac{{\gamma (1 - F)^{2} TD}}{2} + l(1 - \gamma )(1 - F)D + (u + g)wFD \\ & + PI_{e} \frac{{(DM)^{2} }}{2TD} - C_{u} I_{c1} \frac{{(TD - DM)^{2} }}{2TD} \\ \end{array} \right] \end{aligned}$$

By rearranging and soling the parameters, it is obtained as.

$$ \begin{aligned} TP(F,T)& = D(P - C_{u} )\\ & - \left[ \begin{array}{ll} & \frac{O}{T} + C_{s} FD + \beta FD(1 + m)\left[ {\frac{{s_{r} + 2A}}{\beta FTD} + c_{lm} + 2c_{t} + (h_{s} + h_{sc} )\left( {\frac{\beta FTD}{R} + t_{T} } \right)} \right] \\ & + (h + h_{c} )\left[ {\frac{{(1 - \beta )^{2} F^{2} TD}}{2} + \frac{{\beta T(FD)^{2} }}{x}} \right] \\ & + (h_{r} + h_{rc} )\left[ {\beta F^{2} TD - \beta FD\left( {\frac{FTD}{x} + \frac{\beta FTD}{R} + t_{T} } \right) - \frac{{\beta^{2} F^{2} TD}}{2}} \right] \\ & + \pi \frac{{\gamma (1 - F)^{2} TD}}{2} + C_{z} (1 - \gamma )(1 - F)D + (u + g)wFD + + PI_{e} \frac{{(DM)^{2} }}{2TD} - C_{u} I_{c1} \frac{{(TD - DM)^{2} }}{2TD} \\ \end{array} \right]\end{aligned} $$
(A2)

After putting the value of \(C_{z} = (P + l - C_{u} )\), the updated TP can be written as.

$$ \begin{gathered} TP(F,T) = (P - C_{u} )D - C_{z} D(1 - \gamma ) - C_{u} I_{c1} DM \hfill \\ - \left[ \begin{array}{l} \frac{1}{T}\left( {O + (1 + m)(s_{r} + 2A)} \right) - PI_{e} \frac{{DM^{2} }}{2} + C_{u} I_{c1} \frac{{DM^{2} }}{2} - FT(\pi \gamma D) + T(\frac{\pi \gamma D}{2} + \frac{{C_{r} I_{c1} D}}{2}) \hfill \\ + F\left( {C_{s} + \beta (1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )} \right)D \hfill \\ + F^{2} T\left[ \begin{array}{l} \frac{{\left( {(h_{s} + h_{sc} )\beta^{2} D(1 + m)} \right)}}{R} + \frac{{(h + h)(1 - \beta )^{2} }}{2} + \frac{(h + h)\beta D}{x} + (h_{r} + h_{rc} )\beta \hfill \\ - \frac{{(h_{r} + h_{rc} )\beta D}}{x} - \frac{{(h_{r} + h_{rc} )\beta^{2} D}}{R} - \frac{{(h_{r} + h_{rc} )\beta^{2} }}{2} + \frac{\pi \gamma }{2} \hfill \\ \end{array} \right]D \hfill \\ \end{array} \right] \hfill \\ \end{gathered} $$
(A3)

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:

$$ \begin{aligned} Y(F,T) & = \frac{1}{T}\left( {O + (1 + m)(s_{r} + 2A)} \right) - PI_{e} \frac{{DM^{2} }}{2} + C_{u} I_{c1} \frac{{DM^{2} }}{2} - FT(\pi \gamma D) + T\left( {\frac{\pi \gamma D}{2} + \frac{{C_{r} I_{c1} }}{2}} \right) \\ & + F\left( {C_{s} + (1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )} \right)D \\ & + F^{2} T\left[ \begin{gathered} \frac{{\left( {(h_{s} + h_{sc} )\beta^{2} D(1 + m)} \right)}}{R} + \frac{{(h + h_{c} )(1 - \beta )^{2} }}{2} + \frac{{(h + h_{c} )\beta D}}{x} + (h_{r} + h_{rc} )\beta \hfill \\ - \frac{{(h_{r} + h_{rc} )\beta D}}{x} - \frac{{(h_{r} + h_{rc} )\beta^{2} D}}{R} - \frac{{(h_{r} + h_{rc} )\beta^{2} }}{2} + \frac{\pi \gamma }{2} \hfill \\ \end{gathered} \right]D \\ \end{aligned} $$
(A4)

