(20 points) A large department store operates 7 days a week. The manager estimat
ID: 391415 • Letter: #
Question
(20 points) A large department store operates 7 days a week. The manager estimates that the minimum number of salespersons required to provide prompt service is 12 for Monday, 18 for Tuesday, 20 for Wednesday, 28 for Thursday, 32 for Friday, and 40 for each of Saturday and Sunday. Each salesperson works 5 days a week, with the two consecutive off-days staggered thro ughout the week. Develop a Linear programming model to determine how many salespersons should be contracted, and how should their off das be allocated. Hint: define your decision variable as the number of workers that start at day i (Monday, Tuesday, etc.) if the worker starts on Monday, the worker has to rest Saturday and Sunday. If the worker starts on Tuesday, the worker rest on Sunday and Monday.Explanation / Answer
Decision variables:
The seven shifts will be required to schedule employees for 5 consecutive days and next 2 days off. Following the schedule for each shift where 1 represent working day and 0 represent off-day:
Shift
Day
1
2
3
4
5
6
7
# of Employees Required
Monday
1
0
0
1
1
1
1
12
Tuesday
1
1
0
0
1
1
1
18
Wednesday
1
1
1
0
0
1
1
20
Thursday
1
1
1
1
0
0
1
28
Friday
1
1
1
1
1
0
0
32
Saturday
0
1
1
1
1
1
0
40
Sunday
0
0
1
1
1
1
1
40
In scheduling it has to determine how many employees are required in each shift such that the requirement per day is minimized.
Let Xi be the number of employees in ith shift, where i = 1, 2, 3, 4, 5, 6, 7 for the shifts starting from Monday…to... Sunday respectively.
Objective Function:
Min Z = X1 + X2 + X3 + X4 + X5 + X6 + X7
Subject To:
Constraint:
Employees required per day constraint:
Day
Equation
Monday
X1 + 0X2 + 0X3 + X4 + X5 + X6 + X7>= 12
Tuesday
X1 + X2 + 0X3 + 0X4 + X5 + X6 + X7>= 18
Wednesday
X1 + X2 + X3 + 0X4 + 0X5 + X6 + X7>= 20
Thursday
X1 + X2 + X3 + X4 + 0X5 + 0X6 + X7>= 28
Friday
X1 + X2 + X3 + X4 + X5 + 0X6 + 0X7>= 32
Saturday
0X1 + X2 + X3 + X4 + X5 + X6 + 0X7>= 40
Sunday
0X1 + 0X2 + X3 + X4 + X5 + X6 + X7>= 40
Non-negative constraint: Xi >= 0
Shift
Day
1
2
3
4
5
6
7
# of Employees Required
Monday
1
0
0
1
1
1
1
12
Tuesday
1
1
0
0
1
1
1
18
Wednesday
1
1
1
0
0
1
1
20
Thursday
1
1
1
1
0
0
1
28
Friday
1
1
1
1
1
0
0
32
Saturday
0
1
1
1
1
1
0
40
Sunday
0
0
1
1
1
1
1
40
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