Chapter 5 part 1 inventory control

Supply Chain Management

Chapter 5

Inventory Control

� Inventory System Defined� Types of Inventory� Independent vs. Dependent Demand� Inventory System Models� Multi-Period Inventory Models: Basic

Fixed-Order Quantity Models� Inventory Costs� Multi-Period Inventory Models: Basic

Fixed-Time Period Model� Single-Period Inventory Model � Miscellaneous Systems and Issues

OBJECTIVES

Inventory System

� Inventory is the stock of any item or resource used in an organization and can include: raw materials, finished products, component parts, supplies, and work-in-process

� An inventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be

Inventory

One of the most expensive assets of many companies representing as much as 50% of total invested capital

Inventory managers must balance inventory investment and customer service

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

Types of Inventory

Raw materialRaw material

Purchased but not processedPurchased but not processed

WorkWork--inin--processprocess

Undergone some change but not completedUndergone some change but not completed

A function of cycle time for a productA function of cycle time for a product

Maintenance/repair/operating (MRO)Maintenance/repair/operating (MRO)

Necessary to keep machinery and processes Necessary to keep machinery and processes productiveproductive

Finished goodsFinished goods

Completed product awaiting shipmentCompleted product awaiting shipment

Types of Inventory-2

� Cycle Inventory

� Safety Stock Inventory

� Anticipatory Inventory

� Pipeline Inventory

� Inventory that varies directly with lot size.

� Lot size varies with elapsed time between orders.

� The quantity ordered must meet the demand during the ordering period.

� Long gaps in the ordering period will require larger cycle inventory.

� The inventory may vary between order size Q to zero just before the new lot is delivered.

� Average inventory size is therefore Q/2

Cycle Inventory

� Safety stock inventory protects against uncertainties in demand, lead time, and supply.

� It ensures that operations are not disrupted when problems occur.

� To build safety stock an order is placed earlier than the item is needed or the ordered quantity is larger than the quantity required till the next delivery schedule.

Safety Stock Inventory

� Inventory used to absorb uneven rate of demand or supply

� Predictable seasonal demand pattern may justify anticipation inventory.

� Uneven demand often makes the firm to stockpile during low production demand to make better use of production facilities and avoid varying output rates and labor force.

� Uncertainties such as threatened strikes, problem at suppliers facilities etc also justify anticipation inventory.

Anticipation Inventory

� Inventory moving from point to point in the material flow system is called pipeline inventory

- from suppliers to plant, from one operation to the next in processing, from plant to distribution center and from distribution center to retailer

� Pipeline Inventory between two points, can be expressed in terms of lead time and average demand (d) during the lead time (L).

Pipeline Inventory = dL

Pipeline Inventory

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 Models

• Multi-Period Inventory Models- Fixed-Order Quantity ModelsEvent triggered (Example: running out of stock)

- Fixed-Time Period ModelsTime triggered (Example: Monthly sales call by sales representative)

• Multi-Period Inventory Models- Fixed-Order Quantity ModelsEvent triggered (Example: running out of stock)

- Fixed-Time Period ModelsTime triggered (Example: Monthly sales call by sales representative)

• Single-Period Inventory Models- One time purchasing decision (Example:

vendor selling t-shirts at a football game)- Seeks to balance the costs of inventory

overstock and under stock

• Single-Period Inventory Models- One time purchasing decision (Example:

vendor selling t-shirts at a football game)- Seeks to balance the costs of inventory

overstock and under stock

Inventory Models for Independent Demand

� Basic economic order quantity

� Production order quantity

� Quantity discount model

Need to determine when and how much to order

Holding, Ordering, and Setup Costs

� Holding costs - the costs of holding or “carrying” inventory over time

� Ordering costs - the costs of placing an order and receiving goods

� Setup costs - cost to prepare a machine or process for manufacturing an order

Holding Costs

••Housing costs (including rent or depreciation, Housing costs (including rent or depreciation,

operating costs, taxes, insurance)operating costs, taxes, insurance)

••Material handling costs (equipment lease or Material handling costs (equipment lease or

depreciation, power, operating cost)depreciation, power, operating cost)

••Labor costLabor cost

••Investment costs (borrowing costs, taxes, and Investment costs (borrowing costs, taxes, and

insurance on inventory)insurance on inventory)

••Pilferage, space, and obsolescencePilferage, space, and obsolescence

Multi-Period Models:Fixed-Order Quantity Model

Assumptions

� 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

Multi-Period Models:Fixed-Order Quantity Model

Model Assumptions (Contd.)

