• Multiperiod Models
– Data Processing at the IRS
– Arbitrage with Bonds
• Transportation Models
– Gribbin Brewing
Decision Models -- Prof. Juran 2

• 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.
Decision Models -- Prof. Juran 3

• 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.
Decision Models -- Prof. Juran 4

• 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.
Decision Models -- Prof. Juran 5

Managerial Problem Definition
• Determine how many workers should be working and how the workers should allocate their hours during the next 5 weeks.
Decision Models -- Prof. Juran 6

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.
Decision Models -- Prof. Juran 7

Formulation
Decision Variables
Define X ij
= Number of workers on task i during week j.
Define P ij
= Production (forms processed on task i during week j).
i = tasks 1-2, j = weeks 1-5
Objective
Minimize Z =
2 5
 i  1 j  1
X ij
C i
Where C i
is the cost of hiring a worker for task i for one week.
C
1
= C
2
= $500
Decision Models -- Prof. Juran 8

Formulation
Constraints
A new kind of constraint: Balance Equations for each task in each week.
Define I ij
= 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
 P i  1 , j  1
 P ij
Note that
P
0 , 0
P
0 , 1
P
0 , 2



40 , 000
30 , 000
60 , 000
Decision Models -- Prof. Juran 9

3
4
5
6
1
2
7
8
9
10
11
12
13
14
15
28
29
30
31
32
33
34
35
36
37
38
21
22
23
24
25
26
27
16
17
18
19
20
39
Inputs
Data prep form time
Constraints on forms (can't process more than are available for processing)
DP forms available for processing =B14
DP forms processed
DE forms processed
Total cost
Solution Methodology
A B C D E F G
Minutes
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
DE forms available for processing
Constraints on finishing all forms by the end of week 5
DP at end
DE at ent
=B8
=B14
15
10
40
$500
Week1
40000
1
1
1
1
0.25
<=
40
40000
>=
1
1
>=
1
=B15
129995
0
$5,000
Week2
30000
1
1
1
1
0.25
<=
40
69999
>=
1
1
>=
1
=F27-F29
=
=
Week3
60000
1
1
1
1
0.25
<=
40
0
0
0.25
<=
40
=B5*(SUM(B11:F11)+SUM(B12:F12))
0.25
<=
40
0.166667 0.166667 0.166667 0.166667 0.166667
<= <= <= <= <=
40 40 40 40 40
=B27-B29+C8
129998
>=
1
=B31-B33+C29
1
>=
1
Week4
0
1
1
1
1
129997
>=
1
1
>=
1
Week5
0
1
1
1
1
129996
>=
1
1
>=
1
=F14*$B$2/60
=F11*$B$4
=F15*$B$3/60
=F12*$B$4
H
Decision Models -- Prof. Juran 10

Solution Methodology
Decision Models -- Prof. Juran 11

6
7
4
5
8
9
10
1
2
3
Solution Methodology
A B C D E F
Inputs Minutes
Data prep form time
Data entry form time
Hours per worker per week
Pay/week per worker
Number of forms arriving
Decisions
15
10
40
$500
Week1
40000
Week2
30000
Week3
60000
Week4
0
Week5
0
250
0
187.5
0
375
0
0 0
0 541.6667
11
12
13
Workers on data prep
Workers on data entry
14
15
Number of DP forms completed during week
Number of DE forms completed during week
30
31
32
33
26
27
28
29
34
35
36
37
38
16
17
18
19
20
21
22
23
24
25
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
39 Total cost
40000
0
10000
<=
10000
0
<=
0
40000
>=
40000
40000
>=
0
0
0
$677,083
30000
0
7500
<=
7500
0
<=
0
30000
>=
30000
70000
>=
0
=
=
60000
0
15000
<=
15000
0
<=
0
60000
>=
60000
130000
>=
0
0
0
0
<=
0
0
0
0
130000
0
<=
0
0 21666.67
<=
0
<=
21667
0
>=
0
130000
>=
0
0
>=
0
130000
>=
130000
Decision Models -- Prof. Juran 12

Optimal Solution
Minimum total cost is $677,073.
All work could be done in four weeks.
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.
Decision Models -- Prof. Juran 13

Bond
1
2
3
Current Price
$10l.625
$10l.5625
$103.80
Expiration Date
8/15/2000
2/15/2001
2/15/2001
Coupon Rate
6.875
5.5
7.75
Decision Models -- Prof. Juran 14

Formulation
Given the current price structure, the question is whether there is a way to make an infinite amount of money. To answer this, we look for an arbitrage.
An arbitrage exists if there is a combination of bond sales and purchases today that yields
• A positive cash flow today
• Non-negative cash flows at all future dates
Decision Models -- Prof. Juran 15

