College of Business AdministrationCal State San Marcos

Production & Operations ManagementHTM 305

Dr. M. Oskoorouchi

Summer 2006

CHAPTER11

Inventory Management

Types of InventoriesTypes of Inventories

Raw materials & purchased parts Partially completed goods called

work in progress Finished-goods inventories

(manufacturing firms) or merchandise (retail stores)

Functions of InventoryFunctions of Inventory

To meet anticipated demand

To smooth production requirements

To protect against stock-outs

To take advantage of order cycles

To hedge against price increases

To take advantage of quantity discounts

A system to keep track of inventory

A reliable forecast of demand

Knowledge of lead times

Reasonable estimates of

Holding costs

Ordering costs

Shortage costs

Effective Inventory ManagementEffective Inventory Management

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

Ordering costs: costs of ordering and receiving inventory

Shortage costs: costs when demand exceeds supply

Key Inventory TermsKey Inventory Terms

Inventory Counting SystemsInventory Counting Systems

Periodic SystemPhysical count of items made at periodic intervals

Perpetual Inventory System System that keeps track of removals from inventory continuously, thus monitoringcurrent levels of each item

Inventory Counting SystemsInventory Counting Systems

Two-Bin System - Two containers of inventory; reorder when the first is empty

Universal Bar Code - Bar code printed on a label that hasinformation about the item to which it is attached

0

214800 232087768

ABC Classification SystemABC Classification System

Classifying inventory according to some measure of importance and allocating control efforts accordingly.

AA - very important

BB - mod. important

CC - least important Annual $ value of items

AA

BB

CC

High

Low

Few ManyNumber of Items

Economic order quantity model

Economic production quantity model

Quantity discount model

Economic Order Quantity ModelsEconomic Order Quantity Models

Only one product is involved

Annual demand requirements known

Demand is even throughout the year

Lead time does not vary

Each order is received in a single delivery

There are no quantity discounts

Assumptions of EOQ ModelAssumptions of EOQ Model

The Inventory CycleThe 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 frequencyOrder frequency

Annual carrying costAnnual carrying cost

Annual carrying cost =

(average number of inventory)*(holding cost/unit/year)

Average number of inventory = (Q+0)/2 = Q/2

Annual carrying cost = (Q/2)H,

Where

Q = Order quantity in units

H = Holding cost /unit/year

Total CostTotal Cost

Annualcarryingcost

Annualorderingcost

Total cost = +

Q2H D

QSTC = +

Cost Minimization GoalCost Minimization Goal

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

QO

An

nu

al C

os

t

(optimal order quantity)

TCQ

HD

QS

2

Deriving the EOQDeriving 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

Example Example

A toy manufacturer uses approximately 32,000 silicon chips annually. The chips are used at a steady rate during the 240 days a year that the plant operates. Annual holding cost is $3 per chip, and ordering cost is $120. Determine The optimal order quantity. Holding cost, ordering cost, and the total cost. The number of workdays in an order cycle.

Problem Problem

PM assembles security monitors. It purchases 3,600 black-and-white cathode ray tubes a year at $65 each. Ordering costs are $31, and annual carrying costs are 20% of the purchase price. Compute The optimal quantity the total annual cost of ordering and carrying the

inventory.

Problem Problem A large bakery buys flour in 25-pound bags. The bakery uses an

average of 4,860 bags a year. Preparing an order and receiving a shipment of flour involves a cost of $10 per order. Annual carrying costs are $75 per bag

Determine the economic order quantity.

What is the average number of bags on hand.

How many orders per year will there be?

Compute the total cost of ordering and carrying flour.

If ordering costs were to increase in $1 per order, how much would that affect the minimum total annual cost?

