Question

Review, analyze, and complete the Chase Manhattan Bank Case Study below. The workload in many areas of bank operations has the characteristics of a nonuniform distribution with respect to time of day. For example, at Chase Manhattan Bank in New York, the number of domestic money transfer requests received from customers, if plotted against time of day, would appear to have the shape of an inverted U curve with the peak around 1 P.M. For efficient use of resources, the personnel available should, therefore, vary correspondingly. A variable capacity can be achieved effectively by employing part-time personnel. Because part-timers are not entitled to all the fringe benefits, they are often more economical than full-time employees. Other considerations, however, may limit the extent to which part-time people can be hired in a given department. The problem is to find an optimum workforce schedule that would meet personnel requirements at any given time and also be economical. Some of the factors affecting personnel assignment are listed here: 1. By corporate policy, part-time personnel hours are limited to a maximum of 40% of the day’s total requirement. 2. Full-time employees work for 8 hours (1 hour for lunch included) per day. Thus, a full-timer’s productive time is 35 hours per week. 3. Part-timers work for at least 4 hours per day but less than 8 hours and are not allowed a lunch break. 4. Fifty percent of the full-timers go to lunch between 11 A.M. and noon, and the remaining 50% go between noon and 1 P.M. 5. The shift starts at 9 A.M. and ends at 7 P.M. (i.e., overtime is limited to 2 hours). Any work left over at 7 P.M. is considered holdover for the next day. 6. A full-time employee is not allowed to work more than 5 hours overtime per week. He or she is paid at the normal rate for overtime hours—not at one-and-a-half times the normal rate applicable to hours in excess of 40 per week. Fringe benefits are not applied to overtime hours. In addition, the following costs are pertinent: 1. The average cost per full-time personnel hour (fringe benefits included) is $10.11. 2. The average cost per overtime personnel hour for full-timers (straight rate excluding fringe benefits) is $8.08. 3. The average cost per part-time personnel hour is $7.82. The personnel hours required, by hour of day, are given in the following Table. TABLE: Workforce Requirements NUMBER OF PERSONNEL TIME PERIOD REQUIRED 9–10 A.M. 14 10–11 25 11–12 26 12–1 P.M. 38 1–2 55 2–3 60 3–4 51 4-5 29 5-6 14 6-7 9 The bank’s goal is to achieve the minimum possible personnel cost subject to meeting or exceeding the hourly workforce requirements as well as the constraints on the workers listed earlier.

Discussion Questions:

1. What is the minimum-cost schedule for the bank?

Answer 1

Given data

Workforce requirements

NUMBER OF PERSONNEL	TIME PERIOD REQUIRED
9–10 A.M.	14
10–11	25
11–12	26
12–1 P.M.	38
1–2	55
2–3	60
3–4	51
4-5	29
5-6	14
6-7	9

1)

Initially  Develop a grid in Excel with the time periods ( 9:00 to 10:00, 10:00 to 11.00, etc.) down the left side of the grid and all possible full-time and part-time shifts that can be worked across the top. put an "X" in the grid to indicate the times that each of the possible shifts works. all full-time employees start at 9:00. for the full-time employees, half of the employees take lunch from 11"00 A.M to noon and the remainder of them takes lunch from noon to 1:00 P.M. ( hint: there 3 possible full-time shifts and 22 possible part-time shifts.)
   2)

Using the grid from (1), formulate the 10 constraints corresponding to the time periods.

• If that shift works in a given time period, make the coefficient equal to 1.0
• If the shift doesn't work in a given time period, make the coefficient equal to 0.0
• For the lunch periods of full-time employees, make the coefficients equal to 0.5
• The number of employees needed for each time period is given in the problem in the case.

  3)

The final (11th) constraint states that the total number of hours worked by the part-time employees is limited to 40 % of the total hours worked by both full-time and part-time employees. as stated in the problem, each full-time employee works 7 hours out of an 8 hour day.

