Inventory Management; Chapter 6 MGMT 405, POM, 2011/12. Lec Notes © Stevenson, McGraw Hill, 2010-...

Inventory Management; Chapter 6

MGMT 405, POM, 2011/12. Lec Notes © Stevenson, McGraw Hill, 2010- Assoc. Prof. Sami Fethi, EMU, All Right Reserved.

Chapter 6: Inventory Management

Department of Business Administration

FALL 2013-2014


© Stevenson, McGraw Hill, 2010- Assoc. Prof. Sami Fethi, EMU, All Right Reserved. MGMT 405, POM, 2012/13. Lec Notes

2

Outline: What You Will Learn . . .

Define the term inventory and list the major reasons for holding inventories; and list the main requirements for effective inventory management.

Discuss the nature and importance of service inventories

Discuss periodic and perpetual review systems. Discuss the objectives of inventory management. Describe the A-B-C approach and explain how it is

useful.



3

Describe the basic EOQ model and its assumptions and solve typical problems.

Describe the economic production quantity model and solve typical problems.

Describe the quantity discount model and solve typical problems.

Describe reorder point models and solve typical problems. Describe situations in which the single-period model

would be appropriate, and solve typical problems.

Outline: What You Will Learn . . .



4

Inventory management is a core operations management activity.

Good inventory management is important for the successful operation of most businesses and their supply chains.

Operations, marketing, and finance have interests in good inventory management.

Bad inventory management hampers operations, diminishes customer satisfaction, and increases operating costs.

Inventory Management



5

The basic objective of Inventory Management has traditionally been to keep inventory at desired level that will meet product demand and be cost effective.

The major objective of Inventory Management (control system) is to discover and maintain the best possible level of inventory in terms of both unit of product and least possible cost.

In reaching such objectives firms seek out to avoid two common pitfalls.

Management tries to avoid the problem inadequate levels of inventory since too little inventory disrupts production and may results in lost sales.

The existence of too many inventories increases the risk of obsolescence (uselessness or oldness) and create unnecessary cost levels.

Objective of Inventory Management



6

Inventory is a stock or store of goods . Inventory can also be defined as a stock of

materials created to satisfy eventual demand. Inventories are idle resources of any kind that

possess economic value and held for future use. Items in inventory ranges from small things such

as pencils, paper clips, screws nuts and bolts to large items such as machines, trucks, construction equipment and air planes.

Inventory



7

Inventories are present whenever the inputs and outputs of a company are not used as soon as they become available .

Inventory can be thought as a final product waiting to be sold to a retail customer e.g., a new car, canned foods or drinks, and baked goods.

Inventories contain not only finished goods but also raw materials, supplies, and spare parts.

Some very large firms have tremendous amounts of inventory. For example general motor was reported to have as much as $ 40 mn worth of materials, parts, cars and trucks in its supply chain.

The ratio of inventories to sales in manufacturing, wholesale and retail sectors is one measure that is used to determine health of an economy. Inventories may represent a significant proportion of total asset. It is worth to mention that a reduction of inventories can results in a significant increase in return on investment (ROI is profit after taxes diveded by total asset).

Inventory



8

Inventory decisions in service organization can be especially critical. For example , hospital- being out of stock on some important supplies such as drugs imperil the well-being of a patient.

Many inventory items have a limited shelf life so carring large quantities would mean having to dispose of unused, costly supplies.

On-site repair services for computers, printers and fax machines also have to carefully consider which parts to bring to the site to avoid having to make extra trip to obtain parts.

The major revenue for wholesale and retail business is sale of Inventory. In terms of dollar, the inventory of goods held for sale is the one of largest assets of merchandising business.

Inventory



9

Inventory

Independent demand – finished goods, items that are ready to be soldE.g. a computer

Dependent demand – components of finished productsE.g. parts such as chip and system unit that make up the

computer



10

Independent Demand

A

B(4) C(2)

D(2) E(1) D(3) F(2)

Dependent Demand

Independent demand is uncertain. Dependent demand is certain.