The simplified and composed form of Y(T, F) can be given as.

$$ Y(F,T) = \frac{1}{T}(J_{1} ) + T(J_{2} - J_{4} F + J_{4} F^{2} ) + J_{3} F $$
(A5)

(See “Appendix D” for all values).

We can rearrange Eq. (A6) as

$$ Y(F,T) = \frac{1}{T}(J_{1} ) + T\lambda (F) + \alpha F $$
(A6)

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

$$ T* = \sqrt {\frac{{J_{1} }}{\lambda (F)}} $$
(A7)

After substituting T*, the total profit function can be given as.

$$ Y(F) = 2\sqrt {J_{1} \lambda (F)} + \alpha F $$
(A8)

It is observed that the T* is depending on F, where the optimal value of F is formulated as.

$$ F* = \frac{{J_{4} T - J_{3} }}{{2J_{5} T}} $$
(A9)

Substituting the values of J4, J3, and J5, it is obtained as.

$$ F* = \frac{{\pi \gamma T - (C_{s} + \beta (1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )(1 - \beta )}}{{2\left[ {\beta^{2} D\frac{{(1 + m)(h_{s} + h_{sc} )}}{R} + (h + h_{c} )\left( {\frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) - (h_{r} + h_{rc} )\left( {\frac{{\beta^{2} D}}{R} + \frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) + \left( {\frac{{(h + h_{c} ) + \pi \gamma }}{2}} \right)} \right]T}} $$
(A10)

From Eq. (A9)

$$ T* = \sqrt {\frac{{J_{1} }}{{J_{2} - J_{4} F + J_{5} F^{2} }}} $$
(A11)

Putting optimum value of F in Eq. (A12).

$$ T^{*} = \sqrt {\frac{{J_{1} }}{{J_{2} \left( {1 - 2\left( {\frac{{J_{4} T - J_{3} }}{{2J_{5} T}}} \right) + J_{5} \left( {\frac{{J_{4} T - J_{3} }}{{2J_{5} T}}} \right)} \right)}}} = \sqrt {\frac{{J_{1} J_{5} - \frac{1}{4}J_{3}^{2} }}{{J_{2} J_{5} - J_{2}^{2} }}} $$
(A12)

Substituting the values of J1, J2, J3, J4, and J5, it is obtained.

$$ T* = \sqrt {\frac{\begin{gathered} \left( {O + (1 + m)(s_{r} + 2A) + \frac{{C_{u} I_{c1} DM^{2} }}{2} - \frac{{PI_{e} DM^{2} }}{2}} \right)\left( \begin{gathered} \frac{{(1 + m)(h_{s} + h_{sc} )\beta^{2} D}}{R} + (h + h_{c} )\left( {\frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) \hfill \\ - (h_{r} + h_{rc} )\left( {\frac{{\beta^{2} D}}{R} + \frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) + \left( {\frac{{(h + h_{c} ) + \pi \gamma }}{2}} \right) \hfill \\ \end{gathered} \right) \hfill \\ - \frac{D}{4}\left( {C_{s} + \beta (1 + m)(c_{lm} + 2c_{t} + h_{s} t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )(1 - \beta )} \right)^{2} \hfill \\ \end{gathered} }{{\left( {\frac{\pi \gamma }{2} + \frac{{C_{r} I_{c1} }}{2}} \right)\left( \begin{gathered} \frac{{(1 + m)(h_{s} + h_{sc} )\beta^{2} D^{2} }}{R} + (h + h_{c} )D\left( {\frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) \hfill \\ - (h_{r} + h_{rc} )D\left( {\frac{{\beta^{2} D}}{R} + \frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) + \left( {\frac{{(h + h_{c} )D}}{2}} \right) \hfill \\ \end{gathered} \right)}}} $$
(A13)

