Integrated three layer supply chain inventory model for price sensitive and time dependent demand with suggested retail price by manufacturer

Document Type : Research Paper


1 Department of Mathematics and Statistics, Dr. Harisingh Gour Vishwavidyalaya, (A Central University), Sagar M.P. India

2 bDepartment of Mathematics, Ujjain Engineering Collage, Ujjain M.P. India

3 Department of Management Studies (DOM), Indian Institute of Technology (IIT), Madras, India

4 Department of Mathematics, Ujjain Engineering Collage, Ujjain M.P. India


This paper presents an integrated three layer supply chain policy for multi-channel and multi-echelon consisting manufacturer, distributors and retailers as  supply chain members. The demand of retailers end is considered as linear function of  time and retail price. The average net profit function per unit  time is derived for each  supply chain member which are based on demand of retailer's  end. Since  holding cost of goods/inventory is expensive in developed areas, we have introduced a new concept  to share holding cost among distributors and retailers. We have  optimized lot size, retailing price and replenishment time interval for retailers. We have also optimized initial inventory level and wholesale price for distributors and manufacturer respectively. This  study is performed in two different  categories one is decentralized and other  is centralized scenario. The profit function of each supply  chain members has been derived and shown as a concave function with respect to decision variables. More over propositions and results are made to illustrate the proposed model and we have sensitive  analyzed it with numerical example.


