The coronavirus pandemic has affected most aspects of product supply and consumer behaviors and led to transformations in the supply chain. The COVID-19 pandemic and the requirements to reduce its prevalence have led many people to shop online and encouraged many manufacturers to sell their products online. In this study, a manufacturer, who intends to possess an online sales channel, and a retailer, who has an in-person sales channel, are considered. Then, pricing strategies and collaboration mechanisms between them in the health-social dual-channel supply chain are investigated. This study is developed in three models, including centralized, decentralized, and collaborated under Stackelberg game, whereas the optimal price of products in each channel, level of implementation of health and safety protocols in retailers, advertising level, and status of online shopping performance are obtained for improving customer trust. Moreover, the demand is represented as a function of selling prices of products in online and in-person shops, compliance level of health protocols, level of online shopping performance, and advertising in health during the COVID-19 pandemic. Although the centralized model provides more profit for the manufacturer, the collaborated model provides the highest profit for the retailer. Thus, since the supply chain profit of centralized and collaborated models is close, the collaboration model is the best option for members in this situation. Sensitivity analysis is finally performed to evaluate the impact of key parameters, and then according to obtained results, some management insights are suggested for the dual-channel supply chain during the COVID-19 pandemic. Keywords Dual-channel supply chain • COVID-19 • Supply chain collaboration Abbreviations on Manufacture (online sales) of Retailer (in-person sales) SC Supply chain dec Decentralized structure cen Centralized structure CO Collaborated structure

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In the competitive market, applying effective and efficient inventory control and order management methods to reduce costs and improve service efficiency and quality, leading to profit maximization, becomes an absolute necessity at the organizational and supply chain level. Therefore, this study used the optimal control theory to propose an order system and inventory management model. The model is presented from the company’s resourcing stage with a desirable accountability level from the supplier side in a continuous review period with dynamic multi-item demand and budget constraints. In the present model, the order was regarded as a time-dependent function and a control variable. In addition, the need for each item is time-dependent and specified. The present model indicates the order and inventory system as an optimal control problem. In addition, Pontryagin’s maximum principle for optimal control problems, Generalized Reduced Gradient Method (GRG), and Kuhn–Tucker conditions were used to seek the optimal order. The model aimed to maximize the total order profit and inventory management systems. Finally, the results were presented numerically and graphically, and some management decisions were derived.