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:

$$\begin{aligned} TP(F,T) &= P(FD + \gamma (1 - F)D) \\ &- \left[ \begin{array}{ll} & \frac{O}{T} + C_{u} (FD + \gamma (1 - F)D) + C_{s} FD \\ & + \beta FD(1 + m)\left[ {\frac{{s_{r} + 2A}}{\beta FTD} + c_{lm} + 2c_{t} + (h_{s} + h_{sc} )\left( {\frac{\beta FTD}{R} + t_{T} } \right)} \right] \\ & + (h + h_{c} )\left[ {\frac{{(1 - \beta )^{2} F^{2} TD}}{2} + \frac{{\beta T(FD)^{2} }}{x}} \right] \\ & + (h_{r} + h_{rc} )\left[ {\beta F^{2} TD - \beta FD\left( {\frac{FTD}{x} + \frac{\beta FTD}{R} + t_{T} } \right) - \frac{{\beta^{2} F^{2} TD}}{2}} \right] \\ & + \pi \frac{{\gamma (1 - F)^{2} TD}}{2} + l(1 - \gamma )(1 - F)D + (u + g)wFD + PI_{e} \frac{{(DM)^{2} }}{2TD} \\ & - C_{u} I_{c2} \frac{{(TD - DN)^{2} }}{2TD} - C_{u} I_{c1} \frac{D}{T}(NT - N^{2} - MT + MN) - C_{u} I_{c1} \frac{{D(M - N)^{2} }}{2T} \\ \end{array} \right]\end{aligned} $$
(B1)
$$ \begin{aligned} TP(F,T) &= PD(1 - (1 - F)(1 - \gamma ))\\ & - \left[ \begin{array}{ll} & \frac{O}{T} + C_{u} D(1 - (1 - F)(1 - \gamma ) + C_{s} FD \\ & + \beta FD(1 + m)\left[ {\frac{{s_{r} + 2A}}{\beta FTD} + c_{lm} + 2c_{t} + (h_{s} + h_{sc} )\left( {\frac{\beta FTD}{R} + t_{T} } \right)} \right] \\ & + (h + h_{c} )\left[ {\frac{{(1 - \beta )^{2} F^{2} TD}}{2} + \frac{{\beta T(FD)^{2} }}{x}} \right] \\ & + (h_{r} + h_{rc} )\left[ {\beta F^{2} TD - \beta FD\left( {\frac{FTD}{x} + \frac{\beta FTD}{R} + t_{T} } \right) - \frac{{\beta^{2} F^{2} TD}}{2}} \right] \\ & + \pi \frac{{\gamma (1 - F)^{2} TD}}{2} + l(1 - \gamma )(1 - F)D + (u + g)wFD + PI_{e} \frac{{(DM)^{2} }}{2TD} \\ & - C_{u} I_{c2} \frac{{(TD - DN)^{2} }}{2TD} - C_{u} I_{c1} \frac{D}{T}(NT - N^{2} - MT + MN) \\ & - C_{u} I_{c1} \frac{{D(N - M)^{2} }}{2T} \\ \end{array} \right] \end{aligned}$$
(B2)

