Book reference: Ballou, Ronald H. (2004). “Business Logistics/ Supply Chain Management: Planning, Organizing, and Controlling the Supply Chain.” (5th Edition). Original reference of this document: http://wweb.uta.edu/insyopma/prater/ballou09_im.pdf

1. 93
CHAPTER 9
INVENTORY POLICY DECISIONS
1
The probability of finding all items in stock is the product of the individual probabilities.
That is,
(0.95)×(0.93) ×(0.87) ×(0.85) ×(0.94) ×(0.90) = 0.55
2
(a) The order fill rate is the weighted average of filling the item mix on an order. We can
setup the following table.
Order
(1)
Item mix probabilities
(2)
Frequency
of order
(3)=(1)×(2)
Marginal
probability
1 .95×.95×.95×.90×.90 = .69 0.20 0.139
2 .95×.95×.95 = .86 0.15 0.129
3 .95×.95×.90×.90 = .73 0.05 0.037
4 .95×.95×.95×.95×.95×.90×.90 = .62 0.15 0.094
5 .95×.95×.90×.90×.90×.90 = .59 0.30 0.178
6 .95×.95×.95×.95×.95 = .77 0.15 0.116
Order fill rate 0.693
Since 69.3 percent is less than 92 percent, the target order fill rate is not met.
(b) The item service levels that will give an order fill rate of 92 percent must be found by
trial and error. Although there are many combinations of item service levels that can
achieve the desired service level, a service level of 99 percent for items A, B, C, D, E,
and F, and 97 percent to 98 percent for the remaining items would be about right.
The order fill rates can be found as follows.
Order
(1)
Item mix probabilities
(2)
Frequency
of order
(3)=(1) ×(2)
Marginal
probability
1 (.99)3
×(.975)2
= .922 0.20 0.184
2 (.99)3
= .970 0.15 0.146
3 (.99)2
×(.975)2
= .932 0.05 0.047
4 (.99)5
×(.975)2
= .904 0.15 0.136
5 (.99)2
×(.975)4
= .886 0.30 0.266
6 (.99)5
= .951 0.15 0.143
Order fill rate 0.922

2. 94
3
This is a problem of push inventory control. The question is one of finding how many of
120,000 sets to allocate to each warehouse. We begin by estimating the total
requirements for each warehouse. That is,
Total requirements = Forecast + z×Forecast error
From Appendix A, we can find the values for z corresponding to the service level at each
warehouse. Therefore, we have:
Ware-
house
(1)
Demand
forecast, sets
(2)
Forecast
error, sets
(3)
Values
for z
(4)=(1)+(2)×(3)
Total require-
ments, sets
1 10,000 1,000 1.28 11,280
2 15,000 1,200 1.04 16,248
3 35,000 2,000 1.18 37,360
4 25,000 3,000 1.41 29,230
Total 85,000 94,118
We can find the net requirements for each warehouse as the difference between the
total requirements and the quantity on hand. The following table can be constructed:
There is 120,000 − 89,118 = 30,882 sets to be prorated. This is done by assuming that
the demand rate is best expressed by the forecast and proportioning the excess in relation
to each warehouse's forecast to the total forecast quantity. That is, for warehouse 1, the
proration is (10,000/85,000)×30,882 = 3,633 sets. Prorations to the other warehouses are
carried out in a similar manner. The allocation to each warehouse is the sum of its net
requirements plus a proration of the excess, as shown in the above table.
4
(a) The reorder point system is defined by the order quantity and the reorder point
quantity. Since the demand is known for sure, the optimum order quantity is:
Q DS IC*
/ ( , )( ) / ( . )( ) .= = =2 2 3 200 35 015 55 164 78 165, or cases
Ware-
house
(1)
Total
require-
ments
(2)
On hand
quantity
(3)=(1)−(2)
Net require-
ments
(4)
Proration of
excess
(5)=(3)+(4)
Allocation
1 11,280 700 10,580 3,633 14,213
2 16,248 0 16,248 5,450 21,698
3 37,360 2,500 34,860 12,716 47,576
4 29,230 1,800 27,430 9,083 36,513
94,118 89,118 30,882 120,000

3. 95
The reorder point quantity is:
ROP d LT= × = × =( , / ) .3 200 52 15 92 units
(b) The total annual relevant cost of this design is:
TC D S Q I C Q= × + × ×
= +
= +
=
/ /
( , )( ) / . ( . )( )( . ) /
. .
$1, .
*
2
3 200 35 164 78 015 55 164 78 2
679 69 67997
359 66
(c) The revised reorder point quantity would be:
ROP = × =( , / )3 200 52 3 185 units.
The ROP is greater than Q*
. It is possible under these circumstances the reorder
quantity may not bring the stock level above the ROP quantity. In deciding whether
the ROP has been reached, we add any quantities on order or in transit to the quantity
on hand as the effective quantity in inventory. Of course, we start with an adequate
in-stock quantity that is at least equal to the ROP quantity.
5
(a) The economic order quantity formula can be used here. That is,
Q DS IC*
/ ( )( , ) / ( . )( , ) .= = =2 2 300 8 500 010 8 500 775, or 78 students
(b) The number of times that the course should be offered is:
yearperfour timesaboutor,9.35.77/300/ **
=== QDN
6
This is a single-period inventory control problem. We have:
Revenue = $350/unit
Profit = $350 − $250 = $100/unit
Loss = 0.2×250 = $50/unit
Therefore,
CPn =
+
=
100
100 50
0 667.
Developing a table of cumulative frequencies, we have:

4. 96
Quantity Frequency
Cumulative
frequency
50 0.10 0.10
55 0.20 0.30
60 0.20 0.50
65 0.30 0.80 ⇐⇐⇐⇐Q*
70 0.15 0.95
75 0.05 1.00
1.00
CPn lies between quantities of 60 and 65. We round up and select 65 as the optimal
purchase order size.
7
This question can be treated as a single-order problem. We have:
Revenue = 1 + 0.01 = $1.01/$
Cost/Loss = 0.10(2/365) = $0.00055/$ which is the interest expense for two days
Profit = 1.01 − 1.00055 = $0.00945/$
and
CPn =
+
=
0 00945
0 00945 0 00055
0945
.
. .
.
For an area under the normal curve of 0.945 (see Appendix A), z = 1.60.
The planned number of withdrawals is:
Q*
= D + z×σ D = 120 + 1.60(20) = 152.00
The amount of money to stock in the teller machine over two days would be:
Money = Q*
×75 = 152.00×75 = $11,400
8
This is a single-period inventory control problem.
(a) We have:
Profit = 400 − 320
Loss = 320 − 300
Then,

5. 97
CPn =
−
− + −
=
400 320
400 320 320 300
080
( ) ( )
.
We now need to find the sales that correspond to a cumulative frequency of 0.80. In the
following table:
Q*
lies between 1,000 and 1,200 in the cumulative frequency table. We choose to
roundup to Q*
= 1,250 units.
(b) Carrying the excess inventory to next year,
CPn =
+ ×
=
80
80 0 2 320
0556
( . )
.
where the loss is the cost of holding a unit until the next year. The Q*
now lies between
750 and 1,000 units. We choose 1,000 units. Holding the excess units means a potential
loss of 0.2×320 = $64/unit, whereas discounting the excess units represents a loss of only
320 − 300 = $20/unit. Therefore, Cabot will need fewer units if they are held over in
inventory.
9
(a) The optimum order quantity is:
cases556)56)(3.0/()40)(52)(250,1(2/2*
=== ICDSQ
and the reorder point quantity is:
ROP d LT z sd= × + × '
where
s s LTd d
'
.= = =475 25 751
and zP=0 80. = 0.84.
Sales Frequency
Cumulative
frequency
500 0.2 0.2
750 0.2 0.4
1,000 0.3 0.7
1,250 0.2 0.9 ⇐⇐⇐⇐Q*
1,500 0.1 1.0
1.0