Production done in batches or lots Capacity to produce a part exceeds the part’s

usage or demand rate Assumptions of EPQ are similar to EOQ

except orders are received incrementally during production

Economic Production Quantity (EPQ)Economic Production Quantity (EPQ)

Only one item is involved Annual demand is known Usage rate is constant Usage occurs continually Production rate is constant Lead time does not vary No quantity discounts

Economic Production Quantity AssumptionsEconomic Production Quantity Assumptions

EPQ ModelEPQ Model

Economic Run SizeEconomic Run Size

QDS

H

p

p u0

2

Example Example

A toy manufacturer uses 48,000 rubber wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of 800 per day. The toy trucks are assembled uniformly over the entire year. Carrying cost is $1 per wheel a year. Setup cost for a production run of wheels is $45. the firm operates 240 day per year. Determine the Optimal run size Minimum total annual cost for carrying and setup Cycle time for the optimal run Run time.

Problem Problem The Dine Corporation is both a producer and a user of

brass couplings. The firm operates 220 days a year and uses the couplings at a steady rate of 50 per day. Couplings can be produced at a rate of 200 per day. Annual storage cost is $2 per coupling, and machine setup cost is $70 per run. Determine the economic run quantity Approximately how many runs per year will there be? Compute the maximum inventory level. Determine the length of the pure consumption portion of

the cycle.

Problem Problem

The Friendly Sausage Factory (FSF) can produce hot dogs at a rate of 5,000 per day. FSF supplies hot dogs to local restaurants at a steady rate of 250 per day. The cost to prepare the equipment for producing hot dogs is $66. annual holding costs are 45 cents per hot dog. The factory operates 300 days a year. Find The optimal run size. The number of runs per year. The maximum inventory level. The length of run.

Review: EOQ ModelReview: EOQ Model

Cost Minimization GoalCost Minimization Goal

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

QO

An

nu

al C

os

t

(optimal order quantity)

TCQ

HD

QS

2

Holding Costs

Deriving the EOQDeriving 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

Total CostTotal Cost

Annualcarryingcost

Annualorderingcost

Total cost = +

Q2H D

QSTC = +

Total Costs with Purchasing CostTotal Costs with Purchasing Cost

Annualcarryingcost

PurchasingcostTC = +

Q2H D

QSTC = +

+Annualorderingcost

PD +

EOQ models with constant purchase price:

Total Costs with PDTotal Costs with PD

Co

st

EOQ

TC with PD

TC without PD

PD

0 Quantity

Adding Purchasing costdoesn’t change EOQ

Total Costs with quantity discountTotal Costs with quantity discount

Order quantity

Unit Price

1 to 44 $2.00

45 to 69 1.70

70 or more 1.40

Example: Constant Carrying Costs Example: Constant Carrying Costs A mail-order house uses 18,000 boxes a year. Carrying costs

are 60 cents per box a year, and ordering costs are $96. The following price schedule applies:

Determine the optimal order quantity. Determine the number of orders per year.

Number of boxes Price per box

1,000 to 1,999 $1.25

2,000 to 4,999 1.20

5,000 to 9,999 1.15

10,000 or more 1.10

2,400 5,000 10,000

Quantity

TC

Lowest TC

Total Costs with Purchase discountTotal Costs with Purchase discount

Steps for total Cost with Constant Carrying CostsSteps for total Cost with Constant Carrying Costs

Compute the common minimum point.

Only one of the unit prices will have the minimum point in its feasible range since the ranges do not overlap. Identify that range.

If the feasible minimum point is on the lowest price range, that is the optimal order quantity.

Otherwise, compute the total cost for the minimum point and for the price breaks of all lower unit costs. Compare the total costs; the quantity (minimum point or price break) that yields the lowest total cost is the optimal order quantity.

Problem: Constant Carrying Costs Problem: Constant Carrying Costs

The maintenance department of a large hospital uses about 816 cases of liquid cleanser annually. Ordering costs are $12, carrying costs 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 case, and larger orders will cost $16 per case. Determine the optimal order quantity and the

total cost.

Problem: Constant Carrying Costs Problem: Constant Carrying Costs

A jewelry firm buys semiprecious stone to make bracelets and rings. The supplier quotes a price of $8 per stone for quantities of 600 stones or more, $9 per stone for orders of 400 to 599 stones, and $10 per stone for lesser quantities. The jewelry firm operates 200 days per year. Usage rate is 25 stones per day, and ordering costs are $48.

