Zhifang Feng^a,
Zhiping Wang^b,
Yan Chen^a



Highlights

•The consumers’ demand is set to be a function of time.
•We formulate the closed-loop supply chain model with cyclic demand and solve it.
•We study the stability of the model and analyze influence of the price sensitivity.
•Some references of management on economic activity are provided.
•The investment Iki on reverse distribution channels is introduced.



Keywords
Supernetwork; Remanufacture; Time-dependent; Closed-loop supply chain



Abstract
In this paper, we develop a closed-loop supply chain supernetwork model including suppliers, manufacturers, retailers and consumers at demand market, in which the demand for product is seasonal (t), and the sensitivity (w) of demand to price is another key factor which effects consumers’ demand. Moreover, the manufacturers invest the reverse distribution channel for incenting consumers to return more used products. Based on the Evolutionary Variational Inequalities (EVI) theory and Projected Dynamical Systems (PDS), the equilibrium condition of closed-loop supply chain is formulated, and the model is verified reasonably by numerical example.



Article Outline
1. Introduction
2. The closed-loop supply chain supernetwork model with time-dependent parameters
3. The optimization of decision-makers in model
3.1. The optimization of suppliers
3.2. The optimization of manufacturers
3.3. The optimization of retailers
3.4. The optimization of consumers at demand market

4. The equilibrium condition of closed-loop supply chain with time-dependent parameters
5. Numerical example
6. Conclusion
Acknowledgements
References


Figures
   

Fig. 1.

The structure of the closed-loop supply chain supernetwork in continuous time.


Fig. 2.

Cyclical demand with different sensitivities (w).


Fig. 3.

Effects of sensitivity (w) on demand price (ρk).


Fig. 4.

Shipments vary over time.


Fig. 5.

Relationship between the manufactured products and recycled product.


Fig. 6.

Investments from manufacturer 1 under different w.