Inventory



11

Types of Inventories

Raw materials & purchased parts Partially completed goods called

work in progress (WIP) Finished-goods inventories (manufacturing firms)

or merchandise (retail stores) Replacement parts, tools, and supplies Goods-in-transit to warehouse or customers (pipeline

inventory)



12

Functions of Inventory To meet anticipated demand

A customer can be a person who walks in off the street to buy a new stereo system. These inventories are referred to as anticipation stocks because they are held to satisfy expected demand.

To smooth production requirements

Firms that experience seasonal patterns in demand often build up inventories during preseason period to meet overly high requirements during seasonal period. These inventories are aptly named seasonal inventories. i.e., fresh fruits and vegetables or x-mas cards or greeting cards.



13

Functions of InventoryTo decouple operations Firms use inventories as buffers between successive

operations to maintain continuity of production and breakdown of equipment and accidents that cause the operation to shut down temporarily. The buffers permit other operation to continue temporarily while the problem is solved.

To protect against stock-outs

Delayed deliveries and unexpected increase in demand rise the risk of shortages. Delays can occurs because of whether conditions, suppliers out-stocks, deliveries of wrong materials, quality problems and so on. This risk of shortage can be reduced by holding safety stocks.



14

Functions of Inventory To take advantage of order cycles To minimize purchasing and inventory costs, firms

often buy in quantities that exceed immediate requirement. This necessities storing some or all of the purchased amount later use. Similarly, it is usually economical to produce in large rather than small quantities. This inventory storage can used later with demand requirements in short-run.

To help hedge against price increases

Firms will suspect that a substantial price increase is about to occur and purchase larger than normal amounts to beat the rise. The ability to store extra amount of goods also allows firms to take advantage of price discounts for larger orders



15

Functions of Inventory To permit operations The fact that production operations take a certain amount of

time that there will be some work in process including raw materials, semi finished items, unfinished items and finished goods at production site as well as goods stored in warehouse. This leads to pipeline inventories throughout a production-distribution system.

Little’s Law can be useful in quantifying pipeline inventory. Little’s Law: the average amount of inventory in a system is

equal to the product of average rate at which inventory units leave the system and average time a unit is in the system.

To take advantage of quantity discounts

Suppliers may give price discounts for larger orders.



16

Objective of Inventory Control (IC) Inaduqate control of inventories can results in both under and

over stocking of items.

Understocking results in missed deliveries, lost sales, dissatisfied customers and production bottle necks

Overstocking results in unnecessarily ties up funds that might be more productive elsewhere

Objective: To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds

Level of customer service (right time, right quantity)

Costs of ordering and carrying inventory (right place, size)Inventory turnover is the ratio of average cost of goods sold to average inventory investment (performance of IC).



17

Inventory Counting Systems Inventory counting system can be either periodic or perpentual: Periodic System

Physical count of items in inventory made at periodic intervals such as weekly, monthly etc..many small retailers use this approach. i.e., shelves, stock room etc..

Perpetual Inventory System(continual system) A system that keeps track of removals from inventory continuously, thus monitoring current levels of each item

Two-Bin System – It is very elementary system.Two containers of inventory; reorder when the first is empty. Second bin contains enough stock to satisfy expected demand.

Universal Bar Code (UPC) - Bar code printed on a label that has information about the item to which it is attached. i.e., supermarkets, discount stores etc..

Radio frequency Identification (RFID)- keeping track of inventories throughout a supply chain is vitally important for inventory system. i.e. Bar code used via radio waves (agricultural products-fruits-tempt).



18

To be effective, Management has two basic fuctions concerning inventory; 1- establishing system, 2- making decision. The following should be taken into account:

A system to keep track of inventory A reliable forecast of demand Knowledge of lead times Reasonable estimates of

Holding costsOrdering costsShortage costs

A classification system for inventory items.

Effective Inventory Management



19

Lead time: time interval between ordering and receiving the order

Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year

Ordering costs: costs of ordering and receiving inventory

Shortage costs: costs when demand exceeds supply

Key Inventory Terms



20

ABC Classification SystemClassifying inventory according to some measure of importance and allocating control efforts accordingly.A more reasonable approach would be to allocate control efforts according to the relatively importance of various items in inventory.AA - very importantBB - mod. importantCC - least importantVilfredo Pareto (1906); oFirst time, he developed

this concept.H. Ford Dickie (1953);oHe applied this system in the concept

of inventory management.