If carrying costs are $2 per day for each stone, find the order quantity that will minimize total annual cost.

Total Costs with quantity discountTotal Costs with quantity discount

There are two general cases of the model: Carrying costs are constant (for example, $2 per

unit) Carrying costs are stated as a percentage of

purchase price (for example, 20% of purchase price)

Problem: variable Carrying Costs Problem: variable Carrying Costs

A jewelry firm buys semiprecious stone to make bracelets and rings. The supplier quotes a price of $8 per stone for quantities of 600 stones or more, $9 per stone for orders of 400 to 599 stones, and $10 per stone for lesser quantities. The jewelry firm operates 200 days per year. Usage rate is 25 stones per day, and ordering costs are $48.

If annual carrying costs are 30% of unit cost, what is the optimal order size?

Steps for total Cost with variable Carrying CostsSteps for total Cost with variable Carrying Costs

Beginning with the lowest unit price, compute the minimum points for each price range until you find a feasible minimum point (i.e., until a minimum point falls in the quantity range for its price).

If the minimum point for the lowest unit price is feasible, it is the optimal order quantity. If the minimum point is not feasible in the lowest price range, compare the total cost at the price break for all lower prices with the total cost of the feasible minimum point. The quantity which yields the lowest total cost is the optimum.

Problem: variable Carrying CostsProblem: variable Carrying Costs

A manufacturer of exercise equipments purchases the pulley section of the equipment from a supplier who lists these prices: less than 1,000, $5 each; 1,000 to 3,999, $4.95 each; 4,000 to 5,999, $4.90 each; and 6,000 or more, $4.85 each. Ordering costs are $50, annual carrying costs per unit are 40% of purchase cost, and annual usage is 4,900 pulleys. Determine an order quantity that will minimize total cost.

Problem: variable Carrying CostsProblem: variable Carrying Costs

495 497 500 503 1,000 4,000 6,000

Quantity

TC

When to Reorder with EOQ OrderingWhen to Reorder with EOQ Ordering

A simple case: Demand is constant

Lead time is constant

ROP = d * LT

Example: Demand is 20 items per day and lead time is one week. What is the reorder point (ROP)?

When to Reorder with EOQ OrderingWhen to Reorder with EOQ Ordering

Reorder Point - When the quantity on hand of an item drops to this 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.

Determinants of the Reorder PointDeterminants of the Reorder Point

The rate of demand (based on a forecast) The lead time Demand and/or lead time variability Stockout risk (safety stock)

ROP = Expected demand during lead time + safety stocks

Safety StockSafety Stock

LT Time

Expected demandduring lead time

Maximum probable demandduring lead time

ROP

Qu

an

tity

Safety stockSafety stock reduces risk ofstockout during lead time

Reorder PointReorder Point

ROP

Risk ofa stockout

Service level

Probability ofno stockout

Expecteddemand Safety

stock0 z

Quantity

z-scale

The ROP based on a normalDistribution of lead time demand

Reorder PointReorder Point

ROP = Expected demand during lead time + safety stocks

We discuss several models:

An estimate of expected demand during lead time and its standard deviation are available.

Data on lead time demand is not available: we consider three cases: Demand is variable, lead time is constant Lead time is variable, demand is constant Both demand and lead time are variable

ROP for model 1ROP for model 1

An estimate of expected demand during lead time and its standard deviation are available

Expected demand

during lead time dLTROP z

Example Example Given this information:

Expected demand during lead time = 300 units Standard deviation of demand during lead time = 30 units

Determine each of the following, assuming that lead time demand is distributed normally:

The ROP that will provide a 5% risk of stockout during lead time.

The ROP that will provide a 1% risk of stockout during lead time.

The safety stock needed to attain a 1% risk of stockout during lead time.

Cumulative Distribution Function of the Standard Normal Distribution

Example: If Z is standard Normal random variable, then F(1.00) = P(Z 1.00) = .8413z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247–0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

ProblemProblem

Suppose that the manager of a construction supply house determine from historical records that demand for sand during lead time obeys a normal distribution with mean 35 tons and standard deviation of 2.5 tons.

