Machine Learning for Operations Inc. (MLO) is a consulting firm that develops customized machine learning software to support operational decisions (e.g., for forecasting, procurement, scheduling, etc.). On average, MLO receives 1.6 new project requests per week. These arrivals follow a Poisson process.

On each project, the firms consultants work jointly in teams of two a domain expert and a modeler. The same two individuals work as a team throughout the year. On average, each project requires one team working, on average, for 50 business days (from the time the consultants begin working on the project to handing over the results to the client). Historical data reveals that the standard deviation of project work time is 45 business days. MLO assigns projects in first-come-first-served order to the first available consultant team. Each team works on at most one project at a time.

MLO currently employs 36 consultants, working as 18 teams. The firm pays each consultant $ 12,000 per month in salary and benefits, whether or not they are working on a project. On average, MLO charges $ 150,000 per project, based on its consulting rate of $ 3,000 per day of work on the project.

Assume that each week consists of five business days, every month has four working weeks, and consultants work for all 12 months in a year.

(a) Consulting firms such as MLO pay close attention to the percentage of Billable days for their consultants. This percentage, which we call Billable % is the average number of days a consultant works on projects, say, in a quarter, as a proportion of their available time during the quarter. (These firms are very conscientious about billing their clients only for the actual number of days their consultants are working on the project.) What is MLOs current Billable %?

(b) Upon receiving a client request and before knowing which team will work on the project or when the team will begin the project, MLO first needs to inform the client about the time (from now) at which it expects to finish the project and hand over the deliverables. MLO determines this promise time by estimating the average time it takes from the receipt of a clients request to the project completion date, and adding 2 weeks of safety time to this estimated average. (Henceforth, we will refer to average promise time as promise time.)