By putting \(c_{z} = (P + l - c_{u} )\) than the total profit function becomes

$$ \begin{aligned} TP(F,T) &= D(P - C_{u} )\\ & - \left[ \begin{array}{ll} & \frac{O}{T} + C_{s} FD + \beta FD(1 + m)\left[ {\frac{{s_{r} + 2A}}{\beta FTD} + c_{lm} + 2c_{t} + (h_{s} + h_{sc} )\left( {\frac{\beta FTD}{R} + t_{T} } \right)} \right] \\ & + (h + h_{c} )\left[ {\frac{{(1 - \beta )^{2} F^{2} TD}}{2} + \frac{{\beta T(FD)^{2} }}{x}} \right] \\ & + (h_{r} + h_{rc} )\left[ {\beta F^{2} TD - \beta FD\left( {\frac{FTD}{x} + \frac{\beta FTD}{R} + t_{T} } \right) - \frac{{\beta^{2} F^{2} TD}}{2}} \right] \\ & + \pi \frac{{\gamma (1 - F)^{2} TD}}{2} + C_{z} (1 - \gamma )(1 - F)D + (u + g)wFD + PI_{e} \frac{{(DM)^{2} }}{2TD} \\ & - C_{u} I_{c2} \frac{{(TD - DN)^{2} }}{2TD} - C_{u} I_{c1} \frac{D}{T}(NT - N^{2} - MT + MN) - C_{u} I_{c1} \frac{{D(N - M)^{2} }}{2D} \\ \end{array} \right] \end{aligned}$$
(B3)

The profit function can be further simplified as

$$ \begin{gathered} TP(F,T) = (P - C_{u} )D - C_{z} D(1 - \gamma ) - C_{u} I_{c2} DN + C_{u} I_{c1} DN - C_{u} I_{c1} DM \hfill \\ - \left[ \begin{gathered} \frac{1}{T}\left( {O + (1 + m)(s_{r} + 2A)} \right) - \frac{{C_{u} I_{c2} DN^{2} }}{2} - \frac{{PI_{e} DM^{2} }}{2} - C_{u} I_{c1} DN^{2} + C_{u} I_{c1} DMN + \frac{{C_{u} I_{c1} D(N - M)^{2} }}{2} \hfill \\ - FT(\pi \gamma D) + T(\frac{\pi \gamma D}{2} + \frac{{C_{r} I_{c2} D}}{2}) + F\left( {C_{s} + \beta (1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )} \right)D \hfill \\ + F^{2} T\left[ \begin{gathered} \frac{{\left( {(h_{s} + h_{sc} )\beta^{2} D(1 + m)} \right)}}{R} + \frac{{(h + h)(1 - \beta )^{2} }}{2} + \frac{(h + h)\beta D}{x} + (h_{r} + h_{rc} )\beta \hfill \\ - \frac{{(h_{r} + h_{rc} )\beta D}}{x} - \frac{{(h_{r} + h_{rc} )\beta^{2} D}}{R} - \frac{{(h_{r} + h_{rc} )\beta^{2} }}{2} + \frac{\pi \gamma }{2} \hfill \\ \end{gathered} \right]D \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$
(B4)

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:

$$ \begin{aligned} Y(F,T) & = \frac{1}{T}\left( {O + (1 + m)(s_{r} + 2A)} \right) - \frac{{C_{u} I_{c2} DN^{2} }}{2} - \frac{{PI_{e} DM^{2} }}{2} - C_{u} I_{c1} DN^{2} + C_{u} I_{c1} DMN + \frac{{C_{u} I_{c1} D(N - M)^{2} }}{2} \\ & - FT(\pi \gamma D) + T(\frac{\pi \gamma D}{2} + \frac{{C_{r} I_{c2} D}}{2}) \\ & + F\left( {C_{s} + (1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )} \right)D \\ & + F^{2} T\left[ \begin{gathered} \frac{{\left( {(h_{s} + h_{sc} )\beta^{2} D(1 + m)} \right)}}{R} + \frac{{(h + h_{c} )(1 - \beta )^{2} }}{2} + \frac{{(h + h_{c} )\beta D}}{x} + (h_{r} + h_{rc} )\beta \hfill \\ - \frac{{(h_{r} + h_{rc} )\beta D}}{x} - \frac{{(h_{r} + h_{rc} )\beta^{2} D}}{R} - \frac{{(h_{r} + h_{rc} )\beta^{2} }}{2} + \frac{\pi \gamma }{2} \hfill \\ \end{gathered} \right]D \\ \end{aligned} $$
(B5)

The compact form of Y (T, F) can be expressed as:

$$ Y(F,T) = \frac{1}{T}(J_{1} ) + T(J_{2} - J_{4} F + J_{4} F^{2} ) + J_{3} F $$
(B6)

(See “Appendix E” for all values).

