Algebraic modelling of a two level supply chain with defective items

Seliaman, Mohamed E.; Khan, Mehmood; Cárdenas-Barrón, Leopoldo Eduardo

RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 2, pp. 415-427.

M. Khan and M.Y. Jaber, Optimal inventory cycle in a two-stage supply chain incorporating imperfect items from suppliers. $\mathrm{𝐼𝑛𝑡}.𝐽.\mathrm{𝑂𝑝𝑒𝑟}.\mathrm{𝑅𝑒𝑠}.$ $\mathbf{10}$ (2011) 442–457, have addressed a two level supply chain of defective items. They compared three coordination mechanisms, $𝑖.𝑒.$ cycle time; $K$–multiplier cycle time; and $2^K$–multiplier cycle time. This paper proposes a simpler algebraic solution for the $K$–multiplier cycle time mechanism without the use of differential calculus. The two level supply chain with defective items is illustrated with a numerical example. A sensitivity analysis is also provided.

[1] M. Ben–Daya, R. As’Ad and M. Seliaman, An integrated production inventory model with raw material replenishment considerations in a three layer supply chain. Int. J. Prod. Econ. 143 (2013) 53–61 | DOI

[2] L.E. Cárdenas–Barrón, The economic production quantity (EPQ) with shortage derived algebraically. Int. J. Prod. Econ. 70 (2001) 289–292 | DOI

[3] L.E. Cárdenas–Barrón, Optimizing inventory decisions in a multi–stage multi–customer supply chain: A note. Trans. Res. Part E: Logistics Trans. Rev. 43 (2007) 647–654 | DOI

[4] L.E. Cárdenas–Barrón, Optimal manufacturing batch size with rework in a single-stage production system–a simple derivation. Comput. Ind. Eng. 55 (2008) 758–765 | DOI

[5] L.E. Cárdenas–Barrón, A simple method to compute economic order quantities: Some observations. Appl. Math. Model. 34 (2010) 1684–1688 | DOI | MR | Zbl

[6] L.E. Cárdenas–Barrón, The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra. Appl. Math. Model. 35 (2011) 2394–2407 | DOI | MR | Zbl

[7] L.E. Cárdenas–Barrón, A.A. Taleizadeh and G. Trevi No-Garza An improved solution to replenishment lot size problem with discontinuous issuing policy and rework and the multi-delivery policy into economic production lot size problem with partial rework. Exp. Syst. Appl. 39 (2012) 13540–13546 | DOI

[8] L.E. Cárdenas–Barrón, J. Teng, G. Treviño-Garza, H. Wee and K. Lou. An improved algorithm and solution on an integrated production-inventory model in a three-layer supply chain. Int. J. Prod. Econ. 136 (2012) 384–388 | DOI

[9] L.E. Cárdenas–Barrón, B. Sarkar and G. Trevi No-Garza Easy and improved algorithms to joint determination of the replenishment lot size and number of shipments for an EPQ model with rework. Math. Comput. Appl. 18 (2013) 132–138 | Zbl

[10] L.E. Cárdenas–Barrón, B. Sarkar and G. Trevino-Garza An improved solution to the replenishment policy for the EMQ model with rework and multiple shipments. Appl. Math. Model. 37 (2013) 5549–5554 | DOI | MR | Zbl

[11] L.E. Cárdenas–Barrón, K. Chung and G. Trevino-Garza Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris. Int. J. Prod. Econ. 155 (2014) 1–7 | DOI

[12] S.W. Chiu, Production lot size problem with failure in repair and backlogging derived without derivatives. Eur. J. Oper. Res. 188 (2008) 610–615 | DOI | MR | Zbl

[13] C.J. Chung and H.M. Wee, Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives. Eur. J. Oper. Res. 183 (2007) 933–943 | DOI | Zbl

[14] M.A. Darwish, O.M. Odah and S.K. Goyal, Vendor–managed inventory models for items with imperfect quality. Int. J. Oper. Res. 18 (2013), 401–33 | DOI | MR | Zbl

[15] R.W. Grubbström, Modelling production opportunities–an historical overview. Int. J. Prod. Econ. 41 (1995) 1–14 | DOI

[16] F.W. Harris, How many parts to make at once. Factory, The Magazine Manag. 10 (1913) 135–136

