Solve on a spreedsheet using solver. Each day, workers at the Gotham City Police

Question

Solve on a spreedsheet using solver.

Each day, workers at the Gotham City Police Department work two 6-hour
shifts chosen from 12 A.M. to 6 A.M., 6 A.M. to 12 P.M., 12 P.M. to 6 P.M., and 6 P.M. to 12 A.M.
The following number of workers are needed during each shift: 12 A.M. to 6 A.M. { 15 workers; 6 A.M.
to 12 P.M. { 5 workers; 12 P.M. to 6 P.M. { 12 workers; 6 P.M. to 12 A.M. { 6 workers. Workers whose
two shifts are consecutive are paid $12 per hour; workers whose shifts are not consecutive are paid
$18 per hour. Formulate an LP that can be used to minimize the cost of meeting the daily workforce
demands of the Gotham City Police Department.

Explanation / Answer

Decision variables

We are required to find the number of workers joining/working for each shifts from (1) 12AM to 6AM; (2) 6AM to 12PM; (3) 12PM to 6PM and (4) 6PM to 12PM

Workers are required to work for two shifts out of the four scheduled shifts. The likely number of combinations are as follows: (1 & 2), (1 & 3), (1 & 4 ), (2 & 3), (2 & 4) and (3 & 4 ) let us define x1, x2, x3, x4, x5 and x6 represents the number of workers corresponding to these combinations in that order, in other words

x1 for shifts 1&2

x2 for shifts 1&3

x3 for shifts 1&4

x4 for shifts 2&3

x5 for shifts 2&4

x6 for shifts 3&4

Objective function:

As mentioned in the question is to minimize the total cost of hiring workers which is as follows:

Minimize Z = 144x1 + 216x2 + 144x3 + 144x4 + 216x5 + 144x6    ($12 per hour for consecutive shifts- 12*12 and $18 per hour for non consecutive shifts- 18*12)

Constraints:

Shift 1     means x1 + x2 + x3 >= 15

Shift 2      means x1 + x4 + x5 >= 5

Shift 3       means x2 + x4 + x6 >= 12

Shift 4       means x3 + x5 + x6 >= 6

x1, x2,x3,x4,x5 and x6   > = 0 (as all are numbers greater than or equal to zero)

Data table for the solver is as follows:

Solver has given the following solution

Decision variable x1 x2 x3 x4 x5 x6 Value Objective function 144 216 144 144 216 144 0 RHS Constraint shift1 1 1 1 0 0 0 0 15 shift2 1 0 0 1 1 0 0 5 shift3 0 1 0 1 0 1 0 12 shift4 0 0 1 0 1 1 0 6

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