6. 98
Now,
ROP = + =( , )( . ) ( . )( ) ,1 250 25 084 751 3 756 cases
Policy: When the amount of inventory on hand plus any quantities on order or in
transit falls below ROP, reorder an amount Q*
.
(b) For the periodic review system, we first estimate the order review time:
T Q d* *
/ / , .= = =556 1 250 0 44 weeks
The max level is:
M d T LT z sd
* * '
( )= × + + ×
where sd
'
now is:
s s T LTd d
' *
. .= + = + =475 0 44 25 814 cases
Hence,
M*
, ( . . ) . ( ) ,= + + =1 250 0 44 25 084 814 4 359 cases
Policy: Find the amount of stock on hand every 0.44 weeks and place a reorder for
the amount equal to the difference between the quantity on hand and the max level
(M*
) of 4,359 cases.
(c) The total annual relevant cost for these policies is:
QEkDsICzsICQQDSTC zdd /2// )(
''
+++=
For the reorder point system:
TCQ = 1250(52)(40)/556 + .3(56)(556)/2
+ .3(56)(.84)(751) + 10(1250)(52)(751)(.1120)/556
= 4,676.26 + 4,670.40 + 10,598.11 + 98,332.37
= $118,277.14
For the periodic review system:
TCP = 1250(52)(40)/556 + .3(56)(556)/2

7. 99
+ .3(56)(.84)(814) + 10(1250)(52)(814)(.1120)/556
= 4,676.26 + 4,670.40 + 11,487.17 + 106,581.29
= $127,415.12
(d) The actual service level achieved is given by:
SL
s E
Q
d z
= −1
'
( )
For the reorder point system:
SLQ = − = −1
751 01120
556
1 015
( . )
.
or demand is met 85 percent of the time.
For the periodic review system:
SLP = − = −1
814 01120
556
1 016
( . )
.
or demand is met 84 percent of the time.
(e) This requires an iterative approach as follows:
Compute Q DS IC= 2 /
Compute P QIC Dk= −1 / , then z, then E(z)
Compute Q D S ks E ICd z= +2 ( ) /'
( )
Go back and stop when there is no change
in either P or Q
After the initial value of Q = 556.3, the process can be summarized in tabular form.

8. 100
Step Q P z E(z)
1 778.4 0.9856 2.19 0.0050
2 860.0 0.9799 2.06 0.0072
3 889.9 0.9778 2.01 0.0083
4 899.6 0.9777 2.00 0.0085
5 902.8 0.9767 1.99 0.0087
6 902.8 0.9767 1.99 0.0087
Now, for P = 0.9767, z = 1.99
ROP = 1,250(2.5) + 1.99(751) = 4,620 cases
and the total relevant cost is:
TC DS Q ICQ ICzs kDs E QQ d d z= + + +
= +
+
+
=
/ / /
, ( ) / . . ( )( . ) /
. ( )( . )( )
( , )( )( . ) / .
$40,
' '
( )2
65 000 4 9028 03 56 9028 2
03 56 199 751
10 65 000 751 0 0087 9028
275
This is considerably less than the $118,277.14 for the preset P at 0.80.
If you solve this problem using INPOL, you will get a slightly different answer. That
is, Q*
= 858. This simply is because z is carried to two significant digits rather than
the four significant digits used in the above calculations.
10
Refer to the solution of problem 10-9 for the general approach.
(a) Q*
= 556.3 cases
and
ROP d LT z LT s d sd LT= × + × + ×
= + × + ×
= +
=
2 2 2
2 2 2
1 250 25 084 25 475 1 250 05
3125 084 977 08
3 946
, ( . ) . . , .
, . ( . )
, cases
(b) An approximation for T*
= Q*
/d, or
T*
= 556/1,250 = 0.44 weeks
and approximating sd
'
as

9. 101
s T LT s d sd d LT
' *
( )
( . . )( ) , ( . )
,
= + + ×
= + +
=
2 2 2
2 2 2
0 44 25 475 1250 05
1 027 cases
So,
Max d T LT z sd= + + ×
= + +
=
( )
, ( . . ) . ( , )
,
* '
1 250 0 44 25 084 1 027
4 537 cases
(c) According to INPOL,
TCQ = 4,686 + 4,686 + 128,195 + 13,862 = $151,429
TCP = 4,686 + 4,686 + 134,751 + 14,571 = $158,694
(d) According to INPOL,
SLQ = 80.28 percent
SLP = 79.27 percent
(e) According to INPOL,
Q*
= 930 cases, ROP = 5,128 cases,
TCQ = $49,532, SLQ = 99.22 percent
T*
= 0.76 weeks, MAX = 6,257 cases
TCP = $52,894, SLP = 99.18 percent
11
(a) The production run quantity is:
Q
DS
IC
p
p d
p
* ( )( )( )
. ( )
,= ×
−
= ×
−
=
2 2 100 250 250
0 25 75
300
300 10
1 000 units
(b) The production run cycle is:
Qp
*
, / .= =1 000 300 333 days

10. 102
(c) The number of production runs is:
D Qp/ ( ) / ,*
= =100 250 1 000 25 runs per year
12
(a) The order quantity is:
Q DS IC*
/ ( , )( )( . ) / ( . )( )= = =2 2 2 000 250 100 030 35 309 valves
and the reorder point quantity is:
ROP d LT z s LTd= × + ×
but sd = 0. Therefore,
ROP = =( , / )( )2 000 8 1 250 valves
(b) Boxes are set up that contain 309 valves - the optimum order quantity. When an
order arrives from a supplier, 250 valves are set aside in a separate box and are
treated as the backup stock. The residual 309 − 250 = 59 valves are used on the
production line. When the 59 valves at the production line are used up, the backup
box containing 250 valves is brought to the production line and the empty box is sent
to the supplier refilling. One hour later when the order arrives, there will be zero
valves remaining at the production line. Then, 250 valves are set aside and 59 are
sent to the production line. The cycle is then repeated.
This problem approach is similar to that of the KANBAN system. Lead times are
very short so that lead times are virtually certain. Demand is certain, since it is fixed
by the production schedule. Boxes or cards are used to assure movement of the most
economic quantity. KANBAN is essentially classic economic reorder point inventory
control under certainty.
13
(a) The economical quantity of cars to be called for at a time is found by the economic
order quantity formula:
Q DS IC*
/ ( )( )( ) / ( . )( , )( ) / , .= = =2 2 40 52 500 0 25 90 000 30 2 000 785, or 79 cars
(b) This is the reorder point quantity:
ROP d LT z s LTd= × + ×
where z = 1.28 from Appendix A for an area under the curve equal to 0.90.
Therefore,

11. 103
ROP = +
=
40 1 128 333 1
443
( ) . ( . )
. cars, or 44.3(90,000 / 2,000)
= 1,994 tons of soda ash
14
(a) This is a reorder point design under conditions of uncertainty for both demand and
lead-time. We assume that the probability of an out of stock is given. Therefore, the
order quantity is:
Q DS IC*
/ ( )( )( ) / ( . )( ) .= = =2 2 50 365 50 030 45 3677 units
and
ROP d LT z sd= × + × '
where
z = 1.04 (see Appendix A) for the area under the curve equal to 0.85 and
s s LT d sd d LT
'
( ) ( )( ) .= + = + =2 2 2 2 2 2
15 7 50 2 107 6 units
Therefore,
ROP = + =50 7 104 107 6 4619( ) . ( . ) . units
(b) This is the periodic review system design under uncertainty. The complexity requires
us to make some approximations here. The time interval for review of the stock level
is:
T Q d* *
/ . / .= = =367 7 50 735 days
The MAX level is:
MAX d T LT z sd= + + ×( )* '
where z = 1.04 and sd
'
is approximated as:
s T LT s d sd d LT
' *
( )( ) ( )
( . )( ) ( )
.
= + +
= + +
=
2 2 2
2 2 2
735 7 15 50 2
1150 units
Therefore,

