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The paper aims to establish the related backward stochastic dynamic equations on time scales, BS ∇ Es for short, concerning to ∇-integral on time scales. We present the martingale decomposition theorem on time scales and prove the existence and uniqueness theorem of solutions to BS ∇ Es. This work can be considered as a unification and a generalization of similar results in backward stochastic difference equations and backward stochastic differential equations.

The interaction of a hybrid transshipment policy and customer switching behaviour will exacerbate the complexity of the structure of a hybrid transshipment policy. To cope with this problem, a discrete-time dynamic programming model framework with customer switching behaviour is developed. Based on this framework, we demonstrate that the retailer can obtain more profits with a hybrid transshipment than without one. Next, the existence of a reactive and preventive transshipment policy is shown, respectively. We further analyse the structural property of the holdback policy of reactive transshipment and give the threshold of customer switching rate when always rejecting the request. Meanwhile, a dominant preventive transshipment policy is formulated by which the retailer can control the inventory regardless of the influence of the preventive transshipment policy of the other as long as the inventory is observed by developing an easy-to-implement optimal hybrid transshipment strategy. In addition, the existence of an ordering Nash equilibrium of two retailers is proven. Then, we also illustrate the existence of a transshipment area and analyse the impact of the transshipment cost and switching rate on ordering, the hybrid transshipment policy, and profit by using numerical examples. Finally, we find that the retailer is more willing to adjust inventory by ordering when there is a lower transshipment price and adjust inventory by hybrid transshipment when there is a higher transshipment price.