ch12_ism

100

SOLUTIONS TO DISCUSSION QUESTIONS

AND PROBLEMS

12-1. Inventory is an important consideration for managers be-cause as much as 50% of the total assets of a company can be tiedup in inventory. Because of this large investment in inventory,controlling inventory becomes extremely important for most orga-nizations. On the one hand, companies will try to reduce the costof inventory by reducing amounts of inventory on hand. On theother hand, however, companies realize that customer dissatisfac-tion can be increased significantly due to low inventory levels andstockouts. Thus, it is important to reach a fine balance betweenlow and high inventory levels.

12-2. The purpose of inventory control is to regulate the flow ofinventory at the various inventory storage locations within the or-ganization. This can be done by determining how much inventoryis to be ordered and when the inventory should be ordered.

12-3. Buying inventory can be used as a hedge against inflation.When inflation of inventory items is high, purchasing inventory attoday’s prices can be used as a hedge against future inflation of in-ventory items. In using inventory as a hedge against inflation, how-ever, careful consideration should be given to carrying costs. Asmore inventory is purchased as a hedge against inflation, highercarrying costs will be paid to hold inventory over a period of time.

12-4. Storing large quantities of inventory can eliminate short-ages and stockouts. On the other hand, storing large quantities ofinventory can significantly increase the cost of carrying or holdinginventory. Therefore, a delicate balance must be sought betweenincreased carrying costs and shortages and stockouts. In determin-ing how much inventory a company should have on hand to avoidshortages and stockouts, the overall objective is to minimize carry-ing costs and shortage or stockout costs.

12-5. Although there can be many factors to be considered in in-ventory control, there are basically two fundamental decisions thatwill have to be made. These decisions are (1) how much to order,and (2) when to order. The simplest and the most complex inven-tory models must answer both of these questions.

12-6. There are a number of assumptions that are made in usingthe economic order quantity. It is assumed that the cost of theitems, the cost of ordering, the cost of holding inventory, and theannual demand are known and constant. It is also assumed that the time it takes to receive an order is known and constant. In thebasic economic order quantity model, it is assumed that stockoutscan be avoided, there are no quantity discounts, and replenishmentis instantaneous.

12-7. The major costs in determining the economic order quan-tity include (1) the cost of the items, (2) the cost of ordering, (3) the cost of carrying or holding inventory, (4) the cost of safetystock, and (5) the cost of stockouts. Under the basic economicorder quantity model, it is assumed that there are no stockouts;therefore, the cost of stockouts and the cost of safety stock are notincluded in the basic model.

12-8. The most commonly used methods in actually determiningthe equation for the economic order quantity are to use algebra orcalculus. When using basic algebra, expressions for ordering costsand carrying costs are determined. These two costs are then setequal to each other, and the equation is solved for Q, the economicorder quantity. When calculus is being used, an expression is de-veloped for the total cost. This total cost includes ordering costsand carrying costs. Then the first derivative of this equation istaken and set equal to zero. This equation is solved to determinethe economic order quantity. As you would expect, both proce-dures result in the same equation for the economic order quantity.

12-9. The reorder point specifies when an order is to be placedfor new inventory items. When the inventory drops to or below thereorder point, an order is placed.

The reorder point for the basic economic order quantitymodel is determined by multiplying the demand per period timesthe lead time for a new order. In most cases, it is determined bymultiplying the demand per day times the lead time for a neworder in days.

12-10. The purpose of sensitivity analysis is to determine whateffect changes in the annual demand, the ordering cost, and thecarrying cost will have on the economic order quantity. In general,sensitivity analysis is used to determine what effect a change in avariable in the model will have on the optimal quantity, such asthe economic order quantity.

12-11. The assumptions made in the EPQ model are the sameassumptions made in the economic order quantity with the excep-tion that the instantaneous receipt of inventory assumption is elim-inated. Thus, the assumptions are that the demand is known and constant, the lead time is known and constant, quantity dis-counts are not allowed, ordering cost and carrying cost are theonly variable costs, and stockouts and shortages can be completelyeliminated.

12-12. When the daily production rate becomes very large, theEPQ model becomes identical to the economic quantity model.This is because the fraction d/p approaches zero as the productionrate becomes very large.

12C H A P T E R

Inventory Control Models

6634 CH12 UG 8/23/02 2:32 PM Page 100

CHAPTER 12 INVENTORY CONTROL MODELS 101

12-13. In the quantity discount model, the carrying cost is a per-centage of the unit cost. This is due to the fact that the unit cost inthe quantity discount model is allowed to vary or change. Thus,the carrying cost per unit per year is not applicable to quantity dis-count models.

12-14. Solving a quantity discount model involves several steps.The first step is to compute the economic order quantity for eachdiscount range. The second step is to adjust the order quantity de-termined in step one if the order quantity is too low to qualify forthe discount. Furthermore, any economic order quantity valuesgreater than the discount range can be ignored. The third step is tocompute the total cost for every discount range. The fourth step isto select that order quantity from step three which has the lowesttotal inventory cost.

12-15. When the stockout cost is known, the safety stock can bedetermined by comparing the total cost of each safety stock pol-icy. This method requires that we know the probability of demandover lead time and the cost of a stockout in addition to the tradi-tional costs associated with the economic order quantity. When thestockout cost is not known, a service-level policy is established.For this particular model, it is only necessary to know the proba-bility of demand over lead time. This can either be a continuous ora discrete probability function.