[17] W.A. Jauhari, Integrated inventory model for single–vendor single-buyer with imperfect production process. Int. J. Oper. Res. 20 (2014) 283–300 | DOI | MR | Zbl

[18] M. Khan, M. Jaber, A. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items. Int. J. Prod. Econ. 132 (2011) 1–12 | DOI

[19] M. Khan and M.Y. Jaber, Optimal inventory cycle in a two-stage supply chain incorporating imperfect items from suppliers. Int. J. Oper. Res. 10 (2011) 442–457 | DOI | MR | Zbl

[20] M. Khan, M. Hussain and H.M. Saber, Information sharing in a sustainable supply chain. Int. J. Prod. Econ. 181 (2016) 208–214 | DOI

[21] M. Khan, M.Y. Jaber, S. Zanoni and L. Zavanella, Vendor managed inventory with consignment stock agreement for a supply chain with defective items. Appl. Math. Model. 40 (2016) 7102–7114 | DOI | MR | Zbl

[22] M. Khouja, Optimizing inventory decisions in a multi-stage multi-customer supply chain. Trans. Res. Part E: Logistics and Trans. Rev. 39 (2003) 193–208 | DOI

[23] M. Kim and B. Sarkar, Multi–stage cleaner production process with quality improvement and lead time dependent ordering cost. J. Cleaner Prod. 144 (2016) 572–590 | DOI

[24] S. Papachristos and I. Konstantaras, Economic ordering quantity models for items with imperfect quality. Int. J. Prod. Econ. 100 (2006) 148–154 | DOI

[25] M. Salameh and M. Jaber, Economic production quantity model for items with imperfect quality. Int. J. Prod. Econ. 64 (2000) 59–64 | DOI

[26] B. Sarkar, A production–inventory model with probabilistic deterioration in two–echelon supply chain management. Appl. Math. Modell. 37 (2013) 3138–3151 | DOI | MR | Zbl

[27] B. Sarkar, L.E. Cárdenas–Barrón, M. Sarkar and M.L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system. J. Manuf. Sys. 33 (2014) 423–435 | DOI

[28] B. Sarkar and I. Moon, Improved quality, setup cost reduction and variable backorder costs in an imperfect production process. Int. J. Prod. Econ. 155 (2014), 204–13 | DOI

[29] B. Sarkar, Supply chain coordination with variable backorder, inspections and discount policy for fixed lifetime products. Math. Prob. Eng. 14 (2016) 6318737 | MR | Zbl

[30] B. Sarkar, B. Ganguly, M. Sarkar and S. Pareek, Effect of variable transportation and carbon emission in a three-echelon supply chain model. Trans. Res. Part E: Logistics and Trans. Rev. 91 (2016) 112–128 | DOI

[31] B. Sarkar and S. Saren, Product inspection policy for an imperfect production system with inspection errors and warranty cost. Eur . J. Oper. Res. 248 (2016) 263–271 | DOI | Zbl

[32] M. Seliaman, A generalized algebraic model for optimizing inventory decisions in a multi–stage complex supply chain. Trans. Res. Part E: Logistics and Trans. Rev. 45 (2009) 409–418 | DOI

[33] M.E. Seliaman, Using complete squares method to optimize replenishment policies in a four–stage supply chain with planned backorders. Adv. Decision Sci. 2011 (2011) 745896 | MR | Zbl

[34] C. Sivashankari and S. Panayappan, Production inventory model for two levels production with defective items and incorporating multi–delivery policy. Int. J. Oper. Res. 19 (2014) 259–279 | DOI | MR | Zbl

[35] M. Tayyab and B. Sarkar, Optimal batch quantity in a cleaner multi-stage lean production system with random defective rate. J. Cleaner Prod. 139 (2016) 922–934 | DOI

[36] J. Teng, L.E. Cárdenas–Barrón and K. Lou, The economic lot size of the integrated vendor–buyer inventory system derived without derivatives: a simple derivation. Appl. Math. Comput. 217 (2011) 5972–5977 | MR | Zbl

[37] H.M. Wee and C.J. Chung, A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives. Eur. J. Oper. Res. 177 (2007) 1289–1293 | DOI | Zbl

[38] M. Khan, M. Hussain and L.E. Cárdenas–Barrón, Learning and screening errors in an EPQ inventory model for supply chains with stochastic lead time demands. Int. J. Prod. Res. 55 (2017) 4816–4832 | DOI