next page 3 6 5 8 15 8 Problem 3 (LP-Employee Scheduling): SET UP ONLY (You may use the ATTACHED Excel Chart) Manfred Leaks manages a large discount store. His biggest problem has been scheduling cashiers so that he has an adequate number without having too many. The store is open from 9am to 9p.m. each day of the week. Based on historical data, he found that the customer patterns for Monday to Thursday are essentially the same, but those for Friday, Saturday, and Sunday are all different. He divided the day into three four hour segments and estimated at least how many cashiers were needed for each time period for each day of the week. These are given in the following table. Chart on Day of Week Mon Thur Fri Sat Sun 9 am to 1pm 3 11 1 pen to 5 pm 11 5pm to 9pm Employees must work continuous eight-hour shifts beginning at 9 a.m. or 1 pm, and their weekly schedules must be made up of five consecutive days of work with two consecutive days off (and they work the same hours each workday). Manfred would like to devise weekly schedules that will minimize the total number of cashiers needed, but the schedules must be such that the minimum cashier requirements in the table are satisfied Set up Manfred's problem as an LP model with an objective function, the Saturday and Sunday constraints, and the non-negativity constraints. HINTS: 1. There are 14 possible schedules. There is one variable corresponding to each schedule. 2. Use the following notation for this problem: La Xij = # of cashiers who work shift i on weekly schedule j. where i = 1 means 9 a.m. - 5p.m.; and i = 2 means 1 p.m. -9p.m. j = 1 means the cashier works Mon-Fri; j = 2 means the cashier works Tue-Sat: j - 3 means the cashier works Wed Sun; j = 4 means the cashier works Thur-Mon; XL, X12, X 13 .... X 26 X 27 20 j = 5 means the cashier works Fri-Tue; j = 6 means the cashier works Sat-Wed; and j = 7 means the cashier works Sun-Thur. Thus, X11 = # of cashiers who work Monday - Friday from 9am - 5p.m.; X12 = # of cashiers who work Tuesday - Saturday from 9am-5pmand so on. Xai #of cashiers who work Monday - Friday from 1 p.m. - 9p.m.. Xx = # of cashiers who work Tuesday - Saturday from 1p.m. - 9 p.m., and so on. 3. There are a total of 21 constraints apart from the non-negativity constraints. There are three constraints each for Mon Thur, 3 for Friday: 3 for Saturday, and 3 for Sunday. You must write each of the three constraints for ONLY Saturday and Sunday (l.c., a total of 6 constraints). You do NOT have to write the constraints for the other days. That is, I will give you full points (if correct) if you put down these six constraints in addition to the Objective function and the non negativity constraints, min z = next page 3 6 5 8 15 8 Problem 3 (LP-Employee Scheduling): SET UP ONLY (You may use the ATTACHED Excel Chart) Manfred Leaks manages a large discount store. His biggest problem has been scheduling cashiers so that he has an adequate number without having too many. The store is open from 9am to 9p.m. each day of the week. Based on historical data, he found that the customer patterns for Monday to Thursday are essentially the same, but those for Friday, Saturday, and Sunday are all different. He divided the day into three four hour segments and estimated at least how many cashiers were needed for each time period for each day of the week. These are given in the following table. Chart on Day of Week Mon Thur Fri Sat Sun 9 am to 1pm 3 11 1 pen to 5 pm 11 5pm to 9pm Employees must work continuous eight-hour shifts beginning at 9 a.m. or 1 pm, and their weekly schedules must be made up of five consecutive days of work with two consecutive days off (and they work the same hours each workday). Manfred would like to devise weekly schedules that will minimize the total number of cashiers needed, but the schedules must be such that the minimum cashier requirements in the table are satisfied Set up Manfred's problem as an LP model with an objective function, the Saturday and Sunday constraints, and the non-negativity constraints. HINTS: 1. There are 14 possible schedules. There is one variable corresponding to each schedule. 2. Use the following notation for this problem: La Xij = # of cashiers who work shift i on weekly schedule j. where i = 1 means 9 a.m. - 5p.m.; and i = 2 means 1 p.m. -9p.m. j = 1 means the cashier works Mon-Fri; j = 2 means the cashier works Tue-Sat: j - 3 means the cashier works Wed Sun; j = 4 means the cashier works Thur-Mon; XL, X12, X 13 .... X 26 X 27 20 j = 5 means the cashier works Fri-Tue; j = 6 means the cashier works Sat-Wed; and j = 7 means the cashier works Sun-Thur. Thus, X11 = # of cashiers who work Monday - Friday from 9am - 5p.m.; X12 = # of cashiers who work Tuesday - Saturday from 9am-5pmand so on. Xai #of cashiers who work Monday - Friday from 1 p.m. - 9p.m.. Xx = # of cashiers who work Tuesday - Saturday from 1p.m. - 9 p.m., and so on. 3. There are a total of 21 constraints apart from the non-negativity constraints. There are three constraints each for Mon Thur, 3 for Friday: 3 for Saturday, and 3 for Sunday. You must write each of the three constraints for ONLY Saturday and Sunday (l.c., a total of 6 constraints). You do NOT have to write the constraints for the other days. That is, I will give you full points (if correct) if you put down these six constraints in addition to the Objective function and the non negativity constraints, min z =