A construction company is working on two projects, each with their own aggregate composition requirements. The company purchases raw materials from two sources to mix the aggregates needed by the two projects. Shares of coarse and fine materials and sand of aggregates purchased from the two sources are listed in the table below, along with the prices. Source $1$ and $2$ each has $7$ batches available per day.

Project A needs $20-30\%$ coarse and at least $50\%$ fine materials for its aggregates. Project B needs at least $30\%$ coarse and at least $30\%$ sand for its aggregates. Every day, Project A needs at least $4$ batches of aggregates, while the Project B needs at least $6$ batches of aggregates. Formulate, but do not solve, an LP problem to find the least expensive way to meet all the requirements.