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12 – 1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

Inventory ManagementInventory Management12

For For Operations Management, 9eOperations Management, 9e by by Krajewski/Ritzman/Malhotra Krajewski/Ritzman/Malhotra © 2010 Pearson Education© 2010 Pearson Education

PowerPoint Slides PowerPoint Slides by Jeff Heylby Jeff Heyl


Inventory ManagementInventory Management

Inventories are important to all types of organizations They have to be counted, paid for, used in

operations, used to satisfy customers, and managed

Too much inventory reduces profitability Too little inventory damages customer

confidence Inventory trade-offs


ABC AnalysisABC Analysis

Stock-keeping units (SKU) Identify the classes so management can

control inventory levels A Pareto chart Cycle counting


10 20 30 40 50 60 70 80 90 100Percentage of SKUs

Perc

enta

ge o

f dol

lar v

alue

100 —

90 —

80 —

70 —

60 —

50 —

40 —

30 —

20 —

10 —

0 —

Class C

Class A

Class B

ABC AnalysisABC Analysis

Figure 12.1 – Typical Chart Using ABC Analysis


Economic Order QuantityEconomic Order Quantity

The lot size, Q, that minimizes total annual inventory holding and ordering costs

Five assumptions1. Demand rate is constant and known with certainty2. No constraints are placed on the size of each lot3. The only two relevant costs are the inventory holding

cost and the fixed cost per lot for ordering or setup4. Decisions for one item can be made independently of

decisions for other items5. The lead time is constant and known with certainty


Economic Order QuantityEconomic Order Quantity

Don’t use the EOQ Make-to-order strategy Order size is constrained

Modify the EOQ Quantity discounts Replenishment not instantaneous

Use the EOQ Make-to-stock Carrying and setup costs are known and

relatively stable


Calculating EOQCalculating EOQ

Inventory depletion (demand rate)

Receive order

1 cycle

On-

hand

inve

ntor

y (u

nits

)

Time

Q

Averagecycleinventory

Q2

Figure 12.2 – Cycle-Inventory Levels



Annual holding costAnnual holding cost = (Average cycle inventory)

(Unit holding cost) Annual ordering cost

Annual ordering cost = (Number of orders/Year) (Ordering or setup costs)

Total annual cycle-inventory costTotal costs = Annual holding cost

+ Annual ordering or setup cost


Ann

ual c

ost (

dolla

rs)

Lot Size (Q)

Holding cost

Ordering cost

Total cost


Figure 12.3 – Graphs of Annual Holding, Ordering, and Total Costs



Total annual cycle-inventory cost

whereC = total annual cycle-inventory costQ = lot sizeH = holding cost per unit per yearD = annual demandS = ordering or setup costs per lot

C = (H) + (S)Q2

DQ


The Cost of a Lot-Sizing PolicyThe Cost of a Lot-Sizing Policy

EXAMPLE 12.1 A museum of natural history opened a gift shop which

operates 52 weeks per year. Managing inventories has become a problem. Top-selling SKU is a bird feeder. Sales are 18 units per week, the supplier charges $60 per

unit. Ordering cost is $45. Annual holding cost is 25 percent of a feeder’s value. Management chose a 390-unit lot size. What is the annual cycle-inventory cost of the current policy

of using a 390-unit lot size? Would a lot size of 468 be better?



SOLUTIONWe begin by computing the annual demand and holding cost as

D =H =

C = (H) + (S)Q2

DQ

The total annual cycle-inventory cost for the alternative lot size is

= ($15) + ($45)

= $2,925 + $108 = $3,033

3902

936390

The total annual cycle-inventory cost for the current policy is

(18 units/week)(52 weeks/year) = 936 units0.25($60/unit) = $15

C = ($15) + ($45) = $3,510 + $90 = $3,6004682

936468



3000 –

2000 –

1000 –

0 –| | | | | | | |

50 100 150 200 250 300 350 400Lot Size (Q)

Ann

ual c

ost (

dolla

rs)

CurrentQ

Currentcost

Lowestcost

Best Q (EOQ)

Total cost = (H) + (S)

Q2

DQ

Figure 12.4 – Total Annual Cycle-Inventory Cost Function for the Bird Feeder

Ordering cost = (S)DQ

Holding cost = (H)Q2



The EOQ formula:

EOQ = 2DSH

Time between orders

TBOEOQ = (12 months/year)EOQD


Finding the EOQ, Total Cost, TBOFinding the EOQ, Total Cost, TBO

EXAMPLE 12.2For the bird feeders in Example 12.1, calculate the EOQ and its total annual cycle-inventory cost. How frequently will orders be placed if the EOQ is used?