12-16. ABC analysis is the process of categorizing inventoryinto three groups. The A group is very important to the organiza-tion and requires strict monitoring and control. The B group is notas important and selected items from this group are monitored andcontrolled. The C group is not as important as group A or group B,and thus sophisticated inventory control techniques are not used incontrolling inventory levels for these items.

12-17. See file P12-17.XLS. D � 100,000; Co � $10; Ch �$0.005

a.

12-18. See file P12-18.XLS. ROP � 8 days � (500 screws/day)� 4,000 number 6 screws

12-19. . See file P12-19.XLS.

Cost under Lila’s policy

� $0.005 � $100 (Sheet #1)

Q under brother’s policy

Cost under brother’s policy

� $145 (Sheet #2)

Extra cost � $45;no effect on ROP.

12-20. See file P12-20.XLS. D � 4,000 units

Ch � 10% of $90 � $9

Co � $25

a. Q* = =2 4 000 25

9149

( , )( ) units

� �50 000

20 005

,$ .

= ×100 000

50 00010

,

,$

= =100 000

250 000

,,

=100 000

20 00010

20 000

2

,

,$

,� �

TC = +D

Q C QCo h

1

2

Q* = =2 100 000 10

0 00520 000

( , )( )

., number 6 screws

b. ROP � (10 days) � (16 per day) � 160

c. Total cost

� $361,341.64

12-21. Co � $25

Ch � 25% of $100 � $25

Q* � 4,000

12-22. D � 500 sandals; Co � $10If Q* � 100,

Ch � $1, which is 20% of cost.

See file P12-22.XLS. If Ch � 10% of $5 � $0.50,

12-23. See file P12-23.XLS.

When Co � $10, Q* � 20,000 screws

12-24. See file P12-24.XLS. D � 50,000 units; C0 � $10; Ch � $4

a. .

b. ROP � (25 days) � (250 units/day) � 6,250 units

c. Optimal number of orders per year

12-25. See file P12-25.XLS. D � 6,000 units

Co � $10

Ch � 15% of $7 � $1.05

Total cost � $7 � 6,000 � � 1.05 � � 10

� $42,355 (Sheet #1)

If new supplier is used, Ch � 15% of $6.65 � $1

Q � 3,000

Total cost � $6.65 � 6,000 � � 1 � � $10

� $41,416.25 (Sheet #2)

Pampered Pet should use the new supplier and take the discount.

12-26. See file P12-26.XLS. Co � $10; Ch � $10; D � 5,000

Q* = =2 5 000 10

10100

( , )( )motors

6 000

3 000

,

,

3000

2

$ ,

*

6 000

Q

Q*

2

Q* = =2 6 000 10

1 05338

( , )( )

.

= =50 000

500100

,

Q units*( , )( )

= =2 50 000 10

4500

If screws C Q* o = = × =$ , , ,40 20 000 4 40 000

If screws C Q* o = = × =$ , , ,30 20 000 3 34 641

If screws C Q* o = = × =$ , , ,30 20 000 2 28 284

Q* = =2 500 10

0 50141

( )( )

.sandals

1002 500 10

=× ×

Ch or

4 0002 25

25, ,=

×D D �8 million loads of plywood

� � � � � �4 000

14925

149

29 4 000 90

,$ $ ,

6634 CH12 UG 8/23/02 2:32 PM Page 101

102 CHAPTER 12 INVENTORY CONTROL MODELS

Currently, the warehouse can hold .

They should expand the warehouse to 10,000 cubic feet to

hold 100 motors.

Current cost

� $1,250

The expansion would be worth $1,250 � $1,000 � $250 per year.

12-27. See file P12-27.XLS. D � 50,000 units; Co � $10; Ch � $16

a.

b. ROP � (35 days) � (250 units/day) � 8,750 units

12-28. See file P12-28.XLS. D � 12,000; Co � $30; Ch � $2

Currently, number of lawn mowers that can be stored:

� 120 units

Current cost

Optimal cost

To increase the number of units by a factor of 5 (� 600/120), thedepth should also be increased by a factor of 5, that is, from 40 ftto 200 ft, increase depth by 160 ft. They would be willing to pay$3,120 � $1,200 � $1,920 on a per-year basis.

12-29. To begin with, Lisa must determine which costs are notdirectly related to ordering or carrying costs. The cost of newproduct development, product advertising, and research and devel-opment are not related to ordering or carrying cost. Lisa must alsodetermine which costs are related to ordering and carrying costs.See the following table

Ordering CarryingCost Factor Cost Cost

Taxes $2,000Processing and inspection $1,500Bill paying 500Ordering supplies 50Inventory insurance 600Spoilage 750Sending purchasing orders 800Inventory inquiries 450Warehouse supplies 280Purchasing salaries 3,000Warehouse salaries 2,800Inventory theft 800Purchase order supplies 500Inventory obsolescence $0,000 $0,300

$6,800 $7,530

� � � � �600

22

12 000

60030 1 200$

,$ $ ,

� � � � �120

22

12 000

12030 3 120$

,$ $ ,

�60% of 40 ft 25 ft 8 ft

5 ft 4 ft 2 ft

× ×× ×

Q* = =2 12 000 30

2600

( , )( )units

Q* = =2 50 000 10

16250

( , )( )units

� � � �50

210

5 000

5010$

,$

5 000

10050

,

cu ft

cu ftmotors=

Cost� � � � �100

210

5 000

10010 1 000$

,$ $ ,

Next, Lisa needs to determine average ordering cost and car-rying cost. Ordering cost is computed by dividing total orderingcosts by the number of orders per year. Carrying cost is computedby dividing total carrying costs by the number of inventory items.

