1.0 INTRODUCTION
A convenience point to start our discussion in this note is to
provide an answer to the question: what
is an inventory? An inventory is a stock or store of goods. Firms typically stock hundreds or even
thousands of items in inventory, ranging
from small things such as pencils, paper chips to large items such as machines and trucks. Naturally, many of the
items a firm carries in inventory relate
to the kind of business it engages in.
Thus, manufacturing firms carry
supplies of raw materials, purchased parts, partially completed items,
and finished goods, as well as spare
parts for machines, tools and other supplies.
Hospitals stock drugs, surgical supplies, life monitoring equipment etc; supermarket stock fresh and canned foods,
frozen foods etc. To test your understanding
of inventory, try to identify the different types of inventories carried in the following organizations:
Banks, Laboratory, clothing store and
petrol station.
2.0 OBJECTIVES
After completing this note you should be able to:
1) Define the term inventory and list the major reasons for
holding inventories.
2) Contrast independent and dependent demand
3) List the main requirement for effective inventory management
4) Discuss period and
perpetual review system.
5) Describe the A. B. C approach and explain how it is useful
6) Discuss the objectives of inventory management
7) Describe the basic EOQ model and its assumptions and solve
typical problems.
8) Describe the economic
run size model and solve typical problems.
9) Describe the quantity discount model and solve typical
problems.
10) Describe reorder point models and solve typical problems.
11) Describe situation in which the single period model would
be appropriate.
12) Solve typical problems that involve shortage costs and excess
costs.
3.0 MAIN CONTENT
3.1 Purpose of Inventories
To understand why firms have inventories at all, you need to know
something about the various functions of
inventory. Inventories serve a number of
functions. Among the most important are the following:
1. To meet anticipated demand or planned demand.
2. To smooth production requirements – This is true for firms
that experience seasonal patterns in
demand often build up inventories during
off-season periods to meet overly high requirements during certain seasonal periods. For example,
poultry farmers keep inventory of birds
until festival periods when they will be sold. Can you think of examples of firms that keep seasonal
inventories?.
3. To decouple components of the production distribution system
– manufacturing firms have used
inventories as buffers between
successive operations to maintain continuity of production that would otherwise be disrupted by events such as
breakdown of equipment and accidents
that cause a portion of the operation to shut down temporarily. The buffers will permit other operations to
continue temporarily while the problem
is resolved. Similarly, firms can use buffers of raw materials to insulate production from
disruptions in deliveries from
suppliers, and finished goods inventory to buffer sales operations
from manufacturing disruptions.
4. To protect against stock-outs, that is, one can reduce the risk
of shortages – resulting, for example,
from delays due to weather condition –
by holding safety stocks, which are stocks in excess of anticipated demand. Can you identify possible causes of
shortages in raw materials; work in
process and finished goods?
5. To allow economic production and purchase or to take advantage
of order cycles. To minimize purchasing
and inventory costs, a firm can buy in
quantities that exceed immediate requirements. This necessitates storing some or all of the purchased amount
for later use. Similarly, it is usually
economical to produce in large rather than small quantities. Again, the excess output must be stored for
later use. Thus inventory storage
enables a firm to buy and produce in economic lot sizes without having to try to match purchases or
production with demand requirements in
the short run. This results in periodic orders, or order cycles. The resulting stock is known as cycle
stock. You have to know that economic
lot sizes are not the only cause of order cycles. In some instances, it is practical or economical to
group orders and/or to order at fixed
intervals.
6. To hedge against price increases or to take advantage of
quantity discounts. Occasionally, a firm
can suspect that a substantial price
increase is about to be made and therefore purchase larger-than
normal amounts to avoid the increase.
The ability to store extra goods also
allows a firm to take advantage of price discounts for large
orders.
7. To permit operations. The fact that production operations take
a certain amount of time (i.e. they are
not instantaneous) means that there will
generally be some work-in-progress inventory. In addition,
intermediate stocking of goods –
including raw materials, semi-finished items and finished goods at production sites, as well
as goods stored in ware houses, - leads
to pipeline inventories throughout a production – distribution system. As a follow up to
question asked in section 1: What
functions do those inventories identified perform?