SOLUTIONUsing the formulas for EOQ and annual cost, we get

EOQ = =2DSH

= 74.94 or 75 units2(936)(45)15



Figure 12.5 shows that the total annual cost is much less than the $3,033 cost of the current policy of placing 390-unit orders.

Figure 12.5 – Total Annual Cycle-Inventory Costs Based on EOQ Using Tutor 12.2



When the EOQ is used, the TBO can be expressed in various ways for the same time period.

TBOEOQ =EOQ

D

TBOEOQ = (12 months/year)EOQ

D

TBOEOQ = (52 weeks/year)EOQ

D

TBOEOQ = (365 days/year)EOQ

D

= = 0.080 year75936

= (12) = 0.96 month75936

= (52) = 4.17 weeks75936

= (365) = 29.25 days75936


Application 12.1Application 12.1

Suppose that you are reviewing the inventory policies on an $80 item stocked at a hardware store. The current policy is to replenish inventory by ordering in lots of 360 units. Additional information is:

D = 60 units per week, or 3,120 units per yearS = $30 per orderH = 25% of selling price, or $20 per unit per year

What is the EOQ?

EOQ = =2DSH

= 97 units2(3,120)(30)20

SOLUTION


Current Policy EOQ Policy


What is the total annual cost of the current policy (Q = 360), and how does it compare with the cost with using the EOQ?

Q = 360 units Q = 97 units

C = 3,600 + 260

C = $3,860

C = (360/2)(20) + (3,120/360)(30)

C = 970 + 965

C = $1,935

C = (97/2)(20) + (3,120/97)(30)



What is the time between orders (TBO) for the current policy and the EOQ policy, expressed in weeks?

TBO360 =

TBOEOQ =

(52 weeks per year) = 6 weeks3603,120

(52 weeks per year) = 1.6 weeks973,120

SOLUTION


Managerial InsightsManagerial Insights

TABLE 12.1 | SENSITIVITY ANALYSIS OF THE EOQ

Parameter EOQ Parameter Change

EOQ Change

Comments

Demand ↑ ↑ Increase in lot size is in proportion to the square root of D.

Order/Setup Costs ↓ ↓

Weeks of supply decreases and inventory turnover increases because the lot size decreases.

Holding Costs ↓ ↑ Larger lots are justified when holding

costs decrease.

2DSH

2DSH

2DSH


Inventory Control SystemsInventory Control Systems

Two important questions: How much? When?

Nature of demand Independent demand Dependent demand


Inventory Control SystemsInventory Control Systems

Continuous review (Q) system Reorder point system (ROP) and fixed order

quantity system For independent demand items Tracks inventory position (IP) Includes scheduled receipts (SR), on-hand

inventory (OH), and back orders (BO)

Inventory position = On-hand inventory + Scheduled receipts – Backorders

IP = OH + SR – BO


Selecting the Reorder PointSelecting the Reorder Point

Time

On-

hand

inve

ntor

y

TBO TBOL L

TBOL

Orderplaced

Orderplaced

Orderplaced

IP IPIP

R

Q QQ

OH OHOH

Orderreceived

Orderreceived

Orderreceived

Orderreceived

Figure 12.6 – Q System When Demand and Lead Time Are Constant and Certain



The on-hand inventory is only 10 units, and the reorder point R is 100. There are no backorders and one open order for 200 units. Should a new order be placed?

IP = OH + SR – BO = 10 + 200 – 0 = 210

R = 100

SOLUTION

Decision: Place no new order


Placing a New OrderPlacing a New Order

EXAMPLE 12.3Demand for chicken soup at a supermarket is always 25 cases a day and the lead time is always 4 days. The shelves were just restocked with chicken soup, leaving an on-hand inventory of only 10 cases. No backorders currently exist, but there is one open order in the pipeline for 200 cases. What is the inventory position? Should a new order be placed?

