Example 1.3: Scheduling problem A hospital needs to schedule nurses so that it can satisfy the projected weekly demand. Table 1.3 shows the projected number of nurses required for each day of a week. If a nurse is assigned to work, she or he needs to work for three consecutive days. For example, if a nurse is assigned to work on Monday, she or he will be working on Monday, Tuesday, and Wednesday. What is the minimum number of nurses that the hospital needs to hire to satisfy the demand? Table 1.3 Projected number of nurses required for each day of a week Monday Tuesday Wednesday Thursday Friday Saturday Sunday Day Projected Demand X : Number of nurses who are assigned to work on Wednesday x, : Number of nurses who are assigned to work on Thursday xs : Number of nurses who are assigned to work on Friday xe : Number of nurses who are assigned to work on Saturday 7 x+x+x, 26 x + x + x, 215 *, + x + x 210 xs + x + x, 215 X, *., *, *,, X, X, X, 20 and integer Note that since the number of people assigned to work cannot take fractional value, the inte- ger condition is added to the problems decision variables. Using computer, the solution to this optimization problem is x1 =1, x2 =0, x, = 6, x4 = 0, x, =9, x, = 1, X, = 5 and z* = 22. The solution indicates that the hospital needs to hire one nurse on Monday, six nurses on Wednesday, nine nurses on Friday, one nurse on Saturday, and five nurses on Sunday. Therefore, the minimum number of nurses required is 22 to meet the demand requirements. 5 6 7 6 15 10 15 Solution: The Of of this problem is to minimize the total number of nurses hired. Since we can assign nurses to work on any given day, we can define seven variables representing the number of nurses assigned to work on each day of a week. xi: Number of nurses who are assigned to work on Monday *, : Number of nurses who are assigned to work on Tuesday