Inventory_Management.pdf

Inventory Management

Ravindra S. Gokhale

1

Concepts of Inventory

Inventory is the stock of any item used in an organization

Inventory System - Set of policies and controls for:

• Monitoring levels of inventory

• For each item determine: When to order? AND How much

to order?

2

Types of Inventory In Manufacturing Systems

Raw material

Finished products

Component parts

Supplies

Work-in-process

3

Inventory – A Necessary Evil

Arguments in favor of higher inventory

• Higher customer service – To avoid ‘stockouts’

• Lower ordering cost – Minimize the time and money spent for ordering

• Better labor and equipment utilization – As a result of planned stable

production

• Lower transportation cost – As a result of better truckload utilization

• Reduce payments to suppliers – By taking advantage of quantity

discounts

4

Inventory – A Necessary Evil (cont…)

Arguments against higher inventory

• Higher inventory carrying costs

• Requirement of storage space

• Opportunity costs – The capital tied up in inventory can be used to

obtain finance for a more promising project

• Leads to ‘shrinkage’ – (a) pilferage/theft/deterioration

(b) obsolescence

5

Key Terms Associated with Inventory

• EOQ

• p-type of system

• q-type of system

• safety stock

• lead time

• service level

• re-order point

• target inventory level

• ABC analysis

• …

6

Different Costs Associated with Inventory

• Inventory holding (or carrying) cost

Includes costs for storage facilities, handling, insurance,

shrinkage, and opportunity costs

• Ordering cost

Incurred during purchasing of material and includes clerical

expenses (example: stock counting), preparing purchase

orders, tracking of orders

• Shortage cost

Includes cost of a lost order, dissatisfied customer, and

customer waiting costs

7

Independent and Dependent Demand

Independent demand

• Demands for various items are unrelated to each other

• Customer surveys and/or quantitative forecasting techniques are used

to determine their demand

• Since the demand is uncertain, certain amount of inventory has to be

carried

• Leads to the concept of ‘safety stock’

8

Independent and Dependent Demand (cont…)

Dependent demand

• Need for an item is a direct result of need for some other item (usually

a higher level item of which it is a part)

• A relatively straightforward computational concept

• Required quantity is simply computed from the number required in

each higher level item in which it is used

• Additionally, the quantity required for ‘spares’ also needs to be

determined

9

ABC Analysis

Technique of dividing items in three classes based on their sales

(usage) value

Helps the management focus on a few high value items

Three classes:

• A-class items: Approx 20% in numbers but approx 80% in value

• B-class items: Approx 30% in numbers but approx 15% in value

• C-class items: Approx 50% in numbers but approx 5% in value

10

ABC Analysis (cont…)

Procedure

• For each item: Sales value = Quantity x Sales value per piece

• X-axis: Percentage for each item = (100)/(Total number of items)

• Y-axis: Cumulative sales value (either absolute or percentage)

• Based on the guidelines, classify the items as A-class, B-class and C-

class

11

ABC Analysis – Numerical Example

An electronics and computer stores has the data given below on the

average monthly sales. Carry out an ABC analysis for the same.

12

Item Description Unit cost (Rs.) Monthly sales

Television sets 9,000 90

Monitors 8,000 15

Home theatre system 45,000 20

Refrigerators 12,000 180

Laptops 70,000 40

DVD player 5,000 50

Computer hard disks 5,000 10

Blank DVD Box 200 3000

Cameras 4,000 15

Movie CDs 100 1000

ABC Analysis (cont…)

Decisions based on the ABC analysis

• Tight control and frequent review for A-class items

• C-class items have a lower inventory carrying cost and hence

comparatively loose control is tolerated

13

Types of Inventory Models / Systems

Single period inventory model

• Classical example: “Newsboy problem”

Multi period inventory models

• Basic EOQ model and its variants

• Model with quantity discounts

• Fixed order quantity model with safety stock

• Fixed time period model with safety stock

• Hybrid systems

Optional replenishment system

Base stock system

14

Multi Period Inventory Models

Designed to ensure that items will be available on an ongoing

basis throughout the year

Items are usually ordered multiple times throughout the year

• System dictates the actual quantity ordered and the timing of the

order

Principally two kinds of models

• Fixed order quantity – “Event triggered”

• Periodic order quantity – “Time triggered”

15

Multi Period Inventory Models (cont…)

• Fixed order quantity – “Event triggered”

Counting of inventory is perpetual

Generally has a lower average inventory

Most suitable for important i.e. A-class items

• Fixed time period – “Time triggered”