3.2 Inventory Cost Structures
One of the most important prerequisites for effective inventory
management is an understanding of the
cost structure. Inventory cost structures incorporate the following four types of costs:
3.2.1 Item cost
This is the cost of buying or producing the individual inventory
items. The item cost is usually
expressed as a cost per note multiplied by the quantity procured or produced. Sometimes item cost is
discounted if enough notes are purchased
at one time.
3.2.2 Ordering (or set
up) costs
These are costs of ordering and receiving inventory. They include
typing purchase order, expediting the
order, transportation costs, receiving costs, and so on. Ordering costs are generally expressed
in fixed Naira per ordering regardless
of order size. When a firm produces its own inventory instead of ordering it from a supplier, the costs of
machine setup (e.g., preparing equipment
for the job by adjusting the machine, changing cutting tools) are analogous to ordering costs; they are
expressed as a fixed charge per run
regardless of the size of the run.
3.2.3 Carrying (or holding) cost
This is associated with physically having items in storage for a
period of time. Holding costs are stated
in either of two ways: as a percentage of note price, for example, a 15 percent annual holding cost
means that it will cost 15 kobo to hold
N1 of inventory for a year or in Naira per note. The carrying cost usually consists of three
components:
3.2.3.1 Cost of capital
When items are carried in inventory, the capital invested is not
available for other purposes. This
represents a cost of foregone opportunities for other investments, which is assigned to inventory
as an opportunity cost.
3.2.3.2 Cost of storage
This includes variable space cost, insurance, and taxes. In some
cases, a part of the storage cost is
fixed, for example, when a ware house is owned and cannot be used for other purpose. Such fixed costs
should not be included in the cost of
inventory storage. Similarly, taxes and insurance should be included only
if they vary with inventory levels
3.2.3.3 Costs of obsolescence, deterioration, and loss
Obsolescence costs should be assigned to items which have a high
risk of becoming obsolete; the higher
the risk, the higher the costs. Perishable
products such as fresh seafood, meat and poultry and blood should be
charged with deterioration costs when
the item deteriorates over time. The costs of loss include pilferage and breakage costs
associated with holding items in
inventory.
For example, items that are easily concealed (e.g. pocket
cameras, transistor radios, calculators)
or fairly expensive (e.g. cars TVs) are prone to theft.
Stock out or shortage costs result when demand exceeds the supply
of inventory on hand. These costs can
include the sale lost because material is not
on hand, loss of customer goodwill due to delay in delivery of order,
late charges and similar costs. Also, if
the shortage occurs in an item carried for
internal use (e.g. to supply and assembly line), the cost of lost
production or downtime is considered a
shortage cost. Shortage costs are usually difficult to measure, and they are often subjectively
estimated. Estimates can be based on the
concept of foregone profits.
3.3 Independent versus Dependent Demand
A crucial distinction in inventory management is whether demand
is independent or dependent. Dependent
demand items are typically subassemblies
or component parts that will be used in the production of a final or finished product.
Demand (i.e. usage) of subassemblies and component parts is derived from the number of finished
notes that will be produced. A classic
example of this is demand for wheels for new cars. If each car is to have five wheels, then the total number of
wheels required for a production run is
simply a function of the number of cars that are to be produced in that
run. For example, 200 cars would require
200 x 5 = 1,000 wheels. Independent
demand items are the finished goods or end items. Generally these items are sold or at least shipped out
rather than being used in making another
product.
This demand includes an element of randomness. The nature of demand leads to two different
philosophies of inventory management. A
replenishment philosophy, that is, as the stock is used, an order is triggered for more material and
inventory is replenished. A requirements
philosophy, that is, as one stock begins to run out. More materials or ordered only as required by the
need for other higher-level or end
items. The sections that follow
focus on independent demand items.
3.4 Requirements for Effective Inventory Management
Management has two basic functions concerning inventory. One is to
establish a system of keeping track of
items in inventory and other is to make decision about how much and when to order. To be
effective management must have the
following:
1. A system to keep track of the inventory on hand and on
order.
2. A reliable forecast of demand that includes an indication of
possible forecast error.
3. Knowledge of lead times and head time and lead time
variability.
4. Reasonable estimates of
inventory holding costs, ordering costs
and shortage costs.