� Inventory holding cost is based on average inventory

� Ordering or setup costs are constant

� All demands for the product will be satisfied (No backorders 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. You 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

The EOQ Model

QQ = Number of pieces per order= Number of pieces per order

Q*Q* = Optimal number of pieces per order (EOQ)= Optimal number of pieces per order (EOQ)

DD = Annual demand in units for the Inventory item= Annual demand in units for the Inventory item

SS = Setup or ordering cost for each order= Setup or ordering cost for each order

HH = Holding or carrying cost per unit per year= Holding or carrying cost per unit per year

Annual setup cost Annual setup cost == ((Number of orders placed per yearNumber of orders placed per year) )

x (x (Setup or order cost per orderSetup or order cost per order))

Annual demandAnnual demand

Number of units in each orderNumber of units in each order

Setup or order Setup or order

cost per ordercost per order==

= (= (SS))DD

QQ

Annual setup cost = SDQ

The EOQ Model

QQ = Number of pieces per order= Number of pieces per order

Q*Q* = Optimal number of pieces per order (EOQ)= Optimal number of pieces per order (EOQ)

DD = Annual demand in units for the Inventory item= Annual demand in units for the Inventory item

SS = Setup or ordering cost for each order= Setup or ordering cost for each order

HH = Holding or carrying cost per unit per year= Holding or carrying cost per unit per year

Annual holding cost Annual holding cost == ((Average inventory levelAverage inventory level) )

x (x (Holding cost per unit per yearHolding cost per unit per year))

Order quantityOrder quantity

22= (= (Holding cost per unit per yearHolding cost per unit per year))

= (= (HH))QQ

22

Annual setup cost = SD

Q

Annual holding cost = HQ

2

The EOQ Model

Q = Number of pieces per orderQ* = Optimal number of pieces per order (EOQ)

D = Annual demand in units for the Inventory item

S = Setup or ordering cost for each order

H = Holding or carrying cost per unit per year

Optimal order quantity is found when annual setup cost equals annual holding cost

Annual setup cost = SDQ

Annual holding cost = HQ2

DD

QQSS = = HH

QQ

22

Solving for Q*Solving for Q*2DS = Q2H

Q2 = 2DS/H

Q* = 2DS/H

Basic Fixed-Order Quantity (EOQ) Model Formula

H 2

Q + S

Q

D + DC = TC

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual

cost

D =Demand

C =Cost per unit

Q =Order quantity

S =Cost of placing

an order or setup

cost

R =Reorder point

L =Lead time

H=Annual holding

and storage cost

per unit of inventory

TC=Total annual

cost

D =Demand

C =Cost per unit

Q =Order quantity

S =Cost of placing

an order or setup

cost

R =Reorder point

L =Lead time

H=Annual holding

and storage cost

per unit of inventory

Deriving the EOQ The Economic Ordering Quantity (EOQ)

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPTQ =

2DS

H =

2(Annual Demand)(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

EOQ Example-1

Determine optimal number of units to order

D = 1,000 units

S = $10 per order

H = $.50 per unit per year

Q* =Q* =22DSDS

HH

Q* =Q* =2(1,000)(10)2(1,000)(10)

0.500.50= 40,000 = 200= 40,000 = 200 unitsunits

EOQ Example-1aDetermine expected number of orders if:

D = 1,000 units Q* = 200 units

S = $10 per order

H = $.50 per unit per year

= N == N = ==Expected number of

orders

Demand

Order quantity

D

Q**

N N = = 5= = 5 orders per yearorders per year1,000200

EOQ Example- 1b

Determine time between orders if:Determine time between orders if:

D D = 1,000= 1,000 units Q*units Q*= 200= 200 unitsunits

S S = $10= $10 per order Nper order N= 5= 5 orders per yearorders per year

H H = $.50= $.50 per unit/yr working days= 250 days/yrper unit/yr working days= 250 days/yr

= T == T =Expected Expected

time between time between ordersorders

Number of working Number of working days per yeardays per year

NN

T T = = 50 = = 50 days between ordersdays between orders25025055

EOQ Example- 1c

Determine carrying cost if:Determine carrying cost if:

D D = 1,000= 1,000 unitsunits Q*Q* = 200= 200 unitsunits

S S = $10= $10 per orderper order NN = 5= 5 orders per yearorders per year

H H = $.50= $.50 per unit per yearper unit per year TT = 50= 50 daysdays

Total carrying cost = Setup cost + Holding costTotal carrying cost = Setup cost + Holding cost

TCC = S + HTCC = S + HDD

QQQQ

22

TCC TCC = ($10) + ($.50)= ($10) + ($.50)1,0001,000

200200

200200

22

TCC TCC = (5)($10) + (100)($.50) = $50 + $50 = $100= (5)($10) + (100)($.50) = $50 + $50 = $100

Quantity Discount Model orPrice-Break Model

Holding cost is often given as a fraction of unit cost

Cost Holding Annual

Cost) Setupor der Demand)(Or 2(Annual =

iC

2DS = QOPT

Holding cost as a fraction of unit cost

i = percentage of unit cost attributed to carrying inventoryC = cost per unit

Price-Break Model Formula or Quantity Discount Model

Cost Holding Annual

Cost) Setupor der Demand)(Or 2(Annual =

iC

2DS = QOPT

Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:

i = percentage of unit cost attributed to carrying inventoryC = cost per unit

Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value

Price-Break Example- 2Problem Data (Part 1)

A company has a chance to reduce their inventory

ordering costs by placing larger quantity orders using

the price-break order quantity schedule below. What

should their optimal order quantity be if this company

purchases this single inventory item with an e-mail

ordering cost of $4, a carrying cost rate of 2% of the

inventory cost of the item, and an annual demand of

10,000 units?

A company has a chance to reduce their inventory

ordering costs by placing larger quantity orders using

the price-break order quantity schedule below. What

should their optimal order quantity be if this company

purchases this single inventory item with an e-mail

ordering cost of $4, a carrying cost rate of 2% of the

inventory cost of the item, and an annual demand of

10,000 units?

Order Quantity units) Price/unit($)0 to 2,499 $1.202,500 to 3,999 $1.004,000 or more $0.98

Price-Break Example-2 Solution (Part 2)

units 1,826 = 0.02(1.20)

4)2(10,000)( =

iC

2DS = Q OPT

Annual Demand (D)= 10,000 unitsCost to place an order (S)= $4

First, plug data into formula for each price-break value of “C”

units 2,000 = 0.02(1.00)

4)2(10,000)( =

iC

2DS = Q OPT

units 2,020 = 0.02(0.98)

4)2(10,000)( =

iC

2DS = Q OPT

Carrying cost % of total cost (i)= 2%Cost per unit (C) = $1.20, $1.00, $0.98

Interval from 0 to 2499, the Qopt value is feasible

Interval from 2500-3999, the

Qopt value is not feasible

Interval from 4000 & more, the Qopt value is not feasible

Next, determine if the computed Qopt values are feasible or not

Price-Break Example -3 Solution (Part 3)

Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

0 1826 2500 4000 Order Quantity

Total annual costs

So the candidates for the price-breaks are 1826, 2500, and 4000 units

So the candidates for the price-breaks are 1826, 2500, and 4000 units

Because the total annual cost function is a “u” shaped function

Because the total annual cost function is a “u” shaped function


iC 2

Q + S

Q

D + DC = TC

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)= $12,043.82

TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)= $12,043.82

TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

Finally, we select the least costly Qopt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

Finally, we select the least costly Qopt, which is this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

$9849,20


Production Order Quantity Model


� In EOQ Model, We assumed that the entire order was received at one time.