Annual $ value of items

AA

BB

CC

High

Low

Low HighPercentage of Items



21

Cycle Counting

A physical count of items in inventoryThe purpose of Cycle counting is to find out the

real amount between the amount indicated by inventory records and actual quantities of inventory on hand

Cycle counting managementHow much accuracy is needed?When should cycle counting be performed?Who should do it?



22

Example-ABC approach

Using the following annual demand & unit cost and calculate annual dollar value in each raw.

Having computed the annual dollar values, use the concept of ABC classification and array from highest to lowest

Item Numb

erAnnual

demand Unit CostAnnual Dollar

Value Classification

8 1000 $ 4000

5 3900 700

3 1900 500

6 1000 915

1 2500 330

4 1500 100

12 400 300



23

Example-ABC approach

The first two items have relatively high annual dollar value, so it seems reasonable to classify them as A items. The next three items appear to have moderate annual dollar values and should be classified as B items. The remainder are C items due to their relatively low annual dollar value.

Item NumberAnnual

demand Unit CostAnnual Dollar

Value Classification

8 1000 $ 4000 4000000 A

5 3900 700 2730000 A

3 1900 500 950000 B

6 1000 915 915000 B

1 2500 330 825000 B

4 1500 100 150000 C

12 400 300 120000 C



24

The question of how much to order is frequently determined by using an economic order quantity model (EOQ).

(EOQ) models identify the optimal order quantity by minimizing the sum of certain annual cost that vary with order size. Three order size models are described here:

The Basic Economic Order Quantity (EOQ) model

The Economic Production Quantity Model

Quantity Discount Model

Economic Order Quantity Models (Ford Harris, 1915)



25

The basic economic order quantity model (EOQ) is the simplest of the three model. It is used to identify a fixed order size that minimize the sum of annual cost of holding inventory and ordering inventory. The unit purchase price of items in inventory is not generally included in the total cost because the unit cost is unaffected by the order size unless quantity discounts are a factor. If holding costs are specified as a percentage of unit cost, then unit cost is indirectly included in the total cost as a part of holding costs.

Basic Economic Order Quantity Models



26

Only one product is involvedAnnual demand requirements are knownDemand is spread evenly throughout the year so

that the demand rate is reasonably constantLead time does not varyEach order is received in a single deliveryThere are no quantity discounts

Assumptions of the basic EOQ Model



27

The Inventory Cycle

Profile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usage rate

Time

Order size=350 unitsUseage rate= 50 units per dayLead time= 2 daysReorder point= 100 units



28

Total Cost

Annualcarryingcost

Annualorderingcost

Total cost = +

TC = Q2

H DQ

S+

Two basic inventory costs; Ordering Cost: are the basically the costs of getting the items into firm inventory, therefore these costs are the cost of replenishing inventory. Carrying or holding cost: are the basically the costs incurred due to maintenance of inventories or are the costs of holding items in storage.As order size varies, one of type of cost will increase whilst the other decreases. The greater level of inventory over time, the higher the carrying costs exist



29

The relationship between the type of costs

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Carrying Costs

QO

An

nu

al C

os

t

(optimal order quantity)

TCQ

HD

QS

2

Ordering Costs



30

Carring costs are linearly related to order sizeOrdering costs are inversely and non linearly related to

order sizeTotal cost curve is U-shapeThe total cost curve reaches its minimum where the

carrying and ordering costs are equal.

The relationship between the type of costs

Q2

H DQ

S=



31

Deriving the EOQ

Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPT

H

DSQ

H

DSQ

Q

DSH

Q

DSHS

Q

DdH

dQ

dQ

dTC

22

20

2

)(

2

2

2

2



32

Example- EOQ model A local distributor for a national tire company expects to sell

approximately 9600 steel-belted radial tires of a certain size and tread design next year. Annual carring cost is $ 16 per tire, and ordering cost is $ 75. The distributor operates 288 days a year.

(a) What is the EOQ? (b) How many times per year does the store reorder? (c) What is the length of an order cycle? (d) What is the total annual cost if the EOQ quantity is ordered?



