Chapter 12: Inventory Control. Purposes of Inventory 1. To maintain independence of operations 2. To meet variation in product demand 3. To allow flexibility.

Chapter 12: Inventory Control

Purposes of Inventory

1. To maintain independence of operations

2. To meet variation in product demand

3. To allow flexibility in production scheduling

4. To provide a safeguard for variation in raw material delivery time

5. To take advantage of economic purchase-order size

Inventory Costs

Holding (or carrying) costsCosts for capital, storage,

handling, “shrinkage,” insurance, etc

Setup (or production change) costsCosts for arranging specific

equipment setups, etcOrdering costs

Costs of someone placing an order, etc

Shortage costsCosts of canceling an order, etc

E(1)

Independent vs. Dependent Demand

Independent Demand (Demand for the final end-product or demand not related to other items)

Dependent Demand(Derived

demand items for component

parts, subassemblies, raw materials,

etc)

Finishedproduct

Component parts

Inventory Systems Single-Period Inventory Model

One time purchasing decision (Example: vendor selling t-shirts at a football game)

Seeks to balance the costs of inventory overstock and under stock

Multi-Period Inventory ModelsFixed-Order Quantity Models

Event triggered (Example: running out of stock)

Fixed-Time Period Models Time triggered (Example: Monthly

sales call by sales representative)

The Newsvendor Model:

Wetsuit example

Economics:• Each suit sells for p = $180• Seller charges c = $110 per suit• Discounted suits sell for v = $90

The “too much/too little problem”: Order too much and inventory is left over at

the end of the season Order too little and sales are lost. Example: Selling Wetsuits

“Too much” and “too little” costs Co = overage cost (i.e. order “one too many” --- demand < order

amount) The cost of ordering one more unit than what you would have

ordered had you known demand – if you have left over inventory the increase in profit you would have enjoyed had you ordered one fewer unit.

For the example Co = Cost – Salvage value = c – v = 110 – 90 = 20

Cu = underage cost (i.e. order “one too few” – demand > order amount) The cost of ordering one fewer unit than what you would have

ordered had you known demand – if you had lost sales (i.e., you under ordered), Cu is the increase in profit you would have enjoyed had you ordered one more unit.

For the example Cu = Price – Cost = p – c = 180 – 110 = 70

Newsvendor expected profit maximizing order quantity

To maximize expected profit order Q units so that the expected loss on the Qth unit equals the expected gain on the Qth unit:

Rearrange terms in the above equation ->

The ratio Cu / (Co + Cu) is called the critical ratio (CR). We shall assume demand is distributed as the normal

distribution with mean and standard deviation Find the Q that satisfies the above equality use

NORMSINV(CR) with the critical ratio as the probability argument. (Q-)/ = z-score for the CR so Q = + z *

QFCQFC uo 1)(

uo

u

CC

CF(Q)Q}dProb{Deman

Note: where F(Q) = Probability Demand <= Q

Finding the example’s expected profit maximizing order quantity

Inputs:

Empirical distribution function table; p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20

Evaluate the critical ratio: NORMSINV(.7778) = 0.765

Other Inputs: mean = = 3192; standard deviation = = 1181 Convert into an order quantity

Q = + z * = 3192 + 0.765 * 1181 4095

7778.07020

70

uo

u

CC

CFind an order quantity Qsuch that there is a 77.78%prob that demand is Q or lower.

Single Period Model Example A college basketball team is playing in a

tournament game this weekend. Based on past experience they sell on average 2,400 tournament shirts with a standard deviation of 350. They make $10 on every shirt sold at the game, but lose $5 on every shirt not sold. How many shirts should be ordered for the game?

Cu = $10 and Co = $5; P ≤ $10 / ($10 + $5) = .667

Z.667 = .432 (use NORMSINV(.667) therefore we need 2,400 + .432(350) = 2,551 shirts

Hotel/Airline Overbooking The forecast for the number of

customers that DO NOT SHOW UP at a hotel with 118 rooms is Normally Dist with mean of 10 and standard deviation of 5

Rooms sell for $159 per night The cost of denying a room to

the customer with a confirmed reservation is $350 in ill-will and penalties.

Let X be number of people who do not show up – X follows a probability distribution!

How many rooms ( Y ) should be overbooked (sold in excess of capacity)?

Newsvendor setup:Single decision when

the number of no-shows in uncertain.

Underage cost if X > Y (insufficient number of rooms overbooked).

For example, overbook 10 rooms and 15 people do not show up – lose revenue on 5 rooms

Overage cost if X < Y (too many rooms overbooked).