12. 104
MAX = 50(7.35 + 7) + 1.04(115.0)
= 837.1 units
(c) Since the service level is specified, the probability is not set at the optimum level.
Knowing the out-of-stock cost allows us to find the most appropriate service level.
Since this is an iterative process, we use INPOL to carry out the calculations.
The optimized service level yields a reorder point design of
Q*
= 410 units and ROP = 571 units
and the total relevant cost drops from $12,642 in part a to $8,489. The demand in
stock in part a was 97.74 percent, and it now increases to 99.81 percent.
15
(a) Find the common review time:
T O s I C DI i i
*
( ) /
( ) / [( . / )( , )]
.
= +
= + × × + × ×
=
∑∑2
2 100 0 03 52 2 25 2 000 1 90 500
25 weeks
Then,
M d T LT z s T LTA A A dA
* * *
( )= + + × +
where zA = 1.282 for P = 0.90
MA
*
, ( . . ) . ( ) . . ,= + + + =2 000 25 15 1282 100 25 15 8 256 units
and
MB
*
( . . ) . ( ) . . ,= + + + =500 25 15 0842 70 25 15 2118 units
where zB = 0.842 for P = 0.80.
The control system works as follows: the stock levels of both items are reviewed
every 2.5 weeks. The reorder size for A is the difference between the amount on hand
and 8,256 units. The reorder size for B is the difference between the amount on hand
and 2,118 units.
(b) The average amount in inventory is expected to be:

13. 105
AIL d T z s T LTd= × + × +* *
/ 2
For A:
AILA = + + =2 000 25 2 128 100 25 15 2 756, ( . ) / . ( ) . . , units
For B:
AILB = + + =500 25 2 0842 70 25 15 743( . ) / . ( ) . . units
(c) The service level is given by:
SL s E d Td z= − × ×1 '
( )
*
/
For A:
SLA = − + =1 100 25 15 0 0475 2 000 25 0998. . ( . ) / , ( . ) .
For B:
SLB = − + =1 70 25 15 01120 500 25 0987. . ( . ) / ( . ) .
(d) We set T*
= 4 and cycle through the previous calculations. Thus, we have:
M MA B
*
, ,= =11301 2 888units units*
AILA = 4,301 AILB = 1,138
SLA = 0.999 SLB =0 .991
16
This problem is one of comparing the combined cost of transportation and in-transit
inventory. In tabular form, we have the following annual costs:
Cost type Formula Rail Truck
Transportation R×D 6(40,000)(1.25)
= $300,000
11(40,000)(1.25)
= $550,000
In-transit
inventory
ICDT/365 0 25 250 40 000 21
365
. ( )( , )( )
= $143,836
0 25 250 40 00 7
365
. ( )( , )( )
= $47,945
Total $443,836 $597,945
You should select rail.
17

14. 106
The two transport options from the consolidation point are diagrammed in Figure 9-1.
Whether to choose one mode over the other depends more than transportation costs alone.
Because the transport modes differ in the time in transit, the cost of the money tied up in
the goods while in transit must be considered in the choice decision. This in-transit
inventory cost is estimated from
365
ICDt
. The following design matrix can be developed.
Cost type Method Air Ocean
Transportation R×D $180,800 $98,800
In-transit inventory ICDt/365 3,447*
34,467
Total $184,247 $133,267
*
ICDt/365 = 0.17(185)(20,000)(2)/365= 3,447
Ocean appears to be the lowest cost option even when a substantial in-transit inventory
cost is included. The ocean option assumes that the trucking cost to move the goods from
the consolidation point to the Port of Baltimore is included in the ocean carrier rate.
FIGURE 9-1 The
Consolidation Operation for a
Hydraulic Equipment
Manufacturer
18
The demand pattern is definitely lumpy, since sd = 327 > d = 169. To develop the min-
max system of inventory control, we first find Q*
. That is,
Q DS IC*
/ ( )( )( ) / . ( . . ) .= = + =2 2 169 12 10 020 096 0048 4485 units
The ROP is
ROP d LT z s EDd= × + × +'
where
z = 1.04 from Appendix A,
Multiple sourcing points
Baltimore
Sao
Paolo
2 days
20 days
Consolidation
point

15. 107
ED = 8 unitsthe average daily demand rate,
and
s s LT d sd d LT
'
( ) ( . )
.
= +
= +
=
2 2 2
2 2 2
327 4 169 08
6678 units
So,
ROP = 169(4) + 1.04(667.8) + 8
= 1,378.5 units
The max level is:
M*
= ROP + Q*
− ED
= 1,378.5 + 448.5 − 8
= 1,819 units
19
(a) The basic relationship is:
I I nT i=
We know that IT = $5,000,000. If there are 10 warehouses, the amount of inventory in
a single one would be:
I IT1 10 5 000 000 3162 1581139= = =/ , , / . , ,
The inventory in all 10 warehouses would be $1,581,139×10 = $15,811,390.
(b) The inventory in a single warehouse would be:
IT = =1 000 000 9 3 000 000, , , ,
In each of three warehouses, we would have:
I = =3 000 000 3 732 051, , / $1, ,

16. 108
and in all three warehouses, we would have $1,732,051×3 = $5,196,152.
20
(a) The turnover ratio is the annual demand (throughput) divided by the average
inventory level. These ratios for each warehouse and for the total system are shown
in the table below.
Ware-
house
Annual
warehouse
thruput
Average
inventory
level
Turnover
ratio
21 2,586,217 504,355 5.13
24 4,230,491 796,669 5.31 Avg. = 5.59
20 6,403,349 1,009,402 6.34
13 6,812,207 1,241,921 5.49
2 16,174,988 2,196,364 7.36
11 16,483,970 1,991,016 8.28
4 17,102,486 2,085,246 8.20
1 21,136,032 2,217,790 9.53
23 22,617,380 3,001,390 7.54
9 24,745,328 2,641,138 9.37
18 25,832,337 3,599,421 7.18
12 26,368,290 2,719,330 9.70
15 28,356,369 4,166,288 6.81
14 28,368,270 3,473,799 8.17
6 40,884,400 5,293,539 7.72
7 43,105,917 6,542,079 6.59
22 44,503,623 2,580,183 17.25
8 47,136,632 5,722,640 8.24
17 47,412,142 5,412,573 8.76
16 48,697,015 5,449,058 8.94
10 57,789,509 6,403,076 9.03
19 75,266,622 7,523,846 10.00
3 78,559,012 9,510,027 8.26 Avg. = 8.66
5 88,226,672 11,443,489 7.71
818,799,258 97,524,639 8.40
The overall turnover ratio is 8.40. Ranking the warehouses by throughput and
averaging turnover ratios for the top three and the bottom three warehouses shows
that the lowest volume warehouses have a lower turnover ratio (5.59) than the highest
volume warehouses (8.66). There are several reasons why this may be so:
• The larger warehouses contain the higher-volume items such as the A items in the
line. These may carry less safety stock compared with the sales volume.
Conversely, the low-volume warehouses may have more dead stock in them.