33

Example- EOQ model-Answer (a) What is the EOQ?

(b) How many times per year does the store reorder?

(c) What is the length of an order cycle?

(d) What is the total annual cost if the EOQ quantity is order?

tiresH

DSQ 300

16

75)9600(22

32300

9600:

tires

tires

Q

DyearperorderofNumber

yearpertires

tires

D

QcycleorderofLenght

32

1

9600

300:

4800$75300

960016

2

300

2 S

Q

DH

QTC



34

Example- Optimal Quantity Piddling manufacturing assembles security monitors. It

purchases 3600 black and white cathode ray tubes a year at $ 65 each. Ordering costs are $ 31, and annual carring costs are 20 percent of the purchase price.

(a) Compute the optimal quantity (b) What is the carring cost? (c) What is the ordering cost? (d) Calculate the total annual cost ?



35

Example- Optimal Quantity-Answer

(a) Compute the optimal quantity

(b) What is the carring cost?

(c) What is the ordering cost?

(d) Calculate the total annual cost ?

HQ

2

CRTH

DSQHS 131

13

31)3600(2213$)65($

100

2031$

1704$852$852$31131

360013

2

131

2 S

Q

DH

QTC

852$

SQ

D 852$



36

Production done in batches or lots. Even in assembly operations, portions of the work are done in batches.

Capacity to produce a part exceeds the part’s usage or demand rate

As long as production continious, inventory will continue to grow.

This makes sense to periodically produce such items in batches or lots instead of producing continually.

Assumptions of EPQ are similar to EOQ except orders are received incrementally during production

Economic Production Quantity (EPQ)



37

Only one item is involvedAnnual demand is knownUsage rate is constantUsage occurs continuallyProduction rate is constantLead time does not varyNo quantity discounts

Economic Production Quantity Assumptions



38

Setup Cost and Economic Run Size (quantity) In the case of EPQ, there

are no ordering cost, however there are setup costs-the costs required to prepare the equipment for the job, such as cleaning, adjusting changing tools etc.

Setup costs are analogous to ordering costs because they are independent of the lot or run size.

The larger run size, the fewer the number of runs needed as well as the lower the annual setup cost.

D/Q is the number of batches per year.

DS/Q0 is setup cost.2

)(

)(

)(

)(

2

max

))(2

(

)2

(coscos

max

max

max

min

maxmin

IIav

upp

QI

cycletheofphaseproductionthep

QtimeRun

ordersthebetweentimeu

QtimeCycle

ratedemandrateusageU

ratedeliveryorproductionpwhereUP

P

H

DSQquantityruneconomicThe

inventoryimumIwhere

HP

UPQS

Q

DTC

SQ

DH

ItsetuptcarringTC

o

O

O

o

o

o



39

Example- Run size A toy manufacturer uses 48000 rubber wheels per year for its

popular damp truck series. The firm makes its own wheels which it can produce at the rate of 800 per day. The toy trucks are assembled uniformly over the entire year. Carring cost is $ 1 per wheel a year. Setup cost for a production run of wheels is $ 45. The firm operates 240 days per year. Determine

(a) Optimal run size (b) Minimum total annual cost for carring and setup (c) Cycle time for the optimal run size (d) Run time



40

Example- Run size-Answer (a) Optimal run size

(b) Minimum total annual cost for carring and setup

(c) Cycle time for the optimal run size

(d) Run time

wheelsUP

P

H

DSQquantityruneconomicThe 2400

200800

800

1

)45()48000()2(2

wheelsupp

QI o 1800)200800(

800

2400)(max

1800$)1(800

200800)45(

2400

48000))(

2(

1800$)45($2400

48000)1($

2

1800)

2(coscos

min

maxmin

HP

UPQS

Q

DTC

SQ

DH

ItsetuptcarringTC

o

o

o

dayseverymadebewillwheelsofrunaThus

daysu

QtimeCycle O

12

,12200

2400

completetodaysthreerequirewillruneachThus

daysp

QtimeRun O

,

3800

2400



41

The Quantity Discount Model

Annualcarryingcost

PurchasingcostTC = +

Q2 H D

Q STC = +

+Annualorderingcost

PD +

Quantity discount rate are price reductions for large orders offered to customers to induce them to buy in large quantities.

