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, .

* 23 200 35 164 78 015 55 164 78 2

679 69 679 97359 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 =+

=100100 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 009450 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 320400 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* liesroundu (b) Car

CP

where t750 andloss of 320 − invento 9 (a) The *Q and RO whe sd

'

and

S

111

ales

Frequency

Cumulative frequency

500 0.2 0.2 750 0.2 0.4 ,000 0.3 0.7 ,250 0.2 0.9 ⇐⇐⇐⇐ Q*

,500 0.1 1.0 1.0

97

between 1,000 and 1,200 in the cumulative frequency table. We choose to p to Q* = 1,250 units.

rying the excess inventory to next year,

n =+ ×

=8080 0 2 320

0 556( . )

.

he loss is the cost of holding a unit until the next year. The Q* now lies between 1,000 units. We choose 1,000 units. Holding the excess units means a potential

0.2×320 = $64/unit, whereas discounting the excess units represents a loss of only 300 = $20/unit. Therefore, Cabot will need fewer units if they are held over in ry.

optimum order quantity is:

cases 556)56)(3.0/()40)(52)(250,1(2/2 === ICDS

the reorder point quantity is:

P d LT z sd= × + × '

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 5 814 cases 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:

SLs E

Qd z= −1'

( )

For the reorder point system:

SLQ = − = −1 751 01120556

1 015( . ) .

or demand is met 85 percent of the time. For the periodic review system:

SLP = − = −1 814 01120556

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 20 3 56 199 75110 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 53125 0 84 977 083 946

, ( . ) . . , ., . ( . ), 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 51 027 cases

So,

Max d T LT z sd= + + ×

= + +=

( ), ( . . ) . ( , ),

* '

1 250 0 44 2 5 0 84 1 0274 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 DSIC

pp dp

* ( )( )( ). ( )

,= ×−

= ×−

=2 2 100 250 2500 25 75

300300 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 1 128 333 144 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 2115 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 5002 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 21365

. ( )( , )( )

= $143,836

0 25 250 40 00 7365

. ( )( , )( )

= $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

SaoPaolo

2 days

20 days

Consolidationpoint

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 8667 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 )

Aver

age

inve

ntor

y le

vel,

$ (M

illion

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 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 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 2175 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 SI C D

i

i i

* ( ) ( ). [ ( )( ) ( )( ) ( )( )]

. ( , ).

=+

= ++ +

= =

∑∑

2 2 60 00 25 50 100 52 30 300 52 25 200 52

1200 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 6030 000

100 52 70 300 52 60 200 52 1050 10 52 30 30 52 25 20 52

0 25

12030 000

2 340 000988 000

0 25

0 73895 0 25 0 48895

2

2

2

O

D wC D

I

i ii i

Truck capacity

( ),

[ ( )( ) ( )( ) ( )( )]( )( ) ( )( ) ( )( )

.

,, ,

( , ).

. . . Revise T*, the order interval by:

113

TO S

I C Di

i i

* ( )( )

( )( . . )[ ( )( ) ( )( ) ( )( )]

. ( , ).

=+

+= +

+ + +

= =

∑∑

2 2 60 00 25 0 48895 50 100 52 30 300 52 25 200 52

1200 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 dszQAIL ×+=

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 DSIC

* ( )( . ). ( . )

.= = × =2 2 123 52 125019 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

419.0894.771,4

000,2)894.771,4(000,2

)000,180(000,2 7.0

7.0

=

=

===

k

kkkkDI

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 6,000 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 %7.21100340,32

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

466.16

)12117( 3.01

===− x

TODK

α

Next,

90.0)466.1)(400(20.0

)12117(12 3.07.01

===− x

ICKtDX

and with z ≈1.96 from Appendix A

18.0)12117)(466.1(

2)15(96.17.0 ===

xKDLTzsY 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 4572

)15(02.0)25)(000,5(2

units 3542

)15(02.0)25)(000,3(2

2

2

2

2

1

==

==

==

A

A

RS

RS

ICdS

QRS

Product B

units 4452

)30(02.0)25)(500,9(2

units 4082

)30(02.0)25)(000,8(2

2

1

==

==

B

B

RS

RS

Product C

units 6122

)25(02.0)25)(000,15(2

units 5592

)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 8292

)25(02.0)25)(500,27(2

units 6042

)30(02.0)25)(500,17(2

units 5772

)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

unitsSSunitsSS

LTzsSS

A

A

d

000,175.0)700(65.171475.0)500(65.1

2

1

====

=

where z@0.95 = 1.65 from Appendix A Product B

unitsSSunitsSS

B

B

47975.0)335(65.135775.0)250(65.1

2

1

====

Product C

unitsSSunitsSS

LTzsSS

C

C

d

572,375.0)500,2(65.1001,575.0)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 301,4500,2500,3units 418335250

units 860700500

22

22

2222

21

=+==+=

=+=+=

B

B

A

ss

sss

The safety stock is:

units 146,675.)301,4(65.1units 59775.)418(65.1

units 229,175.)860(65.1

======

=

C

B

A

SSSSSS

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(21 ==

units 2115.0)65(65.12

2702 3

11 1

=+=+= LTzsQAIL d

where z@0.95 =1.65 from Appendix A For customer 2

unitsxQ 239)35(2.0

)50)(12333(22 ==

units 1805.0)52(65.12

2392 3

22 2

=+=+= LTzsQAIL d

For customer 3

unitsxQ 218)35(2.0

)50)(12276(23 ==

units 1595.0)43(65.12

2182 3

33 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(28.12000,2

2 D =+=+= LTzsQAILDd

DD

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 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 70050 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.

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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%1007,713,2291,373,545Reduction =×=

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

125

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)

126

(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

127

(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

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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.

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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.