SOLUTIONR =Total demand during lead time = (25)(4) = 100 cases

= 10 + 200 – 0 = 210 casesIP = OH + SR – BO


Continuous Review SystemsContinuous Review Systems

Selecting the reorder point with variable demand and constant lead time

Reorder point =Average demand during lead time + Safety stock=dL + safety stock

whered = average demand per week (or day or months)L = constant lead time in weeks (or days or months)



Time

On-

hand

inve

ntor

y

TBO1 TBO2 TBO3

L1 L2 L3

R

Orderreceived

Q

Orderplaced

Orderplaced

Orderreceived

IP IP

Q

Orderplaced

Q

Orderreceived

Orderreceived

0

IP

Figure 12.7 – Q System When Demand Is Uncertain


Reorder PointReorder Point

1. Choose an appropriate service-level policy Select service level or cycle service level Protection interval

2. Determine the demand during lead time probability distribution

3. Determine the safety stock and reorder point levels


Demand During Lead TimeDemand During Lead Time

Specify mean and standard deviationStandard deviation of demand during lead time

σdLT = σd2L = σd L

Safety stock and reorder point

Safety stock = zσdLT

wherez =number of standard deviations needed to achieve the cycle-service levelσdLT =stand deviation of demand during lead time

Reorder point = R = dL + safety stock


σt = 15

+75

Demand for week 1

σt = 25.98

225Demand for 3-week lead time

+75

Demand for week 2

σt = 15

=75

Demand for week 3

σt = 15


Figure 12.8 – Development of Demand Distribution for the Lead Time



Average demand

during lead time

Cycle-service level = 85%

Probability of stockout(1.0 – 0.85 = 0.15)

zσdLT

R

Figure 12.9 – Finding Safety Stock with a Normal Probability Distribution for an 85 Percent Cycle-Service Level


Reorder Point for Variable DemandReorder Point for Variable Demand

EXAMPLE 12.4Let us return to the bird feeder in Example 12.2. The EOQ is 75 units. Suppose that the average demand is 18 units per week with a standard deviation of 5 units. The lead time is constant at two weeks. Determine the safety stock and reorder point if management wants a 90 percent cycle-service level.


Reorder Point for Variable DemandReorder Point for Variable Demand

SOLUTIONIn this case, σd = 5, d = 18 units, and L = 2 weeks, so σdLT = σd L = 5 2 = 7.07. Consult the body of the table in the Normal Distribution appendix for 0.9000, which corresponds to a 90 percent cycle-service level. The closest number is 0.8997, which corresponds to 1.2 in the row heading and 0.08 in the column heading. Adding these values gives a z value of 1.28. With this information, we calculate the safety stock and reorder point as follows:

Safety stock = zσdLT = 1.28(7.07) = 9.05 or 9 units

2(18) + 9 = 45 unitsReorder point = dL + Safety stock =



Suppose that the demand during lead time is normally distributed with an average of 85 and σdLT = 40. Find the safety stock, and reorder point R, for a 95 percent cycle-service level.

SOLUTIONSafety stock = zσdLT =

Find the safety stock, and reorder point R, for an 85 percent cycle-service level.

R = Average demand during lead time + Safety stock R = 85 + 66 = 151 units

1.645(40) = 65.8 or 66 units

Safety stock = zσdLT = 1.04(40) = 41.6 or 42 unitsR = Average demand during lead time + Safety stockR = 85 + 42 = 127 units


Reorder Point for Variable Demand Reorder Point for Variable Demand and Lead Timeand Lead Time

Often the case that both are variableThe equations are more complicated

Safety stock = zσdLT

whered =Average weekly (or daily or monthly) demandL =Average lead timeσd =Standard deviation of weekly (or daily or monthly) demandσLT =Standard deviation of the lead timeσdLT = Lσd

2 + d2σLT2

R =(Average weekly demand Average lead time) + Safety stock=dL + Safety stock



EXAMPLE 12.5The Office Supply Shop estimates that the average demand for a popular ball-point pen is 12,000 pens per week with a standard deviation of 3,000 pens. The current inventory policy calls for replenishment orders of 156,000 pens. The average lead time from the distributor is 5 weeks, with a standard deviation of 2 weeks. If management wants a 95 percent cycle-service level, what should the reorder point be?



