The bank is working to develop an efficient work schedule for full-time and part
ID: 451741 • Letter: T
Question
The bank is working to develop an efficient work schedule for full-time and part-time tellers. The schedule must provide for efficient operation of the bank including adequate customer service, employee breaks, and so on. On Fridays the bank is open from 9:00 A.M. to 7:00 P.M. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized here.
Each full-time employee starts on the hour and works a 4-hour shift, followed by a 1-hour break and then a 3-hour shift. Part-time employees work one 4-hour shift beginning on the hour. Considering salary and fringe benefits, full-time employees cost the bank $15 per hour ($105 a day), and part-time employees cost the bank $8 per hour ($32 per day).
The bank is working to develop an efficient work schedule for full-time and part-time tellers. The schedule must provide for efficient operation of the bank including adequate customer service, employee breaks, and so on. On Fridays the bank is open from 9:00 A.M. to 7:00 P.M. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized here.
Time Number of Tellers Time Number of Tellers 9:00 a.m.-10:00 a.m. 6 2:00 p.m.-3:00 p.m. 6 10:00 a.m.-11:00 a.m. 4 3:00 p.m.-4:00 p.m. 4 11:00 a.m.-Noon 8 4:00 p.m.-5:00 p.m. 7 Noon-1:00 p.m. 10 5:00 p.m.-6:00 p.m. 6 1:00 p.m.-2:00 p.m. 9 6:00 p.m.-7:00 p.m. 6Each full-time employee starts on the hour and works a 4-hour shift, followed by a 1-hour break and then a 3-hour shift. Part-time employees work one 4-hour shift beginning on the hour. Considering salary and fringe benefits, full-time employees cost the bank $15 per hour ($105 a day), and part-time employees cost the bank $8 per hour ($32 per day).
(a) Formulate an integer programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost. (Hint: Let xi = number of full-time employees coming on duty at the beginning of hour i and yi = number of part-time employees coming on duty at the beginning of hour i.) Min x9 + x10 + x11 + y9 + y10 + y11 + y12 + y1 + y2 + y3 x9 + y9 - Select your answer -- Select your answer -<>=Item 13 Time (9:00 a.m.-10:00 a.m.) x9 + x10 + y9 + y10 - Select your answer -- Select your answer -<>=Item 19 Time (10:00 a.m.-11:00 a.m.) x9 + x10 + x11 + y9 + y10 + y11 - Select your answer -- Select your answer -<>=Item 27 Time (11:00 a.m.-Noon.) x9 + x10 + x11 + y9 + y10 + y11 + y12 - Select your answer -- Select your answer -<>=Item 36 Time (Noon.-1:00 p.m.) x10 + x11 + y10 + y11 + y12 + y1 - Select your answer -- Select your answer -<>=Item 44 Time (1:00 p.m.-2:00 p.m.) x9 x11 + y11 + y12 + y1 + y2 - Select your answer -- Select your answer -<>=Item 52 Time (2:00 p.m.-3:00 p.m.) x9 + x10 + y12 + y1 + y2 + y3 - Select your answer -- Select your answer -<>=Item 60 Time (3:00 p.m.-4:00 p.m.) x9 + x10 + x11 + y1 + y2 + y3 - Select your answer -- Select your answer -<>=Item 68 Time (4:00 p.m.-5:00 p.m.) x10 + x11 + y2 + y3 - Select your answer -- Select your answer -<>=Item 74 Time (5:00 p.m.-6:00 p.m.) x11 + y3 - Select your answer -- Select your answer -<>=Item 78 Time (6:00 p.m.-7:00 p.m.) xi, yj 0 and integer for i = 9, 10, 11 and j = 9, 10, 11, 12, 1, 2, 3 (b) Solve the LP Relaxation of your model in part (a). If required, round your answers to the nearest whole number. x9 x10 x11 y9 y10 y11 y12 y1 y2 y3 Total Cost: $ (c) Solve for the optimal schedule of tellers.Time No. of Full-time
employees No. of Part-time
employees 9:00 a.m.-10:00 a.m. 10:00 a.m.-11:00 a.m. 11:00 a.m.-Noon Noon-1:00 p.m. 1:00 p.m.-2:00 p.m. 2:00 p.m.-3:00 p.m. 3:00 p.m.-4:00 p.m. 4:00 p.m.-5:00 p.m. 5:00 p.m.-6:00 p.m. 6:00 p.m.-7:00 p.m. Comment on the solution. The input in the box below will not be graded, but may be reviewed and considered by your instructor. (d) After reviewing the solution to part (c), the bank manager realized that some additional requirements must be specified. Specifically, she wants to ensure that one full-time employee is on duty at all times and that there is a staff of at least five full-time employees. Revise your model to incorporate these additional requirements, and solve for the optimal solution. If required, round your answers to the nearest whole number. The new optimal solution is as follows:
Time No. of Full-time
employees No. of Part-time
employees 9:00 a.m.-10:00 a.m. 10:00 a.m.-11:00 a.m. 11:00 a.m.-Noon Noon-1:00 p.m. 1:00 p.m.-2:00 p.m. 2:00 p.m.-3:00 p.m. 3:00 p.m.-4:00 p.m. 4:00 p.m.-5:00 p.m. 5:00 p.m.-6:00 p.m. 6:00 p.m.-7:00 p.m. Total Cost: $
Explanation / Answer
As mentioned in the question, xi represents the number of full-time employees and yi represents the number of part-time employees joining duties at the beginning of ith hour. The objective is to have the minimum number of employees, therefore defined as Minimize Sigma (xi+yi) in which i is varying from 9AM to 6PM
Formulation of the problem is as above for the use of excel solver and the solution is as follows:
The above solution is with the objective to have minimum total cost of hiring per day and is obvious to go for part-time employees only.
b)
c)
Solution to part d) is as follows:
Decision Variables x9 x10 x11 y9 y10 y11 y12 y1 y2 y3 Variable values Objective function 1 1 1 1 1 1 1 1 1 1 0 Constraints SumProd. sign RHS 9to10 1 0 0 1 0 ">= 6 10to11 1 1 0 1 1 0 ">= 4 11to12 1 1 1 1 1 1 0 ">= 8 12to1 1 1 1 1 1 1 1 0 ">= 10 1to2 0 1 1 1 1 1 1 0 ">= 9 2to3 1 0 1 1 1 1 1 0 ">= 6 3to4 1 1 0 1 1 1 1 0 ">= 4 4to5 1 1 1 1 1 1 0 ">= 7 5to6 0 1 1 1 1 0 ">= 6 6to7 0 0 1 1 0 ">= 6 Cost per person 105 105 105 32 32 32 32 32 32 32 0Related Questions
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