17. 109
• There may be start-up (fixed) stock in the warehouses, needed to open them, that
becomes less dominant with greater throughput.
(b) A plot of the inventory-throughput data is shown in Figure 9-2. A linear regression
line is also shown fitted to the data. The equation for this line is:
Inventory = 200,168 + 0.1132×Throughput
FIGURE 9-2 Plot of Inventory and Warehouse Thruput for California Fruit
Growers’ Association
(c) The total throughput for the three warehouses is:
Using this total volume and reading the inventory level from Fig. 9-2 or using the
regression equation, we have:
Inventory = 200,168 + .01132(70,121,702)
= $8,137,945
0
2
4
6
8
1 0
1 2
0 2 0 4 0 6 0 8 0 1 0 0
A n n u a l w a re h o u s e th ru p u t, $ (M illio n s )
Averageinventorylevel,$(Millions)
E s tim a tin g lin e
Warehouse Throughput
1 $21,136,032
12 26,368,290
23 22,617,380
Total $70,121,702

18. 110
(d) Warehouse 5 has a throughput of $88,226,672. Splitting this throughput by 30
percent and 70 percent, we have:
0.30×88,226,672 = 26,468,002
0.70×88,226,672 = 61,758,670
88,226,672
Estimating the inventory for each of the new warehouses using the regression
equation, we have:
Inventory = 200,168 + 0.1132×26,468,002 = $3,196,346
and
Inventory = 200,168 + 0 .1132×61,758,670 = $7,191,249
for at total inventory in the two warehouses of $10,387,595
21
The order quantity for each item when there is no restriction on inventory investment is:
Q DS IC*
/= 2
We first find the unrestricted order quantities.
Q
Q
Q
A
B
C
*
*
*
( , )( ) / . ( . ) ,
( , )( ) / . ( . )
( , )( ) / . ( . )
= =
= =
= =
2 51 000 10 0 25 17 1527
2 25 000 10 0 25 325 784
2 9 000 10 025 250 537
units
units
units
The total inventory investment for these items is:
IV C Q C Q C QA A B B C C= + +
= + +
=
( / ) ( / ) ( / )
. ( , / ) . ( / ) . ( / )
$3, .
2 2 2
175 1527 2 325 784 2 250 537 2
28138
Since the total investment limit is exceeded, we need to revise the order
quantities. For each product:
Q DS C I*
/ [ ( )]= +2 α

19. 111
For product A:
QA
*
( , )( ) / [ . ( . )]= +2 51000 10 175 025 α
For product B:
QB
*
( , )( ) / [ . ( . )]= +2 25 000 10 325 0 25 α
For product C:
QC
*
( , )( ) / [ . ( . )]= +2 9 000 10 250 0 25 α
Now, the investment limit must be respected so that:
3 000 2 2 2, ( / ) ( / ) ( / )= + +C Q C Q C QA A B B C C
Expanding we have:
3 000 175 2 51 000 10 175 025
325 2 25 000 10 325 025
250 2 9 000 10 250 025
, . ( , )( ) / [ . ( . )]
. ( , )( ) / [ . ( . )]
. ( , )( ) / [ . ( . )]
= +
+ +
+ +
α
α
α
We now need to find an α value by trial and error that will satisfy this equation. We
can set up a table of trial values.
Investment in
α A B C
Total
inventory
value, $
0.03 1,262.44 1,204.53 633.87 3,100.84
0.04 1,240.48 1,183.58 622.84 3,046.90
0.045 1,229.92 1,173.51 617.54 3,020.97
0.049 1,221.67 1,165.63 613.40 3,000.70
0.05 1,219.63 1,163.69 612.37 2,995.69
0.10 1,129.16 1,077.36 566.95 2,773.47
When the term I+α is the same for all products, as in this case, α may be found
directly from Equation 10-30.
We can substitute the value for α = 0.049 into the equation for Q*
and solve.
Hence, we have:

20. 112
Q
Q
Q
A
B
C
*
*
*
( )( ) / [ . ( . . )] ,
( , )( ) / [ . ( . . )]
( , )( ) / [ . ( . . )]
= + =
= + =
= + =
2 51000 10 175 025 0 049 1396
2 25 000 10 325 025 0 049 717
2 9 000 10 250 025 0 049 491
units
units
units
Checking:
1.75(1,396)/2 + 3.25(717)/2 + 2.50(491)/2 = $3,000
22
We first check to see whether truck capacity will be exceeded. Since three items are to
be placed on the truck at the same time, the items are jointly ordered. The interval for
ordering follows Equation 9-23, or:
T
O S
I C D
i
i i
*
( ) ( )
. [ ( )( ) ( )( ) ( )( )]
. ( , )
.
=
+
=
+
+ +
= =
∑
∑
2 2 60 0
0 25 50 100 52 30 300 52 25 200 52
120
025 988 000
0022 years, or 1.144 weeks
Now, from
D T wi i
i
*
∑ ≤ Truck capacity
[100(70) + 300(60) + 200(25)][1.144] = 34,320 lb.
The truck capacity of 30,000 lb. has been exceeded, and the order quantity or the order
interval must be reduced. Given the revised Equation 9-31, the increment to add to I can
be found. That is,
( )
α =






−
=
+ +





 + +
−






−
= − =
∑
∑
2
2 60
30 000
100 52 70 300 52 60 200 52 10
50 10 52 30 30 52 25 20 52
0 25
120
30 000
2 340 000
988 000
0 25
0 73895 0 25 048895
2
2
2
O
D w
C D
I
i i
i i
Truck capacity
( )
,
[ ( )( ) ( )( ) ( )( )]
( )( ) ( )( ) ( )( )
.
,
, ,
( , )
.
. . .
Revise T*
, the order interval by:

21. 113
T
O S
I C D
i
i i
*
( )
( )
( )
( . . )[ ( )( ) ( )( ) ( )( )]
. ( , )
.
=
+
+
=
+
+ + +
= =
∑
∑
2 2 60 0
025 0 48895 50 100 52 30 300 52 25 200 52
120
073895 988 000
0 01282
α
years, or 0.6667 weeks
Once again, we check that the truck capacity has not been exceeded.
[100(70) + 300(60) + 200(25)][0.66667] = 30,000 lb.
Therefore, place an order every 4.7, or approximately five days.
23
The average inventory for each item is given by:
'
*
2
dsz
Q
AIL ×+=
where s s LTd d
'
= and Q*
is found by Q
DS
IC
*
=
2
. z@ 95% = 1.65 from the normal
distribution in Appendix A. The results of these computations can be tabulated.
Summing the AIL for each product gives a total inventory of 1,022 cases.
24
The peak quantity of an item to appear on a shelf can be approximated as the order
quantity plus safety stock, or
Q z sd+ × ≤'
250 boxes
where z@93% = 1.48 from Appendix A and s s LTd d
'
= = =19 1 19 boxes. The
economic order quantity is
Q
DS
IC
* ( )( . )
. ( . )
.= =
×
=
2 2 123 52 125
019 129
25542 boxes
Checking to see if the shelf space limit will be exceeded by this order quantity
A B C D E
sd
'
7.75 15.49 19.36 11.62 27.11
Q*
188.38 238.28 421.23 361.98 565.14
AIL 106.98 144.70 242.56 200.16 327.30