Counting of inventory is only at the review period

Generally has a higher average inventory

Most suitable for less important i.e. C-class items

16

The Concept of EOQ

EOQ = Economic order quantity

The optimum lot size that minimizes the total annual inventory

costs

17

The Concept of EOQ (cont…)

Assumptions:

• Demand rate constant and deterministic

• No constraints on the size of the lot (example: infinite truck capacity)

• Only two relevant inventory costs: ‘ordering cost’ and ‘carrying cost’

• Decisions made are independently for each item (i.e. no clubbing of items)

• No uncertainty in lead time or supply

18

Basic EOQ Model

• Assumptions: (in addition to the five assumptions discussed previously)

Zero lead time and infinite replenishment rate

No shortages/backorders allowed

• EOQ = Q* =

(D = Annual demand, Co = Ordering cost, Cc = Carrying cost per unit

per year)

• Cycle time = t = Q*/D; Number of orders = 1/t = D/Q*

• Average inventory = Q*/2

• Optimum inventory cost = DCo/Q* + CcQ*/2

19

c

o

C

2DC

This model can be associated with both – ‘fixed order quantity’ and ‘fixed time period’

Numerical Example - Basic EOQ Model

For an item X, the data for inventory is as follows:

• Annual demand: 3000 units,

• Ordering cost: Rs. 200,

• Inventory carrying cost: Rs. 30 per unit per year

Based on this information, determine the following:

Economic order quantity (Q*).

The number of orders per year and the time period between orders.

The average inventory level.

The optimum inventory cost.

20

EOQ Model with Uniform Replenishment (EPQ)

One assumption from the basic EOQ model is relaxed, i.e. now the

replenishment is not instantaneous, but uniform (like a steady production)

EOQ = Q* =

(P = Replenishment or Production Rate)

Cycle time = t = Q*/D; Number of orders (or setups) = 1/t = D/Q*

Maximum Inventory Level = M* =

Average inventory = M*/2

Length of production cycle = Q*/P

Optimum inventory cost =

DCo/Q* + (Cc x Average inventory)

21

D P

P

C

2DC

c

o

P

D1Q*

EOQ Model with Uniform Replenishment (EPQ)

Example:

• Data

Demand: 1250 units per month

Annual production rate: 25000 units

Inventory carrying cost: Re. 1 per unit per week

Setup cost: Rs. 500

• Calculate:

Economic lot size

Maximum inventory level

Average inventory level

Length of time to produce a lot

Length of inventory cycle

Length of time to deplete the maximum inventory

Total annual cost

22

EOQ Model with Shortages

One assumption from the basic EOQ model is relaxed, i.e. now backorders are

allowed and filled immediately after the material is available

EOQ = Q* =

(Cs = Shortage cost per unit per year)

Cycle time = t = Q*/D; Number of orders = 1/t = D/Q*

Maximum Shortage = S* =

Maximum inventory = M* = Q* - S*

23

s

cs

c

o

C

CC

C

2DC

cs

C

CC

CQ*

EOQ Model with Shortages (cont…)

Average (positive) inventory = (Q* - S*)/2

Proportion of time with positive inventory = (Q* - S*)/ Q*

Average (positive) shortage = S*/2

Proportion of time with positive shortage = S*/ Q*

Optimum inventory cost =

DCo/Q* + (Cc x Average inventory x proportion of time with positive inventory)

+ (Cs x Average shortage x proportion of time with positive shortage)

= DCo/Q* + Cc x (Q* - S*)2/2Q* + Cs x (S*)2/2Q*

24

EOQ Model with Planned Shortages (cont…)

Example:

• Data

Demand: 100000 units per year

Inventory carrying cost: Rs. 50 per unit per year

Backorder cost: Re. 15 per unit per year

Ordering cost: Rs. 750

• Calculate:

Economic order quantity

Optimal shortage

Number of orders per year

Length of inventory cycle

Total annual cost

25

Quantity Discounts

In practice, there are some slabs of purchase price of a product depending on

the quantity

• Called as ‘price breaks’ or ‘quantity discounts’

• Higher the quantity, lower the price per unit

In such case, the optimum inventory cost should also consider the material cost

Two types: All-Units Discount and Incremental Quantity Discount

26

All-Units Discounts

Step 1: Beginning with the lowest price, calculate the EOQ for each price level

until a feasible EOQ is found.

(What is a ‘feasible’ EOQ?)