5. A classification system for inventory items. Let’s take a close look at each of these
requirements.
3.4.1 Inventory Counting Systems
Inventory counting system can be periodic or perpetual. Under a
periodic system, a physical count of
items in inventory is made at periodic intervals (e.g., weekly, monthly) in order to know how
much to order of each item. An advantage
of this type of system is that orders for many items occur at the same time, which can result in economies in
processing and shipping orders. There
are also several disadvantages of periodic reviews. One is a lack of
control between reviews. Another is the
need to protect against shortages between
review periods by carrying extra stock. A third disadvantages is the
need to make a decision on order
quantities at each review.
A perpetual inventory system (also known as a continual system)
keeps tracks of removal from inventory
on a continuous basis, so when the system can
provide information on the current level of inventory for each item,
when the amount on hand reaches a pre determined
minimum a fixed quantity, Q, is ordered.
The advantages of this system include;
(i) Continuous monitoring of inventory withdrawals.
(ii) Fixed order quantity that makes it possible for management to
identify an economic order size (discuss
in detail later in the note). The
disadvantages include added cost of record keeping and also a
physical count shall be performed. Bank transactions such as customer deposit
and withdrawals are examples of continuous
recording of inventory changes. An example of perpetual system is in two- bin system that uses two containers
of inventory; reorder is done when the
first is empty. It does not demand record of withdrawal.
Perpetual system can be
batch or on line. In batch system inventory records are collected periodically and entered into the
system. In on-line system the transactions
are recorded instantaneously. The advantage of latter over the former is that they are always up to
date.
3.4.2 Demand Forecasts and Lead Time Information
Since inventories are used to satisfy demand requirement it is
essential to;
(i) have reliable estimates of the amount and timing of
demand
(ii) know how to long it will take for orders to be delivered
(iii) know the extent to which demand and lead time (the time
between submitting an order and
receiving it) might vary.
3.4.3 Classification System
Since items held in
inventory are not of equal importance in terms of naira invested, profit potential, sales, or usage
volume or stock out penalties. They must
be classified in order of their importance to the business. One way you can do this is to employ A- B- C approach
which classifies inventory items
according to some measures of importance, usually annual naira usage
(i.e. naira value per note multiplied by
annual usage rate) and then allocates control
efforts accordingly. Here, A is used for very important items, B for
moderately important and C for least
important. A items generally account for about 15 percent to 20 percent of the items in
inventory but 60 percent to 70 percent of
the naira usage. While C items might account for about 60 percent of
the number of items only abort 10
percent of the items of the naira usage of an
inventory. In most instances A items account for large share of the
value or cost associated with an
inventory; and they should receive a relatively greater share of control efforts. The C items should
receive only loose control and B items
should have controls that lie between the two extremes.
The A. B. C concept is used
by managers in many different settings to
improve operations. For example in customer service, a manager can
focus attention on the most important
aspects of customer service as very important,
or of only minor importance. This is to ensure that he does not
overemphasize minor aspect of customer
service at the expense of major aspects.
A-B- C. concept can also be used as a guide to cycle counting,
which is a physical count of items in
inventory. The purpose of cycle counting is to reduce discrepancies between the amounts indicated
by inventory records and the actual
quantities of inventory on hand. Using A- B- C. concept let us attempt to classify the inventory items contained in the
following table as A, B, or C based on
annual naira value. Item Annual Note
Annual Naira Demand Cost Value
When you look at the
information contained in the table carefully, we can say that the first two items have a relatively
high annual naira value so it seems
reasonable to classify them as A items. The next four items appear to
have moderate annual naira values and should
be classified as B items. The remainders
are C items, based on their low naira value. The key questions concerning cycle counting for management are:
1. How much accuracy is needed
2. When should cycle counting be performed
3. Who should do it?
The American Production and Inventory Control Society (APICS)
recommends the following guideline for
inventory record accuracy ± 0.2 percent for A
items, 1 percent for B items and± 5 percent for C items.
On when cycle counted be
performed, you can decide to do it on periodic
(scheduled) basis or certain events may trigger you do it on a
periodic (scheduled) basis.
An-out-of-stock report written on an item indicated by inventory records to be in stock, an
inventory report that indicate a low or zero
balance of an item and a specified level of activity (e.g. every 2000
notes sold.)
