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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 
 
 
 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 
 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 69 679 97
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 77 5, or 78 students 
 
(b) The number of times that the course should be offered is: 
 
 yearper four timesabout or ,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: 
 
 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
0 945.
. .
. 
 
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, 
 
 
 CPn =
−
− + −
=
400 320
400 320 320 300
0 80
( ) ( )
. 
 
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
0 556
( . )
. 
 
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: 
 
 cases 556)56)(* =Q 
 
 and
 
 RO
 
 whe
 
 sd
'
 
 and
 
 
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 
3.0/()40)(52)(250,1(2/2 == ICDS
 the reorder point quantity is: 
P d LT z sd= × + ×
' 
97
re 
s LTd .= = =475 2 5 751 
 zP=0 80. = 0.84. 
 98
 Now, 
 
 ROP = + =( , )( . ) ( . )( ) ,1 250 2 5 0 84 751 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 2 5 814cases 
 
 Hence, 
 
 M * , ( . . ) . ( ) ,= + + =1 250 0 44 2 5 0 84 814 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 
 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. 
 
 
 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 4 902 8 0 3 56 902 8 2
0 3 56 199 751
10 65 000 751 0 0087 902 8
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 21 250 2 5 0 84 2 5 475 1 250 0 5
3125 0 84 977 08
3 946
, ( . ) . . , .
, . ( . )
, cases
 
 
(b) An approximation for T* = Q*/d, or 
 
 T* = 556/1,250 = 0.44 weeks 
 
 and approximating sd
' as 
 101
 
 
s T LT s d sd d LT
' *( )
( . . )( ) , ( . )
,
= + + ×
= + +
=
2 2 2
2 2 20 44 2 5 475 1 250 0 5
1 027 cases
 
 
 So, 
 
 
Max d T LT z sd= + + ×
= + +
=
( )
, ( . . ) . ( , )
,
* '
1 250 0 44 2 5 0 84 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 dp
* ( )( )( )
. ( )
,= ×
−
= ×
−
=
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 
 
 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 0 30 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 78 5, 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, 
 
 103
 
ROP = +
=
40 1 128 333 1
44 3
( ) . ( . )
. 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 0 30 45 367 7 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 2 215 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 7 35 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 27 35 7 15 50 2
115 0 units
 
 
 Therefore, 
 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 0 3 52 2 25 2 000 1 90 500
2 5 weeks
 
 
 Then, 
 
 M d T LT z s T LTA A A d A
* * *( )= + + × + 
 
 where zA = 1.282 for P = 0.90 
 
 M A
* , ( . . ) . ( ) . . ,= + + + =2 000 2 5 15 1282 100 2 5 15 8 256 units 
 
 and 
 
 MB
* ( . . ) . ( ) . . ,= + + + =500 2 5 15 0 842 70 2 5 15 2 118 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: 
 105
 
 AIL d T z s T LTd= × + × +
* */ 2 
 
 For A: 
 
 AILA = + + =2 000 2 5 2 128 100 2 5 15 2 756, ( . ) / . ( ) . . , units 
 
 For B: 
 
 AILB = + + =500 2 5 2 0 842 70 2 5 15 743( . ) / . ( ) . . units 
 
(c) The service level is given by: 
 
 SL s E d Td z= − × ×1
'
( )
*/ 
 
 For A: 
 
 SLA = − + =1 100 2 5 15 0 0475 2 000 2 5 0 998. . ( . ) / , ( . ) . 
 
 For B: 
 
 SLB = − + =1 70 2 5 15 01120 500 2 5 0 987. . ( . ) / ( . ) . 
 
(d) We set T* = 4 and cycle through the previous calculations. Thus, we have: 
 
 M MA B
* , ,= =11 301 2 888 units 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 
 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 0 20 0 96 0 048 448 5 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
 
 107
 
 ED = 8 unitsthe average daily demand rate, 
 
and 
 
 
s s LT d sd d LT
'
( ) ( . )
.
= +
= +
=
2 2 2
2 2 2327 4 169 0 8
667 8 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 1 581139= = =/ , , / . , , 
 
 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, , 
 
 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. 
 