Step 2: Comparing

• If the first ‘feasible’ EOQ is found at the lowest price level, this quantity is the best

lot size

• Else, calculate the total cost for first feasible EOQ and for larger price break

quantity at each lower price level

The quantity with lowest cost is the optimum lot size

Total Cost = (Material cost) + (Inventory cost)

= (unit cost x D) + (DCo/Q* + CcQ

*/2)

27

Procedure for determining the optimum lot size

Numerical Example – All Units Discounts

A small office consumes a can of packaged drinking water every day, 365 days a year.

Fortunately, a local distributor offers all-units quantity discounts for large orders as

shown in the table below, where the price for each category applies to every can

purchased.

Discount category Quantity purchased Price (per can)

1 1 to 60 Rs. 80

2 61 to 120 Rs. 77

3 121 or more Rs. 75

The distributor charges Rs. 100 per order for delivery, regardless of the size of the order.

The inventory carrying cost is based on an interest rate of 8% per annum.

Determine the optimal order quantity. What is the resulting total cost per year?

With this order quantity, how many orders need to be placed per year? What is the

time interval between orders?

28

Incremental Quantity Discounts

Step 1: Determine an algebraic expression for cost corresponding to each price

interval, and subsequently use that to determine the cost per unit in each

interval.

Step 2: Substitute the cost per unit expressions, into total cost expressions and

compute the optimum value of Q. This is to be done for all the intervals.

Step 3: Select only those minima that are feasible

Step 4: Compute the total cost for each feasible mimimum and select the

quantity corresponding to the total minimum cost.

29

Procedure for determining the optimum lot size

Numerical Example – Incremental Quantity Discounts

A small office consumes a can of packaged drinking water every day, 365 days a year.

Fortunately, a local distributor offers incremental quantity discounts for large orders as

shown in the table below, where the price for each category applies to every can

purchased.

Discount category Quantity purchased Price (per can)

1 1 to 60 Rs. 80

2 61 to 120 Rs. 77

3 121 or more Rs. 75

The distributor charges Rs. 100 per order for delivery, regardless of the size of the order.

The inventory carrying cost is based on an interest rate of 8% per annum.

Determine the optimal order quantity. What is the resulting total cost per year?

With this order quantity, how many orders need to be placed per year? What is the

time interval between orders?

30

Concept of ‘Safety Stock’ and ‘Service Level’

EOQ models assume: “Demand rate is constant and deterministic”

• Not realistic in actual practice

Demand varies from period-to-period

Safety stock: Protection against “stock out” situations

31


(cont…)

Probability approach

• Assume that demand over a period of time is ‘normally distributed’

with mean ‘μ’ and standard deviation ‘σ’ i.e. N ~ (μ, σ2)

• Considers only the probability of ‘running out of stock’ and not ‘how

many units short’

Service Level: Probability of not running out of stock

32

If quantity ordered = μ, then

• Safety stock = 0

• Service level = 50%

The ‘risk period’ = time interval in which one can run out-of-stock

Relation between ‘safety stock’ and ‘service level’

(safety stock) = z x σRisk Period

33


(cont…)

z-value from the standard normal

distribution corresponding to the

required ‘service level’

standard deviation of the demand

during the ‘risk period’

Variation in demand and lead time – four cases

Case 1: Both demand and lead time are constant

Straightforward case, similar to EOQ, but with some positive constant

lead time

Case 2: Demand varied, lead time constant

Case 3: Demand constant, lead time varies

Case 4: Both demand and lead time vary

34

Fixed Order Quantity Model (General)

Ordering quantity (at the time of each order) is fixed

Time period between the orders may vary depending on the

demand rate

The ‘re-order point’ (ROP) is fixed.

When to order?

• What should be the re-order point?

How much to order?

• What should be the fixed ordering quantity?

35

Fixed Order Quantity Model with Safety Stock

Place a new order when the stock reaches Re-order Point

• Order quantity is same as the EOQ, only ordering time changed

What is the ‘risk period’ here?

• In this case: (risk period) = (lead time) [How???]

36

Fixed Order Quantity Model with Safety Stock

(cont…)

When to order?

• Re-order point should be such that it includes estimated demand during

the risk period plus probability of stock-outs during the risk period

Re-order point =

(Average demand over the ‘risk period’) + (Safety stock)

i.e. (Average demand over the ‘lead time’) + (Safety stock)

37

Fixed Order Quantity Model


Case 1: Both are constant - straightforward


• ROP = [average D x LT] + [z x std. dev. of D x (square root of of LT)]


• ROP = [D x (average LT)] + [z x D x (std. dev. of LT)]

Case 4: Both vary

• ROP = [(average D) x (average LT)] +

[z x square root of {(average LT) x Var D + (average D)2 x Var LT}]

38

Numerical Example - Fixed Order Quantity Model with

Safety Stock

One of the largest selling items in a home appliances store is a new model

of refrigerator that is highly energy-efficient. On an average, 40 of these

refrigerators are being sold per month (that is, 1.33 refrigerators per day)

and the demand pattern follows a normal distribution. The variance of the

daily demand is 4. It takes one calendar week for the store to obtain more

refrigerators from a wholesaler. The administrative cost of placing each

order is Rs. 100. For each refrigerator, the holding cost per month is Rs.