22. 114
255.42 + 1.48(19) = 283.54 boxes
The quantity is greater than the 250 allowed. Subtracting the safety stock from the limit
gives 250 − 28 = 222 boxes. The order quantity should be limited to this amount.
25
The plot of average inventory to period facility throughput (shipments) gives an overall
indication of how the company is managing collectively its inventory for all stocked
items. We can see that the relationship is linear with a zero intercept. This suggests that
the company is establishing its inventory levels directly to the level of demand
(throughput). An inventory policy, such as stocking to a number of weeks of demand,
may be in effect.
Overall, the inventory policy seems to be well executed in that the regression line fits
the point for each warehouse quite well. The terminal with an inventory level of $6,000
seems to be an outlier and it should be investigated. If its high turnover ratio were
brought in line with the other terminals, an inventory reduction from $6,000 to $4,000 on
the average could be achieved.
The stock-to-demand inventory policy should be challenged. An appropriate
inventory policy should show some economies of scale, i.e., the inventory turnover ratio
should increase as terminal throughput increases. Whereas the current policy is of the
form DI 012.0= , a better policy would be 7.0
kDI = , where D represents terminal
throughput and I is the average inventory level. The coefficient 0.012 for the current
policy is found as the ratio of 6,000/500,000 = 0.0.12 for the last data point in the plot.
The k value for the improved policy needs to be estimated. From the cluster of the lowest
throughput facilities, the average inventory level is approximately $2,000 with an average
throughput of about $180,000. Therefore, from
419.0
894.771,4
000,2
)894.771,4(000,2
)000,180(000,2 7.0
7.0
=
=
=
=
=
k
k
k
k
kDI
Reading values from the plot, the following table can be developed showing the
inventory reduction that might be expected from revised inventory policy. (Note: If the
inventory-throughput values cannot be adequately read from the plot, the values in the
following table may be provided to the students.)
Terminal
Actual
Inventory, $ Shipments, $
Estimated inventory, $
DI 012.0=
Revised inventory, $
7.0
419.0 DI =
1 2,000 150,000 1,800 1,760
2 1,950 195,000 2,340 2,115
3 2,000 200,000 2,400 2,152
4 2,050 200,000 2,400 2,152

23. 115
5 3,900 320,000 3,840 2,991
6 6,000 330,000 3,960 3,056
7 4,500 390,000 4,680 3,435
8 4,300 410,000 4,920 3,558
9 5,500 500,000 6,000 4,088
Totals 32,200 2,695,000 32,340 25,307
Revising the inventory control policy has the potential of reducing inventory from the
linear policy by %7.21100
340,32
307,25340,32
=
−
x .
26
We can use the decision curves of Figure 9-23 in the text answer this question since it
applies to a fill rate of 95 percent and an α = 0.7. First, determine K for an inventory
throughput curve for the item, which is
466.1
6
)12117( 3.01
===
−
x
TO
D
K
α
Next,
90.0
)466.1)(400(20.0
)12117(12 3.07.01
===
−
x
ICK
tD
X
and with z ≈1.96 from Appendix A
18.0
)12117)(466.1(
2)15(96.1
7.0
===
xKD
LTzs
Y a
The demand ratio r is 42/177 = 0.36. The intersection of r and X lies below the curve Y
(use curve Y = 0.25), so do not cross fill.
27
Regular stock
For two warehouses, estimate the regular stock for the three products.

24. 116
Product A
units457
2
)15(02.0
)25)(000,5(2
units354
2
)15(02.0
)25)(000,3(2
2
2
2
2
1
==
==
==
A
A
RS
RS
IC
dS
Q
RS
Product B
units445
2
)30(02.0
)25)(500,9(2
units408
2
)30(02.0
)25)(000,8(2
2
1
==
==
B
B
RS
RS
Product C
units612
2
)25(02.0
)25)(000,15(2
units559
2
)25(02.0
)25)(500,12(2
2
1
==
==
C
C
RS
RS
Regular system inventory for two warehouses is RS2W = 354 + 457 + 408 + 445 + 559 +
612 = 2,835.
Regular stock for a central warehouse
units829
2
)25(02.0
)25)(500,27(2
units604
2
)30(02.0
)25)(500,17(2
units577
2
)15(02.0
)25)(000,8(2
==
==
==
C
B
A
RS
RS
RS
Total central warehouse regular stock is RS1W =577 + 604 + 828 = 2,009 units.

25. 117
Safety Stock
Product A
unitsSS
unitsSS
LTzsSS
A
A
d
000,175.0)700(65.1
71475.0)500(65.1
2
1
==
==
=
where z@0.95 = 1.65 from Appendix A
Product B
unitsSS
unitsSS
B
B
47975.0)335(65.1
35775.0)250(65.1
2
1
==
==
Product C
unitsSS
unitsSS
LTzsSS
C
C
d
572,375.0)500,2(65.1
001,575.0)500,3(65.1
2
1
==
==
=
System safety stock is SS2W = 714 + 1,000 + 357 + 479 + 5,001 + 3,572 = 11,123 units
For each product, the estimated standard deviation of demand on the central warehouse
is:
units301,4500,2500,3
units418335250
units860700500
22
22
222
2
2
1
=+=
=+=
=+=+=
B
B
A
s
s
sss
The safety stock is:
units146,675.)301,4(65.1
units59775.)418(65.1
units229,175.)860(65.1
==
==
==
=
C
B
A
SS
SS
SS
LTzsSS
Total safety stock in the central warehouse SS1W = 1,229 + 597 + 6,146 = 7,972 units.
Total inventory with two warehouses RS2W + SS2W = 2,835 + 11,123 = 13,958 units and
for a central warehouse RS1W + SS1W = 2,009 + 7,972 = 9,981 units. Centralizing
inventories reduces them by 13,958 – 9,981 = 3,977 units.
28
The solution to this multi-echelon inventory control problem is approached by using the
base-stock control system method. The idea is that inventory at any echelon is to plan its
inventory position plus the inventory from all downstream echelons.

26. 118
First, compute the average inventory levels for each customer. This requires finding
Q and the safety stock. Q is found from the EOQ formula.
For customer 1
units
x
Q 270
)35(2.0
)50)(12425(2
1 ==
units2115.0)65(65.1
2
270
2
3
1
1 1
=+=+= LTzs
Q
AIL d
where z@0.95 =1.65 from Appendix A
For customer 2
units
x
Q 239
)35(2.0
)50)(12333(2
2 ==
units1805.0)52(65.1
2
239
2
3
2
2 2
=+=+= LTzs
Q
AIL d
For customer 3
units
x
Q 218
)35(2.0
)50)(12276(2
3 ==
units1595.0)43(65.1
2
218
2
3
3
3 3
=+=+= LTzs
Q
AIL d
Total customer echelon inventory is AILC = 211 + 180 + 159 = 550 units
For the distributors echelon
unitsQD 000,2=
units120,10.1)94(28.1
2
000,2
2
D =+=+= LTzs
Q
AIL Dd
D
D
where z@0.90 =1.28 from Appendix A
The expected inventory that the distributor will hold is the distributor echelon inventory
less the combined inventory for the customers, or 1,120 - 550 = 570 units.

27. 119
COMPLETE HARDWARE SUPPLY, INC.
Teaching Note
Strategy
Complete Hardware Supply is an exercise involving the control of inventoried items
collectively. Data for a random sample of 30 items from the company's total of 500 items
held in inventory are given. The objective is to manage the total dollar value allowed to
be held as inventory. Several alternatives can be considered for changing inventory
levels, some of which require an investment other than in inventory.
The number of items that must be analyzed and the multiple scenarios that are to be
examined can be computationally time consuming. It is strongly suggested that students
use the INPOL module within LOGWARE to aid analysis. The current database has
been prepared and is available in the LOGWARE software.
The Base Case
We begin with the current data optimized as a reorder point design. The optimum order
quantities and associated inventory levels are found. The base case costs are shown as
follows:
Fixed order quantity policy
Purchase cost $556,912
Transport costa 0
Carrying cost 4,425
Order processing cost 4,425
Out-of-stock cost 0
Safety stock cost 2,529
Total cost $568,291
Total investment $27,801
aIncluded in the purchase cost
We note that optimizing the current design shows that investment of $27,801 exceeds
the allowed investment level of $18,000. Ways need to be explored to reduce this.
Transmit Orders More Rapidly
Instead of mailing orders to vendors, Tim O'Hare can buy a facsimile machine and
transmit orders electronically. This scenario can be tested by reducing the lead times in
the base case by two days, or (2/5) = 0.40 weeks and increasing order processing costs by
two dollars, and then optimizing again. INPOL shows that there will be a slight increase
in operating costs from $568,291 to $568,640, an incremental increase of $349.
Projecting this to all 500 items, we have 349(500/30) = $5,817. Since both operating cost
and inventory investment level increase, there is no economic incentive to implement this
change.
Faster Transportation
Suggesting that vendors who are located some distance (>600 miles) from the warehouse
use premium transportation is a possible way of reducing lead times, and therefore safety