On who should do it, you may use regular stock room personnel
especially during period of slow
activity or give the contract to outside firms to do it on a periodic basis. The latter provides an
independent check on inventory and may
reduce the risk of problem created by dishonest employees.
3.5 Economic Order
Quantity Model The question of how
much to order is frequently determined by using economic order quantity (EOQ) models. EOQ models
identify the optimal order quantity in
terms of minimising order costs. These models can take the following
forms:
1. The economic order quantity model
2. The quantity discount model
3. The economic order quantity model with no instantaneous
delivery.
3.5.1 Basic Economic Order Quantity Model
This basic model assumes the followings:
1. Only one product is involved.
2. Annual demand requirements are known
3. Lead time do not vary
4. Each order is received in a single delivery
5. There are no quantity discount
6. Demand is spread evenly throughout the year so that the demand
rate is reasonably constant.
The exact amount to order will depend on the relative magnitudes
of carrying and ordering cost. Annual
carrying cost is computed by multiplying the average amount of inventory on hand by the cost to
carry one note for one year, even though
any given note would not be held for a year. The average inventory is simply half of the order quantity. Using the
symbol H to represent the average annual
carrying cost per note, the total annual carrying cost is Annual carrying
Q cost = H 2 ……………………. (1)
Annual ordering cost is a function of the number of orders per
year and the ordering cost per order Annual ordering Cost = DS Q
Where
S = ordering cost
D = annual demand
Q = order size
The equation shows that annual ordering cost varies inversely with
respect to order sizes.
The total cost associated with carrying and ordering inventory
when Q notes are ordered each time is
therefore:
TC = Annual carrying cost + Annual ordering cost = QH + DS 2 Q
Where
D = Demand, usually in notes per year
Q = Order quantity, in notes
S = Ordering cost in Naira
H = Carrying cost, usually in Naira per note per year.
If TC is differentiated with respect to Q and equated to zero, and
solving for Q, we will obtain the
expression which we use to determine optimum order quantity, Q0
………………………………… (3)
The minimum total cost is then found by substituting Q0 in total
cost formula. The length of an order
cycle is obtained by dividing optimum quantity (Q0) by annual demand (D).
To illustrate the use of expression (3), suppose a local
distributor for Michelin tyre expect to
sell approximately 9,600 steel-belted radial tires of a certain size and tread designs next year. Annual carrying
costs are N16 per time, and ordering
cost are N 75. The distributor operates 288 days a year
(a) What is the EOQ?
(b) How many times per year does the store reorder?
(c) What is the length of an order cycle?
To answer these question demands that you know the value of D, H
and S.
These are as follows
D= 9,600 tires per
year
H = N 16 per note per year
S = N 75
Having determined these values, answers to those questions are
thus:
(a) Qo = 300 tires
(b) Number of order per year
D/Q0 = 9.600 tires =32
300 tires
(c) (length of order cycle:
Q0/D = 300 tires =
9,600 tires
1/32 of a year, which is 1/32 x 288 or nine workdays.
Now, if your carrying costs are stated as a percentage of the
purchase price of an item rather then as
a naira amount per note, is (3) still appropriate to determine Q0, optimum order size? The answer
is yes as long as you can convert the
percentage in naira equivalent.
Let us illustrate this with
an example: suppose Tijani and Osot. Ltd assembled television sets. It purchases 3,600 black and
white picture tubes a year at N 65 each.
Ordering costs are N 31, and annual carrying costs are 20 percentage of the purchase price. Compute the optimal
quantity and the total annual cost of
ordering and carrying the inventory
Solution
D= 3,600 picture tubes per year
S= N 31
H= 20 (N65) = N13 (since this can be done, Q0 expression is therefore appropriate)
Q0= 2DS = 2 (3,600 (31) = 131
picture tubes
H 13
TC = carrying costs + ordering costs
= (Q0/2) H + (D/Q0) S
= (131/2) 13 + (3.600/13)31
= N852 + N852
= N1, 704
3.5.2 EOQ with Non instantaneous Replenishment
Recall the assumptions of the basic EOQ model discussed in the
last section, it as assumed that each
order is delivered at a single point in time. In some in time instances, however, such as when a firm
is both a producer and user or when
deliveries are spread over time, inventories tend to build up gradually instead of instantaneously. When a company makes the product itself there
are no ordering costs as such.