Formulation
If such a strategy exists, then it is possible to make an infinite amount of money.
For example, if buying 10 units of bond 1 today and selling 5 units of bond 2 today yielded, say,
$1 today and nothing at all future dates, then we could make $k by purchasing 10k units of bond 1 today and selling 5k units of bond 2 today.
Decision Models -- Prof. Juran 16

Formulation
Decision Variables
How much to buy or sell of each bond. (Selling a bond is conceptually the same as buying a negative amount.)
Objective
Maximize cash flow at the end of the first period
(today).
Constraints
Non-negative cash flow at the end of all future periods.
Decision Models -- Prof. Juran 17

Formulation
Decision Variables
Define X i
= quantity of bond i purchased today.
Define C ij
= Cash flow per face value unit for bond i in period j, as shown below.
Bond 1
i = bonds Bond 2
Bond 3
j = periods
0 months from now 6 months from now 12 months from now
-$101.63
-$101.56
-$103.80
$103.44
$2.75
$3.88
$0.00
$102.75
$103.88
Decision Models -- Prof. Juran 18

Formulation
3 3
 i  1 j  1
X i
C ij
= total cash flows from all bonds over all three periods. j
3

 1
X i
C ij
= total cash flows from bond i over all three periods. i
3

 1
X i
C ij
= total cash flows from all bonds in period j.
Decision Models -- Prof. Juran 19

Formulation
Objective
Maximize Z = i
3

 1
X i
C i 1
Constraints i
3

 1
X i
C ij
 0 for j = 2, 3.
Decision Models -- Prof. Juran 20

Solution Methodology
20
21
22
14
15
16
17
18
19
7
8
5
6
3
4
1
2
9
10
11
12
13
A
Data on bonds
Bond
1
2
3
Face value of each bond
Decisions: number of bonds to buy or sell now
Bond
1
2
3
Cash flows
Months from now
Bond 1
Bond 2
Bond 3
Total cash flow
B C D E F G H
Current price Expiration date Coupon rate
$ 101.625
6 0.06875
$
$
101.563
103.800
12
12
0.05500
0.07750
100
Buy
1.00
1.00
1.00
=B3*(C11-B11)
Sell
0.00
0.00
0.00
If we maximize the cash flow in cell B22, without an upper bound constraint on it (as suggested in cells
B23 and B24), the Solver does not converge. To make it converge, we add this upper bound constraint. This lets us make $1 (or any other value you want to put in cell B24).
=IF($C5=D$16,1+$D5/2,IF($C5>D$16,$D5/2,0))*($B13-$C13)*$B$7
0
$ (101.63)
$ (101.56)
$ (103.80)
$ (306.99)
<=
$ 1.00
6
$ 103.44
$ 110.06
>=
$ -
12
$ -
$ 102.75
$ 103.88
$ 206.63
>=
$ -
=SUM(D17:D19)
Decision Models -- Prof. Juran 21

Decision Models -- Prof. Juran 22

Solution Methodology
Decision Models -- Prof. Juran 23

Solution Methodology
This is actually good news! It indicates an “unbounded” problem; one in which there are no constraints that limit the value of the objective function. In the context of this problem, it means that there is no limit on the amount of cash flow in the first period. In other words, there is an arbitrage opportunity.
Unfortunately, because Solver couldn’t solve the problem, we don’t know which bonds to buy and sell.
We can get around this by playing a little trick; we introduce a new constraint limiting the objective function artificially.
Decision Models -- Prof. Juran 24

Decision Models -- Prof. Juran 25

20
21
22
14
15
16
17
18
19
7
8
5
6
3
4
1
2
9
10
11
12
13
Optimal Solution
B A
Data on bonds
Bond
1
2
3
Face value of each bond
Decisions: number of bonds to buy or sell now
Bond
1
2
3
Cash flows
Months from now
Bond 1
Bond 2
Bond 3
Total cash flow
Current price Expiration date Coupon rate
$
$
$
101.625
101.563
103.800
6
12
12
0.06875
0.05500
0.07750
100
Buy
0.21
20.30
0.00
C
Sell
0.00
0.00
20.08
D
0
$ (21.60)
$ (2,061.87)
$ 2,084.48
$ 1.00
<=
$ 1.00
6
$ 21.99
$ 55.83
$ (77.82)
$ -
>=
$ -
12
$ -
$ 2,085.98
$ (2,085.98)
$ -
>=
$ -
Decision Models -- Prof. Juran 26

Conclusions
Buying bonds 1 and 2 today, while selling bond 3, offers an arbitrage opportunity.
Decision Models -- Prof. Juran 27

Back to Reality
Usually bonds are bought at an ask price and sold at a bid price.
Consider the same three bonds and suppose that the ask and bid prices are as listed here.
Bond
1
2
3
Ask Price Bid Price
$101.6563 $101.5938
$101.5938 $101 5313
$103.7813 $103.7188
Decision Models -- Prof. Juran 28