� However, some business firms may receive their orders over a period of time.


� Such cases require a different inventory model. In these cases inventory is being used while new inventory is still being received and the inventory does not build up immediately to its maximum point.

� Instead, it builds up gradually when inventory is received faster than it is being used; then it declines to its lowest level as incoming shipments stop and the use of inventory continues.


� This version of the EOQ model is known as “Noninstantaneous Receipt Model” also referred to as the “Gradual Usage Model ” and “Production Order Quantity Model”. In this, noninstantaneous receipt model, the order quantity is received gradually over time, and the inventory level is depleted at the same time it is being replenished.

� Here, we take into account the daily production rate and daily demand/input rate.


t


Maxi

mum

In

vento

ry

Inventory

Production occurs at a rate of p

Demand occurs at a rate of d

ttime


� Since this model is especially suitable for production environments, It is called Production Order Quantity Model.

� Here, we use the same approach as we used in EOQ model.

� Lets define the following:


� p: Daily Production rate (units / day)

� d: Daily demand rate (units / day)

� t: Length of the cycle in days.

� H: Annual holding cost per unit


� Average Holding Cost = (Average Inventory) . H

= (Max. Inventory / 2) . H


� In the period of production (until the end of each t period):

� Max. Inventory = (Total Produced) – (Total Used)

= p.t - d.t

� Here, Q is the total units that are produced. Therefore,

� Q = p.t t = Q/p


� If we replace the values of t in the Max. Inventory formula:

� Max. Inventory = p (Q/p) - d (Q/p)

= Q - dQ/p = Q (1 – d/p)


� Ann. Holding Cost =(Max. Inventory / 2) . H

� Annual Holding Cost = Q/2 (1 – d/p) . H

Annual Setup Cost = (D/Q) . S


� Now we will set

Annual Holding Cost = Annual Setup Cost

Q/2 (1 – d/p) . H = (D/Q) . S



� This formula gives us the optimum production quantity for the Production Order Quantity Model.

� It is used when inventory is consumed as

it is produced.

Production Order Quantity-Example-5D D == 1,0001,000 unitsunits p p == 88 units per dayunits per day

S S == $10$10 d d == 44 units per dayunits per day

H H == $0.50$0.50 per unit per yearper unit per year

QQ* =* =22DSDS

HH[1 [1 -- ((dd//pp)])]

= 282.8 = 282.8 oror 283 283 unitsunits

QQ* = = 80,000* = = 80,0002(1,000)(10)2(1,000)(10)

0.50[1 0.50[1 -- (4/8)](4/8)]

Miscellaneous Systems and Issues

How Important is the Item?

� Segmentation of Inventory- Not all inventory is created equally

- Different classes of inventory

- Result in different levels of profitability /revenue

- Have different demand patterns and magnitudes

- Require different control policies

� ABC AnalysisCommonly used in practice

Classify items by revenue or value

Combination of usage, sales price, etc.

ABC Analysis

� Identify the items that management should spend time on

� Prioritize items by their value to firm

� Create logical groupings

� Adjust as needed

ABC Analysis

ABC Analysis� What is different between the classes?A Items

Very few high impact items are included Require the most managerial attention and review

Expect many exceptions to be made

B ItemsMany moderate impact items (sometimes most) Automated control w/ management by exception Rules can be used for A (but usually too many exceptions)

C ItemsMany if not most of the items that make up minor impact Control systems should be as simple as possible Reduce wasted management time and attention Group into common regions, suppliers, end users

� But these are arbitrary classifications

Miscellaneous Systems:Bin Systems

Two-Bin System

Full Empty

Order One Bin ofInventory

One-Bin System

Periodic Check

Order Enough toRefill Bin

Maximum Inventory Level, M

Miscellaneous Systems:Optional Replenishment System

MActual Inventory Level, I

q = M - I

I

Q = minimum acceptable order quantity

If q > Q, order q, otherwise do not order any.

