An Integrated Production-Inventory Model for a Dual-Channel Supply Chain

Document Type: Research Paper


Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad


In this paper a dual-channel supply chain with one retailer and one manufacturer is considered. The manufacturer produces in a lot to supply an integer number of the retailer’s order and ships his product to the retailer in equal-sized batches. The manufacturer has also an online channel to meet the demand directly. It is assumed that two members have integrated production-inventory system to minimize the total cost. A procedure is developed to find the optimal solution based on the convexity of the total cost. A numerical example is provided to clarify the characteristics of the model. The results show that the optimal number of the shipments from the manufacturer to the retailer is very sensitive to the holding cost of the manufacturer. When the manufacturer's holding cost is less than that of the retailer, most of the inventory is held in the manufacturer and is delivered to the retailer in low quantities with more shipments. It was also obtained that if the dual-channel supply chain reshapes to single one, the total cost of the supply chain would be minimized and the maximum value of total cost of the supply chain would be happened when the demand is distributed equally between two channels.


Aldurgam, M.,Adegbola, K. and Glock, C.H. (2017). A single-vendor single-manufacturer integrated inventory model with stochastic demand and variable production rate. International Journal of Production Economics, Vol. 191, pp. 335-350.

Banerjee, A. (1986). A joint economic lot size model for purchaser and vendor. Decision Sciences, Vol. 17(3), pp. 292-311.

Batarfi, R., Jaber, M.Y. and Aljazzar, S.M. (2017). A profit maximization for a reverse logistics dual-channel supply chain with a return policy. Computers & Industrial Engineering, Vol. 106, pp. 58-82.

Ben-Daya, M., As’ad, R. and Selaiman, M. (2013). An integrated production inventory model with raw material replenishment considerations in a three layer supply chain. International Journal of Production Economics, Vol. 143(1), pp. 53-61.

Cárdenas-Barrón, L.E. and Treviño-Garza, G. (2014). An optimal solution to a three echelon supply chain network with multi-product and multi-perid. Applied Mathematical Modelling, Vol. 38 (5-6), pp. 1911-1918.

Cárdenas-Barrón, L.E. and Treviño-Garza, G. Widyadana, G.A. and Wee, H.M. (2014). A constrained multi-products EPQ inventory model with discrete delivery order and lot size. Applied Mathematics and Computation, Vol. 230, pp. 359-370.

Cárdenas-Barrón, L.E. and Sana, S.S. (2015). Multi-item EOQ inventory model in a two-layer supply chain while demand varies with a promotional effort. Applied Mathematical Modelling, Vol. 39 (21), pp. 6725-6737

Chiang, W.-y. K., Chhajed, D. and Hess, J.D. (2003). Direct marketing, indirect profits: A strategic analysis of dual-channel supply chain design. Management science, Vol. 49(1), pp. 1-20.

Chiang, W.-y. K. and Monahan, G.E. (2005). Managing inventories in a two-echelon dual-channel supply chain. European Journal of Operational Research, Vol. 162(2), pp. 325-341.

Dumrongsiri, A., Fan, M., Jain, A. and Moinzadeh, K. (2008). A supply chain model with direct and retail channels. European Journal of Operational Research, Vol. 187(3), pp. 691-718.

Fakhrzad, M.B., Pourfereidouni, H. and Pourfereidouni, M. (2018). Dual-channel Supply Chain Synchronization with Deterministic and Stochastic Demand under Cost-sharing Contract. International Journal of Supply and Operatins Management, Vol. 5(1), pp. 28-41.

Glock, C.H. (2012). The joint economic lot size problem: A review.  International Journal of Production Economics, Vol. 135(2), pp. 671-686.

Goyal, S. (1977). An integrated inventory model for a single supplier-single customer problem. The International Journal of Production Research,Vol. 15(1), pp. 107-111.

Goyal, S. K. (1988). A joint economic lot size model for purchaser and vendor: A comment, Decision Sciences, Vol. 19(1), pp. 236-241.

Hill, R. M. (1997). The single-vendor single-buyer integrated production-inventory model with a generalised policy. European Journal of Operational Research, Vol. 97(3), pp. 493-499.

Hoque, M.A. (2013). A vendor–buyer integrated production–inventory model with normal distribution of lead time. International Journal of Production Economics, Vol. 144(2), pp. 409-417.

Khan, M., Hussain, M. and Cárdenas-Barrón, L.E. (2017). Learning and screening errors in an EPQ inventory model for supply chains with stochastic lead time demands. International Journal of Production
, Vol. 55(16), pp. 4816-4832.

Mashud, A.M. and Hasan, M.R. (2019). An Economic Order Quantity model for Decaying Products with the Frequency of Advertisement, Selling Price and Continuous Time Dependent Demand under Partially Backlogged Shortage. International Journal of Supply and Operations Management, Vol. 6(4), pp. 296-314.

MohsenzadehLedari, A. and ArshadiKhamseh, A. (2018). Optimal decisions in a dual-channel supply chain for the substitute products with the special orders under disruption risk and brand consideration.International Journal of Engineering-Transactions B: Applications, Vol. 31(5), pp. 759-769.

Moriarty, R. T. and Moran, U. (1990). Managing Hybrid Marketing Systems, Harvard Business Review.

Najafi, M., Ghodratnama, A. and Pasandideh, H.R. (2018). Solving a deterministic multi product single machine EPQ model withpartial backordering, scrapped products and rework. International Journal of Supply and Operations Management, Vol. 5(1), pp. 11-27.

Ouyang, L.-Y., Ho, C.-H., Su, C.-H. and Yang. C.-T. (2015). An integrated inventory model with capacity constraint and order-size dependent trade credit. Computers & Industrial Engineering, Vol. 84, pp. 133-143.

Panda, S., Modak, N.M., Sana, S.S. and Basu, M. (2015). Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease. Applied Mathematics andComputation, Vol. 256, pp. 913-929.

Rasouli, N. and Kamalabadi, I.N. (2014). Joint pricing and inventory control for seasonal and substitutable goods mentioning the symmetrical and asymmetrical substitution. International Journal of Engineering-Transactions C: Aspects, Vol. 27(9), pp. 1385-1394.

Sahin, F. and Robinson Jr, E.P. (2005). Information sharing and coordination in make-to-order supply chains. Journal of operations management, Vol. 23(6), pp. 579-598.

Sarkar, B. (2013). A production-inventory model with probabilistic deterioration in two-echelon supply chain management. Applied Mathematical Modelling, Vol. 37(5), pp. 3138-3151.

Seliaman, M.E., Khan, M. and Cardenas-Barron, L.E. (2018). Algebraic modelling of a two level supply chain with defective items. Rairo Operations Research, Vol. 52(2), pp. 415-427.

Takahashi, K., Aoi, T., Hirotani, D. and Morikawa, K. (2011). Inventory control in a two-echelon dual-channel supply chain with setup of production and delivery.  International Journal of Production Economics, Vol. 133(1), pp. 403-415.

Taleizadeh, A.A, Cárdenas-Barrón, L.E. and Sohani, R. (2019). Coordinating the supplier-retailer supply chain under noise effect with bundling and inventry strategies. Journal of Industrial and Management
, Vol. 15(4), pp. 1701-1727.

Yan, R. and Wang, J. (2009). Product choice and channel strategy for multi-channel retailers.Interanational Journal of E-Business Research, Vol. 5(3), pp. 78-99.

Yazdi, A.A and Honarvar, M. (2015). A two stage stochastic programming model of the price decision problem in the dual-channel closed-loop supply chain. International Journal of Engineering-Transactions B: Applications, Vol. 28(5), pp. 738-745.