Ch12

1M. Verma: October 23-31, 2006 Faculty of Business Administration, Memorial University of Newfoundland

Inventory Management

Chapter 12


Independent and Dependent Demand

• Dependent Demand– Used in the production of a final or finished product.

– Is derived from the number of finished units to be produced.

– E.g. For Each Car: 4-wheels. Wheels are the dependent demand.

• Independent Demand– Sold to someone.

– Precise determination of quantity impossible due to randomness.

– Forecasting the number of units that can be sold.

Focus on management of Independent Demand Items.


Types of Inventory

Raw material Components Work-in-progress Finished goods Distribution Inventory Maintenance/repair/operating supply (MRO)


1. To meet anticipated demand.

2. To smooth production requirements.

3. To decouple operations.

4. To protect against stockouts.

5. To take advantage of economic lot size.

6. To hedge against price increases or to take advantage of quantity discounts.

Functions of Inventory


Objectives of Inventory ControlInventory Management has two main concerns:

Level of Customer Service. Cost of ordering and carrying inventories.

Achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds. 2 fundamental

decisions:

--Timing of Orders

--Size of Orders

Measures of effective Inventory Management: Customer Satisfaction. Inventory Turnover = COGS / Average Inventory. Days of Inventory: expected number of days of sales that can

be supplied from existing inventory.

i.e. when to order and how much to order.


Characteristics of Inventory Systems

• Demand

-- Constant Vs Variable

-- Known Vs Random

• Lead time - Known Vs Random

• Review Time - Continuous Vs Periodic

• Excess demand - Backordering Vs Lost Sales

• Changing Inventory


Periodic Vs Continuous Review Systems

• Periodic Review System• Inventory level monitored at constant intervals.• Decisions:

• To order or not.• How much to order?

• Realize economies in processing and shipping.• Risk of stockout between review periods. • Time and cost of physical count.

• Continuous Review System • Inventory level monitored continuously.• Decisions:

• When to order? • How much to order?

• Shortages can be avoided.• Optimal order quantity can be determined. • Added cost of record keeping.

More appropriate for valuable items


Relevant Inventory Costs• Item price (Cost of an item).

• Holding costs:– Variable costs dependent upon the amount of inventory held (e.g.: capital &

opportunity costs, storage & insurance, risk of obsolescence).

– Expressed either as a percentage of unit price or as a dollar amount per unit.

• Ordering & setup costs:– Fixed cost of placing an order (e.g.: clerical accounting & physical handling) or

setting up production (e.g.: lost production to change tools & clean equipment).

– Expressed as a fixed dollar amount per order regardless of order size.

• Shortage costs:– Result when demand exceeds the supply of inventory on-hand. (stock-out)

– Lost profit, expediting & back ordering expenses.

– Are sometimes difficult to measure, & they may be subjectively estimated.


Why Control Inventory

Cost Vs Service• Under-stock: Frequent stock-outs

– Lost sales, loss of customer goodwill Low level of customer service

• Over-stock: Excess inventory– Costs of ordering and carrying inventory increase

• Objective: Establish an inventory control system to find a balance between cost and service: – When to order?

– How much to order?


• Lot-for-lot: • Order exactly what is needed.

• Fixed order quantity:• Order a predetermined amount each time.

• Min-max system:• When inventory falls to a set minimum level, order up to

the predetermined maximum level.

• Order enough for n periods:• The order quantity is determined by total demand for the

item for the next n-periods.

• Periodic review:• At specified intervals, order up to a predetermined target

level.

Ordering Quantity Approaches


• Divides on-hand inventory into 3 classes: • A class, B class, C class (very --> moderate --> least: important).

• Basis is usually annual $ volume: • $ volume = Annual demand x Unit cost

• Class A: 15% -20% of items but 70%- 80% dollar usage.• Class B: 30% -35% of items but 15% -20% dollar usage.• Class C: 50% -60% of items but 5% -10% dollar usage.

• Policies based on ABC analysis:• Develop class A suppliers more• Give tighter physical control of A items• Forecast A items more carefully

Classification System


Example

Item Annual Demand x Unit Cost = Annual $ Value1 1000 4300 $4,300,0002 5000 720 $3,600,0003 1900 500 $950,0004 1000 710 $710,0005 2500 250 $625,0006 2500 192 $480,0007 400 200 $80,0008 500 100 $50,0009 200 210 $42,00010 1000 35 $35,00011 3000 10 $30,00012 9000 3 $27,000

Total = $10,929,000

Classify the inventory items as A,B and C based on annual dollar value.

