# Mathematical Modelling In Supply Chains

Problem 1.

A service firm will need to modify its staff timetable from eight-hour shifts to twelve-hour shifts. Before the change, the staff was working eight hours a day for five days per week. After the change, for the first week, the staff will be working for three days and then will be off for four days. For the following second week, they will be working for four days and will be off for four days. This means overall the company staff will be working eighty-four hours every 2 weeks.

The highest demand period is between 5 A.M. – 7 A.M. and between 5 P.M. – 7 P.M. To accommodate this peak demand, the following twelve-hour shifts must be arranged:

• Shift: E & E [alt], Working hours: 5 A.M. to 5 P.M., and Payment rate per week: \$757
• Shift: F & F [alt], Working hours: 7 A.M. to 7 P.M., and Payment rate per week: \$841
• Shift: G & G [alt], Working hours: 5 P.M. to 5 A.M., and Payment rate per week: \$881
• Shift: H & H [alt], Working hours: 7 P.M. to 7 A.M., and Payment rate per week: \$923

The last and most desired times to commence and finish work will be used as the basis for the shift pay differentials. In any one week, staff on shift E might work Sun to Tues, while staff on shift E [alt] would work at the same times, but on Wednesday to Saturday. In the following week, staff on shift E would work Sunday to Wednesday, while staff on shift E [alt] would work the corresponding Thursday to Saturday. So, this results in scheduling the same number of staff for shift E as for shift E[alt].

Table 1 shows the firm’s staff requirement for the twenty-four-hour day. (1) Formulate this example as an LP model to determine the most economical schedule? (2) Use Microsoft Solver to solve the formulated LP model in (1).