Nonetheless, with every run there are setup costs. Setup costs are
similar to ordering cost hence they are
treated in (3) in exactly the same way. In this
case, the number of runs is D/Qo and the annual setup cost is equal to
the number of runs per year times the
setup cost per run: (D/Qo)S
Total cost is TCmh =
carrying cost + setup cost
= (Imax) H + (D/Q0)S ----------------- (4)
Where
Imax = maximum inventory
The economic run quantity is
Where
P = production or delivery rate
U= usage rate
The maximum and average inventories are Imax= Q0 (P-U) and Iaverage = Imax p 2
The cycle time (the time between orders or between the beginning
of runs) for the economic run size is
dependent on the run size and use (demand) rate: Cycle time = Q0 U
Similarly, the run time (the production phase of the cycle) is
dependent on the run size and the
production rate: Run time = Q0 P Now
let us illustrate our discussion in this section with an example: A toy manufacturer uses 48,000 rubber wheels
per year for its popular dump truck
series. The firm makes its own wheels which it can produce at a rate of 800 per day. The toy trucks are assembled
uniformly over the entire year. Carrying
cost for a production run of wheel is 45. The firm operates 240 days per year. Determine each of the
following:
(a) optimal run size
(b) minimum total annual cost for carrying and setup
(c) cycle time for the optimal run size (d) run time
Solution
D= 48,000 wheels per year
S= N45 H= N 1 per wheel per
year
P= 800 wheels per day
U = 48,00 wheels per 240 days or 200 wheel per day 
Thus each run will require 3 days.
3.5.3 Quantity Discounts This section
discusses the third variant of EOQ model. This requires that the assumption of no quantity discounts is
relaxed. A convenient point to start our
discussion in this section is to understand what quantity discounts
mean. We would define quantity discounts
as a price reduction for large orders offered to customers to induce them to buy in large
quantities.
The buyer’s goal with discount is to select the order quantity
that will minimize total cost, which is
the sum of carrying cost, ordering cost, + purchasing cost:
TC = Carrying cost +
ordering cost of purchasing
= (Q) H + (Q)S + PD
2 D
Where
P = note price
Recall that in the basic EOQ model, determination of order size
does not involve the purchasing cost.
The rationale for not including note price is that under the assumption of no quantity discounts,
price per note is the same for all order
sizes.
There are two general cases of the model. In one, carrying costs
are constant (e.g. N20 per note) in the
other, carrying costs are stated as a percentage of purchase price (e.g. 20 percent of note
price).
The procedure for determining the overall EOQ differs slightly,
depending on which of these two cases is
relevant. For carrying cost that is constant, the procedure is as follows:
(1) Compute the common EOQ
(2) Only one of the note price will have the EOQ in its feasible
range since the ranges do not overlap.
Identify that range
(a) if the feasible EOQ is on the lowest price range, that is the
optimum order quantity.
(b) If the feasible EOQ is in any other range, compute the total
cost for the EOQ and for the price break
of all lower note cost. Compare the total
costs: the quantity (EOQ or the price break) that yield the lowest total
is the optimum order quantity.
3.5.4 When to Reorder with EOQ Ordering
EOQ models answer the question of how much to order but not the
question of when to order. The latter is
the function of models that identity the reorder point (ROP) in terms of a quantity: the
reorder point occur when the quantity on
hand drop to a predetermine amount. The amount generally includes expected demand during lead time and perhaps
an extra cushion of stock, which serves
to reduce the probability of experiencing a stock out during lead time. There are four determinants of the
reorder point quantity.
(1) The rate of demand (usually based on a forecast).
(2) The length of lead time.
(3) The extent of demand and/or lead time variability.
(4) The degree of stock-out risk acceptable to management.
If demand and lead time are both constant, the reorder point is
simply: ROP = D x LT
Where
D = demand per day or week
LT = lead time in days or weeks
Note: Demand and lead time must be in the same notes.