Number of orders 100Number of inventory items 10,000

Ordering cost per order $68.00Carrying cost per unit per year $0.753

Given an annual demand of 1,000 for the new product, the EOQ of 424.98 can be computed using ExcelModules. (See file P12-29.XLS)

12-30. Melinda can solve this problem by determining the proba-bility distribution for ordering cost. This is done by finding the totalof the frequency of ordering cost and dividing each number by thetotal. Melinda can also determine the EOQ value for each possibleordering cost value using ExcelModules. In order to determine theEOQ for the average or expected ordering cost, Melinda can multi-ply the probability of each ordering cost by the EOQ for the order-ing cost. This is displayed in the following table under the EXPcolumn. Summing this column will give us the resulting EOQ re-quested by Melinda’s boss. The results are shown in the table thatfollows. As you can see, the economic order quantity is 1,891 units.

Order Cost Frequency Probability EOQ EXP

$40 24 0.049 1,789 8841 34 0.070 1,811 12742 44 0.091 1,833 16643 56 0.115 1,855 21444 76 0.156 1,876 29345 66 0.136 1,897 25846 64 0.132 1,918 25347 45 0.093 1,939 18048 44 0.091 1,960 17749 23 0.047 1,980 9450 410 0.021 2,000 41

486 1.000 1,891

See file P12-30.XLS for the EOQ calculations.

12-31. See file P12-31.XLS. D � 8,000; d � 40; p � 150; Cs � $100; Ch � $0.30

12-32. See file P12-32.XLS. D � 10,000; d � 50; p � 500; Co � $40; Ch � $0.60

12-33. See file P12-33.XLS. D � 1,000; unit cost � $50; Co � $40;

Ch � 0.25 � unit cost

With discount, unit cost � (1 � 0.03) � $50 � $48.50

Qd* ( , )( )

. ..=

×=

2 1 000 40

0 25 48 5081 22

Q = =2 1 000 40

0 25 5080

( , )( )

. ( )

Q*( , )( )

. ( / ),=

−=

2 10 000 40

0 60 1 50 5001 217 wheel bearings

Qp* ( , )( )

. ( / ),=

−=

2 8 000 100

0 3 1 40 1502 697 scissors

6634 CH12 UG 8/23/02 2:32 PM Page 102


which should be adjusted to minimum orderable quantity (i.e.,200).

Original total cost

� $51,000

Discount cost

Therefore, North Manufacturing should take the discount.

12-34. Cc � $40; Ch � $5; ROP � 60 units � safety stock.The expected stockout cost is $50 per stockout � 7 orders per

year � the number of units short. For this problem, there will besix alternatives. Alternative 1 is to have a reorder point of 40, al-ternative 2 is a reorder point of 50, alternative 3 is a reorder pointof 60, and so on. The additional carrying cost is equal to $5 � thenumber of additional inventory items. There are six states of na-ture or events for this problem. Event or state of nature 1 is a de-mand over lead time of 40 units. Event 2 is a demand of 50, event3 is a demand of 60, and so on. The solution for this problem ispresented below. As you can see, the best decision is alternative 6,which is to have the reorder point plus safety stock equal to 90. Ifthe normal reorder point is 60, the safety stock is 30 units.

� � � �200

20 25 48 50 49 912 50. . $ , .

� � � �1 000 48 501 000

20040, .

,

� � �80

20 25 50.

� � � �1 000 501 000

8040,

,

12-35. m � 60; � � 7. See file P12-35.XLS.

Safety stock for 90% service level

� sZ (at 0.90) � 7 � 1.282 � 8.97 � 9

12-36. See file P12-36.XLS.

The item that needs strict control is 33CP. Items that should not bestrictly controlled are XX1, B66, 3CP0, R2D2, and RMS.

12-37. stockout cost � $50/unit; ROP �650; number of orders � 5

For this problem, the expected stockout cost is $50 per stock-out � 5 times per year � the number of units short. There are 11 al-ternatives. Alternative 1 is to have a reorder point plus safety stockof 600, alternative 2 is a reorder point of 650, alternative 3 is a re-order point of 700, and so on. The additional carrying cost is equalto $10 � the number of additional inventory items. There are 11

Co � $60; Ch � $10;

DEMAND

Event 1 Event 2 Event 3 Event 4 Event 5 Event 6

Probability 0.10 0.20 0.20 0.20 0.20 0.10

Alternative 1 $ 0 $3,500 $7,000 $10,500 $14,000 $17,500Alternative 2 50 0 3,500 7,000 10,500 14,000Alternative 3 100 50 0 3,500 7,000 10,500Alternative 4 150 100 50 0 3,500 7,000Alternative 5 200 150 100 50 0 3,500Alternative 6 250 200 150 100 50 0