28. 120
stock levels. Of course, the increase in transportation cost for those affected vendors is
likely to lead to a price increase to cover these costs. This scenario is tested by reducing
the lead-time in weeks to 2.2 for those vendors over 600 miles from the warehouse. For
these same vendors, a five percent price increase is made.
Compared with the base case, there is little change in the inventory investment
($27,801 vs. $27,746); however, operating costs increase. The total costs now are
$585,490 compared with the base case of $568,291, an increase of $27,199. The major
portion ($17,159) of this comes from the increase in price. We conclude that this is not a
good option for Tim.
Reduce Forecast Error
Reducing the forecast error involves reducing the standard deviation of the forecast error.
Testing this option requires taking 70 percent of the base-case forecast error standard
deviations and optimizing the design once again.
These changes have a positive impact on operating costs and inventory investment.
Operating cost now is $567,529 and inventory investment is $24,739. This is a saving in
operating costs of $762 per year. For all 500, we can project the savings to be
762(500/30) = $12,700. Based on a simple return on investment, we have:
ROI =
12 700
50 000
0 25
,
,
. , or 25% / year
This would appear to be attractive since carrying costs are 25 percent per year and the
company's return on investment probably makes up about 80 percent of this value.
Reduce Customer Service
At this point, we have only accepted the idea of reducing the forecast error. However,
inventory investment remains too high. We can now try to reduce it by reducing the
service levels. This is tested by dropping the service index from its current 0.98 level to a
level where inventory investment approximates $18,000. This is done, assuming the
forecast software will be purchased and the forecast error reduced by 30 percent. By trial
and error, the service index is found to be 0.54, which gives an investment level of
$18,028. The revised service level compared with the base case is summarized below for
the 30 items.

29. 121
Notice how little the service level changes, even with a substantial reduction in the
service index.
Conclusions
Tim can make a good economic argument for purchasing software that will reduce the
forecast error. The only questions here are whether the software can truly produce at
least the error reduction noted and whether a 25 percent return on investment is adequate
for the risks involved.
Arguing to accept a service reduction in order to lower the investment level is a little
less obvious since we do not know the effect that service levels have on sales. However,
Tim may point out that the service levels need to be changed so little that it is unlikely
that customers will detect the change. He might also raise the question as to whether
customer service levels were too high initially, and suggest that customers be surveyed as
to the service levels that they do need.
Item
Base
case Revised Item
Base
case Revised
1 99.88% 96.26% 16 99.98% 99.56%
2 99.92 98.02 17 99.90 97.57
3 99.96 98.54 18 99.95 97.81
4 99.98 99.15 19 99.89 95.96
5 99.98 99.45 20 99.97 98.15
6 99.96 98.60 21 99.69 89.53
7 99.97 98.84 22 99.97 98.96
8 99.96 98.61 23 99.97 98.96
9 99.92 97.29 24 99.96 97.58
10 99.98 99.26 25 99.92 99.33
11 99.99 99.70 26 99.97 96.68
12 99.99 99.43 27 99.93 97.45
13 99.92 97.30 28 99.89 98.78
14 99.98 99.14 29 99.97 96.92
15 99.96 98.84 30 99.91 96.78

30. 122
AMERICAN LIGHTING PRODUCTS
Teaching Note
Strategy
American Lighting Products is a manufacturer of fluorescent lamps in various sizes for
industrial and consumer use. As frequently happens in business, top management has
requested that inventories be reduced across the board, but it does not want to sacrifice
customer service. Sue Smith and Bryan White have been asked to eliminate 20 percent
of the finished goods inventory. Their plan is to reduce the number of stocking locations
and, thereby, eliminate the amount of inventory needed. Of course, they must recognize
that with fewer stocking points, transportation costs are likely to increase and customer
delivery times may increase as well. On the other hand, facility fixed cost may be
reduced.
The purpose of this case is to allow students to examine inventory policy and
planning through aggregate inventory management procedures. They also can see the
connection between location and inventory levels.
Answers to Questions
(1) Evaluate the company’s current inventory management procedures.
The company’s procedures for controlling inventory levels are at the heart of whether
inventory reductions are likely to be achieved through inventory consolidation. The
company appears to be using some form of reorder point control for the entire system
inventory, but it is modified by the need to produce in production lot sizes. It is not clear
how the reorder point is established. If it is based on economic order quantity principles,
then the effect of the principles becomes distorted by the need to produce to a lot size that
is different from the economic order quantity. Therefore, average inventory levels in a
warehouse will not be related to the square root of the warehouse’s throughput (demand),
i.e., throughput raised to the 0.5 power.1
Rather, the throughput will be raised to a higher
exponent between 0.5 and 1.0.
The above ideas can be verified by plotting the data given in Table 1 of the case and
then fitting a curve of the form I TP= α β
. Note: The curve can be found from standard
linear regression techniques when the equation is converted to a linear form through a
logarithmic transformation, i.e., lnI = lnα + βlnTP. The results are shown in Figure 1.
The inventory curve is I TP= 299 0 816
. .
with r = 0.86, where I and TP are in lamps. The
projected inventory reduction can be calculated by using this formula.
From the plot of the inventory data, we can see that there is substantial variation
about the fitted inventory curve. There is not a consistent turnover ratio between the
warehouses. This probably results from the centralized control policy. On the other
hand, improved control may be achieved by using a pull procedure at each MDC. The
data available in the case do not let us explore this issue.
1
Based on the economic order quantity formula, the average inventory level (AIL) for an item held in
inventory can be estimated as AIL Q DS IC= =/ / /2 2 2. Collecting all constants into K, we have
AIL=K(D)0.5
, where D is demand, or throughput.

31. 123
FIGURE 1 Plot of
MDC average
inventory vs. annual
throughput.
(2) Should establishing the LOC be pursued?
One of the ideas proposed in the case is to consolidate all Consumer product line items
into one large order center (LOC). Evaluating the impact of the LOC on inventory
reduction requires that an assumption be made as to how much demand and associated
inventory of the total belongs to Consumer products. Table 2 of the case gives the order
and back order breakdown by sales channel. Using this data, total consumer demand is
312,211 line items, or 33.4 percent of the total line items. The assumption is that the
same percentage applies to total demand. Hence, Consumer demand is
33.4%×169,023,000 = 56,453,682 lamps. From the inventory-throughput curve, we can
estimate the amount of inventory needed at the single LOC. That is, I =
2.997(56,453,682)0.816
= 6,339,684 lamps. If Consumer products account for 33.4% of
total inventory, then there are 33.4%×23,093,500 = 7,713,229 lamps in Consumer
inventory. The reduction that can be projected is 7,713,229 − 6,339,684 = 1,373,545
lamps for a reduction of
17.8%100
7,713,229
1,373,545
Reduction =×=
in Consumer inventory levels, but only a 6 percent reduction in overall inventory levels.
The 20 percent reduction goal is not achieved. Other alternatives need to be explored.
(3) Does reducing the number of stocking locations have the potential for reducing
system inventories by 20 percent? Is there enough information available to make a good
inventory reduction decision?
The second alternative proposed in the case is to reduce the number of MDCs from eight
to a smaller number. In order to evaluate this proposal, it needs to be determined which
MDCs will be consolidated and the associated total demand flowing through the
consolidated facilities. The inventory-throughput relationship can then be used to
estimate the resulting inventory levels. For example, if the Seattle and Los Angeles
MDCs are combined, the consolidated demand would be 4,922,000 + 21,470,000 =
26,392,000 lamps. The combined inventory is projected to be I = 2.997(26,392,000)0.816
=