The following example illustrates this concept: Osot takes Two – a
Day vitamins, which are delivered to his
home by salesman seven days after an
order is called in. At what point should Osot telephone his order
in? Usage = 2 vitamins per day
Lead time = 2 days ROP
= Usage x lead time
= 2 vitamins per day x 7 days
= 14 vitamins
Thus, Osot should reorder when 14 vitamin tablets are left. Now
let us look at a scenario where demand
or lead time is not constant as earlier assumed. If this is the case, there is the possibility that
actual demand will exceed expected
demand. It therefore becomes necessary to carry additional inventory
called safety stock, to reduce the risk
of running out of inventory (a stock-out) during lead time. The reorder point then increased
by the amount of the safety stock.
ROP = Expected demand + safety stock during lead time. For example, if expected demand during lead
time is 100 units and the desire amount
of safety stock is 10 units the ROP would be 110 units.
Service Level: Because it cost money to hold safety stock, a
manager must carefully weigh the
cost of carrying safety stock against the reduction in stock – out risk it provides, since the service
level increases as the risk of stock-out
decreases. Order cycle service level can be defined as the probability
that demand will not exceed supply
during lead time (i.e., that amount of stock on
hand will be sufficient to meet demand) Hence a service level of 95
percent implies a probability of 95
percent that demand will not exceed supply during lead time.
An equivalent statement that demand will be satisfied in 95 percent of
such instance does not mean that as percent
of demand will be satisfied. The risk of
a stock out is the compliment of service level; a customer service level
of 95 percent implies a stock-out risk
of 5 percent. That is service level = 100
percent – stock-out risk. Later you will see how the order cycle service
level relates to annual service
level. The amount of safety stock that
is appropriate for a given situation depends on
the following factors:
(1) The average demand rate & average lead time.
(2) Demand and lead time variability.
(3) The desire service level.
For a given order cycle, service level the greater the variability
in either demand rate or lead time, the
greater the amount of safety stock that will be
needed to achieve that service level. Similarly, for a given amount of
variation in demand rate or head time,
achieving an increase in the service level will
require increasing the amount of safety stock. Selection of a service
level may reflect stock out costs (e.g.
lost sales, customer dissatisfaction) or it might simply be a policy variable (e.g. manager
wanting to achieve a specified service
level for a certain item). Several models will be described that can be used in cases when variability is present.
The first model can be used if an
estimate of expected demand during lead time and its standard deviation
are available. The formula: ROP = expected demand + Z dLT during lead
time.
Where
Z = Number of standard deviations
dLT = The standard deviation of lead time demand. The models generally assume that any
variability in demand rate or lead time
can be adequately described by a normal distribution. However, this is
not a strict requirement; the models
provide approximately reorder points even
where actual distribution departs from normal.
The value of Z, used in a particular instance depends on the
stock-out risk that the manager is
willing to accept. Generally, the smaller the risk the manager is willing to accept, the greater the value of
Z. Let us illustrate this with an
example: Suppose that the manager
of a construction supply house determined from
historical records that the lead time demand for sand averaged 50 tons.
In addition, suppose the manager determined
the 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 the
following questions assuming that the manager
is willing to accept a stock out 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?
Expected lead time demand = 50 tons
dLT = 5 tons
Risk = 3 percent
(a) From normal deviate table, using a service level of 1 – 0.3
=.9700 you obtain a value of Z =
+1.82.
(b) Safety stock = Z dLT =
1.88 (5) = 9.40 tons
(c) ROP = expected lead time demand + safety stock = 50 + 9.40
= 59.40 tons
If data are available, a manager can determine whether demand
and/or lead time is variable, and if
variability exist in one or both, the related standard deviation. For those situations, one of the
following formulae can be used. If only
demand is variable, then d LT = 1LT d
and the reorder point is
ROP = - d X LT + Z LT d ----------------------------- (1)
Where -
d = Average daily or weekly demand
d = standard deviation of demand per day or week
LT = lead time in days or weeks if only lead time is variable,
than
dLT =d dLT and the reorder
point is ROP = d x LT + Z dLT ----------
2)
Where
d = Daily or weekly demand
LT = Average lead time in days or week
dLT = Standard deviation of lead-time in days or weeks. If both demand and lead-time are variables,
then. 2 LT = LT 2 d + d 2 LT
and the reorder point is 2 + d2
LT2 …………………. (3)
ROP = d1 x L1T1 + Z LT d
Note: each of these models assumes that demand and time are
independent. Let us illustrate the use
of these formulas with the following.