SOLUTION

From the Normal Distribution appendix for 0.9500, the appropriate z value = 1.65. We calculate the safety stock and reorder point as follows:

σdLT = Lσd2 + d2σLT

2 =

We have d = 12,000 pens, σd = 3,000 pens, L = 5 weeks, and σLT = 2 weeks

(5)(3,000)2 + (12,000)2(2)2

= 24,919.87 pens

Safety stock = zσdLT =

Reorder point = dL + Safety stock =

(1.65)(24,919.87) = 41,117.79 or 41,118 pens

(12,000)(5) + 41.118 = 101,118 pens



Grey Wolf lodge is a popular 500-room hotel in the North Woods. Managers need to keep close tabs on all of the room service items, including a special pint-scented bar soap. The daily demand for the soap is 275 bars, with a standard deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier is 5 days, with a standard deviation of 1 day. The lodge is open 365 days a year.What should the reorder point be for the bar of soap if management wants to have a 99 percent cycle-service?



SOLUTION

d =275 barsL =5 days

σd =30 barsσLT =1 day

283.06 barsσdLT = Lσd2 + d2σLT

2 =

From the Normal Distribution appendix for 0.9900, z = 2.33. We calculate the safety stock and reorder point as follows;

Safety stock = zσdLT = Reorder point + safety stock = dL + safety stock

(2.33)(283.06) = 659.53 or 660 bars

= (275)(5) + 660 = 2,035 bars



Two-Bin system Visual system An empty first bin signals the need to place an order

Calculating total systems costs

Total cost = Annual cycle inventory holding cost + Annual ordering cost + Annual safety stock holding cost

C = (H) + (S) + (H) (Safety stock)Q2

DQ



The Discount Appliance Store uses a continuous review system (Q system). One of the company’s items has the following characteristics:

Demand = 10 units/wk (assume 52 weeks per year)Ordering and setup cost (S) = $45/orderHolding cost (H) = $12/unit/yearLead time (L) = 3 weeks (constant)Standard deviation in weekly demand = 8 unitsCycle-service level = 70%



SOLUTIONWhat is the EOQ for this item?

D = 10/wk 52 wks/yr = 520 units

EOQ = =2DSH

= 62 units2(520)(45)12

What is the desired safety stock?

σdLT = σd L = 8 3 = 14 units

Safety stock = zσdLT = 0.525(14) = 8 units



What is the desired reorder point R?

R = Average demand during lead time + Safety stockR =

What is the total annual cost?

3(10) + 8 = 38 units

($12) + ($45) + 8($12) = $845.42622

52062C =



Suppose that the current policy is Q = 80 and R = 150. What will be the changes in average cycle inventory and safety stock if your EOQ and R values are implemented?

Reducing Q from 80 to 62Cycle inventory reduction = 40 – 31 = 9 units

Safety stock reduction = 120 – 8 = 112 units

Reducing R from 150 to 38


Periodic Review System (Periodic Review System (PP))

Fixed interval reorder system or periodic reorder system

Four of the original EOQ assumptions maintained No constraints are placed on lot size Holding and ordering costs Independent demand Lead times are certain

Order is placed to bring the inventory position up to the target inventory level, T, when the predetermined time, P, has elapsed


Periodic Review System (Periodic Review System (PP))

P P

T

L L L

Protection interval

Time

On-

hand

inve

ntor

y

IP3

IP1

IP2

OrderplacedOrderplaced

Orderplaced

Orderreceived

Orderreceived

Orderreceived

IP IPIP

OH OHQ1

Q2

Q3

Figure 12.10 – P System When Demand Is Uncertain


How Much to Order in a How Much to Order in a PP System System

EXAMPLE 12.6A distribution center has a backorder for five 36-inch color TV sets. No inventory is currently on hand, and now is the time to review. How many should be reordered if T = 400 and no receipts are scheduled?

SOLUTIONIP = OH + SR – BO

That is, 405 sets must be ordered to bring the inventory position up to T sets.

= 0 + 0 – 5 = –5 sets

T – IP = 400 – (–5) = 405 sets



The on-hand inventory is 10 units, and T is 400. There are no back orders, but one scheduled receipt of 200 units. Now is the time to review. How much should be reordered?

