case study, you need to submit only the model (Variable Definition, Expression for objective, Mathematical Expressions for Constraints, etc.) - do not try to solve it using SOLVER yet! In many situations like Call Centers, Banks, and Hospitals, the manpower planning is a non-trivial task as the staffing needs varies over time (hour to hour or shift to shift). In such cases, one would like to achieve a cost efficient staffing without sacrificing the customer service. Consider for example, hospitals where scheduling of nursing staff is a very critical task. It is crucial that each shift is fully staffed (based on projected needs), and no person is given an unreasonable schedule. A local hospital in consultation with their nursing staff has decided to follow four basic policies in scheduling -- (1) Each person should have at least every other weekend off, (2) On an average, a person should work 5 days a week, (3) a nurse should not work more than 5 days without an off day, and (4) there will not be any shift change. The policy (4) is restrictive, but makes the scheduling task much simpler, as each shift now can be scheduled independent of the other. Also, based on the historical data, this hospital has observed that the demand for nursing staff in a particular shift varies from day to day within a week, but does not vary too much from week to week. The administration expects this observation to be true in near future. Based on this observation and policy (1), each person can be assigned a two-week rolling schedule. In accordance with policies (1) through (4), and some other considerations, the administration has enumerated 15 acceptable schedules as shown in following table. In this table, a 0 indicates an off day, and I a working day. For example, schedule 1 (S1) in row 1 implies that over next two weeks, the person will be off on the first Monday, then off for the weekend (Saturday and Sunday), and off on Wednesday of the second week. This two-week schedules then repeats for subsequent weeks. $1 S2 S3 $4 $5 S6 S7 S8 S9 $10 $11 $12 $13 S14 $15 wk1- wk1-wk1- wk1- wk1-wk1- wk1- wk2-wk2-wk2-wk2-wk2-wk2-k2- mon tue wed thur fri sat sun mon tue wed thur fri sat sun 0 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 ol 0 D 0 1 1 1 1 1 1 1 1 1 11 1 11 1 Minimum number of nurses required 1 1 D 0 0 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 0 0 1 0 0 o 0 0 0 0 0 0 1 1 of 1 1 0 C 0 of 1 1 1 1 1 1 1 1 1 1 0 0 1 1 of 1 al 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 Mon Tue Wed Thur. Fri. Sat. Sun. of 1 0 1 1 1 1 1 1 1 1 1 D 1 1 1 1 0 1 1 1 1 ol 1 0 d 1 1 7 1 1 1 1 1 1 1 1 1 1 0 0 C 0 C C 0 C 1 0 1 1 1 1 1 1 0 0 0 To schedule the nursing staff for the third shift at the rehab floor of this hospital, the historical data projects the following demand for nurses for each day of the week. 0 0 20 10 | 12 | 15 | 17 | 14 | 10 | 11 The hospital currently has two grades of nurses -- registered nurses (RN) or practitioner nurses (PN). However, hospital has a rule that requires that, at all times, there should be at least 1 registered nurse for every two practitioner nurses (that is ratio of RN to PN every day should be at least 1/2). Assuming that an RN is paid 20% more than a PN, develop a mathematical model to decide how many RN and PN should be assigned to each of the 15 schedules, so that each day is properly staffed. The Hospital will like to achieve this with the least cost. (Hint: Define a variable for each type of nurse (RN and PN) to be assigned to each schedule. Hence you will have 30 variables and 28 constraints - a demand and a ration constraint (at least 1 to 2 ratio between registered nurses and practitioner nurses) for each day in a two week cycle. This model is very similar to the one developed in class for the ambulance staffing problem where table above is the "time chart")