Inventory Accuracy and Cycle Counting

� Inventory accuracy refers to how well the inventory records agree with physical count

� Cycle Counting is a physical inventory-taking technique in which inventory is counted on a frequent basis rather than once or twice a year

Question

On average, I sell 150,000 units a year, which I obtain from a wholesaler. I estimate that the cost to me of placing an order is $50 with the average inventory storage cost being 20% per year of the cost of a unit ($5).

1. What would be the optimal order quantity?

2. I currently order 5 times a year. How much would I save by switching to the optimal order quantity as compared with my current policy of ordering 5 times a year?


Inventory Control Part 2

Safety Stock, Fixed Period Model and Single Period Model

Planned Shortages with Back-Orders

� Shortage: when customer demand cannot be met

� Planned shortages could be beneficial

Cost of keeping item is more expensive than the profit from selling it e.g. car

Uncertain Demand

Uncertain Demand- Safety Stock

� Buffer added to on hand inventory during lead time

� Extra reserved stock

� To prevent stock-out under uncertain demand

� Safety stock will not normally be used, but it is available under uncertain demand

How much safety stock should we hold? Judgment on service level

Service Level

� A target for the proportion of demand that is met directly from stock

� The maximum acceptable probability that a demand can be met from the stock

� For example 90% service level90% chance of meeting demand during lead time or 10% chance of not meeting demand (having back-order or lost sales)

Probabilistic Models

� So far we assumed that demand is constant and uniform.

� However, In Probabilistic models, demand is specified as a probability distribution.

� Uncertain demand raises the possibility of a stock out (or shortage).

Probabilistic Models

� One method of reducing stock outs is to hold extra inventory (called Safety Stock).

� In this case, we change the ROP formula to include that safety stock (ss).

Reorder Level

� Reorder Level (ROL) = LT x D

� Reorder Level (ROL) = (LT x D) + Safety Stock

Safety Stock

Safety Stock Example

� ROP = 50 units Stock-out cost = $40 per unit

� Orders per year = 6 Carrying cost = $5 per unit per year

Number of Units Probability

30 0.2

40 0.2

ROP 50 0.360 0.2

70 0.1

1.0

Safety Stock Example

� ROP = 50 units Stock-out cost = $40 per unit

� Orders per year = 6 Carrying cost = $5 per unit per year

A safety stock of 20 units gives the lowest total costROP = 50 + 20 = 70 units

Example

Probabilistic Demand

Reorder Point for a Service Level

Using the Standard Normal Probability Table

Using the Standard Normal Probability Table

=

Probabilistic DemandDemand is variable and lead time is constant

� Safety stock, SS:

= Z × standard deviation of lead time

= Z × σ × √LT

= Z × σdlt

� Reorder level:

ROL = lead time demand + safety stock

= LT × D + Z × σ × √LT

� where σ = standard deviation of demand per day and

� σdlt = σ × √LT Standard deviation of demand during lead time

Example- 3: Safety Stock

� Daily usage at a drug store follows a normal distribution with a mean of 500 gm and a standard deviation of 50 gm. If the lead time for procurement is 7 days and the drug store wants a risk of only 2% determine

a) reorder point and b) safety stock necessary

Example-3: Safety Stock� Mean daily demand, D =500 gm/day

� Lead Time, LT = 7days

� Standard deviation, σ = 50 gm/day

� Service level required = 98% or 0.98

� From normal distribution level Z is determined as z =2.05

ROL = (LT × D)+ z σ √LT

= (500 x 7)+ 2.05 * 50 * √7

= 3771 gm

� Safety Stock = z σ √LT

= 2.05 * 50 * √7 = 271 gm

Example: Safety Stock using Z-Score

� Mean Demand in lead period, µL =3500 gm

� Standard deviation, σ = 50 gm/day

� σL = σ √ Lt= 50 √7 gm

� Z= 2.05 from Table

where X is a normal random variable

� X=3771gm

� Safety stock = 3771 gm- 3500 gm =271 gm

σ

µXZ

−=

L

L


Inventory Control

Periodic Review System

Maximum Inventory Level, M

Periodic Review System

MActual Inventory Level, I

q = M - I

I

P-System: Periodic Review System

� In this system, costs are not explicitly considered and order quantity is not fixed.