32. 124
3,408,852 lamps, compared with the inventory for the two locations of 4,626,333, as
shown in Table 1. This yields a 26.3 percent reduction from current levels.
Table 1 shows other possible MDC consolidations and the resulting inventory
reductions that can be projected.
TABLE 1 Inventory Reduction for Selected MDC Combinations, in Lamps
MDC combination
Combined
demand
Combined
inventory
Inventory
reduction
Seattle/Los Angeles 26,392,000 3,408,852 1,217,481
Kansas City/Dallas 29,194,000 3,701,403 50,181
Chicago/Ravenna 49,174,000 5,664,257 -557,590
Atlanta/Dallas 39,314,000 4,718,862 1,224,721
Kansas City/Chicago 39,271,000 4,714,650 -933,900
Ravenna/Hagerstown 64,046,000 7,027,231 1,715,607
K City/Dallas/Chicago 52,515,000 5,976,377 -36,377
Ravenna/H’town/Chicago 87,367,000 7,508,054 3,423,196
Atlanta/Dallas/K City 55,264,000 5,242,351 2,293,566
From the MDC combinations in Table 1, proximity to each other is a primary
consideration in order to not increase transportation costs or jeopardize delivery service
any more than necessary. Several options can be identified that yield a 20 percent
inventory reduction. These are:
Option MDC combinations
Inventory
reduction,
lamps
Total
inventory
reduction
1 LA/Seattle 1,217,481
Ravenna/H’town/Chicago 3,423,196
Total reduction 4,640,677 20.1%
2 LA/Seattle 1,217,481
Kansas City/Hagerstown 1,224,721
Ravenna/Hagerstown 1,715,602
Total reduction 4,157,804 18.0%
3 LA/Seattle 1,217,481
Ravenna/Hagerstown 1,715,602
Atlanta/Dallas/K City 2,293,566
Total reduction 5,226,649 22.6%
Options 1 and 3 achieve the 20 percent reduction goal, although other MDC
combinations not evaluated may also do so. The maximum reduction would be achieved
with one MDC. The total inventory would be I = 2.997(169,023,000)0.816
= 15,512,812
lamps, for a system reduction of 32.8 percent. However, we must recognize that as the
number of warehouses is decreased, outbound transportation costs will increase. Inbound
transportation costs to the combined MDC will remain about the same, since

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replenishment shipments are already in truckload quantities. Some difference in cost will
result from differences in the length of the hauls to the warehouses. On the other hand,
outbound costs may substantially increase, since the combined MDC locations are likely
to be more removed from customers then they are at present. Outbound transportation
rates will be higher, as they are likely to be for shipments of less-than-truckload
quantities. If the sum of the inbound and outbound transportation cost increases is
greater than the inventory carrying cost reduction, then the decision to reduce inventories
must be questioned.
Calculating all transportation cost changes is not possible, since the case study does
not provide sufficient data on outbound transportation rates. However, they should be
determined before and after consolidation to assess the tradeoff between inventory
reduction and transportation costs increases. On the other hand, inbound transportation
costs can be found, as shown below for option 1, where the consolidation points are Los
Angeles and Hagerstown.
Location
TL rate,
$/TL
Annual
demand,
lamps
Transport
cost, $
Combined
annual
demand, lamps
Transport
cost, $
Seattle 1800 4,922,000 253,131a
Los Angeles 1800 21,470,000 1,104,171 26,392,000 1,357,302
Ravenna 250 25,853,000 184,664
Hagerstown 475 38,193,000 518,334 87,367,000 1,185,695
Chicago 350 23,321,000 233,210
Total 113,759,000 2,293,510 113,759,000 2,542,997
a
(4,922,000/35,000)×1800 = 253,131
There will be a net increase in inbound transportation costs of $2,542,997 − 2,293,510 =
$249,487 for option 1.
In addition, the annual fixed costs for the MDCs will be less, since the total space
needed in the consolidated facilities should be less than that for the existing facilities.
Again, the case study does not estimate the fixed costs for existing or potential locations.
We do know that taking them into account would favor consolidation.
In summary, the costs associated with option 1, that just meets the 20 percent
inventory reduction goal, would be:
Although Sue and Bryan could report a substantial savings in inventory related costs,
they should be encouraged to include fixed costs and transportation costs so as to report
the true benefits of the inventory reduction plan.
Cost type Cost savings, $
Inventory carrying cost reduction 0.20×0.882×4,640,677 = 818,615
Warehouse cost 0.10×4,640,677 = 464,068
Warehouse fixed cost Unknown, but may be included in warehouse cost
Outbound transportation cost Unknowndata not given
Inbound transportation cost (249,487)

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(4) How might customer service be affected by the proposed inventory reduction?
The general effect of inventory consolidation is to reduce the number of stocking points
and make them more remote from customers. That is, the delivery distance will be
increased if inventory consolidation is implemented. Therefore, delivery customer
service may be jeopardized and must be considered before deciding to consolidate
inventories.
From Table 3 of the case, it can be seen that customer lead times remain constant for
a variety of locations with the exception of Kansas City. Since consolidation points will
be selected among the existing locations, outbound lead times will remain unaffected.
Customer service due to location should be constant, at least for a moderate degree of
consolidation.
Customer service due to stock availability will be affected if safety stock levels are
reduced after consolidation. Although the inventory-throughput relationship projects
adequate safety stock to maintain the current first-time delivery levels, it does not account
for any increase in lead times that may occur between the current system of MDCs and
the consolidated ones. By comparing the weighted inbound lead times for the existing
distribution system and option 1, as shown in Table 2, the average inbound lead-time is
slightly reduced through consolidation. Lead-time variability is usually related to
average lead-time. This should have a favorable affect on inventory levels since
uncertainty is reduced. First-time deliveries should not be adversely affected by
consolidation, according to option 1.
TABLE 2 A Comparison of Inbound Lead Times for the Existing Distribution
System and a Consolidated Distribution System (Option 1)
(a) Current Distribution System
Master Distribution Center Shipments
Inbound
lead time,
days
Weighted
lead time,
days
Atlanta 26,070,000 2 0.308
Chicago 23,321,000 1 0.138
Dallas 13,244,000 3 0.235
Hagerstown 38,193,000 1 0.226
Kansas City 15,950,000 2 0.094
Los Angeles 21,470,000 5 0.635
Ravenna 25,853,000 1 0.153
Seattle 4,922,000 6 0.175
Total 169,023,000 1.964

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(b) Consolidation Option 1
Master Distribution Centera
Shipments
Inbound
lead time,
days
Weighted
lead time,
days
Atlanta 26,070,000 2 0.308
Dallas 13,244,000 3 0.235
H’town/Ravenna/Chicago 87,367,000 1 0.517
Kansas City 15,950,000 2 0.094
Los Angeles/Seattle 26,392,000 5 0.781
Total 169,023,000 1.935
a
Consolidation is assumed to take place at the MDC with the largest number of current shipments.

