Formulate the following question. Do not give solution using excel. Just FORMULA
ID: 464891 • Letter: F
Question
Formulate the following question. Do not give solution using excel. Just FORMULATE (express in mathematical expression) problem. I don't know how to formulate scheduling problems. not good in precalculus too. please explain well.
DON'T JUST PULL CONTRAINTS OUT OF THE BLUE. PLEASE EXPLAIN HOW YOU GOT THEM. I know the monday constraint is x4 + x5 + x6 + x7. But I dont know how and why.
Problem statement: a post office requires different numbers of full-time employees on different days of the week. Each full-time employee must work five consecutive days and then receive two days off.
Step1: define the decision variables
Step 2: set up the objective function
Step 3: identify the constraints
Day number of full time employees required 1= momday 17 2= tuesday 13 3= wednesday 15 4= thursday 19 5= friday 14 6= saturday 16 7= sunday 11Explanation / Answer
Let i represents number shifts required such that employee in i the shift works for consecutively 5 days and take s off on next two days. As the post office requires to employees to work on seven days a week, post office will have seven shifts with varying off days.
i = 1, represent shift 1, in which employees work from Monday to Friday and take off on Saturday and Sunday.
i = 2, represent shift 2, in which employees work from Tuesday to Saturday and take off on Sunday and Monday.
i = 3, represent shift 3, in which employees work from Wednesday to Sunday and take off on Monday and Tuesday and so on.
The arrangement of shifts is tabulated as follows:
Shift 1
Shift 2
Shift 3
Shift 4
Shift 5
Shift 6
Shift 7
Monday
W
O
O
W
W
W
W
Tuesday
W
W
O
O
W
W
W
Wednesday
W
W
W
O
O
W
W
Thursday
W
W
W
W
O
O
W
Friday
W
W
W
W
W
O
O
Saturday
O
W
W
W
W
W
O
Sunday
O
O
W
W
W
W
W
Decision variables:
In scheduling it has to determine how many employees are required in each shift such that the minimum requirement per day is fulfilled.
Let Xi be the number of employees in ith shift, where i = 1,2,3,4,5,6,7
X1 = number of employees in shift 1 (sat and sun off)
X2 = number of employees in shift 2 (sun and mon off)
X3 = number of employees in shift 3 (mon and tue off)
X4 = number of employees in shift 4 (tue and wed off)
X5 = number of employees in shift 5 (wed and thus off)
X6 = number of employees in shift 1 (thus and fri off)
X7 = number of employees in shift 1 (fri and sat off)
Define Objective Function:
While scheduling the objective is to minimizes the total requirement of employees. For Post office the objective function is to minimize total requirement of employees in each shift. Objective function is written as follows:
Minimize, Z = X1 + X2 + X3 + X4 + X5 + X6 + X7
Constraint:
The major constraint is the minimum requirement of the employees per day.
For Monday minimum requirement of employees is 17.
On Monday employees of Shift 1, 4, 5, 6, and 7, thus constraint for Monday is written as:
X1 + 0X2 + 0X3 + X4 + X5 + X6 + X7 >= 17
For Tuesday minimum requirement of employees is 13.
On Tuesday employees of Shift 1, 2, 5, 6, and 7, thus constraint for the day is written as:
X1 + X2 + 0X3 + 0X4 + X5 + X6 + X7 >= 13
For Wednesday minimum requirement of employees is 15.
On Wednesday employees of Shift 1, 2, 3, 6, and 7, thus constraint for the day is written as:
X1 + X2 + X3 + 0X4 + 0X5 + X6 + X7 >= 15
Similarly,
For Thursday: X1 + X2 + X3 + X4 + 0X5 + 0X6 + X7 >= 19
For Friday: X1 + X2 + X3 + X4 + X5 + 0X6 + 0X7 >= 14
For Saturday: 0X1 + X2 + X3 + X4 + X5 + X6 + 0X7 >= 16
For Sunday: 0X1 + 0X2 + X3 + X4 + X5 + X6 + X7 >= 11
Nonnegative constraint:
The number of scheduling employees cannot be negative,
Xi >= 0,for each i.
Shift 1
Shift 2
Shift 3
Shift 4
Shift 5
Shift 6
Shift 7
Monday
W
O
O
W
W
W
W
Tuesday
W
W
O
O
W
W
W
Wednesday
W
W
W
O
O
W
W
Thursday
W
W
W
W
O
O
W
Friday
W
W
W
W
W
O
O
Saturday
O
W
W
W
W
W
O
Sunday
O
O
W
W
W
W
W