An integrated vendor–buyer model with stochastic demand, lot-size dependent lead-time and learning in production

Subject Areas : Mathematical Optimization

Anindita Mukherjee ¹ (Department of Mathematics, Jadavpur University, Kolkata, India)
Oshmita Dey ² (Department of Mathematics, Techno India University, Kolkata, India)
B. C. Giri ³ (Department of Mathematics, Jadavpur University, Kolkata, India)

Received: 2020-10-09 Accepted : 2020-10-09 Published : 2020-09-01

Keywords: Lead time, investment, Inventory, Process quality, Learning, Vendor, Buyer model,

Abstract :

In this article, an imperfect vendor–buyer inventory system with stochastic demand, process quality control and learning in production is investigated. It is assumed that there are learning in production and investment for process quality improvement at the vendor’s end, and lot-size dependent lead-time at the buyer’s end. The lead-time for the first batch and those for the rest of the batches are different. Undern-shipment policy, the annual expected total cost of the system is derived. An algorithm is suggested to derive the optimal values of the number of shipments, the lot-size, the percentage of defective produced per batch and the safety stock factor so as to minimize the annual expected total cost of the system. The solution procedure is illustrated through numerical examples. The benefit of investment for reducing the defect rate is shown numerically. It is also observed that learning in production has significant effect on the annual expected total cost of the integrated system.

References:

Banerjee A (1986) A joint economic-lot-size model for purchaser and vendor. Decis Sci 17:292–311
Ben-Daya M, Hariga M (2000) Economic lot scheduling problem with imperfect production process. J Oper Res Soc 51:875–881
Ben-Daya M, Hariga M (2004) Integrated single vendor single buyer model with stochastic demand and variable lead-time. Int J Prod Econ 92:75–80
Dey O, Giri BC (2014) Optimal vendor investment for reducing defect rate in a vendor–buyer integrated system with imperfect production process. Int J Prod Econ 155:222–228
Dey O, Giri BC (2018) A new approach to deal with learning in inspection in an integrated vendor-buyer model with imperfect production process. Comput Ind Eng. https://doi.org/10.1016/j.cie.2018.12.028
Elmaghraby SE (1990) Economic manufacturing quantities under conditions of learning and forgetting (EMQ/LaF). Prod Plan Control 1(4):196–208
Ghasemy Yaghin R, Fatemi Ghomi SMT, Torabi SA (2014a) Enhanced joint pricing and lotsizing problem in a two-echelon supply chain with logit demand function. Int J Prod Res 52:4967–4983
Ghasemy Yaghin R, Fatemi Ghomi SMT, Torabi SA (2014b) Pricing and lot-sizing decisions in retail industry: a fuzzy chance
constraint approach. In: Chakraverty S (ed) Mathematics of uncertainty modeling in the analysis of engineering and science problems. IGI Global, pp 268–289. https://doi.org/10.4018/978-1-4666-4991-0.ch013.
Giri BC, Bardhan S (2015) A vendor–buyer JELS model with stockdependent demand and consigned inventory under buyer’s space constraint. Oper Res 15(1):79–93
Giri BC, Glock CH (2017) A closed-loop suply chain with stochastic product returns and worker experience under learning and forgetting. Int J Prod Res 55:6760–6778
Glock CH (2009) A comment: “Integrated single vendor-single buyer model with stochastic demand and variable lead time”. Int J Prod Econ 122:790–792
Glock CH (2012) Lead time reduction strategies in a single-vendor–single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. Int J Prod Econ 136:37–44
Glock CH, Jaber MY (2013) A multi-stage production-inventory model with learning and forgetting efects, rework and scrap. Comput Ind Eng 64(2):708–720
Goyal SK (1976) An integrated inventory model for a single supplier single customer problem. Int J Prod Res 15:107–111
Goyal SK (1988) A joint economic-lot-size model for purchaser and vendor: a comment. Decis Sci 19:236–241
Ha D, Kim SL (1997) Implementation of JIT purchasing: an integrated approach. Prod Plan Control 8:152–157
Hill RM (1997) The single-vendor single-buyer integrated production-inventory model with a generalized policy. Eur J Oper Res 97:493–499
Hill RM (1999) The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory problem. Int J Prod Res 37:2463–2475