� Time is taken into account and given more emphasis

� Inventory is periodically reviewed at fixed intervals and any difference between the present and the last review is made up by replenishment order.

� The order quantity is thus equal to replenishment level minus actual inventory on hand.

P-System: Periodic Review System-2

� In this system, we are interested in actual and average consumption over a period of time i.e. time between two reviews and lead time. Order quantity can be computed as follows:

If L< R then Q= M - I If L> R then Q= M – I - Q ord

Where

L= Lead Time R = Review Period

M= Replenishment Level in Units I = Inventory on hand in Units

Q =Quantity to be Ordered Qord= Quantity on order (in pipeline)

Example: Fixed Period Inventory Control System (P-System)

The average monthly consumption of an item is 40 units, Safety Stock is 20 units, review time is 1 month and lead time is 15 days, calculate replenishment level M

Safety Stock=B

R

Replen. Lvl. = M

LT

20

40

60

1 2 3

Example-Solution: P-System

L<R

Replenishment Level, M =

Safety Stock (B)+ consumption, D* (Review Time+ Lead Time)

M= 20+ 40(1+0.5) = 80 Units

Inventory on Hand, I = B + consumption/2

I = 20+ 40/2 = 40 units

The Order Quantity, Q = M – I

Q= 80- 40 = 40 Units

Example 2: (P-System)

� Consider a case where Lead time > review time

� Buffer/safety stock= 50 units D= 100 units/month

� Review Time= 1 month L= 2 months

� M= replinsh. Lvl. = B +D (1+2) = 50+ 100*3

M= 350 Units

� I = B+D/2 = 50+50 = 100 units

� Order Qty Q= M – I = 250 units

� If Qty already on order is 100 units (review after 1 mth)

� Q= M-I- Qord= 150 units

Fixed Order Vs. Periodic Review

� Fixed-order quantity models–when holding costs are high (usually expensive items or high deprecation rates), or when items are ordered from different sources.

� Fixed-time period models—when holding costs are low (i.e., associated with low-cost items, low-cost storage), or when several items are ordered from the same source (saves on order placement and delivery charges).

Fixed Order Vs. Periodic Review

� A fixed-order quantity system can operate with a perpetual count (keeping a running log of every time a unit is withdrawn or replaced) or through a simple two-bin or flag arrangement wherein a reorder is placed when the safety stock is reached

� The main disadvantage of a fixed-time period inventory system is that inventory levels must be higher to offer the same protection against stockout as a fixed-order quantity system.

� It also requires a periodic count and closer surveillance than a fixed-order quantity system.

Single-Period Inventory Model

Decision under uncertainity & risk

In inventory control, sometimes management has to take risk under uncertainity, though wanting to keep the risk factor to a minimum.

In inventory control, sometimes management has to take risk under uncertainity, though wanting to keep the risk factor to a minimum.

• How many World Cup shirts to produce, when the shirts will be of little or no value after the Cup.

• How many suits to stock for Eid or Xmas season, profit margin is high but the leftover stock will probably be of no value

• How many World Cup shirts to produce, when the shirts will be of little or no value after the Cup.

• How many suits to stock for Eid or Xmas season, profit margin is high but the leftover stock will probably be of no value

Single-period inventory model Applies in these cases


This model states that we should stock up to the point where incremental gain (IG) is equal to incremental loss

(IL)

This model states that we should stock up to the point where incremental gain (IG) is equal to incremental loss

(IL)

IG is the profit per item times the probability of

selling ‘x’ items

IG= m. P(x)

IG is the profit per item times the probability of

selling ‘x’ items

IG= m. P(x)

IL is the cost per item times

the probability that ‘x’ items will not be sold

IL= C. [1-P(x)].

Equating IG& IL and solving the equation we get:

IL is the cost per item times

the probability that ‘x’ items will not be sold

IL= C. [1-P(x)].

