7. A hospital division consists of three large nursing units…

7. A hospital division consists of three large nursing units, A, B, and C. The registered nurses (RN’s) in units A, B, and C earn $33, $37, and $41 per hour, respectively. Overtime is paid at double time. The division wants to develop a 4-month staffing plan. The minimum number of labor hours required in each unit each month are:   January February March April Unit A 2700 2300 2100 2500 Unit B 2400 2800 2900 2200 Unit C 2100 2600 2300 2800 The number of regular-time hours allocated for any individual unit should be the same in each of the 4 months (e.g., the number of regular-time hours assigned for A in January must be the same number in A for Feb, March, and April). Overtime can be used as needed in any unit but cannot exceed 15% of regular-time hours in the unit in any month. In any given month, up to 18% of the total number of regular-time plus overtime hours available from nurses assigned to unit B can be used to meet nursing requirements in unit A. Likewise, in any given month, up to 18% of the total number of regular-time plus overtime hours available from nurses assigned to unit C can be used to meet nursing requirements in unit B. Develop a linear programming formulation that will determine the assignment of regular-time and overtime nursing hours to each nursing unit each month. The objective of the model is to minimize total staffing costs for the division during the 4-month period. Constraints of the model should guarantee that nursing hours meet or exceed the labor hours required in each unit each month, and that the overtime and substitution (i.e., using hours assigned to B to meet requirements in A and/or hours assigned to C to meet requirements in B) conditions are satisfied. 

6. A project requires the completion of eight activities. Th…

6. A project requires the completion of eight activities. The immediate predecessors (predecessors), normal activity time in weeks (normal time), maximum crashing time in weeks (max crash time), and per week crashing cost (crash cost) are shown in the table below. Activity 1 2 3 4 5 6 7 8 predecessors none none none 1,2 1,2,3 3 5,6 4,7 normal time 6 8 7 9 7 7 4 6 max crash time 3 2 3 4 1 2 2 3 crash cost $650 $550 $700 $465 $500 $385 $720 $810 Using the “normal time” values, find and report the early start times, early finish times, late start times, late finish times, and slack for each activity. Identify the critical path and project completion time.