September 5, 2017 This example is from the textbook -Introduction to Mathematica

Question

September 5, 2017 This example is from the textbook -Introduction to Mathematical Programming" by Winston and Venkataramanan. A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given in the table below. Union rules state that each full-time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday through Friday must be off on Saturday and Sunday. The post office wants to meet its daily scheduling requirements using only full-time employees. Formulate an IP that the post office can use to minimize the number of full-time employees who must be hired. TABLE 4 Requirements for Post Office Day 1-Monday 2 = Tuesday 3 Wednesday 4 Thursday 5 = Friday 6 = Saturday 7 Sunday Number of Full-time Employees Required 17 13 15 19 14 16

Explanation / Answer

Let x1,x2,x3,x4,x5,x6,x7 be the number of full time employees who start work on Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday respectively;
Thus, the objective function is to Minimise: x1 + x2 + x3 + x4 + x5 + x6 + x7
Thus on Monday: The following employees will be working : x1 , x4, x5, x6, x7
Thus, the constraint for Monday will be : x1 + x4 + x5 + x6 + x7 >= 17

Similarly on Tuesday, The following employees will be working : x1 , x2, x5, x6, x7
Thus, the constraint for Monday will be : x1 + x2 + x5 + x6 + x7 >= 13

Thus on Wedensday: The following employees will be working : x1 , x2, x3, x6, x7
Thus, the constraint for Monday will be : x1 + x2 + x3 + x6 + x7 >= 15

Thus on Thursday: The following employees will be working : x1 , x2, x3, x4, x7
Thus, the constraint for Monday will be : x1 + x2 + x3 + x4 + x7 >= 19

Thus on Friday: The following employees will be working : x1 , x2, x3, x4, x5
Thus, the constraint for Monday will be : x1 + x2 + x3 + x4 + x5 >= 14

Thus on Saturday: The following employees will be working : x6 , x2, x3, x4, x5
Thus, the constraint for Monday will be : x6 + x2 + x3 + x4 + x5 >= 16

Thus on Sunday: The following employees will be working : x7, x6, x3, x4, x5
Thus, the constraint for Monday will be : x7 + x6 + x3 + x4 + x5 >= 11

Thus, this LPP has one objective function to minimise and 7 (one for each day) man power specific constraints to abide by; Also that, each of x1, x2, x3, x4, x5, x6 and x7 are >= 0 ;

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