We can re-write Eq. (B6) as

$$ Y(F,T) = \frac{1}{T}(J_{1} ) + T\lambda (F) + \alpha F $$
(B7)

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

$$ T* = \sqrt {\frac{{J_{1} }}{\lambda (F)}} $$
(B8)

The minimum value for the total cost by substituting T* in the cost equation is

$$ Y(F) = 2\sqrt {J_{1} \lambda (F)} + \alpha F $$
(B9)
$$ F* = \frac{{J_{4} T - J_{3} }}{{2J_{5} T}} $$
(B10)

After substituting J4, J3, and J5 in Eq. (B10)

$$ F* = \frac{{\pi \gamma T - (C_{s} + \beta (1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )(1 - \beta )}}{{2\left[ {\beta^{2} D\frac{{(1 + m)(h_{s} + h_{sc} )}}{R} + (h + h_{c} )\left( {\frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) - (h_{r} + h_{rc} )\left( {\frac{{\beta^{2} D}}{R} + \frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) + \left( {\frac{{(h + h_{c} ) + \pi \gamma }}{2}} \right)} \right]T}} $$
(B11)

From Eq. (B8)

$$ T* = \sqrt {\frac{{J_{1} }}{{J_{2} - J_{4} F + J_{5} F^{2} }}} $$
(B12)

By putting the value of F, then it is obtained.

$$ T^{*} = \sqrt {\frac{{J_{1} }}{{J_{2} - J_{4} \left( {\frac{{J_{4} T - J_{3} }}{{2J_{5} T}}} \right) + J_{5} \left( {\frac{{J_{4} T - J_{3} }}{{2J_{5} T}}} \right)^{2} }}} $$
(B13)

Substituting J1, J2, J3, J4, and J5, then it is obtained.

$$ T* = \sqrt {\frac{\begin{gathered} \left( {O + (1 + m)(s_{r} + 2A) + \frac{{C_{u} I_{c1} DM^{2} }}{2} - \frac{{PI_{e} DM^{2} }}{2}} \right)\left( \begin{gathered} \frac{{(1 + m)(h_{s} + h_{sc} )\beta^{2} D}}{R} + (h + h_{c} )\left( {\frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) \hfill \\ - (h_{r} + h_{rc} )\left( {\frac{{\beta^{2} D}}{R} + \frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) + \left( {\frac{{(h + h_{c} ) + \pi \gamma }}{2}} \right) \hfill \\ \end{gathered} \right) \hfill \\ - \frac{D}{4}\left( {C_{s} + \beta (1 + m)(c_{lm} + 2c_{t} + h_{s} t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta - C_{z} (1 - \gamma )(1 - \beta )} \right)^{2} \hfill \\ \end{gathered} }{{\left( {\frac{\pi \gamma }{2} + \frac{{C_{r} I_{c1} }}{2}} \right)\left( \begin{gathered} \frac{{(1 + m)(h_{s} + h_{sc} )\beta^{2} D^{2} }}{R} + (h + h_{c} )D\left( {\frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) \hfill \\ - (h_{r} + h_{rc} )D\left( {\frac{{\beta^{2} D}}{R} + \frac{{\beta^{2} }}{2} + \frac{\beta D}{x} - \beta } \right) + \left( {\frac{{(h + h_{c} )D}}{2}} \right) \hfill \\ \end{gathered} \right)}}} $$
(B14)