20. The store’s inventory manager has decided to use continuous-review

model with a service level of 0.8 (that is, 80%).

Determine the order quantity, re-order point and safety stock.

What will be the average number of stock outs per year with this

inventory policy?

39

Expected number of units short per cycle:

• E(n) = E(z) x std. dev. of demand during lead time

[Note: E(z) is obtained from the normalized table]

Expected annual shortage

• E(N) = E(n) x (D/Q)

Expected annual service level

• SLannual = 1 – [E(z) x std. dev. of demand during lead time / Q]

40

Fixed Order Quantity Model

Relation between service level and shortages

Fixed Time Period Model (General)

Ordering quantity (at the time of each order) may vary depending

on the demand rate

Time period between the orders is fixed (i.e. constant)

The ‘target inventory level’ (also called as ‘order up to level’) is

constant

No concept of Re-order Point

When to order?

• What should be the fixed time period between orders?

How much to order?

• What should be the target inventory level?

41

Fixed Time Period Model with Safety Stock

The review period is equal to the time between orders that is

obtained by considering the model as EOQ model.

What is the ‘risk period’ here?

• In this case: (risk period) = (review period) + (lead time) [[How???]

42

Fixed Time Period Model with Safety Stock

(cont…)

How much to order?

• Quantity ordered should be such that it includes estimated demand during

the risk period plus probability of stock-outs during the risk period minus

the current level of inventory

(Quantity ordered) = (Target Inventory Level) – (Current Inventory)

= (Average demand over risk period) + (Safety stock) – (Current Inventory)

43

Fixed Time Period Model


Case 1: Both are constant - straightforward


• TIL = [average D x (T + LT)] + [z x std. dev. of D x (square root of {T

+ LT})]


• TIL = [D x (T + average LT)] + [z x D x (std. dev. of LT)]

Case 4: Both vary

• TIL = [(average D) x {T + (average LT)}] +

[z x square root of {(T + average LT) x Var D + (average D)2 x Var LT}]

44 Note: TIL = Target Inventory Level, T = Review Period, LT = Lead Time

Numerical Example - Fixed Time Period Model with

Safety Stock

One of the largest selling items in a home appliances store is a new model

of refrigerator that is highly energy-efficient. On an average, 40 of these

refrigerators are being sold per month (that is, 1.33 refrigerators per day)

and the demand pattern follows a normal distribution. The variance of the

daily demand is 4. It takes one calendar week for the store to obtain more

refrigerators from a wholesaler. The administrative cost of placing each

order is Rs. 100. For each refrigerator, the holding cost per month is Rs.

20. The store’s inventory manager has decided to use the periodic review

model (with a review period equal to that obtained from an ideal EOQ

model) with a service level of 0.8 (that is, 80%).

What will be the review period? What is the risk period in this case?

What is the safety stock and the corresponding target inventory level?

45

Inventory Control Systems – P and Q

46

A comparison

Parameter Fixed Order Quantity

System

(Q – System)

Fixed Time Period

System

(P – System)

Time between order Varies Constant

Quantity ordered Constant Varies

Risk period Lower Higher

Safety stock required Lower Higher

Monitoring Continuous Periodic (not continuous)

Operating costs Higher Lower

Advantages • Lowe inventory

carrying cost

• Ease of operation

• Combine multiple

orders

Recommended for • A-class items • C-class items

Hybrid Systems

Two types:

• Optional replenishment system

• Base stock model

Optional Replenishment System (‘s-S system’ or ‘min-max system’)

• Similar to the fixed order period model

• If inventory has dropped below a prescribed level (similar to the re-order

point) at the review time

An order is placed

Otherwise, no order is placed

• Protects against placing very small orders

• Attractive when review and ordering costs are both large

47

Hybrid Systems (cont…)

Base stock model

• Start with a certain inventory level

• Whenever a withdrawal is made

An order of equal size is placed

• Ensures that inventory maintained at an approximately constant level

• Appropriate for very expensive items that are fast moving but with small

ordering costs

48

Single Period Inventory Model

Decision has to be taken only for a single period: How much to

order?