As prices decrease order of quantities increase. The buyer’s goal with quantity discount rate is to select

the order quantity that minimize total cost in the following equation where p is unit price



42

The Quantity Discount Model In the basic EOQ model, determination of order

size does not involve the purchasing cost. The rationale for not including unit price is that under the assumption of no discount, price per unit is the same for all order sizes.

Inclusion of unit price in the total cost computation in that case would merely increase the total cost by the amount P times D

A graph of total annual purchase cost versus quantity would be a horizontal line. Hence, including purchasing costs would only raise the total cost curve by the same amount (PD) at every point. In the following graph, that would not change the EOQ.



43

Total Costs with PDC

ost

EOQ

TC with PD

TC without PD

PD

0 Quantity

Adding Purchasing costdoesn’t change EOQ



44

The Quantity Discount ModelWhen quantity discounts are offered, there is a

separate U-shaped total cost curve for each unit price. Again, including unit prices merely raises each curve by a constant amount.

However, because the prices are all different, each curve is raised by a different amount.

Smaller unit prices will raise a total cost curve less than large unit prices

In the following graph, no one curve applies to the entire range of quantities and each curve applies to only a portion of the range. Hence the applicable or feasible total cost is initially on the curve with highest unit price and then drops down curve by curve at the price breaks which are the minimum quantities needed to obtain the discount.



45

Total Cost with Constant Carrying Costs

OC

EOQ Quantity

To

tal C

os

t

TCa

TCc

TCbDecreasing Price

CC a,b,c

Comparison of TC curves for constant carrying costs and carrying costs that are a percentage of unit costs. When carrying costs are constant, all curves have their minimum points at the same quantity. When carrying costs are stated as percentage of unit price, the minimum points do not line up.



46

Example- Discount Model The maintenance department of a large hospital uses about

816 cases of luquid cleanser annually. Ordering costs are $ 12, Carring cost are $ 4 per case a year and the new price schedule indicates that orders of less than 50 cases will cost $ 20 per case, 50 to 79 cases will cost $ 18 per case, 80 to 99 cases will cost $ 17 per cases and larger orders will cost $ 16 per case. Determine

(a) the common minimum point for EOQ (b) the total cost if the feasible minimum point is on the lowest

price range, that is the optimal order quantity. (c) the total cost if the feasible minimum point is in any other

price range. Range Price

1 to 49 $ 20

50 to 79 18

80 to 99 17

100 or more 16



47

Example- Discount Model -Answer

(a) the common EOQ

(b) total annual cost

(c) total annual cost

Range Price

1 to 49 $ 20

50 to 79 18

80 to 99 17

100 or more 16

casesH

DSQ 7097.69

4

12)816(22

The cases can be bought at $ 18 per case because 70 falls in the range of 50 to 79 cases.

14968$)816(181270

8164

2

70

270 PDSQ

DH

QTC

o

TC = Carrying cost + Order cost + Purchase cost

13354$)816(1612100

8164

2

100

2100 PDSQ

DH

QTC

o

14154$)816(171280

8164

2

80

280 PDSQ

DH

QTC

o



48

Example 2- Discount Model

Surge Electric uses 4000 toggle switches a year. Switch are priced in the following table. It costs approximately $ 30 to prepare an order and receive it, and carring costs are 40 percent of purchase price per unit on an annual basis.

Range Price

1 to 449 $ 0.90

500 to 999 0.85

1000 or more 0.80

Determine (a) the common EOQ (b) the total cost if the feasible minimum point is on the lowest

price range, that is the optimal order quantity. (c) the total cost if the feasible minimum point is in any other

price range.



49

Example- Discount Model -Answer

(a) the common minimum point or EOQ

(b) total annual cost

(c) total annual cost

switches86632.0

30)4000(2280.0

H

DSQ

3686$)4000(85.030840

400034.0

2

840

2840 PDSQ

DH

QTC

o

TC = Carrying cost + Order cost + Purchase cost

3480$)4000(80.0301000

400032.0

2

1000

21000 PDSQ

DH

QTC

o

Range Price

1 to 449 $ 0.90

500 to 999 0.85

1000 or more 0.80

H= 0.40 PH= 0.40 (0.80)=0.32H= 0.40 (0.85)=0.34

feasibleisThisH

DSQ .switches840

34.0

30)4000(2285.0



50

When to Reorder with EOQ Ordering

Reorder Point (ROP) – This point occurs, when the quantity on hand of an item drops to a predetermined amount, the item is reordered.

Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.

Service Level - Probability that demand will not exceed supply during lead time.



51

Determinants of the Reorder Point

The rate of demandThe lead timeDemand and/or lead time variabilityStock out risk (safety stock)

If demand and lead time are both constant, the reorder point is simply defined as ROP=(d) x (LT)

Where d = demand rate units per day or week

LT= lead time in days or weeks



52

Example - reorder

Tingly takes two a day vitamins which are delivered to his home by routeman seven days after an order is called in. At what point should Tingly reorder?.

Usage is 2 vitamins per day

Lead time is 7 days

ROP=(2) (7)= 14 vitamins

Thus, Tingly should reorder when 14 vitamin tables are left.



53

Safety Stock

LT Time

Expected demandduring lead time

Maximum probable demandduring lead time

ROP

Qu

an

tity

Safety stock

Safety stock reduces risk of stockout during lead time

ROP= Expected demand in (LT) + safety stock



54

Reorder Point

ROP

Risk ofa stockout

Service level

Probability ofno stockout

Expecteddemand Safety

stock0 z

Quantity

z-scale

The ROP based on a normal Distribution of lead time demand

ROP= Expected demand in (LT) + ZαdLT Where z is the number of standard deviationm αdLT is the standard deviation lead time demand



55

Example – ROP and Z

A manager of a construction supply house determined from historical records that demand for sand during lead time averages 50 tons. In addition, the manager determined that demand during lead time could be described by a normal distribution that has a mean of 50 tons and a standard deviation of 5 tons as well as the manager is willingly to accept a stockout risk of no more than 3 percent.

(a) What value of Z is appropriate?

(b) How much safety stock should be held?

(c) What reorder point should be used?



56

Answer– ROP and Z

(a) What value of Z is appropriate? Expected lead time is 50 tons α dLT=5 tons

risk =3 percent 1-0.03=0.97 from z table (lead time), z= 1.88 (page 569, table 12.3)

(b) How much safety stock should be held?

Safety stock = Z αdLT= (1.88) (5)= 9.40 tons

(c) What reorder point should be used?

ROP= Expected demand in (LT) + ZαdLT =50+9.40= 59.40 tonsROP= Expected demand in (LT) + ZαdLT =50+9.40= 59.40 tons



57

Table 12.3, page 569



58

Example – shortage and service leadExcluded from the exam topics

Suppose standard deviation of lead time demand is known to be 20 units. Lead time demand is approximately normal.

(a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle.

(b) What lead time service level would an expected shortage of 2 units imply?



59

Answer – shortage and service leadExcluded from the exam topics

(a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle.

αdLT= 20 units

lead time service level is 0.90 from z table (lead time), (E) z= 0.048 (page 569, table 12.3)

E(n) =(E)z αdL= (0.048) (20)= 0.96 or about 1 unit.

or E(N) =E (n) (D/Q)

(b) What lead time service level would an expected shortage of 2 units imply?

E(n) = 2 E(n) =(E)z αdL or (E)z = E(n) / αdL =(2)/(20)= 0.100 from the

table, lead time service level is 81.06 percent or 81%



60

Orders are placed at fixed time intervalsOrder quantity is for next intervalSuppliers might encourage fixed intervalsMay require only periodic checks of inventory levelsRisk of stockoutFill rate – the percentage of demand filled by the stock

on hand

Fixed-Order-Interval ModelExcluded from the exam topics



61

Tight control of inventory items Items from same supplier may yield savings in:

OrderingPackingShipping costs

May be practical when inventories cannot be closely monitored

Fixed-Interval BenefitsExcluded from the exam topics



62

Requires a larger safety stock Increases carrying costCosts of periodic reviews

Fixed-Interval DisadvantagesExcluded from the exam topics



63

Example – Amount to orderExcluded from the exam topics

Given the following information:

(a) determine the amount to order

dayservalorderOIdaysLT

unitstimereorderathandonamountdayperunitdemandofdevst

percentislevelservicedesireddayperunitdemanddailyaveraged

7)int(2

713

9930

ALTOIzLTOIdATO d )(

Amount to order = expected time during the production interval + safety stock - Amount on hand at reorder time

unitsATO 2207127)3(33.2)27(30



64

Single period model: model for ordering of perishables and other items with limited useful lives

Shortage cost: generally the unrealized profits per unit

Excess cost: difference between purchase cost and salvage value of items left over at the end of a period

Single Period ModelExcluded from the exam topics



65

Continuous stocking levels Identifies optimal stocking levelsOptimal stocking level balances unit shortage and

excess cost

Discrete stocking levelsService levels are discrete rather than continuousDesired service level is equaled or exceeded

Single Period ModelExcluded from the exam topics



66

Optimal Stocking LevelExcluded from the exam topics

Service Level

So

Quantity

Ce Cs

Balance point

Service level =Cs

Cs + CeCs = Shortage cost per unitCe = Excess cost per unit



67

Example-optimal stocking levelExcluded from the exam topics

Sweet cider is delivered weekly to Cindy’s bar. Demand varies uniformly between 300 lt and 500 lt per week. Cindy pays 20 cents per liter and charges 80 cent per liter for it. Unsold cider has no salvage value and cannot be carried over into the week due to spoilage.

Find the optimal stocking level and its stockout risk for that quantity

where Ce = Cost per unit- Salvage value per unit

=$0.20 -0= $0.20 per unit Cs = Revenue per unit-Cost per unit

=$0.80-$0.20=$0.60 per unit



68

Answer-optimal stocking levelExcluded from the exam topics

Ce = $ 0.20 per unit Cs = $ 0.60 per dozen Service level = Cs/(Cs+Ce) =

0.60/(0.60+0.20) Service level = .75 S=300+ 0.75(500-300)=450 lt

Service Level = 75%

Quantity

Ce Cs

Stockout risk = 1.00 – 0.75 = 0.25



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Example-Discrete-optimal stocking levelExcluded from the exam topics

Demand for jelly doughnuts on Saturdays at Don’s Doughnut Shoppe is shown in the following table. If labor, materials, and overhead are estimated to be $ 3.20 per dozen, doughnuts are sold for $ 4.80 per dozen, and left over doughnuts at the end of each day are sold the next day at half price:

(a) Determine the shortage and excess costs (b) Determine the service level(c) Calculate the stock-out risk factor(d) list the cumulative tables for demand(e) Compute the optimal number of doughnuts?



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Answer-Discrete-optimal stocking levelExcluded from the exam topics

a) Ce = Cost per unit- Salvage value per unit Ce = 3.20-2.40=$ 0.80 per dozen Cs = Revenue per unit-Cost per unit Cs = 4.80-3.20=$ 1.60 per dozen b) Service level = Cs/(Cs+Ce) =

1.60/(1.60+0.80) Service level = .67

Demand (Dozen) Relative.Freq.

19 0.01

20 0.05

21 0.12

22 0.18

23 0.13

24 0.14

25 0.10

26 0.11

27 0.10

28 0.04

29 0.02



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c)

Service Level = 67%

Quantity

Ce Cs

Stockout risk = 1.00 – 0.67 = 0.33

Demand (Dozen) Relative.Freq.

19 0.01

20 0.05

21 0.12

22 0.18

23 0.13

24 0.14

25 0.10

26 0.11

27 0.10

28 0.04

29 0.02



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d)Demand

(Dozen)Relative Freq. Cumulative Freq.

19 0.01 0.01

20 0.05 0.06

21 0.12 0.18

22 0.18 0.36

23 0.13 0.49

24 0.14 0.63

25 0.10 0.73

26 0.11 0.84

27 0.10 0.94

28 0.04 0.98

29 0.02 1.00



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(e) Since the service level falls between the cumulative probabilities of 0.63 (24) and 0.73 (25), this means that the company should stock 25 dozen doughnuts because 0.67 is closer to 0.73.



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