What value of z is appropriate if the manager is willing to accept a stockout risk of no more than 4%.

How much safety stock should be held?

What reorder point should be used?

ROP for variable demand and constant lead timeROP for variable demand and constant lead time

Only demand is variable

Average daily or weekly demand

Standard deviation of demand per day or week

Lead time in days or weeks

d

d

ROP d LT z LT

where

d

LT

Example Example

A restaurant uses an average of 50 jar of a special sauce each week. Weekly usage of sauce has a standard deviation of 3 jars. The manager is willing to accept no more than a 10% risk of stockout during lead time, which is two weeks. Assume the distribution of usage is normal.

Determine the value of z

Determine the ROP

ROP for variable lead time and constant demandROP for variable lead time and constant demand

Only lead time is variable

aily or weekly demand

Standard deviation of lead time per day or week

Average lead time in days or weeks

LT

LT

ROP d LT zd

where

d D

LT

Problem Problem The housekeeping department of a motel uses approximately 600

bars of soap per day, and this tends to be fairly constant. Lead time for soap delivery is normally distributed with a mean of six days and standard deviation of two days. A service lead of 98% is desired. Find the ROP.

ROP for variable demand and variable lead timeROP for variable demand and variable lead time

Both demand and lead time are variable

2 2 2d LTROP d LT z LT d

ProblemProblem

The motel replaces broken glasses at a rate of 25 per day. In the past, this quantity has tend to vary normally and have a standard deviation of 3 glasses per day. Glasses are ordered from a Cleveland supplier. Lead time is normally distributed with an average of 10 days and a standard deviation of 2 days. What ROP should be used to achieve a service level of 95%?

Problem Problem

The manager of FSF has determined that demand for hot dogs obeys a normal distribution with an average of 5000 per week and standard deviation of 150 per week.

The manager is willing to accept 3% stockout risk during lead time which is one week.

Determine the reorder point

Problem Problem

The manager of FSF has determined that demand for hot dogs is approximately 5000 per week and it is fairly constant.

The lead time obeys a normal distribution with an average of one week and standard deviation of 0.5 weeks.

Determine the reorder point if the manager is willing to accept 8% stockout risk.

Problem Problem The manager of FSF has determined that demand for

hot dogs obeys a normal distribution with an average of 5000 per week and standard deviation of 150 per week.

In the past the lead time had a normal distribution with mean of one week and standard deviation of 0.5 weeks

The manager is willing to accept 3% stockout risk during lead time.

Determine the reorder point

Shortages per cycleShortages per cycle

The expected number of units short can be very useful to management.

This quantity can easily be determined using the information for ROP and Table 11.3 in your textbook.

E(n), expected number of units short per order cycle:

( ) ( ) dLTE n E z

( ) Standardized number of units short from Table 11.3

standard deviation of demand during lead timedLT

E z

Example Example

Suppose the standard deviation of demand during lead time is known to be 50 units. Lead time demand is approximately normal. For the lead time service level of 95%, determine

the expected number of units short for any order cycle.

What lead time service level would an expected shortage of 3 units imply?

Shortages per yearShortages per year

Expected number of units short per year =

(Expected number of units short per cycle) *( Number of cycles)

( ) ( )D

E N E nQ

Example Example

Given the following information, determine the expected number of units short per year.

Demand = 48,000 units per year Order quantity (Q) = 1,200 units Service level for the lead time = 90% Std. of demand during lead time = 150 units

How the expected number of units short would change, if the service level is increases from 90% to 99%?

Annual service level (fill rate)Annual service level (fill rate) Annual service level (fill rate) is the percentage of demand

filled directly from inventory.

Example Demand = 100 units/year, Annual shortage = 10 units

Annual service level = 90/100 = 90%

General formula:

( )1annual

E NSL

D

Example Example

Given A lead time service level of 85% D = 48,000 Q = 1,200 Std. of demand during lead time = 150

Determine the annual service level

Problem Problem The manager of a store that sells office supplies has decided to

set an annual service level of 96% for a certain model of answering machine. The store sells approximately 300 of this model a year. Holding cost is $5 per unit annually. Ordering cost is $25, and the standard deviation of demand during lead time is 7.