72% of total value and

17% of items: Class A.

25% of value and

33% of items: Class B.

3% of total value and

50% of items: Class C.


1. Basic Economic Order Quantity (EOQ).

2. Economic Production Quantity (EPQ)

(EOQ with non-instantaneous delivery).

3. Quantity Discount Model.

Economic-Order Quantity Models


1. Only one product is involved.2. Demand is known & constant - no safety stock is

required.3. Lead time is known & constant.4. No quantity discounts are available.5. Ordering (or setup) costs are constant.6. All demand is satisfied (no shortages).7. The order quantity arrives in a single shipment.

EOQ Model Assumptions


Inventory Level (Cycle)

Q

ROP

Qu

antity on

han

d

Receive OrderPlace Order Receive Order

Lead Time


Total annual costs = Annual ordering costs + Annual holding costs

EOQ Model


Minimize the TC by ordering the EOQ:

H

QS

Q

DTCEOQ 2

H

DSEOQ

2

EOQ: Total Cost Equations

D = Annual Demand.H= Annual Inventory Holding/Carrying Cost per Unit.S= Ordering/Setup Cost per order.

Length of an Order Cycle (time between orders) = (Q/D) years.


Example1. A local distributor for a national tire company expects to sell

approximately 9,600 belted radial tires for a certain size and tread design next year. Annual carrying 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 order?

C. What is the length of an order cycle?

D. What is the total annual cost if the EOQ quantity is ordered?

2. Pidding Manufacturing assembles security monitors. It purchases 3,600 black-and-white cathode ray tubes a year at $65 each. Ordering cost is $31 per order, and annual carrying cost per unit is 20% of the purchase price. Compute the optimal order quantity and the total annual cost of ordering and carrying the inventory.


Assumptions: Same as the EOQ except: inventory arrives in increments & is drawn down as it arrives.

Economic Production Quantity Model


• Adjusted total cost:

• Maximum inventory:

• Adjusted order quantity:

• Cycle Time: Q/d.

• Run Time (production phase of the cycle): Q/p.

where, d = demand (usage rate); p = production rate; S = ordering / setup cost;

D = Annual demand and H = Annual Holding cost.

H

IS

Q

DTC MAX

EPQ 2

2 ;1 .

MAXavgMAX

II

p

dQI

dp

p

H

DS

p

dH

DSEPQ

2

1

2

EPQ Equations


ExampleA 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 days per year. Determine the:

A. Optimal run size.

B. Minimum total annual cost for carrying and setup.

C. Cycle time for the optimal run size.

D. Run time.


• Same as the EOQ, except:• Unit price depends upon the quantity ordered.

• Adjusted total cost equation:

PDHQ

SQ

DTCQD

2

Quantity Discount Model


Total Costs with Purchasing Cost (PD)

Co

st

EOQ

TC with PD

TC without PD

Annual purchasing cost=(PD)

0 Quantity

Adding Purchasing costdoesn’t change EOQ


Quantity Discount Representation

Order Price

Quantity per Box

1 to 44 $2.00

45 to 69 $1.70

70 or more $1.40

PD @ $2.00 each

PD @ $1.70 each

PD @ $1.40 each

045 70

Quantity

Total

Cost

TC@ $1.70 each

TC@ $2.00 each

TC@ $1.40 each


Quantity Discount Model

• The objective is to identify ‘an order quantity’ that will represent the lowest total cost for the entire set of curves.

• 2 General Cases:– Carrying Cost is constant.

• Single EOQ.

• Same for all total cost curves.

– Carrying Cost is a percentage of the Price.• Different EOQ for different prices.

• Lower carrying cost means larger EOQ.

• As unit price decreases, each curve EOQ will be to the right of the next higher curves’ EOQ.


• Calculate the EOQ at the lowest price.• Determine whether the EOQ is feasible at that

price • Will the vendor sell that quantity at that price.

• If yes, Stop – if no, Continue.• Check the feasibility of EOQ at the next higher

price• Continue until you identify a feasible EOQ.• Calculate the total costs (including purchase

price) for the feasible EOQ model.• Calculate the total costs of buying at the minimum

quantity allowed for each of the cheaper unit prices.

• Compare the total cost of each option & choose the lowest cost alternative.