Overbook 10 rooms and 5 do not show up; pay penalty on 5 rooms

Overbooking solution Underage cost:

if X > Y then we could have sold X-Y more rooms… … to be conservative, we could have sold those rooms at the low

rate, Cu = rL = $159

Overage cost: if X < Y then we bumped Y - X customers … … and incur an overage cost Co = $350 on each bumped customer.

Optimal overbooking level:

Critical ratio:

( ) .u

o u

CF Y

C C

1590.3124

350 159u

u o

C

C C

Optimal overbooking level

Suppose distribution of “no-shows” is normally distributed with a mean of 10 and standard deviation of 5

Critical ratio is:

z = NORMSINV(.3124) = -0.4891 Y = + z = 10 -.4891 * 5 = 7.6 Overbook by 7.6 or 8

Hotel should allow up to 118+8 reservations.

1590.3124

350 159u

u o

C

C C

Multi-Period Models:Fixed-Order Quantity Model Model

Assumptions (Part 1) Demand for the product is

constant and uniform throughout the period

Lead time (time from ordering to receipt) is constant

Price per unit of product is constant

Inventory holding cost is based on average inventory

Ordering or setup costs are constant

All demands for the product will be satisfied (No back orders are allowed)

Basic Fixed-Order Quantity Model and Reorder Point Behavior

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Time

Numberof unitson hand

1. You receive an order quantity Q.

2. Your start using them up over time. 3. When you reach down

to a level of inventory of R, you place your next Q sized order.

4. The cycle then repeats.

Cost Minimization Goal

Ordering Costs

HoldingCosts

Order Quantity (Q)

COST

Annual Cost ofItems (DC)

Total Cost

QOPT

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs

Deriving the EOQUsing calculus, we take the first derivative of the total cost function

with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

Q = 2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPTQ =

2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPT

Reorder point, R = d L_

Reorder point, R = d L_

d = average daily demand (constant)

L = Lead time (constant)

_We also need a reorder point to tell us when to place an order

We also need a reorder point to tell us when to place an order

Basic Fixed-Order Quantity (EOQ) Model

Formula

H 2Q

+ S QD

+ DC = TC H 2Q

+ S QD

+ DC = TC

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual costD =DemandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

TC=Total annual costD =DemandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

EOQ Class Problem 1

Dickens Electronics stocks and sells a particular brand of PC. It costs the firm $450 each time it places and order with the manufacturer. The cost of carrying one PC in inventory for a year is $170. The store manager estimates that total annual demand for computers will be 1200 units with a constant demand rate throughout the year. Orders are received two days after placement from a local warehouse maintained by the manufacturer. The store policy is to never have stockouts. The store is open for business every day of the year. Determine the following:

Optimal order quantity per order. Minimum total annual inventory costs (i.e.

carrying plus ordering – ignore item costs). The optimum number of orders per year (D/Q*)

Problem 1Demand 1200

Ordering Cost 450Carrying Cost 170

Lead Time 2 Rounding up …

Q* 79.7 80Ordering + Carrying= 13,549.91$ 13,550.00$

Orders/year = 15.1

EOQ Problem 2

The Western Jeans Company purchases denim from Cumberland textile Mills. The Western Jeans Company uses 35,000 yards of denim per year to make jeans. The cost of ordering denim from the Textile Mills is $500 per order. It costs Western $0.35 per yard annually to hold a yard of denim in inventory. Determine the following:

a. Optimal order quantity per order.b. Minimum total annual inventory costs (i.e. carrying plus

ordering).c. The optimum number of orders per year.      

Demand 35000Ordering Cost 500Carrying Cost 0.35

a. Optimal order quantity per order.Q* 10000.0

b. Minimum total annual inventory costs (i.e. carrying plus ordering).Ordering + Carrying= 3,500.00$

c. The optimum number of orders per year.Orders/year = 3.5

Problem 3A store specializing in selling

wrapping paper is analyzing their inventory system. Currently the demand for paper is 100 rolls per week, where the company operates 50 weeks per year.. Assume that demand is constant throughout the year. The company estimates it costs $20 to place an order and each roll of wrapping paper costs $5.00 and the company estimates the yearly cost of holding one roll of paper to be 50% of its cost.

a) If the company currently orders 200 rolls every other week (i.e., 25 times per year), what are its current holding and ordering costs (per year)?

per yearD 5000S $20.00c $5.00i 0.5

H $2.50

part a

Ordering $500.00

Holding $250.00Total $750.00

Problem 3b) The company is considering

implementing an EOQ model. If they do this, what would be the new order size (round-up to the next highest integer)? What is the new cost? How much money in ordering and holding costs would be saved each relative to their current procedure as specified in part a)?