EXPECTED COST TABLE

Alternative Expected Cost

1 $8,7502 5,6053 3,1704 1,4455 4306 125 k The best alternative

Solution for Problem 12-34 (See file P12-34.XLS)

Total Cost �Unit Cost �

Code Demand

XX1 $7,008B66 $5,9943CP0 $1,003.5233CP $82,292.16R2D2 $2,220RMS $1,998.88

Total cost � $100,516.5670% of total cost � $70,347.92

6634 CH12 UG 8/23/02 2:32 PM Page 103


states of nature or events in this problem. Event or state of nature 1is a demand over lead time of 600 units. Event 2 is a demand of 650,event 3 is a demand of 700, and so on. The solution for this problemis presented in the table. As you can see, the best decision is alterna-tive 10, which is to have the reorder point plus safety stock equal to1,050 units. If the normal reorder point is 650 units, the safety stockis 400 units. The total expected cost is $3,425.

Event 1 Event 2 Event 3 Event 4 Event 5 Event 6

Probability 0.300 0.200 0.100 0.100 0.050 0.050

Alternative 1 $ 0 $12,500 $ 25,000 $ 37,500 $ 50,000 $62,500Alternative 2 500 0 12,500 25,000 37,500 50,000Alternative 3 1,000 500 0 12,500 25,000 37,500Alternative 4 1,500 1,000 500 0 12,500 25,000Alternative 5 2,000 1,500 1,000 500 0 12,500Alternative 6 2,500 2,000 1,500 1,000 500 0Alternative 7 3,000 2,500 2,000 1,500 1,000 500Alternative 8 3,500 3,000 2,500 2,000 1,500 1,000Alternative 9 4,000 3,500 3,000 2,500 2,000 1,500Alternative 10 4,500 4,000 3,500 3,000 2,500 2,000Alternative 11 5,000 4,500 4,000 3,500 3,000 2,500

Event 7 Event 8 Event 9 Event 10 Event 11

Probability 0.050 0.050 0.050 0.030 0.020

Alternative 1 $75,000 $87,500 $100,000 $112,500 $125,000Alternative 2 62,500 75,000 87,500 100,000 112,500Alternative 3 50,000 62,500 75,000 87,500 100,000Alternative 4 37,500 50,000 62,500 75,000 87,500Alternative 5 25,000 37,500 50,000 62,500 75,000Alternative 6 12,500 25,000 37,500 50,000 62,500Alternative 7 0 12,500 25,000 37,500 50,000Alternative 8 500 0 12,500 25,000 37,500Alternative 9 1,000 500 0 12,500 25,000Alternative 10 1,500 1,000 500 0 12,500Alternative 11 2,000 1,500 1,000 500 0

EXPECTED COST TABLE

Alternative Expected Cost

1 $33,3752 24,7753 18,7754 14,0755 10,6756 7,9257 5,8258 4,3759 3,575

10 3,425 k The best alternative11 3,665

Table for Problem 12-37 (See file P12-37.XLS)

6634 CH12 UG 8/23/02 2:32 PM Page 104


12-38. D � 5,000; Co � $15; Ch � $0.50; d � 100; t � 3

a. (See file P12-38.XLS)

b. The ordering cost is still , but the carrying cost

will be reduced because it arrives over three weeks.

Maximum inventory level � total order � total usedduring lead time

� Q � 3 � 100

� Q � 300

Carrying cost

Order cost

Setting the two equal

Q*2 � 300Q* � 300,000 � 0

Q* � 717.9

c. Total cost for

instantaneous delivery � $273.86Total cost for

installment delivery � 0.50(717.9 � 300)

� 0.50(417.9)

� $208.95

Note: Total cost � ordering cost � carrying cost. Since orderingcost � carrying cost, total cost � 2 � carrying cost � Q* � Ch.

Go for installment delivery.

12-39. See file P12-39.XLS. Round up safety stock values. Ch � $0.50; � � 600; � � 7

Safety stock for 90% service level � 9

Carrying cost � 9 � 0.5 � $4.50

Safety stock for 95% service level � 7 � 1.65 � 12

Carrying cost � $6.00

Safety stock for 98% service level � 7 � 2.05 � 15

Carrying cost � $7.5012-40. Maximum inventory level � Q � 5 � 100

� Q � 500

Carrying cost

Ordering cost

Setting the two equal,

Q*2 � 500Q* � 300,000 � 0

Q* � 852Total cost � (852 � 500) � 0.50 � $176

Note: Total cost � 2 � carrying cost because ordering cost � carrying cost.

Q*Q*

− =500300 000,

= ×5 000

15,

Q

= − ×12

500 0 50( ) .Q

Or,, ( )( )

.

, Q*

Q* Q*− = =300

5 000 15 2

0 50

300 000

12

300( )Q*Q*

− =CD

Ch o

=D

CoQ*

= −12

300( )Q Ch

D

QCo

Q*( , )( )

..= =

2 5 000 15

0 50547 7

12-41. Item 4 should be carefully controlled; See file P12-41.XLS for the ABC analysis.

