Purpose.　In　order　to　study　the　impact　of　grey　delivery　time　uncertainty　on　customer　satisfaction　and　delivery　costs,　a　vehicle　routing　problem　with　grey　delivery　time　windows　and　multiobjective　constraints　is　defined.　Method.　The　paper　first　defines　the　uncertainty　of　the　delivery　vehicle＇s　arrival　time　to　the　customer　as　grey　uncertainty　and　then　whitens　the　grey　time　windows;　at　the　same　time,　the　customer＇s　hard　time　windows　is　expanded　into　a　soft　time　windows　to　measure　customer　satisfaction　when　the　vehicle　arrives.　Experiment.　In　order　to　verify　the　validity　of　the　established　model,　numerical　experiments　are　carried　out　in　two　groups　based　on　the　Solomon　example,　and　the　solution　is　solved　based　on　the　improved　quantum　evolution　algorithm.　Analysis.　Distribution　cost　fluctuations　and　customer　satisfaction　fluctuations　with　grey　time　windows　are　relatively　small;　under　different　satisfaction　threshold　conditions,　the　distribution　cost　is　increased　gently　with　the　satisfaction　threshold.　Conclusion.　The　grey　delivery　time　windows　have　certain　advantages　in　solving　the　random　travel　time　vehicle　routing　problem.