Liner　shipping　is　an　important　branch　of　maritime　transportation.　As　bunker　fuel　consumption　causes　high　operating　cost　and　harmful　gas　emissions,　bunker　fuel　management　is　a　great　challenge　and　a　hot　research　topic　in　liner　shipping.　Bunker　charging　can　be　achieved　at　ports　with　diverse　prices,　and　it　is　recognized　that　appropriately　managing　bunker　fuel　and　sailing　speed　can　improve　liner　shipping　performance　and　reduce　environmental　pollution.　Most　existing　works　assume　that　the　container　demand　is　deterministic.　However,　in　practice,　it　is　usually　difficult　to　exactly　estimate　the　volume　of　containers　to　be　shipped　due　to　various　factors.　This　paper　studies　a　liner　ship　bunkering　and　speed　optimization　problem　under　uncertain　container　demand.　For　the　problem,　a　two-stage　stochastic　and　non-linear　programming　formulation　is　proposed.　To　split　the　complexity　of　the　problem,　the　complicated　bunker　consumption　function　is　approximated　by　piecewise　linear　ones.　To　solve　the　problem,　a　classic　sample　average　approximation　(SAA)　method,　and　the　SAA　based　on　scenario　reduction,　and　an　L-shaped　method　are　developed　and　compared.　Numerical　results　show　that　the　L-shaped　method　outperforms　the　two　SAA　methods,　in　terms　of　solution　quality　and　computational　time.