Solution to IRS Data Processing1
To process income tax forms, the Internal Revenue Service (IRS) first sends each
form through the data preparation (DP) department, where information is coded
for computer entry. Then the form is sent to data entry (DE), where it is entered
into the computer. During the next 3 weeks, the following numbers of forms will
arrive: week 1, 40,000; week 2, 30,000; week 3, 60,000. All employees work 40
hours per week and are paid $500 per week. Data preparation of a form requires
15 minutes, and data entry of a form requires 10 minutes. Each week an
employee is assigned to either data entry or data preparation. The IRS must
complete processing all forms by the end of week 5 and wants to minimize the
cost of accomplishing this goal. Assume all employees are capable of performing
data preparation or data entry, but must be assigned to one task for an entire
week at a time.
Determine how many workers should be working and how the workers should
allocate their hours during the next 5 weeks.
Managerial Problem Formulation
Decision Variables
Numbers of workers for two tasks over five weeks (10 decisions) and numbers of
forms completed on each task in each week (10 decisions).
Objective
Minimize total cost.
Constraints
Forms arrive at a fixed schedule.
All work must be completed in five weeks.
Data prep task cannot begin until the forms arrive.
Data entry task cannot begin until data prep task is finished.
The plan can’t call for more labor than is available for either task in any week.
Based on 4-89 (p. 177) in Practical Management Science (2nd ed., Winston and Albright, 2001
Duxbury Press). Based on Lanzenauer et al. (1987). Solution by David Juran, 2001.
1
Mathematical Formulation
Decision Variables
Define Xij = Number of workers on task i during week j.
Define Pij = Production (forms processed on task i during week j).
i = tasks 1-2, j = weeks 1-5
Objective
Minimize Z =
2
5
 X C
i 1 j 1
ij
i
Where Ci is the cost of hiring a worker for task i for one week.
C1 = C2 = $500
Constraints
A new kind of constraint: Balance Equations for each task in each week.
Define Iij = Inventory (work ready to be processed on task i at the end of week j).
Ending “inventory” = beginning inventory + new forms arriving – current
period processing
I ij  I i , j 1  Pi 1 , j 1  Pij
Note that
P0 ,0  40,000
P0 ,1  30,000
P0 ,2  60,000
B60.2350
2
Prof. Juran
Spreadsheet Model
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
B
Minutes
15
10
40
$500
Inputs
Data prep form time
Data entry form time
Hours per worker per week
Pay/week per worker
C
D
E
F
G
Week1
40000
Week2
30000
Week3
60000
Week4
0
Week5
0
Decisions
Workers on data prep
Workers on data entry
1
1
1
1
1
1
1
1
1
1
Number of DP forms completed during week
Number of DE forms completed during week
1
1
1
1
1
1
1
1
1
1
0.25
<=
40
0.25
<=
40
0.25
<=
40
0.25
<=
40
0.25
<=
40
Number of forms arriving
Constraints on labor (need sufficient workers to complete forms)
Hours for DP needed
=F14*$B$2/60
Hours for DP available
Hours for DE needed
Hours for DE available
=B8
0.166667 0.166667 0.166667 0.166667 0.166667
<=
<=
<=
<=
<=
40
40
40
40
40
=B27-B29+C8
Constraints on forms (can't process more than are available for processing)
DP forms available for processing
=B14
DP forms processed
H
40000
>=
1
=B14
69999
>=
1
1
>=
1
1
>=
1
DE forms available for processing
DE forms processed
=B15
Constraints on finishing all forms by the end of week 5
DP at end
DE at ent
129995
0
Total cost
$5,000
129998
>=
1
129997
>=
1
129996
>=
1
1
>=
1
1
>=
1
1
>=
1
=B31-B33+C29
=F11*$B$4
=F15*$B$3/60
=F12*$B$4
=F27-F29
=
=
0
0
=B5*(SUM(B11:F11)+SUM(B12:F12))
Note that the balance equations are not constraints in the usual sense (i.e.
specified in Solver). We build them into the model, linking the tasks and weeks
together.
B60.2350
3
Prof. Juran
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
B
Minutes
15
10
40
$500
Inputs
Data prep form time
Data entry form time
Hours per worker per week
Pay/week per worker
Number of forms arriving
Decisions
Workers on data prep
Workers on data entry
Number of DP forms completed during week
Number of DE forms completed during week
Constraints on labor (need sufficient workers to complete forms)
Hours for DP needed
Hours for DP available
Hours for DE needed
Hours for DE available
Constraints on forms (can't process more than are available for processing)
DP forms available for processing
DP forms processed
DE forms available for processing
DE forms processed
Constraints on finishing all forms by the end of week 5
DP at end
DE at ent
Total cost
B60.2350
C
D
E
F
Week1
40000
Week2
30000
Week3
60000
Week4
0
Week5
0
250
0
187.5
0
375
0
40000
0
30000
0
60000
0
0
0
0
130000
10000
<=
10000
7500
<=
7500
15000
<=
15000
0
<=
0
0
<=
0
0
<=
0
0
<=
0
0
<=
0
40000
>=
40000
30000
>=
30000
60000
>=
60000
0
>=
0
0
>=
0
40000
>=
0
70000
>=
0
130000
>=
0
130000
>=
0
130000
>=
130000
0
0
=
=
0
0
0
0
0 541.6667
0 21666.67
<=
<=
0
21667
$677,083
4
Prof. Juran
Conclusions
Minimum total cost is $677,073.
All work could be done in four weeks.
B60.2350
5
Prof. Juran