Appendix C

$$ \begin{aligned} J_{1} & = O + (1 + m)(s_{r} + 2A) \\ J_{2} & = \frac{\pi \gamma D}{2} + \frac{{PI_{e} D}}{2} \\ J_{3} & = C_{s} D + \beta D(1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta D - C_{z} D(1 - \gamma )(1 - \beta ) \\ J_{4} & = 2J_{2} = \pi \gamma D \\ J_{5} & = \frac{{\left( {(h_{s} + h_{sc} )\beta^{2} D^{2} (1 + m)} \right)}}{R} + \frac{{(h + h)D(1 - \beta )^{2} }}{2} + \frac{{(h + h)\beta D^{2} }}{x} + (h_{r} + h_{rc} )\beta D \\ & - \frac{{(h_{r} + h_{rc} )\beta D^{2} }}{x} - \frac{{(h_{r} + h_{rc} )\beta^{2} D^{2} }}{R} - \frac{{(h_{r} + h_{rc} )D\beta^{2} }}{2} + \frac{\pi \gamma D}{2} \\ \end{aligned} $$

Appendix D

$$ \begin{aligned} J_{1} & = O + (1 + m)(s_{r} + 2A) + \frac{{C_{u} I_{c1} DM^{2} }}{2} - \frac{{PI_{e} DM^{2} }}{2} \\ J_{2} & = \frac{\pi \gamma D}{2} + \frac{{C_{r} I_{c1} D}}{2} \\ J_{3} & = C_{s} D + \beta D(1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta D - C_{z} D(1 - \gamma )(1 - \beta ) \\ J_{4} & = 2J_{2} = \pi \gamma D \\ J_{5} & = \frac{{\left( {(h_{s} + h_{sc} )\beta^{2} D^{2} (1 + m)} \right)}}{R} + \frac{{(h + h)D(1 - \beta )^{2} }}{2} + \frac{{(h + h)\beta D^{2} }}{x} + (h_{r} + h_{rc} )\beta D \\ & - \frac{{(h_{r} + h_{rc} )\beta D^{2} }}{x} - \frac{{(h_{r} + h_{rc} )\beta^{2} D^{2} }}{R} - \frac{{(h_{r} + h_{rc} )D\beta^{2} }}{2} + \frac{\pi \gamma D}{2} \\ \end{aligned} $$

Appendix E

$$ \begin{aligned} J_{1} & = O + (1 + m)(s_{r} + 2A) + \frac{{C_{u} I_{c2} DN^{2} }}{2} - \frac{{PI_{e} DM^{2} }}{2} \\ & \quad - C_{u} I_{c1} DN^{2} + C_{u} I_{c1} DMN + \frac{{C_{u} I_{c1} D(N - M)^{2} }}{2} \\ J_{2} & = \frac{\pi \gamma D}{2} + \frac{{C_{r} I_{c2} D}}{2} \\ J_{3} & = C_{s} D + \beta D(1 + m)(c_{lm} + 2c_{t} + (h_{s} + h_{sc} )t_{T} ) - (h_{r} + h_{rc} )t_{T} \beta D - C_{z} D(1 - \gamma )(1 - \beta ) \\ J_{4} & = 2J_{2} = \pi \gamma D \\ J_{5} & = \frac{{\left( {(h_{s} + h_{sc} )\beta^{2} D^{2} (1 + m)} \right)}}{R} + \frac{{(h + h)D(1 - \beta )^{2} }}{2} + \frac{{(h + h)\beta D^{2} }}{x} + (h_{r} + h_{rc} )\beta D \\ & - \frac{{(h_{r} + h_{rc} )\beta D^{2} }}{x} - \frac{{(h_{r} + h_{rc} )\beta^{2} D^{2} }}{R} - \frac{{(h_{r} + h_{rc} )D\beta^{2} }}{2} + \frac{\pi \gamma D}{2} \\ \end{aligned} $$

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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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