part b

EOQ 282.8 283

Ordering $353.55

Holding $353.55

Total $707.11

$$ Saved: $42.89

c) The vendor says that if they order only twice per year (i.e., order 2500 rolls per order), they can save 10 cents on each roll of paper – i.e., each roll would now cost only $4.90. Should they take this deal (i.e., compare with part b’s answer) [Hint: For c]. calculate the item, holding, and ordering costs in your analysis.]

part c

Ordering $40.00

Holding $3,062.50

Total $3,102.50

I tem Cost Holding Ordering Net Total

Part b) option 25,000.00$ $353.55 $353.55 $25,707.11

Part c) option $24,500.00 $3,062.50 $40.00 $27,602.50

Do not accept the deal

Safety Stocks

Probability of stockout(1.0 - 0.85 = 0.15)

Cycle-service level = 85%

Average demand

during lead time

zL

R

Suppose that we assume orders occur at a fixed review period and that demand is probabilistic and we want a buffer stock to ensure that we don’t run out

Suppose that we assume orders occur at a fixed review period and that demand is probabilistic and we want a buffer stock to ensure that we don’t run out

Safety Stock Formula

Reorder Point = Average demand + Safety stockReorder Point = Average demand + Safety stock

Reorder Point = Demand during Lead Time + Safety Stock

Demand during lead time = daily demand * L = d*LSafety stock = Zservice~level * L

Where L = square root of L*2, where is the standard deviation of demand for one day

Problem 4A large manufacturer of VCRs sells 700,000

VCRs per year. Each VCR costs $100 and each time the firm places an order for VCRs the ordering charge is $500. The accounting department has determined that the cost of carrying a VCR for one year is 40% of the VCR cost. If we assume 350 working days per year, a lead-time of 4 days, and a standard deviation of lead time of 20 per day, answer the following questions.

a) How many VCRs should the company order each time it places an order?

b) If the company seeks to achieve a 99% service level (i.e. a 1% chance of being out of stock during lead time), what will be the reorder point? How much lower will be the reorder point if the company only seeks a 90% service level?

D 700,000.00 c 100.00$ S 500.00$ i 40%

days 350L 4

sigma(d) 20

part a 4183.3 4184

part b

d(L) 8000For 99%

z(.99) 2.33sigma(L) 40

Safety Stock 93.05

ROP 8093.1 8094

For 90%

z(.90) 1.282sigma(L) 40

Safety Stock 51.26

D ROP 41.79 42

Fixed-Time Period Model with Safety Stock Formula

order) on items (includes level inventory current = I

time lead and review the over demand of deviation standard =

yprobabilit service specified a for deviations standard of number the = z

demand daily average forecast = d

days in time lead = L

reviews between days of number the = T

ordered be to quantitiy = q

:Where

I - Z + L)+(Td = q

L+T

L+T

order) on items (includes level inventory current = I

time lead and review the over demand of deviation standard =

yprobabilit service specified a for deviations standard of number the = z

demand daily average forecast = d

days in time lead = L

reviews between days of number the = T

ordered be to quantitiy = q

:Where

I - Z + L)+(Td = q

L+T

L+T

q = Average demand + Safety stock – Inventory currently on hand

q = Average demand + Safety stock – Inventory currently on hand

Multi-Period Models: Fixed-Time Period Model: Determining the Value of T+L

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

The standard deviation of a sequence of random events equals the square root of the sum of the variances

Example of the Fixed-Time Period Model

Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The

daily demand standard deviation is 4 units.

Given the information below, how many units should be ordered?Given the information below, how many units should be ordered?

Example of the Fixed-Time Period Model: Solution (Part 1)

T+ L d2 2 = (T + L) = 30 + 10 4 = 25.298 T+ L d

2 2 = (T + L) = 30 + 10 4 = 25.298

The value for “z” is found by using the Excel NORMSINV function.

The value for “z” is found by using the Excel NORMSINV function.

or 644.272, = 200 - 44.272 800 = q

200- 298)(1.75)(25. + 10)+20(30 = q

I - Z+ L)+(Td = q L+T

units 645

or 644.272, = 200 - 44.272 800 = q

200- 298)(1.75)(25. + 10)+20(30 = q

I - Z+ L)+(Td = q L+T

units 645

ABC Classification System Items kept in inventory are not of

equal importance in terms of:

dollars invested

profit potential

sales or usage volume

stock-out penalties

0

30

60

30

60

AB

C

% of $ Value

% of Use

So, identify inventory items based on percentage of total dollar value, where “A” items are roughly top 15 %, “B” items as next 35 %, and the lower 65% are the “C” items