Equating IG& IL and solving the equation we get:

m= margin of profit itemP (x)= probability of selling the itemC= Cost of the item

m= margin of profit itemP (x)= probability of selling the itemC= Cost of the item

P(x) = Cm+C

P(x) = Cm+C

Single Period Model Example-4

� Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. How many shirts should we make for the game?

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

Z.667 = .432

therefore we need 2,400 + .432(350) = 2,551 shirts


uo

u

CC

CP

+

≤

sold be unit will y that theProbabilit

estimatedunder demand ofunit per Cost C

estimatedover demand ofunit per Cost C

:Where

u

o

=

=

=

P

This model states that we should continue to increase the size of the inventory so long as the probability of

selling the last unit added is equal to or greater than the ratio of: Cu/Co+Cu

This model states that we should continue to increase the size of the inventory so long as the probability of

selling the last unit added is equal to or greater than the ratio of: Cu/Co+Cu

Example-5 (Solution)

uo

u

CC

CP

+

≤

sold be unit will y that theProbabilit

estimatedunder demand ofunit per Cost C

estimatedover demand ofunit per Cost C

:Where

u

o

=

=

=

P

Co = Rs 1.5 [(Cost) Loss if demand is overestimated]

Cu = Rs 2.5 [(Cost) Profit Loss if demand is underestimated]

P ≤ [2.5/(2.5+1.5)]

P ≤ 0.625

Probability of meeting demand is 0.65 at 700 buns. The baker should make 700 buns.

Example-6� Ahmed Juices makes a variety of juices for on-the-

counter sales. Ahmed uses ice, which he grates in making these drinks. Ice is supplied to Ahmed in large blocks, each costing Rs 10. Ice blocks not used during a day gets wasted as the ice melts and cannot be used the next day. If Ahmed is short of ice blocks on any day, he buys them from elsewhere, but at a premium of Rs5 per block. Each block of ice can be used for 20 glasses of juice. The probability distribution for the demand of ice blocks is as follows

� What is the least cost stocking policy for Ahmed Juices?

� x ice blocks: 20 21 22 23 24 25 26 27 28

� p Probability 0 0.05 0.10 0.20 0.25 0.20 0.15 0.05 0

The end

Practice Numerical

Example

A plant makes monthly shipments of electric drills to a wholesaler in average lot sizes of 280 drills. The wholesaler’s average demand is 70 drills a week and the lead time from the plant is 3 weeks. The wholesaler must pay for the order the moment it leaves the plant.

If the wholesaler is willing to increase its purchase quantity to 350 units, the plant will guarantee a lead time of two weeks. What is effect on cycle and pipeline inventories?

A plant makes monthly shipments of electric drills to a wholesaler in average lot sizes of 280 drills. The wholesaler’s average demand is 70 drills a week and the lead time from the plant is 3 weeks. The wholesaler must pay for the order the moment it leaves the plant.

If the wholesaler is willing to increase its purchase quantity to 350 units, the plant will guarantee a lead time of two weeks. What is effect on cycle and pipeline inventories?

Example (Contd.)The present cycle and pipeline inventories are:

Cycle Inventory = Q/2 = 280/2 = 140 drills

Pipeline Inventory, dL= (70 drills/week)* (3 weeks) = 210 drills

The present cycle and pipeline inventories are:


Pipeline Inventory, dL= (70 drills/week)* (3 weeks) = 210 drills

Under the new offer, cycle and pipeline inventories are:


Pipeline Inventory, dL= (70 drills/week)* (2 weeks)

= 140 drills

Under the new offer, cycle and pipeline inventories are:


Pipeline Inventory, dL= (70 drills/week)* (2 weeks)

= 140 drills

Under the new offer, cycle inventory increases by 25% but pipeline inventories reduce by 33% (Decision Point)

Under the new offer, cycle inventory increases by 25% but pipeline inventories reduce by 33% (Decision Point)

Benefit of Better Inventory Control

� A firm's inventory turnover (IT) is 4 times on a cost of goods sold (COGS) of $800,000. Through better inventory control, inventory turnover is improved to 8 times while the COGS remains the same, a substantial amount of funds is released from inventory. What is the amount released?

