The childcare centre is a non-profit community and has been in the business for over 20 years. It is located in Melbourne managed by a local university to provide high quality care for children of university staff and students. Its aim is to provide a safe, supportive environment for children to meet the needs of children and families. The centre has four floors with five rooms. Rooms 13 are for children under the age of 3. Room 4 is mainly for children at the age of 3 but this room has some 4-year-old children mixed in due to a transition to a kindergarten. Room 5 is mainly for children at the age of 4. However, some 5-year-old children still attend because they are waiting for a term to open in a kindergarten. According to the Assistant Coordinator, the current numbers of children attending each day in each room and the maximum room capacities are summarised in the table below:

According to the National Quality Framework (NQF), the minimum ratio of educators to children is 1:4 for 035 months and 1:11 for 36+ months. Although the NQF ratios are calculated across the service, and not by individual rooms, for practical and safety reasons, the centre calculates the ratio on a room-by-room basis. Generally, there are two levels of qualifications required to work as educators: Certificate III and Diploma. To comply with the NQF and to offer high quality care, in each room, the centre requires one educator must have a Diploma qualification for every three educators. For example, if Room 1 requires 3 educators, one of them must have a Diploma and the other two may have either a Diploma or a Certificate III qualification. However, if Room 1 requires 4 educators, two of them must have a Diploma qualification. The centre has 16 permanent educators which are summarised in the table below:

In addition, the centre has a contract with the agency. If the centre requires additional qualified educators, either Certificate III or Diploma, they can contact the agency for an arrangement. The agency educators are paid only when needed, and there is no on-going cost required. Currently, the centre has booked eight educators on standby through the agency. The centre, however, can ask the agency to provide more educators if needed. The centre pays the wages to the permanent educators according to the Childrens Services Award 2010. The wages depend on the experience and qualifications of the educators which are summarised below:

Level-3 employees are Certificate-III qualified educators, and Level-4 employees are Diploma qualified educators. All permanent educators in this centre have more than two years of experience working in this industry. The hourly wage to hire an agency educator, whether Certificate III or Diploma, is $51.77; and they will work for the full eight hours. Since one of the duties of educators is to engage and familiar with children, when they are assigned to supervise one room, they will stay in the same room for the whole week. The centre opens at 06:30 and closes at 18:30. Due to the availability of the permanent educators, the centre also hires casual educators to cover for the permanent ones during the opening time, lunch break, and closing time, which are approximately a few hours per day. As a result, the management has decided to ignore the available time of each permanent educator and focus on the main problem which is how to schedule the permanent educators for the whole week. To simplify the problem, the centre does not group children from different rooms into a single room when there are only few of them present (i.e. in the early morning and in the evening). The centre also assumes that all children are present for the whole day. With all the details above, the Centre Manager, Joan Gilmore, wants to create a roster for the educators with a minimum-cost plan. She wants the plan that (1) each permanent educator will be given a job to do and (2) in case that there are not sufficient educators, an agency educator will be given a job: a) Formulate and implement the mathematical model to solve this problem b) How much will it cost for the plan? c) Which educator will be assigned to which room? d) Is there any available place for a new child to attend? Which room? Which day?

Max Capacity Age Monday Tuesday Wednesday Thursday Friday Room 1 Room 2 Room 3 Room 4 Room 5 12 16 16 16 22