Consider a DISTRIBUTION NETWORK comprising three production plants with capacities of $600,500,$ and $700$ units per week, respectively. These plants ship their products through two warehouses to meet the demands of three distribution centers, where the weekly requirements are $350,600,$ and $800$ units. The unit shipping costs for each route are detailed in the provided image. Your task is to determine the optimal production levels at each plant and devise a distribution plan that fulfills the demand at all centers while minimizing the overall shipping costs.