$ 100,000 is released

Example -1

� Fleming sells distributor rebuild kits used on Ford V-8 engines. Fleming purchases these kits for $20 and sells about 250 kits a year. Each time Fleming places an order, it costs him $25 to cover paperwork. He estimated that the cost of holding a rebuild kit in inventory is about $3.5 per kit per year.

a) What is the economic order quantityb) How many times per year will Fleming place an order?

Example -1 (Contd.)

� Qopt= √ [2 S D/H]

� Qopt= √ [(2*25*250)/3.5]

= 59.75 round to 60 kits

� Orders per year =D/Qopt = 250/59.75

= 4.18

� S = Cost of placing order = $ 25 D= Annual demand = 250 units/year

� H= Annual per-unit carrying cost =$3.5 per kit/year

� Q = order quantity

EOQ Example (2) Problem Data

Annual Demand = 1,000 unitsDays per year considered in average

daily demand = 365Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $15

Given the information below, what are the EOQ and reorder point?

Given the information below, what are the EOQ and reorder point?

EOQ Example (2) Solution

Q = 2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or O PT 90 un its

d = 1,000 units / year

365 days / year = 2.74 units / day

Reorder po int, R = d L = 2.7 4units / d ay (7d ays) = 1 9.18 or _

20 u nits

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

EOQ Example (3) Problem Data

Annual Demand = 10,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = 10% of cost per unitLead time = 10 daysCost per unit = $15

Determine the economic order quantity and the reorder point given the following…

Determine the economic order quantity and the reorder point given the following…

EOQ Example (3) Solution

Q =2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 units

d =10,000 units / year

365 days / year= 27.397 units / day

R = d L = 27.397 units / day (10 days) = 273.97 or _

274 un its

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Example- 8

� Demand for Deskpro computer at Best Buy is 1000 units per month. Best Buy incurs a fixed order placement, transportation and receiving cost of $4000 each time an order is placed. Each computer costs Best Buy $500 and the retailer has a holding cost of 20%. Evaluate the number of computers that the store manager should order in each replenishment lot.

� Annual Demand, D = 1000 x12 = 12000 units

� Order cost per lot, S = $4000

� Unit Cost per computer, C =$500

� Holding cost per year as a fraction of the inv. Value, h = 0.2

Example-8 Solved

� Q opt = √ 2 * 12000 * 4000

0.2 * 500

= 980 units

Other InfoCycle Inventory = Qopt/2 = 980/2 = 490

No. of orders/year = D/Q = 12000/980 = 12.24

Annual ordering & holding costs =(D/Q)*S + (Q/2)hC

= $97,980

Average Flow time= Q/2D = 490/12000 = 0.041year

= 0.49 months

Example-9

� In the above example, the manager at Best Buy would like to reduce the lot size from 980 to 200. For this lot size to be optimal, the store manager wants to evaluate how much the order cost per lot should be reduced.

� Desired Qopt = 200 units

� Annual Demand, D = 1000 x12 = 12000 units

� New Order cost per lot, S = ?

� Unit Cost per computer, C =$500

� Holding cost per year as a fraction of the inv. Value, h = 0.2

Example-9 (Contd.)

� S = H [Qopt]2/2D

� H =hC= 0.2*500

� S = [0.2*500* 2002]/ [2*12000]

S = $166.7

THUS THE STORE MANAGER AT BEST BUY WOULD HAVE TO REDUCE THE ORDER COST PER LOT FROM $4000 TO $166.7 FOR A LOT SIZE OF 200 TO BE OPTIMAL

Problem-10

� The Acer Co. sells 10,000 units per year. The cost of placing one order is $50 and it costs $4 per year to carry one unit of inventory. What is Acer’s EOQ?

Chapter 5 part 1 inventory control

Documents

items inventory

constant inventory

costs of inventory overstock

demand anticipatory

pipeline inventory independent

objectives inventory

average inventory size

basic inventory control