36. 128
AMERICAN RED CROSS: BLOOD SERVICES
Teaching Note
Strategy
The American Red Cross Blood Services has a mission to provide the highest quality
blood components at the lowest possible cost. High quality blood products are provided
to regional hospitals, but managing the inventory to meet demand as it occurs is a
difficult problem. Blood is considered a precious product, especially by those who give it
voluntarily. So, managing this perishable product carefully is a foremost concern.
Blood is a vital product to those in need of it for emergencies and a precious product
to those requiring it for elective surgery and other treatments. The goal is to always have
what is needed but never so much that this perishable product has to outdated. Managing
the blood inventory is quite difficult because (1) forecasting demand is not particularly
accurate, (2) the planning horizon for collections can be up to a year long with uncertain
yields, (3) the life of blood products ranges from 42 days to as short as five days, (4) once
scheduled, blood donors are never turned away except for medical reasons, and (5) there
is a limited opportunity to sell blood outside of the local region if too much is on hand.
Overall, this situation has many characteristics of a “supply driven” inventory
management problem, which requires inventory management techniques different from
those for typical consumer products.
The intended purpose of this case study is for students to examine an inventory
situation where there is limited control over the amount of the product flowing into
inventory. This supply-driven inventory situation is likely to be quite different from that
discussed on the introductory level. Students are encouraged to consider the various
elements that affect inventory levels of individual products and how they interact. These
elements are (1) demand forecasting, (2) collections, (3) decision rules for creating blood
derivatives, (4) product prices, and (5) inventory policy. It is expected that students will
be able to make general suggestions for improvement.
Questions
(1) Describe the inventory management problem facing blood services at the American
Red Cross.
One of the major problems facing the American Red Cross (ARC) is that the availability
of blood is supply-driven, meaning that quantities of blood received for processing to
meet demand in the short term are unknown, yet they must be placed in inventory if
demand is less than the collected quantities. Blood availability is a function of number of
factors that cannot be well-controlled by the regional blood center in the short run,
causing wide variability in supply. The usage of blood at hospital blood banks, which
creates the demand on ARC’s blood inventories, is also uncertain and varies from day to
day and between hospital facilities.
The yield of blood at the point of collection is random and does not necessarily give
the product mix needed to meet demand. Different blood types can only be known by a
probability distribution as to the percentage of the blood types that exist in the general
population. In the short term, the demand for blood types may differ from the collected

37. 129
quantities, resulting in a potential for under- and over-stocking, since blood is drawn
from all qualified donors as they arrive at collection sites.
Forecasting demand for blood products will likely be reasonably accurate for a base
load. Surgery loads on hospitals are scheduled in advance so that blood needs will be
known with a fair degree of certainty, although each operation will not typically use the
full amount of blood allocated to it. However, emergency blood needs are not well
predicted, and they can cause spikes in demand and unplanned draws on inventory. A
problem is establishing how much accuracy is needed for good inventory management.
Inventory policy for managing inventory levels is a mixed strategy of product pricing,
derivative product selection for processing at the time of collection, conversion to other
products later in the product life cycle, product sell off, emergency supply (call for
blood), discount pricing, and stocking rules for hospitals. Although there are many
avenues to controlling inventory levels, shortages and outdating cannot always be
avoided. It is not clear that these procedures lead to an optimal control of inventory
levels.
Competition from local independent blood banks that sell selected blood products at
low prices makes it difficult for ARC to cover costs. ARC provides a wider range of
products, but it has difficulty-differentiating price among derivative products so that it
might compete effectively. Given pressures for hospitals to increase efficiency, they will
shop around for the lowest-priced blood products. ARC is having difficulty maintaining
its position as the dominant supplier of blood products in the region, which results in the
greater uncertainty in managing inventory levels.
In summary, blood is a precious product given by volunteers for the benefit of others.
Donors have the right to expect that their contribution will be handled responsibly. To
ARC, this means managing the blood supply so that recipients receive a high-quality
product at the lowest possible price. To achieve this goal, ARC manages the blood
supply through four inter-connected elements: (1) estimating the blood product needs
over time, (2) planning the collection of whole blood, (3) deciding which derivative
products and their amounts should be created from whole blood, and (4) controlling the
inventory levels to avoid outdating. The volunteer nature of the blood giving and donor
attitudes surrounding it, long planning lead times and the associated uncertainties, rising
competition among some products from local blood banks, and the uncertainties of blood
needs all make blood supply management a unique inventory management problem.
(2) Evaluate the current inventory management practices in light of ARC’s mission.
Performance of blood management can be evaluated on two levels: customer service and
cost. Tables 8 and 9 of the case show that in March standards were not quite met overall.
Within specific product types, there was up to an eight percent deficit. Both order fill
rate and item fill rate were less than 100 percent for most products. There would seem to
be some room for improvement, especially in managing the variation among product
types.
From a cost standpoint, it is not known how efficiently the blood supply is managed
since no costs are reported. In addition, the revenue that the blood products generate is
not known. We would like to know how prices of the various products are set so that
revenues might be maximized, considering competition among some of the product line.

38. 130
We do expect that demand is price elastic, since hospitals do shop around for blood
products that are available from local, commercial, and community blood banks. On the
other hand, ARC is the sole regional supplier of certain products such as platelets.
Setting product fill-rate standards at various levels can influence costs. We do not
know this effect.
Setting inventory levels by a “number of days of inventory” rule of thumb is simple
but not as effective as planning inventory levels based on the uncertainties that occur in
demand forecasts and supply lead times. The number-of-days of inventory rule does tend
to lead to too much inventory or to too many out-of-stock situations.
The plan for evaluation, if enough data were available, would be to establish a base
case of cost and service. This, then, would provide a basis for evaluating the effect of
change in the supply procedures.
(3) Can you suggest any changes in ARC’s inventory planning and control practices that
might lead to cost reduction or service improvement?
Suggestions for improvement in blood supply management stem from a basic
understanding of the nature of the demand-supply relationship. When supply is uncertain
and all supply must be taken that is available, there is the possibility that significant
excess inventory will occur. The goal is to “manage” the demand in the short run to
reduce inventory levels when overstocking occurs, rather than focusing on managing
supply. Several approaches for doing this are:
• Aggressively price selected products that are in excess supply and are nearing their
expiration dates, e.g. run a sale or offer price discounts.
• Sell off excess supply to secondary demand sources or other regions of the ARC.
• Temporarily adjust return rules for hospitals.
• Bring demand more in line with supply by converting products into derivative ones
that have excess demand, e.g., reprocess whole blood into plasma.
• Encourage hospitals to buy certain products in excess supply for a more favorable
status in buying other products that are in short supply, such as phersis platelets and
rare whole blood types.
• Try to create excess demand for all products, especially those items that are
available from local blood banks, through promotion of ARC’s distinct advantages,
such as quality, high service levels, and a wide range of blood derivative products.
• Offer “two-for-one” sales, such that if a hospital buys one blood product, it may
receive another at a favorable price.
• Pool the risk of uncertain demand by maintaining a central inventory for all
hospitals, or managing the inventories at all hospitals, as well at ARC, collectively.
Provide quick deliveries or transfers among inventory locations.
ARC should attempt to be the premier provider of blood products and leverage the
advantage. This will allow it to maintain a degree of control over the demand for blood.
Effectively controlling demand in turn allows it to control its costs and avoid product
outdating.

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(4) Is pricing policy an appropriate mechanism to control inventory levels? If so, how
should price be determined?
From the previous discussion, it can be seen that price plays a role in controlling demand.
Since there appears a relationship between demand and price for some products,
especially among those products offered by local blood banks that compete with ARC
blood products, price may be an effective weapon to meet competition. Rather than
setting price based on the cost of production, ARC might consider raising the price on
products for which it is the sole provider, such as platelets, and then meeting the price of
competitors on whole blood. Although ARC strives to be a nonprofit organization, the
increased volume that an effective pricing strategy promotes would allow more of the
fixed costs to be covered. This may lead to lower overall average prices for ARC’s
products.
Blood could also be priced as a function of its freshness at two or more levels.
Although blood that has been donated within 42 days legally can be utilized, the quality
of blood does not remain the same for the entire 42-day period. A chemical compound
found in blood, called 2,3-DPG, decreases with the age of the stored blood, and is
believed to be important in oxygen delivery. For this reason, certain procedures such as
heart transplants and neonatal procedures require that blood be fresh, usually donated
within 10 days or less. Thus, a simple pricing policy could be to charge a higher price for
blood that is less than 10 days old, and a lower price for blood that is between 10 and 42
days old. Price differences here are based on product quality.