 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 il l io n s )
Av
er
ag
e 
in
ve
nt
or
y 
le
ve
l, 
$ 
(M
illi
on
s)
E s t im a t in g lin e
Warehouse Throughput 
 1 $21,136,032 
12 26,368,290 
23 22,617,380 
 Total $70,121,702 
 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 quantityfor 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 1 527
2 25 000 10 0 25 325 784
2 9 000 10 0 25 2 50 537
 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 1 527 2 325 784 2 2 50 537 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 
 
 
 
 111
 For product A: 
 
 QA
* ( , )( ) / [ . ( . )]= +2 51 000 10 175 0 25 α 
 
 For product B: 
 
 QB
* ( , )( ) / [ . ( . )]= +2 25 000 10 325 0 25 α 
 
 For product C: 
 
 QC
* ( , )( ) / [ . ( . )]= +2 9 000 10 2 50 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 0 25
325 2 25 000 10 325 0 25
2 50 2 9 000 10 2 50 0 25
, . ( , )( ) / [ . ( . )]
. ( , )( ) / [ . ( . )]
. ( , )( ) / [ . ( . )]
= +
+ +
+ +
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: 
 
 112
 
Q
Q
Q
A
B
C
*
*
*
( )( ) / [ . ( . . )] ,
( , )( ) / [ . ( . . )]
( , )( ) / [ . ( . . )]
= + =
= + =
= + =
2 51000 10 175 0 25 0 049 1 396
2 25 000 10 325 0 25 0 049 717
2 9 000 10 2 50 0 25 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
0 25 988 000
0 022 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 0 48895
2
2
2
O
D w
C D
I
i i
i i
Truck capacity
( )
,
[ ( )( ) ( )( ) ( )( )]
( )( ) ( )( ) ( )( )
.
,
, ,
( , )
.
. . .
 
Revise T*, the order interval by: 
 113
 
T
O S
I C D
i
i i
* ( )
( )
( )
( . . )[ ( )( ) ( )( ) ( )( )]
. ( , )
.
=
+
+
=
+
+ + +
= =
∑
∑
2 2 60 0
0 25 0 48895 50 100 52 30 300 52 25 200 52
120
0 73895 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 d
szQAIL ×+= 
 
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
255 42 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 
 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 7.0kDI = , 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, $ 
7.0419.0 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 
 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 6,0004,088 
 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
DK
α
 
 
Next, 
 
 90.0
)466.1)(400(20.0
)12117(12 3.07.01
===
− x
ICK
tDX 
 
and with z ≈1.96 from Appendix A 
 
 18.0
)12117)(466.1(
2)15(96.1
7.0 === xKD
LTzsY 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. 
 
 116
Product A 
units 457
2
)15(02.0
)25)(000,5(2
units 354
2
)15(02.0
)25)(000,3(2
2
2
2
2
1
==
==
==
A
A
RS
RS
IC
dS
QRS
 
 
Product B 
units 445
2
)30(02.0
)25)(500,9(2
units 408
2
)30(02.0
)25)(000,8(2
2
1
==
==
B
B
RS
RS
 
 
Product C 
units 612
2
)25(02.0
)25)(000,15(2
units 559
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 
 
units 829
2
)25(02.0
)25)(500,27(2
units 604
2
)30(02.0
)25)(500,17(2
units 577
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. 
 
 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: 
 
units 301,4500,2500,3
units 418335250
units 860700500
22
22
222
2
2
1
=+=
=+=
=+=+=
B
B
A
s
s
sss
 
 
The safety stock is: 
units 146,675.)301,4(65.1
units 59775.)418(65.1
units 229,175.)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. 
 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 
 
unitsxQ 270
)35(2.0
)50)(12425(2
1 == 
 
units 2115.0)65(65.1
2
270
2 3
1
1 1
=+=+= LTzsQAIL d 
 
where z@0.95 =1.65 from Appendix A 
 
For customer 2 
 
unitsxQ 239
)35(2.0
)50)(12333(2
2 == 
 
units 1805.0)52(65.1
2
239
2 3
2
2 2
=+=+= LTzsQAIL d 
 
For customer 3 
 
unitsxQ 218
)35(2.0
)50)(12276(2
3 == 
 
units 1595.0)43(65.1
2
218
2 3
3
3 3
=+=+= LTzsQAIL d 
 
Total customer echelon inventory is AILC = 211 + 180 + 159 = 550 units 
 
For the distributors echelon 
 
unitsQD 000,2= 
 
units 120,10.1)94(28.1
2
000,2
2 D
=+=+= LTzsQAIL
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. 
 
 
 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 
 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 
operatingcosts 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. 
 
 
 
 
 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 
 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= 2 99 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. 
 
 
1Based 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.
 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
1,373,545Reduction =×= 
 
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= 
 
 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 
aConsolidation is assumed to take place at the MDC with the largest number of current shipments. 
 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 
 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. 
 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 isthe 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. 
 
 131
(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. 
	CHAPTER 9