Optimization and coordination of decentralized supply chains with vertical cross-shareholding

We consider a decentralized supply chain in which an upstream firm sells a product to a downstream firm, who faces a random demand. The upstream firm and the downstream firm are linked by vertical cross-shareholding, which enables one firm to share the other firm's profit. The two firms are both independent decision makers, and seek to maximize their own profits (i.e., profits after dividends). We investigate two distinct supply chains: push supply chain and pull supply chain. We first derive the optimal wholesale price and production quantity decisions of the two firms, and show that the impacts of vertical cross-shareholding on efficiencies of the push and pull supply chains are very different. We then study supply chain coordination. We find that to achieve a win-win coordination, one firm may need to operate at a loss first, and then get revenue by sharing its partner's profit. We propose schemes to coordinate supply chains. The proposed schemes each contains three parameters. We show that the proposed schemes can coordinate supply chains with vertical cross-shareholding, and result in a win-win situation.