In suburban areas where the demand is not high enough, the fixed-route buses usually operate with infrequent services, which increase the cost of waiting transit services for fixedroute service users. For those low demand regions, flexible-route bus operations with doorstep services, which operate with flexible routes, flexible schedules, and smaller vehicles, can be a feasible alternative. This study formulates the social welfare for flexible-route bus operations under various financial constraints, seeking to jointly optimize the headway, fare, and zone size. Elastic demand functions are constructed as a part of problem formulations. Unlike many previous studies, this study does not presume the shape of the zone, so that the flexible-route bus services can be designed according to the geographical conditions in the regions. Moreover, the amount of subsidy for financially constrained policy is calculated based on the actual demand although many previous studies used the potential demand to calculate the subsidy amount. Numerical analyses are explored to evaluate and compare system-wide welfares for different financial perspectives, i.e., no financial constraint, break-even policy, and subsidized policy. The solution optimality for constrained nonlinear optimization problems is verified using analytical optimization or the Karush-Kuhn-Tucker (KKT) conditions. It is found that the welfare of subsidized transit operations is located between the welfare amounts of unconstrained and break-even cases; this finding of welfare variations over subsidy amounts can be a helpful resource for transportation mobility analysts in determining the appropriate service quality and the corresponding amount of subsidy. Sensitivity analyses are presented to explore the system-wide welfare with respect to critical input parameters, including a comprehensive analysis of the unit cost of bus operations, which offers implications toward Mobility-as-a-Service (MaaS) planning.

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