(Sheet #2)

The other items contribute together about 15% of total revenues.They do not need strict quantitative control. If however, items 1and 2 are also controlled using EOQ:

(Sheet #3)

(Sheet #4)

Items 3, 4, and 5 are definitely in category C.

12-42. See file P12-42.XLS. Co � $45; I � 20%; D � 100

Optimal order quantity would be 51.

� 1,725 � 88.24 � 87.98

� $1,901.22

12-43. This is a typical quantity discount problem. It is compli-cated, however, by the fact that there are drawings for computersand trips, which must be considered as part of the quantity dis-count. When this is done, a quantity discount table can be devel-oped and used to determine the best inventory policy. The quantitydiscount table is shown below.

AverageProgram Cans Bonus Discount Cost

1 0–199 0 $9.902 200–299 10 cans 9.393 300–399 30 cans 8.904 400–499 40 cans 8.89

Here is how the quantity discount table was determined. Dis-count 1 represents a quantity ranging from 0 to 199 units. There isno discount, and therefore the cost is simply $9.90. For discountnumber two, 10 free cans of product are offered. This has a totalvalue of $99. In addition, it is possible to receive a personal com-puter valued at $3,000. Since there are 1,000 companies that are eligible, the expected monetary value for the personal computerdrawing is $3 (3 � 3,000/1,000). This represents a total discount of$102. For 200 cans of product, this represents a 51-cent discount(0.51 � 102/200). Therefore, the discount price is $9.39. The sametype of computations can be made for discount number three. The30 cans of free product have a value of $297, and the personal com-puter drawing has an expected value of $3. The total discount is$300 or $1 per unit. Therefore, the average discount price is $8.90.For discount number four, there is also a drawing for a free trip.

TC� � �100 17 25100 45

51

51 0 2 17 25

2( . )

( ) ( . )( . )

Q32 100 45

0 2 17 2551 1* ( )( )

. ( . ).= =

Q22 100 45

0 2 17 5050 7* ( )( )

. ( . ).= =

Q12 100 45

0 2 1850* ( )( )

. ( )= =

Q22 450 30

0 25 11 0099* ( )( )

. ( . )= =

Q12 600 40

0 2 10 6151* ( )( )

. ( . )= =

Q42 560 40

0 15 15045* ( )( )

. ( )= =

6634 CH12 UG 8/23/02 2:32 PM Page 105

a


This trip has a value of $5,000 and 800 businesses are eligible forthe drawing. This represents a $6.25 value ($6.25 � $5,000/800).Adding this to the $396 value for the 40 free cans and the $3 ex-pected monetary value for the personal computer drawing, the totaldiscount is approximately $405. The average discount therefore is$1.01. This represents a discount cost of $8.89. This information,along with the standard information for inventory control, can beused in ExcelModules (See file P12-43.XLS) to compute a quantitydiscount. The computer output reveals that the optimal strategy is toorder 300 units at a total cost of $9,083.

12-44. This safety stock problem can be solved using decisionmaking under risk. The cost of a stockout is $13.05 ($13.05 �$45.95 � $32.90). Carrying cost is $7 per unit per year. The deci-sion table shows that the best policy is to have a reorder point in-cluding safety stock of 600 units. This corresponds to a safetystock of 200 units with a normal reorder point of 400 units. Mini-mum EMV is $993, which corresponds to the alternative of 600 units for the reorder point including safety stock. See file P12-44.XLS for the calculations.

12-45. a. (See file P12-45.XLS) This is a typical quantity dis-count problem. The data and results are presented below. The opti-mal quantity is 1,500 disks.

Data

Demand rate (D) 2,000Setup/Ordering cost (S) 250Holding cost (H) 1

Price Ranges From To Price

1 500 $10501 1,000 9.95

1,001 1,500 9.91,501 2,000 9.85

Results

Optimal order quantity (Q*) 1,501Average inventory 750.50Orders per period (year) 1.33Annual Setup cost $333.11Annual Holding cost $750.50Unit costs (PD) $19,700Total Cost $20,783.61

b. Given a different quantity discount schedule, we can computethe optimal order policy using the same approach. The resultsare shown below. See sheet #2 in file P12-45.XLS.

Data

Demand rate (D) 2,000Setup/Ordering cost (S) $250Holding cost (H) $1

Price Ranges From To Price

1 500 $10501 1,000 9.99

1,001 1,500 9.981,501 2,000 9.97

Results

Optimal order quantity (Q*) 1,001Average inventory 500.50Orders per period (year) 2Annual Setup cost $499.50Annual Holding cost $500.50Unit costs (PD) $19,960.00Total Cost $20,960.00

12-46. See file P12-46.XLS. If the mean DDLT is 36 and thestandard deviation is 15, we need to carry 19 TVs as safety stockto achieve a 90% service level. The reorder point is 55.

12-47. See file P12-47.XLS. If the mean DDLT is 50 and thestandard deviation is 5, we get the following values:

a. Z value to be applied = 1.88

b. Safety stock = 9 drives

c. Reorder point = 59 drives

12-48. See file P12-48.XLS. We can solve this problem using a decision analysis. The alternatives as well as the states of natureare the 5 possible reorder points (0 to 400). For each case, we cancompute the total stockout or inventory carrying cost. We canthen compute the expected value for each decision alternative.Based on the analysis, we should set a reorder point of 400 units.The expected monetary value of this alternative is $3,000. If thereorder point without safety stock is 200 units, this implies that weshould carry a safety stock of 400 � 200 � 200 units.