4)

Disregard the 6th factor(constraint) listed in the case.

5)

The coefficients in the objective function is the hourly wage multiplied by hours worked. assume that full-time employees are paid for lunch ( they are paid for 8 hours but only work 7 hours).

6)

Using the linear programming module in QM software, input the problems (25 decision variables and 11 constraints) and solve the problem.

7)

Copy the formulation from the linear programming module and paste it into the integer programming module. verify that all of the constraints copy over correctly! solve the problem. if your answer for the integer program has no full-time employees, reverse the sign in the 11th constraint.

Requirements:
Grid from (1)
solution output from a linear programming module
  solution output from linear programming module solution and the integer program solution.
                                           
Employee Type  Shift Worked   Linear Program Solution  Integer Program solution
  Full time          9:00am to 5:00pm  
  Full time          9:00am to 6:00pm
  Full time           9:00am to 7:00pm
  part time          9:00am to 1:00pm
  part time          9:00am to 2:00pm
  part time          9:00am to 3:00pm
  part time          9:00am to 4:00pm
  part time          10:00am to 2:00pm
  part time          10:00am to 3:00pm
  part time          10:00am to 4:00pm
  part time          10:00am to 5:00pm
  part time          11:00am to 3:00pm
  part time          11:00am to 4:00pm
  part time          11:00am to 5:00pm
  part time          11:00am to 6:00pm
  part time          Noon to 4:00 pm
  part time          Noon to 5:00 pm
  part time          Noon to 6:00 pm
  part time          Noon to 7:00 pm
  part time          1:00pm to 5:00pm
  part time          1:00pm to 6:00pm
  part time          1:00pm to 7:00pm
  part time          2:00pm to 6:00pm
  part time          2:00pm to 7:00pm
  part time          3:00pm to 7:00pm
  Total Cost  

1. What is the minimum-cost schedule for the bank?

Objective Function:Minimize total personnel cost: $80.88F + $31.28(P1+P2+P3+P4+P5+P6+P7) + $16.16FO

Let:

Ffull-time tellers where F≤35FO

full-time tellers working OT where FO ≤ 5P1

part-timers (PT) start 9am, leave between 1pmP2

PT start at 10am, leave between 2pmP3

PT start at 11am, leave between 3pmP4

PT start at 12pm, leave between 4pmP5

PT start at 1pm, leave between 5pmP6

PT start at 2pm, leave at 6pm

Limitations:Part time workers cannot exceed 40% of total hours required each day,

which is the sum of the tellers needed each hour

4 (P1 + P2 + P3 + P4 + P5 + P6 + P7) ≥ 0.40 (14 + 25 + 26 + 38 + 55 + 60 + 51 + 29 + 14 + 19)

4P1+4P2+4P3+4P4+4P5+4P6+4P7 ≤ .40 (331)

Variables FP1P2P3P4P5P6P7FO

Values 29.270.000.0011.3721.730.000.000.000.00

Cost$ 80.88 $ 31.28 $ 31.28 $ 31.28 $ 31.28 $ 31.28 $ 31.28 $ 31.28 $ 16.16

Shift Covers Time Period (1=yes, 0=no         

Constraints                                                                                                                                                                                                  LHS

9am-10am	1	1							1	29.27	≥
10am-11a	1	1	1							29.27	≥
11am-12p	0.5	1	1	1						26.00	≥
12pm-1pm	0.5	1	1	1	1					47.73	≥
1pm-2pm	1		1	1	1	1				62.37	≥
2pm-3pm	1			1	1	1	1			62.37	≥
3pm-4pm	1				1	1	1	1		51.00	≥
4pm-5pm	1					1	1	1		29.27	≥
5pm-6pm	1						1	1		29.27	≥
6pm-7pm	1							1		29.27	≥
Max full	1									29.27	≤
Total PT hour		4	4	4	4	4	4	4		132.40	≤

Total Cost of Labor$ 3,402.46

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