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This type of fallacy would argue that "A well-run office is like a machine. You should be able to replace people without disturbing the office’s efficiency, just as you can replace a bolt or a gear in a machine."
What is an "invalid analogy"?
*As for an office being like a machine, every organization has an informal organization that compensates for problems with the official organization. For instance, a receptionist may compensate for a company officer’s inability to keep to deadlines. Replace that receptionist, and the office may become disorganized.