To produce in Shandong, you must register a company in Shandong, or be a branch of your Hebei company. Of course, the corresponding production-related taxes will be paid in Shandong. If you don't register with the industry and commerce, Shandong Industry and Commerce will seal you up within three to five days after you start work.
Products should be marked with the place of origin. Specifically, we can see the requirements of the product quality law. You can produce it in Shandong, and then all the products are sold by the company in Hebei. Of course, the taxes paid in the middle are different, so you need your company's accountant to make a good tax plan.
Your situation is different from that of construction enterprises, and there is no tax division problem for off-site operations. If the factory in Shandong is a branch of Hebei company, all income taxes will be paid by Hebei company, and all other taxes will be paid in Shandong. So you don't have to worry about taxes, and you can learn about your business in different places.