Problem 12-44 Decision Table (See file P12-44.XLS) TotalFrequency 1 2 2 3 4 5 4 4 3 2 2 32

Probability 0.03 0.06 0.06 0.09 0.12 0.15 0.12 0.12 0.09 0.06 0.06

STATE OF NATURE

Alternative 300 350 400 450 500 550 600 650 700 750 800 EMV

300 0 653 1,305 1,958 2,610 3,263 3,915 4,568 5,220 5,873 6,525 3,466350 350 0 653 1,305 1,958 2,610 3,263 3,915 4,568 5,220 5,873 2,845400 700 350 0 653 1,305 1,958 2,610 3,263 3,915 4,568 5,220 2,287450 1,050 700 350 0 653 1,305 1,958 2,610 3,263 3,915 4,568 1,791500 1,400 1,050 700 350 0 653 1,305 1,958 2,610 3,263 3,915 1,389550 1,750 1,400 1,050 700 350 0 653 1,305 1,958 2,610 3,263 1,112600 2,100 1,750 1,400 1,050 700 350 0 653 1,305 1,958 2,610 993650 2,450 2,100 1,750 1,400 1,050 700 350 0 653 1,305 1,958 998700 2,800 2,450 2,100 1,750 1,400 1,050 700 350 0 653 1,305 1,129750 3,150 2,800 2,450 2,100 1,750 1,400 1,050 700 350 0 653 1,353800 3,500 3,150 2,800 2,450 2,100 1,750 1,400 1,050 700 350 0 1,641

minimum EMV

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12-49. See file P12-49.XLS. It appears from the analysis thatitems A and D are category A items and should be closely moni-tored. Items B and E are category B items, while item C is a cate-gory C item.

12-50. See file P12-50.XLS. It appears from the analysis thatitems G2 and F3 are category A items and should be closely moni-tored. Items A2, C7, and D1 are category B items. All other itemsare category C items.

12-51. See file P12-51.XLS. Several classifications are possiblein this case. For example, we could classify items D23, V75, andU11 as category A items that should be closely monitored. ItemsE102 and S107 could be classified as category B items. All otheritems could be classified as category C items.

SOLUTION TO STURDIVANT SOUND SYSTEMS CASE

The optimal order quantity is (See sheet #1 in file P12-Sturdivant.XLS).

where

Q � optimal order quantity

D � annual demand (5,000)

Co � procurement costs ($20)

Ch � carrying costs ($6)

d � average daily demand (20)

L � lead time in days (10)

P � cost per unit ($60)

The reorder point is

ROP � dL

� 20(10)

Q = = =2 5 000 20

633 333 183

( , )( ), units

QDC

Co

h=

2

� 200 (meaning they are actually ordering one cycle inadvance when there are 17 units on hand)

Total annual cost under present system (current order level is 400units): (See sheet #2 in file P12-Sturdivant.XLS).

Cost of equipment � 5,000 units � $60 � $300,000Procurement costs � $20 per order � 12.5 orders � 250

Carrying costs � (400/2) units � $6/unit � $301,200

$301,450

Total annual cost per optimal procurement policy:

� 300,000 � 546.45* � 549*

� $301,095.45

*Procurement costs and carrying costs are not equal, due to round-ing to an even number of units for Q.

Cost savings:

$ 301,450.00

�301,095.45

$ 354.55

The typical costs associated with procurement of materialsinclude costs of preparing requisitions, writing purchase orders,receiving merchandise, inspecting goods, storage, updating inven-tory records, and so on. These costs are usually fixed, regardless ofthe size of the order. While a large order may require more time,the increase in procurement costs is minimal. As lot size increases,the number of orders decreases (assuming a constant requirementlevel). Consequently, procurement costs will decrease with in-creases in lot sizes.

MARTIN-PULLIN BICYCLE CORPORATION

1. Inventory plan for Martin-Pullin Bicycle Corporation. Theforecasted demand is summarized in the following table.

� � �5 000 605 000

18320

183

26, ( )

,( ) ( )

TC� � �DPD

QC

QCo h

2

Average demand per month � 439/12 � 36.58 bicycles. The stan-dard deviation of the monthly demand � 24.58 bicycles.

The inventory plan is based on the following costs and values.

Order Cost � $65/orderCost per bicycle � $102.00Holding cost � ($102.00) � (1%) � 12 per year

per bicycle� $12.24 per year per bicycle

Service level � 95%, with corresponding Zvalue of 1.6425

Lead time � 1 month (4 weeks)Total demand/year � 439 units of bicycles

The solution shown uses the simple EOQ model with reorder pointand safety stock. It ignores the seasonal nature of the demand. Thefluctuation in demand is dealt with by the safety stock based on

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Total

8 15 31 59 97 60 39 24 16 15 28 47 439

6634 CH12 UG 8/23/02 2:32 PM Page 107


the variation of demand over the planning horizon (See file P12-MP.XLS).

Economic order quantity (Q*) is given by:

where the Total demand and the Holding Cost are calculated onthe same time unit (monthly, yearly, etc.).