SOLUTION

IP = OH + SR – BO

The decision is to order 190 units

= 10 + 200 – 0 = 210

T – IP = 400 – 210 = 190


Periodic Review SystemPeriodic Review System

Selecting the time between reviews, choosing P and T

T = d(P + L) + safety stock for protection interval

Selecting T when demand is variable and lead time is constant

IP covers demand over a protection interval of P + L

The average demand during the protection interval is d(P + L), or

Safety stock = zσP + L , where σP + L = LPd


Calculating Calculating PP and and TT

EXAMPLE 12.7Again, let us return to the bird feeder example. Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system developed in Example 12.4 called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent. What is the equivalent P system? Answers are to be rounded to the nearest integer.



SOLUTIONWe first define D and then P. Here, P is the time between reviews, expressed in weeks because the data are expressed as demand per week:

D = (18 units/week)(52 weeks/year) = 936 units

P = (52) =EOQD

(52) = 4.2 or 4 weeks75936

With d = 18 units per week, an alternative approach is to calculate P by dividing the EOQ by d to get 75/18 = 4.2 or 4 weeks. Either way, we would review the bird feeder inventory every 4 weeks.



We now find the standard deviation of demand over the protection interval (P + L) = 6:

Before calculating T, we also need a z value. For a 90 percent cycle-service level z = 1.28. The safety stock becomes

Safety stock = zσP + L = 1.28(12.25) = 15.68 or 16 units

We now solve for T:

= (18 units/week)(6 weeks) + 16 units = 124 units

T = Average demand during the protection interval + Safety stock= d(P + L) + safety stock

units 12.2565 LPdLP


Periodic Review SystemPeriodic Review System

Use simulation when both demand and lead time are variable

Suitable to single-bin systems Total costs for the P system are the sum of the

same three cost elements as in the Q system Order quantity and safety stock are calculated

differently

C = (H) + (S) + HzσP + LdP2

DdP



Return to Discount Appliance Store (Application 12.4), but now use the P system for the item. Previous information

Demand = 10 units/wk (assume 52 weeks per year) = 520EOQ = 62 units (with reorder point system)Lead time (L) = 3 weeksStandard deviation in weekly demand = 8 unitsz = 0.525 (for cycle-service level of 70%)

Reorder interval P, if you make the average lot size using the Periodic Review System approximate the EOQ.



SOLUTIONReorder interval P, if you make the average lot size using the Periodic Review System approximate the EOQ.

P = (EOQ/D)(52) = (62/529)(52) = 6.2 or 6 weeks

Safety stock

Target inventory

T = 10(6 + 3) + 13 = 103 unitsT = d(P + L) + safety stock for protection interval

Safety stock = LPd

units 13 or 12.63680.525



Total cost

C = (H) + (S) + HzσP + L

dP2

DdP

= ($12) + ($45) + (13)($12) = $906.0010(6)

2520

10(6)


Comparative AdvantagesComparative Advantages

Primary advantages of P systems Convenient Orders can be combined Only need to know IP when review is made

Primary advantages of Q systems Review frequency may be individualized Fixed lot sizes can result in quantity discounts Lower safety stocks


Optional replenishment systems Optimal review, min-max, or (s,S) system, like the P

system Reviews IP at fixed time intervals and places a variable-

sized order to cover expected needs Ensures that a reasonable large order is placed

Hybrid systemsHybrid systems

Base-stock system Replenishment order is issued each time a withdrawal is

made Order quantities vary to keep the inventory position at R Minimizes cycle inventory, but increases ordering costs Appropriate for expensive items


Solved Problem 1Solved Problem 1

Booker’s Book Bindery divides SKUs into three classes, according to their dollar usage. Calculate the usage values of the following SKUs and determine which is most likely to be classified as class A.

SKU Number Description Quantity Used per Year

Unit Value ($)

1 Boxes 500 3.002 Cardboard

(square feet)18,000 0.02

3 Cover stock 10,000 0.754 Glue (gallons) 75 40.005 Inside covers 20,000 0.056 Reinforcing tape

(meters)3,000 0.15

7 Signatures 150,000 0.45



SOLUTIONThe annual dollar usage for each item is determined by multiplying the annual usage quantity by the value per unit. As shown in Figure 12.11, the SKUs are then sorted by annual dollar usage, in declining order. Finally, A–B and B–C class lines are drawn roughly, according to the guidelines presented in the text. Here, class A includes only one SKU (signatures), which represents only 1/7, or 14 percent, of the SKUs but accounts for 83 percent of annual dollar usage. Class B includes the next two SKUs, which taken together represent 28 percent of the SKUs and account for 13 percent of annual dollar usage. The final four SKUs, class C, represent over half the number of SKUs but only 4 percent of total annual dollar usage.



