Moores is a large clothing store in Ottawa that operates 7 days per week.The sto

Question

Moores is a large clothing store in Ottawa that operates 7 days per week.The store needs the following number of full-time employees working each day of the week:

Each employee must work 5 consecutive days each week and then have 2 days off. For example, any employee who works Sunday through Thursday has Friday and Saturday off. The store currently has a total of 60 employees available to work. Moores has developed the following set of prioritized goals for employee scheduling:

1. The store would like to avoid hiring any additional employees

2. The most important days for the store to be fully staffed are Saturday and Sunday

3. The next most important day to be fully staffed is Friday

4. The store would like to be fully staffed the remaining 4 days in the week Formulate a goal programming model to determine the number of employees who should begin their 5-day workweek each day of the week to achieve the store’s objectives.

Formulate the problem (write variables, objective function, constraints). Use prioritization and re-write the goals as constraints. (Hint: remember deviation variables)

Day # of employees Sunday 47 Monday 22 Tuesday 28 Wednesday 35 Thursday 34 Friday 43 Saturday 53

Explanation / Answer

Decision variables:

The seven shifts will be required to schedule employees for 5 consecutive days and next 2 days off.

Shift

Day

1

2

3

4

5

6

7

# of Employees Required

Sunday

1

0

0

1

1

1

1

47

Monday

1

1

0

0

1

1

1

22

Tuesday

1

1

1

0

0

1

1

28

Wednesday

1

1

1

1

0

0

1

35

Thursday

1

1

1

1

1

0

0

34

Friday

0

1

1

1

1

1

0

43

Saturday

0

0

1

1

1

1

1

53

In scheduling, 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 for the shifts starting from Sunday, Monday….Saturday respectively.

First define constraints

Constraint:

The maximum available employees at store:

X1 + X2 + X3 + X4 + X5 + X6 + X7 <= 60

Employees required per day constraint:

Day

Equation

Sunday

X1 + 0X2 + 0X3 + X4 + X5 + X6 + X7 >= 47

Monday

X1 + X2 + 0X3 + 0X4 + X5 + X6 + X7 >= 22

Tuesday

X1 + X2 + X3 + 0X4 + 0X5 + X6 + X7 >= 28

Wednesday

X1 + X2 + X3 + X4 + 0X5 + 0X6 + X7 >= 35

Thursday

X1 + X2 + X3 + X4 + X5 + 0X6 + 0X7 >= 34

Friday

0X1 + X2 + X3 + X4 + X5 + X6 + 0X7 >= 43

Saturday

0X1 + 0X2 + X3 + X4 + X5 + X6 + X7 >= 53

To convert above LPP in Goal programming, lets define deviational variables for each constraint:

d+j and d-j be the deviational variables for the for constraints j

j = 1,2..8

For total number of workers available

X1 + X2 + X3 + X4 + X5 + X6 + X7 + d-1 – d+1 = 60

# of employees required on Sunday

X1 + 0X2 + 0X3 + X4 + X5 + X6 + X7 + d-2 – d+2 = 47

# of employees required on Monday

X1 + X2 + 0X3 + 0X4 + X5 + X6 + X7 + d-3 – d+3 = 22

# of employees required on Tuesday

X1 + X2 + X3 + 0X4 + 0X5 + X6 + X7 + d-4 – d+4 = 28

# of employees required on Wednesday

X1 + X2 + X3 + X4 + 0X5 + 0X6 + X7 + d-5 – d+5 = 35

# of employees required on Thursday

X1 + X2 + X3 + X4 + X5 + 0X6 + 0X7 + d-6 – d+6 = 34

# of employees required on Friday

0X1 + X2 + X3 + X4 + X5 + X6 + 0X7 + d-7 – d+7 = 43

# of employees required on Saturday

0X1 + 0X2 + X3 + X4 + X5 + X6 + X7 + d-8 – d+8 = 53

Non-negative constraint

All d+j and d-j >= 0 for all j.

Let,

P1 = Priority 1st: avoid hiring any additional employees (minimize d+1)

P2 = Priority 2nd: Fully staffed on Saturday and Sunday (minimize d-2 and d-8)

P3 = Priority 3rd: Fully staffed on Fridays (minimize d-7)

P4 = Priority 4th: Fully staffed from Monday to Thursday (minimize d-3, d-4, d-5, and d-6)

The objective function is given as follows:

Min Z = P1(d+1) + P2(d-2 + d-8) + P3(d+7) + P4(d-3 + d-4 + d-5 + d-6)

The formulation can be summarized as follows:

Objective Function

Min Z = P1(d+1) + P2(d-2 + d-8) + P3(d+7) + P4(d-3 + d-4 + d-5 + d-6)

Subject to:

For total number of workers available

X1 + X2 + X3 + X4 + X5 + X6 + X7 + d-1 – d+1 = 60

# of employees required on Sunday

X1 + 0X2 + 0X3 + X4 + X5 + X6 + X7 + d-2 – d+2 = 47

# of employees required on Monday

X1 + X2 + 0X3 + 0X4 + X5 + X6 + X7 + d-3 – d+3 = 22

# of employees required on Tuesday

X1 + X2 + X3 + 0X4 + 0X5 + X6 + X7 + d-4 – d+4 = 28

# of employees required on Wednesday

X1 + X2 + X3 + X4 + 0X5 + 0X6 + X7 + d-5 – d+5 = 35

# of employees required on Thursday

X1 + X2 + X3 + X4 + X5 + 0X6 + 0X7 + d-6 – d+6 = 34

# of employees required on Friday

0X1 + X2 + X3 + X4 + X5 + X6 + 0X7 + d-7 – d+7 = 43

# of employees required on Saturday

0X1 + 0X2 + X3 + X4 + X5 + X6 + X7 + d-8 – d+8 = 53

Non-negative constraint

All d+j and d-j >= 0 for all j.

Shift

Day

1

2

3

4

5

6

7

# of Employees Required

Sunday

1

0

0

1

1

1

1

47

Monday

1

1

0

0

1

1

1

22

Tuesday

1

1

1

0

0

1

1

28

Wednesday

1

1

1

1

0

0

1

35

Thursday

1

1

1

1

1

0

0

34

Friday

0

1

1

1

1

1

0

43

Saturday

0

0

1

1

1

1

1

53

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