Thus,

2. The reorder point is calculated by the following relation:

Reorder point (ROP) � average demand during the lead time(�) � z � (standard deviation of thedemand during the lead time (�))

Therefore, (See sheet #2 in file P12-MP.XLS)

ROP � 36.58 � 1.6425(24.581) � 77 bicycles

Safety stock (ss) is given by

ss � z� � 1.6425(24.581) � 40 bicycles

Inventory cost is calculated as follows:

� $416.00 � $489.60 � 416.00 � $1321.60

This case can be made more interesting by asking students to tracethe inventory behavior with the above plan (assuming that the fore-cast figures are accurate and ignoring the forecast errors) and to seethe amount of total stockout, if any. Students then can calculate thelost profit due to stockout and add it to the total cost.

3. A plot of the nature of the demand clearly shows that it is not alevel demand over the planning horizon. An EOQ, for the entireyear, therefore, may not be appropriate. Students should try to seg-ment the planning horizon in a way so that the demand is moreevenly distributed and come up with an inventory plan for each ofthese segments (e.g., quarterly inventory planning). The challenge isthen to manage the transition from one planning period to the next.

SOLUTION TO INTERNET CASESProfessional Video Management 1. To determine the reorder points for the two suppliers, dailydemand for the videotape systems must be determined. Since eachvideo system requires two videotape systems that are connected toit, the demand for the videotape units is equal to twice the numberof complete systems.

The demand for the complete video system appears to be rel-atively constant and stable. The monthly demand for the past few months can be averaged, and this value can be used for theaverage monthly demand. The average monthly sales is equal to(7,970 � 8,070 � 7,950 � 8,010)/4 � 8,000. Therefore, the

�Total

Ordering Demand

Q* Cost( )

�1

2Q* Cost Cost( ) ( )Holding ss Holding+

Total annual Annualinventory � holding � Annual ordering cost

cost cost

Q* units of bicycles=× ×

≈2 439 65

12 2468

.

Q* demand Cost

Holding Cost=

× ×2 ( ) ( )Total Ordering

average monthly demand of the videotape systems is 16,000 units,because two tape units are required for every complete system.Annual demand is 192,000 units (192,000 � 12 � 16,000).

We will assume that there are 20 working days per month. Inother words, there are 5 working days per week. Making this as-sumption, we can determine the average daily sales to be equal tothe average monthly sales divided by 20. In other words, the dailysales is equal to 800 units per day (800 � 16,000/20).

To determine the reorder point for Toshiki, we must knowthe lead time. For Toshiki, it takes 3 months between the time anorder is placed and when the order is actually received. In otherwords, the lead time is 3 months. Again, assuming 20 workingdays per month, the lead time for Toshiki is 60 days (60 � 20 �3). To determine the reorder point, we multiply the demand ex-pressed as units per day times the lead time in days. For Toshiki,the reorder point is equal to 48,000 units (48,000 � 800 � 60).Because the reorder point will be greater than the EOQ (see num-ber 2 for EOQ calculations), the lead time will likely be more im-portant for ordering more inventory.

For Kony, the reorder point can be computed in the same man-ner. Assuming again that there are 5 working days per week, we cancompute the lead time in days. For Kony, it takes 2 weeks betweenthe time an order is placed and when it is received. Therefore, thelead time in days is equal to 10 days (10 � 2 � 5). With the lead timeexpressed in days, we can compute the reorder point for Kony. Thisis done by multiplying the lead time in days times the daily demand.Therefore, the reorder point for Kony is 8,000 (8,000 � 800 � 10).

2. To make a decision concerning which supplier to use, total in-ventory cost must be considered for both Toshiki and Kony. Bothcompanies have quantity discounts. Because there are two suppli-ers, we had to make two separate quantity discount calculations.The first was for Toshiki. The second was for Kony. Toshiki hadthe lowest total cost of $40,950,895.50. The EOQ for the mini-mum cost inventory policy was 20,001. Kony had a cost of$42,406,569. See sheets #1 and #2 in file P12-PVM.XLS for thecalculations.

3. Each alternative that Steve is considering would have a directimpact on the quantity discount model and the results. The firststrategy is to sell the components separately. If this is done, thedemand for videotape systems could change drastically. In addi-tion to selling the videotape units along with the complete system,additional tape units could be demanded. An increase in demandcould change the outcome of the quantity discount model. Thesecond strategy would also have an impact on the results of theanalysis. At this time, two videotape systems are used for everycomplete system. If other videotape systems can be used as well,there will be fewer videotape systems ordered when obtaining thecomplete system. At this time, exactly two videotape systems aresold with every complete system. Implementing the second strat-egy would cause this ratio to drop below two. Again, this willchange the annual demand figures.

Drake Radio1. In order to figure out the reorder points for the two suppliers,daily demand for the FM tuner must be derived. Since one FMtuner is required for each DR-2000 (stereo system), demand fortuners is equal to 1 � (demand for DR-2000).