• Assume: No on-hand inventory

• Assume: Demand distribution is known

Let,

• c = Cost of purchasing each unit

• p = Selling price per unit

• h = Salvage value of each unit (may be positive, zero, or negative)

Therefore,

• Cost of ‘under ordering’ = cu = p – c = Loss of opportunity

• Cost of ‘over ordering’ = co = c - h

49

Single Period Inventory Model (cont…)

P(Demand < Stock) <= (cu)/(cu + co)

This is called as the critical fractile

One must order as much as the ‘critical fractile’ to achieve a trade

off between ‘under ordering’ and ‘over ordering’

50

Single Period Inventory Model (cont…)

Typical applications

• Perishable products (example: newspaper, magazines)

• One time event (example: selling T-shirts for Finals of a tournament)

• Service industry where cancellations are allowed (example: airline

tickets, hotel bookings)

51

Single Period Inventory Model –

Numerical Example 1

A newspaper vendor sells daily Financial Times. He purchases

copies of this newspaper from its distributor at the beginning of

each day for Rs.2.50 per copy, sells it for Rs.4 each, and then

receives a refund of Rs.1 from the distributor the next morning for

each unsold copy. The number of requests for this newspaper

range from 15 to 18 copies per day. The newspaper vendor

estimates that there are 15 requests on 40 percent of the days, 16

requests on 20 percent of the days, 17 requests on 30 percent of

the days, and 18 requests on the remaining days. What is the

optimal ordering quantity?

52


Numerical Example 1a

If the newspaper vendor from Numerical Example 1 wants to

achieve a service level of 90%, what should be his/her ordering

quantity?

53


Numerical Example 1b

If the newspaper vendor from Numerical Example 1 suppose has cu

= 11 and co = 9, then verify that critical fractile is 0.45 and also

verify that the optimum ordering quantity is 16 (and not 15). For

verification of optimum ordering quantity, you need to calculate the

Expected Cost for each demand quantity.

54


Numerical Example 2

A wholesaler stocks special high-quality kites for selling them to small size

shops, every year around December. The season lasts for approximately 2

months. Each kite sold by the wholesaler yields him a profit of Rs. 4. At the

end of the season, the wholesaler has to dispose off all the kites for making

room for other goods. There is no salvage value for the unsold kites, and

in fact the wholesaler has to spend Re. 1 per kite to dispose it off properly.

Assume that purchase cost of the price for the wholesaler is Rs. 6 per kite.

Years of data has shown that the demand for kites during this season for

that region, follows a normal distribution with a mean of 10000 and a

variance 6000. What is the best stocking quantity for the wholesaler at the

beginning of the season?

55


Numerical Example 2a

If the wholesaler from Numerical Example 2 wants to achieve a

service level of 95%, what should be his ordering quantity?

56


Numerical Example 3

The management of Quality Airlines has an over booking policy, since the cancellations

(i.e. ‘no-shows’) are common in this industry. The policy now needs to be applied to a

new flight from Indore to Raipur. The airplane has 125 seats available for a fare of Rs.

2500 each. However, since there commonly are a few no-shows, the airline should

accept a few more than 125 reservations. On those occasions when more than 125

people arrive to take the flight, the airline will find volunteers who are willing to

sacrifice their journey plan in return for being given points worth Rs. 2000 (in addition

to giving full refund of any booking amount, if collected) toward any future travel on

this airline. Based on previous experience with similar flights, it is estimated that the

relative frequency(proportion of the number of no-shows will be as shown below.

57

# of no shows 0 1 2 3 4 5 6 7 8

Proportion 5% 10% 15% 15% 15% 15% 10% 10% 5%

How many overbooking reservations should Quality Airlines accept for this flight?


Numerical Example 4

A social club organizes an annual dinner party for its members. The organizers give

food and beverage orders to a local caterer. The caterer charges on basis of per plate

(that is, per person). This year, the rate per plate has been decided as Rs. 500/-. The

organizers need to inform the caterer in advance, the number of people expected. If

the actual number of people attending the dinner exceeds the expected number, then

the caterer can still provide food for them at a very short notice, but charges Rs. 150

extra per plate. At the end of the party, if the actual number of people attending is

less than the expected number, the caterer gives a discount of Rs.100/- per plate only

for the number of people not attended. Based on the data available with the

organizers, the distribution of the number of people attending the dinner is given as

follows:

58

# of people attending 120 125 130 135 140 145 150

Probability 5% 10% 10% 15% 20% 25% 15%

Determine how many plates should the organizers order?

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