Game model for online and offline retailers under buy-online and pick-up-in-store mode with delivery cost and random demand

Online retailers are increasingly adding buy-online and pick-up-in-store (BOPS) modes to order fulfilment. In this paper, we study a system of BOPS by developing a stochastic Nash equilibrium model with incentive compatibility constraints, where the online retailer seeks optimal online sale prices and an optimal delivery schedule in an order cycle, and the offline retailer pursues a maximal rate of sharing the profit owing to the consignment from the online retailer. By an expectation method and optimality conditions, the equilibrium model is first transformed into a system of constrained nonlinear equations. Then, by a case study and sensitivity analysis, the model is validated and the following practical insights are revealed. (I) Our method can reliably provide an equilibrium strategy for the online and offline retailers under BOPS mode, including the optimal online selling price, the optimal delivery schedule, the optimal inventory and the optimal allocation of profits. (II) Different model parameters, such as operational cost, price sensitivity coefficient, cross-sale factor, opportunity loss ratio and loss ratio of unsold goods, generate distinct impacts on the equilibrium solution and the profits of the BOPS system. (III) Optimization of the delivery schedule can generate greater consumer surplus, and makes the offline retailer share less sale profit from the online retailer, even if the total profit of the BOPS system becomes higher. (IV) Inventory subsidy is an indispensable factor to improve the applicability of the game model in BOPS mode. doi:10.1017/S144618112000012

Partially Smoothing and Gradient-Based Algorithm for Optimizing the VMI System with Competitive Retailers under Random Demands

Vendor managed inventory (VMI) is an improved sustainable inventory management system, but it is difficult to establish and solve an integrated Stackelberg game model under the complicated practical environment. In this paper, a bilevel programming model is proposed to formulate the VMI system by taking into account the uncertainty of demand, the competition among retailers, the cooperative advertising, the shortage and holding costs, and the practical constraints. For the established stochastic model being associated with continuously random demands, a deterministic mathematical program with complementarity constraints (MPCC) is first derived by expectation method and the first-order optimality conditions of the lower-level problem. Then, with a partially smoothing technique, the MPCC is solved by transforming it into a series of standard smooth optimization subproblems. Finally, owing to complexity caused by evaluating the integrals with unknown decision variables in the objective function, an efficient algorithm is developed to solve the problem based on the gradient information of model. Sensitivity analysis has been employed to reveal a number of managerial implications from the constructed model and algorithm. (1) The participation rate depends on advertising expenditures from both the manufacturer and the retailer. There exists an optimal threshold of participation rate for the manufacturer, which can be provided by the intersection point of the manufacturer and retailer’s cost-profit curves. (2) The manufacturer’s advertising policy is less sensitive to uncertainty of demand than the change of the retailer’s advertising policy. (3) The manufacturer in the VMI system should concern about the differences caused by symmetric or asymmetric retailers