SKU Number Description

Quantity Used per

YearUnit Value

($)Annual Dollar

Usage ($)

1 Boxes 500 3.00 = 1,500

2 Cardboard (square feet)

18,000 0.02 = 360

3 Cover stock 10,000 0.75 = 7,500

4 Glue (gallons) 75 40.00 = 3,000

5 Inside covers 20,000 0.05 = 1,000

6 Reinforcing tape (meters)

3,000 0.15 = 450

7 Signatures 150,000 0.45 = 67,500

Total 81,310



Percentage of SKUs

Perc

enta

ge o

f Dol

lar V

alue

100 –90 –80 –70 –60 –50 –40 –30 –20 –10 –

0 –10 30 40 50 60 70 80 90 10020

Class C

Class A

Class B

Figure 12.11 – Annual Dollar Usage for Class A, B, and C SKUs Using Tutor 12.2



Nelson’s Hardware Store stocks a 19.2 volt cordless drill that is a popular seller. Annual demand is 5,000 units, the ordering cost is $15, and the inventory holding cost is $4/unit/year.

a. What is the economic order quantity?b. What is the total annual cost for this inventory item?

SOLUTIONa. The order quantity is

EOQ = =2DSH

2(5,000)($15)$4

= 37,500 = 193.65 or 194 drills

b. The total annual cost is

C = (H) + (S) =Q2

DQ ($4) + ($15) = $774.60194

25,000194



A regional distributor purchases discontinued appliances from various suppliers and then sells them on demand to retailers in the region. The distributor operates 5 days per week, 52 weeks per year. Only when it is open for business can orders be received. Management wants to reevaluate its current inventory policy, which calls for order quantities of 440 counter-top mixers. The following data are estimated for the mixer:

Average daily demand (d) = 100 mixersStandard deviation of daily demand (σd) = 30 mixersLead time (L) = 3 daysHolding cost (H) = $9.40/unit/yearOrdering cost (S) = $35/orderCycle-service level = 92 percentThe distributor uses a continuous review (Q) system



a. What order quantity Q, and reorder point, R, should be used?b. What is the total annual cost of the system?c. If on-hand inventory is 40 units, one open order for 440

mixers is pending, and no backorders exist, should a new order be placed?



SOLUTIONa. Annual demand is

The order quantity is

D = (5 days/week)(52 weeks/year)(100 mixers/day)= 26,000 mixers/year

EOQ = =2DSH

2(26,000)($35)$9.40

= 193,167 = 440.02 or 440 mixers



The standard deviation of the demand during lead time distribution is

A 92 percent cycle-service level corresponds to z = 1.41

σdLT = σd L = 30 3 = 51.96

Safety stock = zσdLT = 1.41(51.96 mixers) = 73.26 or 73 mixers

100(3) = 300 mixersAverage demand during lead time = dL =

Reorder point (R) = Average demand during lead time + Safety stock

= 300 mixers + 73 mixers = 373 mixers

With a continuous review system, Q = 440 and R = 373



b. The total annual cost for the Q systems is

C = (H) + (S) + (H)(Safety stock)Q2

DQ

C = ($9.40) + ($35) + ($9.40)(73) = $4,822.384402

26,000440

c. Inventory position = On-hand inventory + Scheduled receipts – Backorders

IP = OH + SR – BO = 40 + 440 – 0 = 480 mixers

Because IP (480) exceeds R (373), do not place a new order



Suppose that a periodic review (P) system is used at the distributor in Solved Problem 3, but otherwise the data are the same.a. Calculate the P (in workdays, rounded to the nearest day)

that gives approximately the same number of orders per year as the EOQ.

b. What is the target inventory level, T? Compare the P system to the Q system in Solved Problem 3.

c. What is the total annual cost of the P system?d. It is time to review the item. On-hand inventory is 40 mixers;

receipt of 440 mixers is scheduled, and no backorders exist. How much should be reordered?


Solved Problem 4Solved Problem 4SOLUTIONa. The time between orders is

P = (260 days/year) = EOQD (260) = 4.4 or 4 days440

26,000

b. Figure 12.12 shows that T = 812 and safety stock = (1.41)(79.37) = 111.91 or about 112 mixers. The corresponding Q system for the counter-top mixer requires less safety stock.

