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I recall this one from the mid-1980s. I read it in one of the sci-fi magazines of the day, (i.e. Asimov, Analog, etc.), though I do not recall which one. It was presented in parts, over several issues of the magazine.
The story is about an interstellar starship carrying colonists in search of a suitable planet. All the passengers and crew are kept in cryogenic suspended animation, except when a particular member is needed to handle something the automated systems cannot.
The protagonist is a human male who is, IIRC, either the captain or other senior authority. He's revived several times by the ship's computer. I can recall three specific instances where he is revived:
He is revived to check on a malfunction in another passenger's cryo-pod. The compartment is currently at near absolute zero temperatures, so the protagonist has to put on a bulky foam envirosuit. He finds a young woman, who was frozen half-outside of her sleeping pod. He revives her, and discovers she was caught outside the pod because she wanted to spend a few more moments up and about, and ignored the warnings that the automatic freezing process was about to begin. The woman is dying, but the man comforts her and lies to her that she will be fine. IIRC, he also violates the rules on food rationing and heats her up some hot soup to comfort her in her final moments.
Another occasion, the man is revived because the sensors have detected a possible colonization site. The planet is an Earth-type moon in orbit around a supergiant planet. It looks like they have found a new home, until the man notices the computer's note that the moon would make an acceptable temporary settlement. It is then that the man examines the telemetry and realizes the moon is in a decaying orbit, and will plunge into the supergiants atmosphere within the next 200 years.
The final occasion involves the ship passing through a solar system with a hostile alien race. The inhabitants of the system launch interplanetary range nuclear missiles at the vessel. The man is trying to use the ship's engines to outrun the missiles, or something like that.
That's all I have. I never read the final installment, so I do not know the ending of the story. Any guesses? Thanks in advance.