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Despite their different designation on paper, Trade and Mine ships in practice , are essentially interchangeable in their roles. It is therefore necessary to consider things a little before assigning them to a Manager to ensure they do what you intend. If you would like them to operate in their defined roles (trading vs. mining), this is achieved through equipping/removing suitable mining drones to them or not. Without mining drones, they will only trade; with mining drones they will only mine (unless the station is full and you un-assign and re-assign them - then they may begin to trade).
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Once a Trading ship's Captain has had their commander assigned as is assigned to a Station Manager, the Captain will begin to try to complete trade offers that the Manager advertises (taking into consideration the Manager's "Trading operational range" setting & current budget). Take note of trade ship cargo type restrictions: it is pointless assigning a trader tradeship to the manager that manager if it cannot transport any of the wares involved used/produced by the factory. Once you have a larger more self-sufficient supply chain it might only be necessary to supply the factory with a trader for its product type, as the inputs will be covered by the traders from other factories or directly by mining ships.
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Any fighter (S/M Fight ships) assigned to a station's DO will be used to patrol the local zone or escort medium Trade/Mine ships assigned to Managers. If you assign new medium Trade/Mine ships to the manager, you'll need to relocate escorting fighters back to the DO if you'd like the DO to share them out again. Given their fragility, you are likely to have to assign new fighters to the DO on a semi-regular basis, so beyond the early game- game , you may prefer to use capital ships instead (sturdier and self-repairing).
Capital ships assigned to DOs will only go on patrol in the station's zone (any capital, even if they have Trade/Mine/Build designation). If you'd d like combat-capital ships to escort Trade/Mine ships assigned to Managers, you'll need to do that manually. Ordering combat capital ships to patrol particular zones is another alternative.
An additional means of defending a station against hostile fighters is with combat drones, which can be replaced automatically via the Supply Options menu.