14
15
16
17
18
19
20
21
22
7
8
5
6
3
4
1
2
9
10
11
12
13
Optimal Solution
B A
Data on bonds
Bond
1
2
3
Face value of each bond
Decisions: number of bonds to buy or sell now
Bond
1
2
3
Cash flows
Months from now
Bond 1
Bond 2
Bond 3
Total cash flow
Ask price
100
Buy
0
0
0
0
$ -
$ -
$ -
$ -
C
Sell
0
0
0
6
$ -
$ -
$ -
$ -
>=
$ -
D
12
>=
E
Bid price Expiration date Coupon rate
6
12
12
0.06875
0.055
0.0775
Decision Models -- Prof. Juran 29

Conclusions
This result indicates that no arbitrage opportunity exists.
The only way to have non-negative cash flows in the first period and zero cash flows in all future periods is not to invest at all.
Decision Models -- Prof. Juran 30

Regional brewer Andrew Gribbin distributes kegs of his famous beer through three warehouses in the greater News York City area, with current supplies as shown:
Warehouses Supply
Hoboken
Bronx
Brooklyn
80
145
120
Decision Models -- Prof. Juran 31

On a Thursday morning, he has his usual weekly orders from his four loyal customers, as shown :
Bars
Der Ratkeller
McGoldrick's Pub
Night Train Bar & Grill
Stern Business School
Demand
80
65
70
85
Decision Models -- Prof. Juran 32

Tracy Chapman, Gribbin’s shipping manager, needs to determine the most cost-efficient plan to deliver beer to these four customers, knowing that the costs per keg are different for each possible combination of warehouse and customer:
Hoboken
Bronx
Brooklyn
Ratkeller McGoldrick's Night Train Stern
$4.64
$3.52
$9.95
$5.13
$4.16
$6.82
$6.54
$6.90
$3.88
$8.67
$7.91
$6.85
Decision Models -- Prof. Juran 33

a) What is the optimal shipping plan?
b) How much will it cost to fill these four orders?
c) Where does Gribbin have surplus inventory?
d) If Gribbin could have one additional keg at one of the three warehouses, what would be the most beneficial location, in terms of reduced shipping costs? e) Gribbin has an offer from Lu Leng Felicia, who would like to sublet some of Gribbin’s Brooklyn warehouse space for her tattoo parlor. She only needs 240 square feet, which is equivalent to the area required to store 40 kegs of beer, and has offered Gribbin $0.25 per week per square foot. Is this a good deal for Gribbin? What should Gribbin’s response be to
Lu Leng?
Decision Models -- Prof. Juran 34

Managerial Problem Formulation
Decision Variables
Numbers of kegs shipped from each of three warehouses to each of four customers (12 decisions).
Objective
Minimize total cost.
Constraints
Each warehouse has limited supply.
Each customer has a minimum demand.
Kegs can’t be divided; numbers shipped must be integers.
Decision Models -- Prof. Juran 35

Mathematical Formulation
Decision Variables
Define X ij
= Number of kegs shipped from warehouse i to customer j.
Define C ij
= Cost per keg to ship from warehouse i to customer j.
i = warehouses 1-3, j = customers 1-4
Decision Models -- Prof. Juran 36

Mathematical Formulation
Objective
Minimize Z =
3 4
 i  1 j  1
X ij
C ij
Constraints
Define S i
= Number of kegs available at warehouse i.
j
4

 1
X ij
 S i
Define D j
= Number of kegs ordered by customer j.
i
3

 1
X ij
 D j
Do we need a constraint to ensure that all of the X ij are integers?
Decision Models -- Prof. Juran 37

7
8
9
10
11
12
13
14
4
5
6
1
2
3
A
Total Cost
Costs
Hoboken
Bronx
Brooklyn
B
74.97
C D E
=SUMPRODUCT(B4:E6,B12:E14)
Shipping Plan Ratkeller McGoldrick's Night Train Stern
Hoboken
Bronx
Brooklyn
1
1
1
1
1
1
1
1
1
1
1
1
3
=
80
3
=
65
3
=
70
3
=
85
$
$
$
4.64
3.52
9.95
$
$
$
5.13
4.16
6.82
$
$
$
6.54
6.90
3.88
F G H I
=SUM(B4:E4)
4 <= 80
4 <= 145
4 <= 120
=SUM(E4:E6)
Decision Models -- Prof. Juran 38

