This　paper　investigates　an　optimal　operation　problem　for　on-demand　bus　services　within　an　advanced　demand-responsive　service　platform,　where　passengers　reserve　their　trip　requests　online.　Each　request　specifies　the　passenger＇s　pickup　and　dropoff　locations,　service　time　window,　and　number　of　passengers.　To　enhance　bus　utilization,　the　operator　incentivizes　passengers　within　the　same　request　to　accept　split　pickup　or　delivery　services　and　tolerate　limited　time　window　violations　by　offering　specific　fare　discounts.　The　problem　is　formulated　as　a　mixed-integer　model　that　jointly　optimizes　the　route　and　schedule　for　each　bus,　with　the　objective　of　minimizing　the　total　system　cost.　To　address　this　problem,　a　time-window-backward-induction　approach　is　initially　introduced　to　efficiently　formulate　the　bus　schedule　while　accounting　for　time　window　flexibility.　Building　on　this　approach,　a　variable　neighborhood　search　with　customized　operators　is　developed　to　optimize　bus　routes　while　adhering　to　bus　capacity　and　time　window　constraints,　particularly　in　scenarios　involving　passenger　split　pickups　and　deliveries.　Algorithm　comparisons　across　extended　benchmark　instances　demonstrate　the　superior　performance　of　the　proposed　algorithm　in　scenarios　characterized　by　narrow　time　windows,　high　request　volumes,　and　limited　fleet　sizes.　Subsequently,　large-scale　experiments　based　on　real-world　ride-hailing　data　are　conducted　to　provide　managerial　insights　and　guide　operators　in　designing　a　cost-efficient　on-demand　bus　system.　©　2025　Elsevier　Ltd