Hosseini Z, Ghasemy Yaghin R, Esmaeili M (2013) A multiple objective approach for joint ordering and pricing planning problem with stochastic lead times. J Ind Eng Int 9:29. https://doi.org/10.1186/2251-712X-9-29
Hsiao Y-C (2008a) A note on integrated single vendor single buyer model with stochastic demand and variable lead time. Int J Prod Econ 114:294–297
Hsiao Y-C (2008b) Integrated logistic and inventory model for a twostage supply chain controlled by the reorder and shipping points with sharing information. Int J Prod Econ 115:229–235
Huang CK (2004) An optimal policy for a single-vendor single-buyer integrated production-inventory problem with process unreliability consideration. Int J Prod Econ 91:91–98
Jaber MY, Bonney M (1996) Production breaks and the learning curve: the forgetting phenomenon. Appl Math Model 20(2):162–169
Jaber MY, Bonney M (1999) The economic manufacture/order quantity (EMQ/EOQ) and the learning curve: past, present, and future. Int J Prod Econ 59(1–3):93–102
Jaber MY, Bonney M (2003) Lot sizing with learning and forgetting in set-ups and in product quality. Int J Prod Econ 83(1):95–111
Khan M, Jaber MY, Ahmed AR (2014) An integrated supply chain model with errors in quality inspections and learning in production. Omega 42(1):16–24
Lin HJ (2012) An integrated supply chain inventory model. Yugosl J Oper Res. https://doi.org/10.2298/YJOR110506019L
Li J, Min KJ, Otake T, Voorhis TM (2008) Inventory and investment in setup and quality operations under return on investment maximization. Eur J Oper Res 185:593–605 
Mukherjee A, Dey O (2018) An integrated imperfect productioninventory model with lot-size-dependent lead-time and quality control. In: Kar S, Maulik U, Li X (eds) Operations research and optimization. FOTA 2016. Springer Proceedings in Mathematics and Statistics, vol 225. Springer, Singapore, pp 303–314
Mukherjee A, Dey O, Giri BC (2019) An integrated imperfect production–inventory model with optimal vendor investment and backorder price discount. In: Chandra P, Giri D, Li F, Kar S, Jana D (eds) Information technology and applied mathematics. Advances in intelligent systems and computing, vol 699. Springer, Singapore, pp 187–203
Noori-Daryan M, Taleizadeh AA, Jolai F (2019) Analyzing pricing, promised delivery lead time, supplier-selection, and ordering decisions of a multi-national supply chain under uncertain environment. Int J Prod Econ 209(2019):236–248
Otake T, Min KJ (2001) Inventory and investment in quality improvement under return on investment maximization. Comput Oper Res 28:113–124
Otake T, Min KJ, Chen C (1999) Inventory and investment in setup operations under return on investment maximization. Comput Oper Res 26:883–899
Ouyang LY, Wu KS, Ho CH (2006) Analysis of optimal vendor–buyer integrated inventory policy involving defective items. Int J Adv Manuf Technol 29:1232–1245
Pan JC-H, Yang JS (2002) A study of an integrated inventory with controllable lead time. Int J Prod Res 40:1263–1273
Porteus EL (1986) Optimal lot sizing, process quality improvement and setup cost reduction. Oper Res 36:137–144
Salameh MK, Jaber MY (2000) Economic production quantity model with for items with imperfect quality. Int J Prod Econ 64:59–64 
Shu H, Zhou X (2013) An optimal policy for a single-vendor and a single-buyer integrated system with setup cost reduction and processquality improvement. Int J Syst Sci. https://doi.org/10.1080/00207721.2013.786155
Taleizadeh AA, Noori-daryan M (2016) Pricing, inventory and production policies in a supply chain of pharmacological products with rework process: a game theoretic approach. Oper Res Int J
16:89–115
Taleizadeh AA, Noori-daryan M, Tavakkoli-Moghaddam R (2015) Pricing and ordering decisions in a supply chain with imperfect quality items and inspection under buyback of defective items. Int J Prod Res 53:4553–4582
Wee H, Lo C, Hsu P (2009) A multi-objective joint replenishment inventory model of deteriorated items in a fuzzy environment. Eur J Oper Res 197:620–631
Wright TP (1936) Factors afecting the cost of airplanes. J Aeronaut Sci 3(2):122–128