Competition and Integration in Closed-Loop Supply Chain Network with Variational Inequality

Table 1

Notation for the closed-loop supply chain network prior to horizontal merger.

Notation	Definition
𝐿𝑖^1𝑁	The set of all forward links for the Firm i, where i = 1,…,I
𝐿𝑖^1𝑅	The set of all reverse links for the Firm i, where i = 1,…,I
p	A path of the network
𝑃^1𝑁𝑖𝑗	The set of all forward paths connecting the first layer node i with demand market j, where i = 1,…,I, j = 1,…,J
𝑃^1𝑅𝑖𝑗	The set of all reverse paths connecting the node i with demand market j, where i = 1,…,I, j = 1,…,J
𝑃𝑗^1𝑁	The set of all forward paths between all I nodes with demand market j, where j = 1,…,J
𝑃𝑗^1𝑅	The set of all reverse paths between all I nodes with demand market j, where j = 1,…,J
𝑃𝑖^1𝑁	The set of all forward paths between the node i with all demand markets, where i = 1,…,I
𝑃𝑖^1𝑅	The set of all reverse paths between the node i with all demand markets, where i = 1,…,I
𝑃^1𝑁	The set of all forward paths
𝑀^1𝑁	The number of all forward paths
𝑃^1𝑅	The set of all reverse paths
𝑀^1𝑅	The number of all reverse paths
𝑥^1𝑝	The nonnegative remanufactured product flow on the path p (𝑝∈𝑃^1𝑁)
𝑥^1	𝑀^1𝑁-dimensional vector of all remanufactured product flows on the paths in 𝑝∈𝑃^1𝑁
𝑦^1𝑝	The nonnegative manufactured product flow on the path p (𝑝∈𝑃^1𝑁)
𝑦^1	𝑀^1𝑁-dimensional vector of all manufactured product flows on the paths in 𝑝∈𝑃^1𝑁
𝑧^1𝑝	The nonnegative recovered EOL product flow on the path p (𝑝∈𝑃^1𝑅)
𝑧^1	𝑀^1𝑅-dimensional vector of all EOL product flows on the paths in 𝑝∈𝑃^1𝑅
𝑋^1	The set of all product flows, 𝑋^1≡𝑥^1,𝑦^1,𝑧^1