Figure 12.12 –OM Explorer Solver for Inventory Systems



c. The total annual cost of the P system is

C = (H) + (S) + (H)(Safety stock)dP2

DdP

C = ($9.40) + ($35) + ($9.40)(1.41)(79.37)100(4)

226,000100(4)

= $5,207.80d. Inventory position is the amount on hand plus scheduled

receipts minus backorders, or

IP = OH + SR – BO = 40 + 440 – 0 = 480 mixersThe order quantity is the target inventory level minus the inventory position, or

Q = T – IP = An order for 332 mixers should be placed.

812 mixers – 480 mixers = 332 mixers



Grey Wolf Lodge is a popular 500-room hotel in the North Woods. Managers need to keep close tabs on all room service items, including a special pine-scented bar soap. The daily demand for the soap is 275 bars, with a standard deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier is 5 days, with a standard deviation of 1 day. The lodge is open 365 days a year.a. What is the economic order quantity for the bar of soap?b. What should the reorder point be for the bar of soap if

management wants to have a 99 percent cycle-service level?c. What is the total annual cost for the bar of soap, assuming a

Q system will be used?



SOLUTIONa. We have D = (275)(365) = 100,375 bars of soap; S = $10; and

H = $0.30. The EOQ for the bar of soap is

EOQ = =2DSH

2(100,375)($10)$0.30

= 6,691,666.7 = 2,586.83 or 2,587 bars



b. We have d = 275 bars/day, σd = 30 bars, L = 5 days, and σLT = 1 day.

σdLT = Lσd2 + d2σLT

2 = (5)(30)2 + (275)2(1)2 = 283.06 bars

Consult the body of the Normal Distribution appendix for 0.9900. The closest value is 0.9901, which corresponds to a z value of 2.33. We calculate the safety stock and reorder point as follows:

Safety stock = zσdLT = (2.33)(283.06) = 659.53 or 660 bars

Reorder point = dL + Safety stock = (275)(5) + 660 = 2,035 bars



c. The total annual cost for the Q system is

C = (H) + (S) + (H)(Safety stock)Q2

DQ

C = ($0.30) + ($10) + ($0.30)(660) = $974.052,587

2100,375

2,587



Zeke’s Hardware Store sells furnace filters. The cost to place an order to the distributor is $25 and the annual cost to hold a filter in stock is $2. The average demand per week for the filters is 32 units, and the store operates 50 weeks per year. The weekly demand for filters has the probability distribution shown on the left below. The delivery lead time from the distributor is uncertain and has the probability distribution shown on the right below. Suppose Zeke wants to use a P system with P = 6 weeks and a cycle-service level of 90 percent. What is the appropriate value for T and the associated annual cost of the system?



Demand Probability

24 0.15

28 0.20

32 0.30

36 0.20

40 0.15

Lead Time (wks) Probability

1 0.05

2 0.25

3 0.40

4 0.25

5 0.05



SOLUTIONFigure 12.13 contains output from the Demand During the Protection Interval Simulator from OM Explorer.

Figure 12.13 – OM Explorer Solver for Demand during the Protection Interval



Given the desired cycle-service level of 90 percent, the appropriate T value is 322 units. The simulation estimated the average demand during the protection interval to be 289 units, consequently the safety stock is 322 – 289 = 33 units.The annual cost of this P system is

C = ($2) + ($25) + (33)($2)6(32)

250(32)6(32)

= $192.00 + $208.33 + $66.00 = $466.33



Consider Zeke’s inventory in Solved Problem 6. Suppose that he wants to use a continuous review (Q) system for the filters, with an order quantity of 200 and a reorder point of 140. Initial inventory is 170 units. If the stockout cost is $5 per unit, and all of the other data in Solved Problem 6 are the same, what is the expected cost per week of using the Q system?

SOLUTIONFigure 12.14 shows output from the Q System Simulator in OM Explorer. Only weeks 1 through 13 and weeks 41 through 50 are shown in the figure. The average total cost per week is $305.62. Notice that no stockouts occurred in this simulation. These results are dependent on Zeke’s choices for the reorder point and lot size. It is possible that stockouts would occur if the simulation were run for more than 50 weeks.



Figure 12.14 – OM Explorer Q System Simulator

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