What average number of units short per year will be consistent with the specified annual service level?

What average number of units short per cycle will provide the desired annual service level?

What lead time service level is necessary for the 96% annual service level?

In EOQ/ROP, order size is fixed and time to reorder varies.

In FOI model, orders are placed at fixed time intervals (weekly, monthly, etc.)

Order quantity for next interval?

If demand is variable, order size may vary from cycle to cycle

Fixed-Order-Interval (FOI) ModelFixed-Order-Interval (FOI) Model

Suppliers might encourage fixed intervals

May require only periodic checks of inventory levels

Risk of stockout

FOI must have stockout protection for lead time plus the next order cycle

Reasons for using FOI modelReasons for using FOI model

Fixed-Order-Interval ModelFixed-Order-Interval Model

Expected demandAmount Safety Amount on hand

= during protection +to order stock at reorder time

interval

= ( ) + dd OI LT z OI LT A

Where

OI = Order interval (length of time between orders)

A = Amount on hand at reorder time

Example Example

A lab orders a number of chemicals from the same supplier every 30 days. Lead time is 5 days. The assistant manager of the lab must determine how much of one of these chemicals to order. A check of stock revealed that eleven 25-ml jars are on hand. Daily usage of the chemicals is approximately normal with a mean of 15.2 ml per day and a standard deviation of 1.6 ml per day. The desired service level for this chemical is 95%.

How many jars of the chemical should be ordered?

What is the average amount of safety stock of the chemicals?

Single period model: model for ordering of perishables (vegetables, milk, …) 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 ModelSingle Period Model

Example: Uniform distribution Example: Uniform distribution

Sweet potatoes are delivered weekly to Caspian restaurant. Demand varies uniformly between 30 to 50 pounds per week. Caspian pay 25 cents per pound for potatoes and charges 75 cent per pound for it as a side dish. Unsold potatoes have no salvage value can cannot be carried over into the next week. Find

The optimal stocking level.

Its stockout risk.

Example: Normal distribution Example: Normal distribution

Caspian restaurant also serves apple juice. Demand for the juice is approximately normal with a mean of 35 liters per week and std of 6 liters per week. If shortage cost is 50 cents and excess cost is 25 cents per liter, find

The optimal stocking level.

Its stockout risk.

• A newsboy can buy the Wall Street Journal newspaper for 20 cents and sell them for 75 cents.

• If he buys more paper he can sell, he disposes of the excess at no additional cost.

• If he does not buy enough paper, he loses potential sales

For simplicity, we assume that the demand distribution is

P{demand = 0} = 0.1 P{demand = 1} = 0.3

P{demand = 2} = 0.4 P{demand = 3} = 0.2

Example: Discrete distributionExample: Discrete distribution

Problem Problem

Skinner’s Fish Market buys fresh Boston blue fish daily for $4.20 per pound and sells it for $5.70 per pound. At the end of each business day, any remaining blue fish is sold to a producer of cat food for $2.40 per pound. Daily demand can be approximated by a normal distribution with a mean of 80 pounds and a standard deviation of 10 pounds. What is the optimal stocking level?

Problem Problem

Burger Prince buys top-grade ground beef for $1.00 per pound. A large sign over the entrance guarantees that the meat is fresh daily. Any leftover meat is sold to the local high school cafeteria for 80 cents per pound. Four hamburgers can be prepared from each pound of meat. Burgers sell for 60 cents each. Labor, overhead, meat, buns, and condiments cost 50 cents per burger. Demand is normally distributed with a mean of 400 pounds per day and a standard deviation of 50 pounds per day.

What daily order quantity is optimal?

Continuous stocking levels

Identifies optimal stocking levels

Optimal stocking level balances unit shortage and excess cost

Discrete stocking levels

Service levels are discrete rather than continuous

Desired service level is equaled or exceeded

Single Period ModelSingle Period Model