Erratum to: Research on closed loop supply chain with reference price effect

1. The inaccurate Figs. 1–4 should be replaced with a table (Table 1). The reason is that the optimal performance for the M-, R-, and 3P-managed models is so close to each other numerically, and the subtle difference was not illustrated in the figures. However, Table 1 in this erratum shows the actual difference and is more helpful in understanding Corollaries 1–3. 2. Secondly, the more accurate Figs. 9, 10, 11, 12, 13, 14, 16 and 17 are displayed with this erratum (r = 30, cm = 20, cr = 10, A = 5, B= 1,000, φ = 100, β = 0.5).