Explanation

No improvement is required. (The original print reads ‘everybody had gone home’ — the word ‘gone’ appears dropped in this transcription.) ‘By the time he arrived’ is the correct time clause, and the past perfect ‘had gone home’ correctly marks the earlier completed action. None of the substitutions (‘when he arrived’, ‘at which he arrived’, ‘by which he arrive’) improves the sentence. Hence (d) is correct.