Decision Models -- Prof. Juran 39

1
4
5
6
2
3
7
8
9
10
11
12
13
14
A
Total Cost
Costs
Hoboken
Bronx
Brooklyn
1469.55
$
$
$
B
4.64
3.52
9.95
C
$
$
$
5.13
4.16
6.82
D
$
$
$
6.54
6.90
3.88
$
$
$
E
8.67
7.91
6.85
F G H
Shipping Plan Ratkeller McGoldrick's Night Train Stern
Hoboken
Bronx
Brooklyn
0
80
0
0
65
0
0
0
70
35 35 <= 80
0 145 <= 145
50 120 <= 120
80
=
80
65
=
65
70
=
70
85
=
85
Decision Models -- Prof. Juran 40

20
21
22
23
24
25
26
27
28
29
30
3
4
1
2
5
6
7
12
13
14
15
8
9
10
11
16
17
18
19
A B C D
Microsoft Excel 15.0 Sensitivity Report
Worksheet: [03a-03-trans.xls]Model
Variable Cells
Cell
Constraints
Cell
$F$4 Hoboken
$F$5 Bronx
$F$6 Brooklyn
Name
$B$4 Hoboken Ratkeller
$C$4 Hoboken McGoldrick's
$D$4 Hoboken Night Train
$E$4 Hoboken Stern
$B$5 Bronx Ratkeller
$C$5 Bronx McGoldrick's
$D$5 Bronx Night Train
$E$5 Bronx Stern
$B$6 Brooklyn Ratkeller
$C$6 Brooklyn McGoldrick's
$D$6 Brooklyn Night Train
$E$6 Brooklyn Stern
Name
$B$7 Ratkeller
$C$7 McGoldrick's
$D$7 Night Train
$E$7 Stern
E F G H
Final Reduced Objective Allowable Allowable
Value Cost Coefficient Increase Decrease
80
65
0
0
0
35
0
0
70
50
0
0
0.15
0
0.84
0
0
0
2.17
0.21
7.28
3.51
0
0
4.64
5.13
6.54
8.67
3.52
4.16
6.9
7.91
9.95
6.82
3.88
6.85
1E+30
0.15
1E+30
0.21
0.15
0.21
1E+30
1E+30
1E+30
1E+30
0.84
1.82
0.15
0.21
0.84
1.82
1E+30
0.15
2.17
0.21
7.28
3.51
1E+30
0.84
Final Shadow Constraint Allowable Allowable
Value Price
80
65
70
85
35
145
120
4.49
5.13
5.7
8.67
0
-0.97
-1.82
R.H. Side
80
65
70
85
80
145
120
Increase
45
45
45
45
1E+30
0
35
Decrease
35
35
0
0
45
45
45
Decision Models -- Prof. Juran 41

Where does Gribbin have surplus inventory?
The only supply constraint that is not binding is the
Hoboken constraint. It would appear that Gribbin has
45 extra kegs in Hoboken.
Decision Models -- Prof. Juran 42

If Gribbin could have one additional keg at one of the three warehouses, what would be the most beneficial location, in terms of reduced shipping costs?
Decision Models -- Prof. Juran 43

According to the sensitivity report,
•One more keg in Hoboken is worthless.
•One more keg in the Bronx would have reduced overall costs by $0.76.
•One more keg in Brooklyn would have reduced overall costs by $1.82.
Decision Models -- Prof. Juran 44

Gribbin has an offer from Lu Leng Felicia, who would like to sublet some of Gribbin’s Brooklyn warehouse space for her tattoo parlor. She only needs 240 square feet, which is equivalent to the area required to store 40 kegs of beer, and has offered Gribbin $0.25 per week per square foot.
Is this a good deal for Gribbin?
What should Gribbin’s response be to Lu Leng?
Decision Models -- Prof. Juran 45

Assuming that the current situation will continue into the foreseeable future, it would appear that Gribbin could reduce his inventory in Hoboken without losing any money (i.e. the shadow price is zero).
However, we need to check the sensitivity report to make sure that the proposed decrease of 40 kegs is within the allowable decrease.
This means that he could make a profit by renting space in the Hoboken warehouse to Lu Leng for $0.01 per square foot.
Decision Models -- Prof. Juran 46

Lu Leng wants space in Brooklyn, but Gribbin would need to charge her more than $1.82 for every six square feet
(about $0.303 per square foot), or else he will lose money on the deal.
Note that the sensitivity report indicates an allowable decrease in Brooklyn that is enough to accommodate Lu
Leng.
Decision Models -- Prof. Juran 47

As for the Bronx warehouse, note that the allowable decrease is zero. This means that we would need to rerun the model to find out the total cost of renting Bronx space to Lu Leng.
A possible response from Gribbin to Lu Leng:
“I can rent you space in Brooklyn, but it will cost you
$0.35 per square foot. How do you feel about
Hoboken?”
Decision Models -- Prof. Juran 48

• Multiperiod Models
– Data Processing at the IRS
– Arbitrage with Bonds
• Transportation Models
– Gribbin Brewing
Decision Models -- Prof. Juran 49