Quantity Discount Procedure


1. The maintenance department of a large hospital uses about 816 cases of liquid cleaner 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.

2. Surge Electric uses 4,000 toggle switches a year. Switches are priced as follows: 1 to 499, 90 cents each; 500 to 999, 85 cents each; and 1,000 or more, 80 cents each. It costs approximately $30 to prepare an order and receive it, and carrying costs are 40 percent of purchase price per unit on an annual basis. Determine the optimal order quantity and the total annual cost.

Examples


ROP = d(LT)

When to Order?

where LT = Lead time (in days or weeks)d = Daily or weekly demand rate


An office supply store sells floppy disk sets at a fairly constant rate of 6,000 per year. The accounting dept. states that it costs 8$ to place an order and annual holding cost are 20% of the purchase price 3$ per unit. It takes 4 days to receive an order. Assuming a 300-day year, find:

a) Optimal order size and ROP.b) Annual ordering cost, annual carrying cost.c) How many orders are given a year and what is the time between the orders?

Example


What if Demand is Uncertain?

ROP

Time

Qu

antity on

han

d


Uncertain Demand (Lead Time)

Safety Stock Models: • Use the same order quantity (EOQ) based on expected

(average) annual demand.

• Determine ROP to satisfy a target Service Level:– Probability that demand will not exceed supply during lead

time (Lead time service level).

– Percent of annual demand immediately satisfied (Annual service level or fill-rate).

– Equals: 1- stock-out risk

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


• Demand variability.• Lead time variability.• Order-cycle service level:

• From a managerial standpoint, determine the acceptable probability that demand during lead time won’t exceed on-hand inventory.

• Risk of a stockout: 1 – (service level).

Adding Safety Stock


R = reorder pointd = average daily demandLT = lead time in daysz = number of standard deviations associated with desired service

levelσ = standard deviation of demand during lead time

(Assumes that any variability in demand rate or lead time can be adequately described by a normal distribution)

LTzLTdR )(

Adjusted Reorder Point Equation


ExampleSuppose that the manager of a construction supply house determined from historical records that demand for sand during lead time averages 50 tons. In addition, suppose 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. Answer these questions, assuming that the manager is willing 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?


Reorder Point -Continued

• When data on lead time demand is not readily available, cannot use the standard formula.– Use the daily or weekly demand and the length of the lead time

to generate lead time demand.• If only demand is variable, then use ,

and the ROP is:

dLTzLTdROP

ddLT LT

or weeks. daysin timelead

or week.day per demand ofdeviation standard

demand.or weekly daily average

LT

d

d


• If only lead time is variable, then use , and the ROP is:

• If both demand and lead times variable, then

LTzdLTdROP

LTdLT d

222

222

:ispoint reorder theand

LTd

LTddLT

dLTzLTdROP

dLT

or weeks. daysin timelead average

or weeks. daysin timelead ofdeviation standard

demand.or weekly daily average

LT

d

LT


ExampleA restaurant uses an average of 50 jars 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 percent risk of stockout during lead time, which is two weeks. Assume the distribution of usage is normal.

A. Which of the above formula is appropriate for this situation? Why?

B. Determine the value of z.

C. Determine the ROP.


Shortage and Service Level

dLTzEnE )()(

demand timelead ofdeviation standard

13.3 tablefrom obtainedshort units of # edstandardiz )(

cycle.order per short units of # expected )(

dLT

zE

nE

E.g.: Suppose the standard deviation of lead time demand is known to be 20 units and lead time demand is approximately Normal.

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

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


Q

DnENE )()(

year.per short units ofnumber expected )( NE

Q

zE

D

NESL dLT

annual)(

1)(

1

Given the following information, determine the expected number of units short per year. D=1,000; Q=250; E (n)=2.5.

Given a lead time service level of 90%, D=1,000, Q=250, and σdLT=16, determine (a) the annual service level, and (b) the amount of cycle safety stock that would provide an annual service level of .98 (Given: E (z) = 0.048 for 90% lead time service level).


Fixed-Order Interval Model• Order groupings can produce savings in ordering and shipping costs.• Can have variations in demand, lead time, or in both. • Our focus is only on demand variability, with constant lead times.