[1] A. Arefmanesha and M. Abbaszadehb, On the natural stabilization of convection diffusion problems using LPIM meshless method, Int. J. Nonlinear Anal. Appl. 8 (2017) 9–22.
[2] A. Bahiraiea, B. Abbasia, F. Omidia, N.A. Hamzahb and A.H. Yaakubb, Continuous-time portfolio optimization, Int. J. Nonlinear Anal. Appl. 6 (2015) 103–112.
[3] G.P. Cachon and M.A. Lariviere, Developed supply chain coordination with revenue sharing contracts model, Manag. Sci. 51 (2005) 30–44.
[4] L.E. Cardenas-Barron, G. Trevino-Garza and H.M. Wee, A simple and better algorithm to solve three vendor managed inventory control system of multi-product multi-constraints economic order quantity model, Expert Syst. Appl. 39 (2012) 3888–3895.
[5] L.E. Cardenas-Barron, J.T. Teng, G. Trevino-Garza, H.M. Wee and K.R. 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.
[6] L.E. Cardenas-Barron and S.S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams’ initiatives, Int. J. Prod. Econ. 155 (2014) 249–258.
[7] L.E. Cardenas-Barron and G. Trevino-Garza, An optimal solution to a three echelon supply chain network with multi-period, Appl. Math. Model. 38 (2014) 1911–1918.
[8] L.E. Cardenas-Barron, K.J. 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.
[9] L.E. Cardenas-Barron and S. S. Sana, Multi-item EOQ inventory model in a two-layer supply chain while demand varies with a promotional effort, Appl. Math. Model. 39 (2015) 6725–6737.
[10] C.T. Chung, L.Y. Ouyang and J.T. Teng, An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Appl. Math. Model. 27 (2003) 983–996.
[11] K.J. Chung and J.J. Liao, Lot-sizing decisions under trade credit depending on the ordering quantity, Comput. Oper. Res. 31 (2004) 909–928.
[12] B. Moh and R.D. As’ad and S. Mohammed, An integrated production inventory model with raw material replenishment consideration in three-layer supply chain, Int. J. Prod. Econ. 143 (2013) 53–61.
[13] D. Ding and J. Chen, Coordinating a three level supply chain with flexible return policies, Omega 36 (2008) 865–876.
[14] A. Gunasekaran and B. Kobu, Performance measures and metrics in logistics and supply chain management: a review of recent literature (1995 -2004) for research and applications, Int. J. Prod. Res. 45 (2007) 2819–2840.
[15] Y.F. Huang, Optimal retailer’s ordering policies in the EOQ model under trade credit financing, J. Oper. Res. Soc. 54 (2003) 1011–1015.
[16] J. Jain, G.S. Dangayach, G. Agrawal and S. Banerjee, Supply chain management: Literature Review and some issues, J. Studies Manuf. 1 (2010) 11–25.
[17] R.S. Kadadevaramath, J.C.H. Chen, B.L. Shankar and K. Rameshkumar, Application of particle swarm intelligence algorithms in supply chain network architecture optimization, Expert Syst. Appl. 39 (2012) 10160–10176.
[18] R. Kamalia and A. Davarib, A necessary condition for multiple objective fractional programming, Int. J. Nonlinear Anal. Appl. 8 (2017) 201–207.
[19] M.R. Karim and K. Suzuki, Analysis of warranty claim data: A literature review, Int. J. of Qual. Reliab. Manag. 22 (2005) 667–686.
[20] U.K. Khedlekar and D. Shukla, Dynamic pricing model with logarithmic demand, OPSEARCH 50 (2013) 1—13.
[21] U.K. Khedlekar, D. Shukla and A. Namdeo, Pricing policy for declining demand using item preservation technology, Springer Plus 5 (2016) 1–11.
[22] U.K. Khedlekar, A. Namdeo and A.R. Nigwal, Production inventory model with disruption considering shortages and time proportional demand, Yugosl. J. Oper. Res. 28 (2018) 123–139.
[23] A.H.L. Lau and H.S. Lau, Effects of demand curve’s-shape on the optimal solution of a multi-echelon inventory/pricing model, Eur. J. Oper. Res. 147 (2002) 530–548.
[24] J. Li and L. Liu, Supply chain coordination with quantity discount policy, Int. J. Prod. Econ. 101 (2006) 89–98.
[25] N.M. Modak, S. Panda, S.S. Sana and B. Manjusri, Corporate social responsibility, coordination and profit distribution in a dual-channel supply chain, Pac. Sci. Rev. 16 (2014) 235–249.
[26] N.M. Modak, S. Panda and S.S. Sana, Pricing policy and coordination for a distribution channel with manufacturer suggested retail price, Int. J. Syst. Sci. Oper. Logist. 3 (2015) 92–101.
[27] N.M. Modak, S. Panda and S.S. Sana, Managing a two-echelon supply chain with price warranty and quality dependent demand, Congent Bus. Manag. 2 (2015) 1–13.
[28] M. Nadjafikhah and S. Shagholi, Mathematical modeling of optimized SIRS epidemic model and some dynamical behaviors of the solution, Int. J. Nonlinear Anal. Appl. 8 (2017) 125–134.
[29] B. Pal, S.S. Sana and K. Chaudhuri, Three stage trade credit policy in a three-layer supply chain-a production inventory model, Int. J. - Syst. Sci. 45 (2014) 1844–1868.
[30] S. Panda, M.N. Modak and L.E. Cardenas-Barron, Coordination and benefit sharing in a three-echelon distribution channel with deteriorating product, Comput. Ind. Eng. 113 (2017) 630–645.
[31] S. Panda, N.M. Modak, and L. E. Cardenas-Barron, Coordinating a socially responsible closed-loop supply chain with product recycling, Int. J. Prod. Econ. 188 (2017) 11–21.
[32] M. Parlar and Q. Wang, Discounting decision in a supply-buyer relationship with a linear buyer’s demand, IIE Trans. 26 (1994) 34–41.
[33] T. Russel Crook and J. G. Combs, Sources and consequences of bargaining power in supply chains, J. Oper. Manag. 25 (2007) 546–555.
[34] M. Safi and S.M. Ghasemi, Uncertainty in linear fractional transportation problem, Int. J. Nonlinear Anal. Appl. 8 (2017) 81–93.
[35] D. Shukla, and U.K. Khedlekar, Inventory model for convertible item with deterioration, Comm. Stat. Theo. Meth. 45 (2016) 1137–1147.
[36] P. Trkman, M.L. Stemberger and M. Jaklic, Information transfer in supply chain management, Issues in Info. Sci. Inf. Tech. 2 (2006) 559-573.
[37] Z.K. Weng, Modeling quantity discounts under general price sensitive demand functions: Optimal policies and relationships, Eur. J. Oper. Res. 86 (2000) 34–41.
[38] N. Zhao and L. Chen, Price decision models of a manufacturer-retailer supply chain based on game theory, AASRI Int. Conf.  Ind. Electron. Appl. IEA, 2015.
Volume 12, Issue 1
May 2021
Pages 1135-1152
  • Receive Date: 20 April 2020
  • Revise Date: 05 August 2020
  • Accept Date: 26 November 2020