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Demand for DR-2000’s appears to be fairly constant and sta-ble, based on the figures from Figure 1. An average monthly de-mand can be calculated as follows:

(Demand for Jan. � Demand for Feb. � Demand for Mar. �Demand for Apr.)/4

= (801 � 807 � 795 � 797)/4 � 800 per month

Assuming that there are 20 working days per month, dailydemand can be estimated as follows:

Avg. Monthly Demand � # days/months � Avg. daily demand800 � 20 � 40 units

The reorder point is equal to daily demand times the lead time.ROP � dL

For Collins, lead item is 2 weeks which (following the previ-ous assumption about working days per month) is equal to 10 days.Therefore,

ROP � 40 units/day � 10 days � 400 units,

meaning that if Drake Radio is being supplied by Collins, the firmshould reorder stock when inventory reaches a level of 400 units.

For Nitobitso, the lead time is 2 months or 40 days. Therefore,

ROP � 40 units/day � 40 days � 1,600 units,

meaning that if Drake Radio is being supplied by Nitobitso, thefirm should reorder stock when the inventory falls to a level of1,600 units.

2. To make a sound recommendation, total inventory costs forboth Collins and Nitobitso must be determined. Both companieshave quantity discounts.

Annual demand is estimated to be 9,600 units (800 units/month � 12 months/yr.).

The first step in determining inventory costs is to determinewhat the economic order is; then total costs can be derived.

Collins: (See sheet #1 in file P12-Drake.XLS)Using total costs, an evaluation of the price breaks due to the

quantity discount can be done:

TCP2 � (9,600/400)(50) � (400/2)6 � 24(9,600) � $232,800

TCP3 � (9,600/501)(50) � (501/2)(5.5) � 22(9,600) � $213,535.83

The lowest total inventory cost for Collins is $213,535.83 withEOQ of 501 units.

Nitobitso (See sheet #2 in file P12-Drake.XLS)

TCP2 � (96,00/577.85)(100) � (577.85/2)(5.75) � 23(9,600)

� $224,122.65

TCP3 � (9,600/801)(100) � (801/2)(5.5) � 22(9,600)

� $214,601.25

TCP4 � (9,600/2001)(100) � (2,001/2)(5.25) � 21(9,600)

� $207,332.39

The lowest total cost for Nitobitso is $207,332.39 with an EOQ of2,001 units.

A comparison of the two lowest total cost figures indicatesthat using Nitobitso as supplier would be the least costly of thetwo. Ordering costs decreases and price breaks far outweigh anycarrying cost increases in this case.

3. Everything else being equal, Collins would be the best sup-plier of FM tuners in the event of fluctuating demand. Collins’

lead time is substantially less than Nitobitso’s. Should high de-mand occur during the time when a shipment is expected, stock-outs could occur. With a short lead time, lost sales are kept to aminimum. Thus, a supplier with a shorter lead time is less of a riskto a purchaser whose product demand fluctuates a great deal.

LaPlace Power and Light Co.The optimal order quantity is given by: (see sheet #1 in file P12-LaPlace.XLS.

Q* � 34.74 thousand feetThe reorder point is given by:

ROP � Daily demand * Lead time

ROP � 115.27 thousand feet

Currently, the company is committed to take 1/12th of its annualneed every month. Therefore, each month the storeroom issues apurchase requisition for 41,625 feet of cable. With TC � total in-ventory cost,

Present TC

� (499.5)(414)

� 600 � 861.64 � 209,793

� $208,254.64 (See sheet #2)

Optimal TC

� (499.5)(414)

� 718.91 � 719.12 � 206,793

� $208,231.03 (See sheet #1)

Savings � Present TC � Optimal TC � $23.61Ordering costs are a linear function because no matter how largean order is or how many orders are sent in, the cost to order anymaterial is $50 per order.

The student should recognize that it is doubtful the firm will orshould alter any current ordering policy for a savings of only $23.

Western Ranchman OutfittersThe EOQ for a yearly demand of 2,000, order cost of $10.00 andholding cost of 0.12 (10.05) � $1.206 is (See file P12-Western.XLS)

The solution recommends 2,000/182 � 11 orders to be sub-mitted per year; WRO orders monthly. The EOQ is about 182pairs, as compared to 167 ordered monthly. The annual cost differ-ence is miniscule. (See sheet #2 in P12-Western.XLS)

There is one remaining problem which the model doesn’tsolve, but which Mr. Randell has. That is the problem of the unreliability of the supplier. By ordering one extra time (twelve

EOQ = =2 10 2 000

1 206182 12

( )( , )

..

=

+

499 5

34 7450

34 74

241 4

.

.( )

.( . )

=

+

499 5

41 62550

41 625

241 4

.

.( )

.( . )

=

499 5

260

. (60)

QDS

H*

( . ) *

.= =

2 2 499 5 50

41 4

6634 CH12 UG 8/23/02 2:32 PM Page 109


orders per year instead of eleven) and by ordering extra quantitiesjudiciously, Mr. Randell has managed to keep WRO almost totallysupplied with the requisite number of Levi 501s. Further, since theactual solution is so close to the model solution, and since we haveseen that the EOQ is a robust model, Mr. Veta can feel that he iskeeping his inventory goals close to the minimum while still meet-ing his goal of avoiding stockouts.

The conclusion is that the model has been shown to be practi-cally valid with minor adjustments which compensate for the un-reliability of the manufacturer.

This case differs from most in that the EOQ is just a startingpoint for discussion. Students must then develop their own ap-proach and reasoning for why the current policy is acceptable orunacceptable.

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