LTOIzLTOIdI

on hand - Amount I Order size

d

)(max

max

OI = Order Interval (length of time between orders)

Imax = Maximum amount of inventory (also called order-up-to-level point)

= Expected demand during protection interval + Safety stock

E.g.: Given the following information, determine the amount to order.

days 2

dayper units 3

day per units 30

LT

d

ddays 7 OI

units 71 mereorder tiat handon Amount

percent 99 level service Desired


Single-Period Models

Goal is to identify the order quantity, or stocking level, that will minimize the long-run total excess and shortage cost.

unit value per - Salvaget per unit Original C C

t per unitunit - Cosvenue per CC

eexcess

sshortage

cos

Re

• Used for order perishables.

• Analysis focus on two costs: Shortage and Excess.

2 kinds of problems:

a) Demand can be approximated using a continuous distribution.

b) Demand can be approximated using a discrete distribution.


Continuous Stocking Levels:

Discrete Stocking Levels:

es

s

CC

C SL vel Service le

E.g.: Sweet cider is delivered weekly to Cindy’s Cider Bar. Demand varies uniformly between 300 liters and 500 liters per week. Cindy pays 20 cents per liter for the cider and charges 80 cents per liter for it. Unsold cider has no salvage value and cannot be carried over into the next week due to spoilage. Find the optimal stocking level and its stockout risk for that quantity.

E.g.: Cindy’s Cider Bar also sells a blend of cherry juice and apple cider. Demand for the blend is approximately Normal, with a mean of 200 liters per week and a standard deviation of 10 liters per week. Cs=60 cents per liter, and Ce=20 cents per liter. Find the optimal stocking level for the apple cherry blend.

E.g.: Historical records on the use of spare parts for several large hydraulic presses are to serve as an estimate of usage for spares of a newly installed press. Stockout costs involve downtime expenses and special ordering costs. These average $4,200 per unit short. Spares cost $800 each, and unused parts have zero salvage. Determine the optimal stocking level. # of Spares Used Relative Frequency Cumulative Frequency

0 0.20 0.201 0.40 0.602 0.30 0.903 0.10 1.004 or more 0.00

1.00


ExamplesA 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 $4 per order. Annual carrying costs are $30 per bag.

A. Determine the economic order quantity.

B. What is the average number of bags on hand?

C. Compute the total cost of ordering and carrying flour.

D. If ordering cost were to increase by $1 per order, how much would that affect the minimum total annual

ordering and carrying cost?


ExamplesA large law firm uses an average of 40 packages of copier paper a day. Each package contains 500 sheets. The firm operates 260 days a year. Storage and handling costs for the paper are $1 a year per pack, and it costs approximately $6 to order and receive a shipment of paper.

a) What order size would minimize total annual ordering and carrying costs?

b) Compute the total annual cost using your order size from part a.

c) Except for rounding, are annual ordering and carrying costs always equal at the EOQ?

d) The office manager is currently using an order size of 400 packages. The partners of the firm expect the office to be managed “in a cost-efficient manner.” Would you recommend that the office manager use the optimal order size instead of 400 packages? Justify your answer.


Examples

A chemical form produces sodium bisulphate in 100-kg bags. Demand for this product is 20 tons per day. The capacity for producing the product is 50 tons per day. Setup costs $100, and storage and handling costs are $50 per ton per year. The firm operates 200 days a year. (Note: 1 ton = 1,000 kg)

a) How many bags per run are optimal?

b) What would the average inventory be for this lot size?

c) Determine the approximate length of a production run, in days.

d) About how many runs per year would there be?

e) How much could the company save annually if the setup cost could be reduced to $25 per run?


ExamplesA company is about to begin production of a new product. The manager of the department that will produce one of the components for the product wants to know how often the machine to be used to produce the item will be available for other work. The machine will produce the item at a rate of 200 units a day. Eighty units will be used daily in assembling the final product. Assembly will take place five days a week, 50 weeks a year. The manager estimates that it will take almost a full day to get the machine ready for a production run, at a cost of $300. Inventory holding costs will be $10 per unit a year.

a) What run quantity should be used to minimize total annual costs?b) What is the length of a production run in days?c) During production, at what rate will inventory build up?d) If the manager wants to run another job between runs of this item, and

needs a minimum of 10 days per cycle for the other work, will there be enough time.


Examples

A mail-order company uses 18,000 boxes a year. Carrying costs are 20 cents per box per year, and ordering costs are $32 per order. The following quantity discount is available. Determine:

a) The optimal order quantity.

b) The number of orders per year.

Number of Boxes Price per Box1,000 to 1,999 $1.252,000 to 4,999 $1.205,000 to 9,999 $1.1810,000 or more $1.15


ExamplesA jewelry firm buys semi-precious stones to make bracelets and rings. The supplier quotes a price of $8 per stone and 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 cost is $48 per order.A. If carrying cost are $2 per year for each stone, find the order quantity that will minimize total annual cost.

B. If annual carrying cost are 30 percent of unit cost, what is the optimal order size?

C. If lead time is six working days, at what point should the company reorder?


ExamplesThe housekeeping department of a motel uses approximately 400 washcloths per day. The actual amount tends to vary with the number of guests on any given night. Usage can be approximated by a normal distribution that has a mean of 400 and a standard deviation of 9 washcloths per day. A linen supply company delivers towels and washcloths with a lead time of three days. If the motel policy is to maintain a stockout risk of 2 percent, what is the minimum number of washcloths that must be on hand at reorder point, and how much of that amount can be considered safety stock?

The motel in the preceding example uses approximately 600 bars of soap each day, and this tends not to vary by more than a few bars either way. Lead time for soap delivery is normally distributed with a mean of six days and a standard deviation of two days. A service level of 90 percent is desired. Find the ROP.


ExamplesA distributor of large appliances needs to determine the order quantities and reorder points for the various products it carries. The following data refers to a specific refrigerator in its product line:

Cost to place an order: $100

Holding Cost: 20 percent of product cost per year.

Cost of refrigerator: $500 each.

Annual demand: 500 refrigerators.

Standard deviation during lead time: 10 refrigerators.

Lead time: 7 days.

Consider an even daily demand and a 365-day year.A. What is the economic order quantity?

B. If the distributor wants a 97% service probability, what reorder point R should be used? What is the corresponding safety stock?

C. If the current reorder point is 26 refrigerators, what is the possibility of stock-out?


Examples

A local service station is open 7 days a week, 365 days per year. Sales of 10W40 grade premium oil average 20 cans per day. Inventory holding costs are $0.50 pre can per year. Ordering costs are $10 per order. Lead time is two weeks. Backorders are not practical -- the motorist drives away.A. Based on these data, choose the appropriate inventory model and

calculate the economic order quantity and reorder point. ( Demand is deterministic).B. The boss is concerned about this model because demand really varies. The standard deviation of demand was determined from a data sample to be 6.15 cans per day. The manager wants a 99.5% service

probability. Determine the new reorder point? Use Qopt from Part-A.


ExamplesA small copy centre uses five boxes of copy paper a day. Each box contains 10 packages of 500 sheets. Experience suggests that usage can be well approximated by a Normal distribution with a mean of five boxes per day and a standard deviation of one-half box per day. Two days are required to fill an order for paper. Ordering cost is $10 per order, and annual holding cost is $10 per box per year.

a) Determine the economic order quantity, assuming 250 work days a year.

b) If the copy center reorders when the supply on hand is 12 boxes, compute the risk of a stockout.

c) If fixed interval of seven days, instead of ROP, is used for reordering, what risk does the copy center incur that it will run out of stationery before this order arrives if it orders 36 boxes when the amount on hand is 12 boxes?


Examples

Regional Supermarket is open 360 days per year. Daily use of cash register tape averages 10 rolls. Usage appears Normally distributed with a standard deviation of 2 rolls per day. The cost of ordering tape is $5 per order, and carrying costs are 40 cents per roll a year. Lead time is three days.

a) What is the EOQ?

b) What ROP will provide a lead time service level of 96 percent?

c) What is the expected number of units short per cycle with 96 percent service level? Per year?

d) What is the annual service level?


ExamplesA depot operates 250 days a year. Daily demand for diesel fuel at the depot is Normal with an average of 250 liters and a standard deviation of 14 liters. Holding cost for the fuel is $0.30 per liter per year, and it costs $10 in administrative time to submit an order for more fuel. It takes one day to receive a delivery of diesel fuel.

a) Calculate the EOQ.

b) Determine the amount of safety stock that would be needed if the manager wants.

i. An annual service level of 99.5 percent.

ii. The expected amount of fuel short per order cycle to be no more than 5 liters.


Examples

A drugstore uses fixed-order-interval model for many of the items it stocks. The manager wants a service level of 0.98. Determine the order size that will be consistent with this service level for the items in the following table for an order interval of 14 days and a lead time of 2 days.

Item Average Daily Demand Daily Standard Deviation Quantity on HandK033 60 5 420K144 50 4 375L700 8 2 160

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Ch12