Example
Suppose a 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 a stock-out
during lead time, which is two weeks.
Assume the distribution of usage is normal.
(a) Which of the above formulas is appropriate for this situation?
Why?
(b) Determine the value of Z
(c) Determine the ROP
Solution
d = 50 jars per week
LT = 2 weeks
d = 3 jars per week
Acceptable risk = 10 percent, so service level is .90
(a) Because only demand is variable (i.e., has a standard deviation)
formula (l) is appropriate
(b) From the normal distribution table, using a service level of.
9000, you obtain Z = + 1.28.
(c) ROP = d X LT + Z LT d
= 50 X 2 + 1.28 2 (3)
= 100 + 5.43
= 105.43.
3.6 How Much To Order: Fixed –Order-Interval Model.
When inventory replenishment is based on EOQ /ROP model, fixed
quantities of items are ordered at
varying time interval. Just the opposite occurs under the fixed-order-interval (FOI) model orders for
varying quantities are placed at fixed
time intervals (e.g. weeks, every 20 days).
3.6.1 Reasons for Using the Fixed-Order-Interval Model
In some cases, a supplier policy might encourage orders at fixed
interval. Grouping orders for items from
the same supplier can produce saving in
shipping costs. Furthermore some situations to not readily lend
themselves to continuous monitoring of
inventory levels. Many retail operator (e.g. drug stores) falls into this category. The
alternative for them is to use fixed-intervalordering, which requires only periodic checks on
inventory levels.
3.6.2 Determining the
Amount to Order
If both the demand rate and lead time are constant, the fixed
interval model and the fixed quantity
model function identically. The difference in the two models becomes apparent only when examined under
condition of variability. Like the ROP
model, the two models can have variation in demand only, in lead time only, or in both demand and lead time.
However, for the sake of simplicity ad
because it is perhaps the most frequently encountered situation, the
discussion here will focus on variable
demand and constant lead time. Order
size in the fixed-interval model is determined by the following computation:
Amount = Expected demand during protection interval + safe stock
– Amount on hand at reorder time = d (OI + LT) + Z d O1 + LT - A
Where 01 = order interval
(length of time between order) A =
Amount on hand at reorder time As in
previous models, it is assumed that demand during the protection interval is normally distributed. Given the following information determine the
amount to order d = 30 note per day
Desired service = 99 percent d = 3 notes
per day LT = 2 days Amount on hand at
reorder time = 71 notes 01 = 7 days
Solution
Z = 2.33 for 99 percent service level Amount =d (01 + LT) + Z d 01 + LT - A = 30 (7+2) + 2.33 (3) 7+ 2 – 71 = 220 units
3.6.2 Benefits and Disadvantages
The fixed-interval system result in the tight control need for A
items in an A-BC classification due to
the periodic review it requires. In addition, when two or more items come from the same supplier,
grouping orders can yield saving in
ordering, packing and shipping costs. Moreover, it may be the only
practical approach if inventory
withdrawal cannot be closely monitored.
On the negative side, the fixed system necessitate a large amount of
safely stock for a given risk of
stock-out because of the need to protect against shortage during an entire order interval plus
lead time (instead of lead time only)
and this increases the carrying cost. Also, there are the costs of the periodic reviews.
3.7 The Single-Period Model.
The single-period is used to handle ordering of perishable (e.g.
fresh fruits, vegetables, seafood, cut
flowers) and items that have a limited useful life (e.g. newspaper’s magazines, spare parts for
specialized employment.) The period for
parts is the life of the equipment (assuming that the part cannot be used
for other equipment) what sets unsold or
unused goods apart is that they are not
typically carried over from one period to the next, at least not without
penalty.
Day-old baked goods, for instance, are often sold at reduced
prices, left over seafood may be
discarded, and out of-date magazines may be offered to used book stores at bargain rates. At times, there
may even be some cost associated with
disposing of left over goods. Analysis
of single – period situation generally focuses on two costs: Shortages and excess shortage cost may include a charge
for loss of customer goodwill as well as
the opportunity cost of lost sales. Generally shortage cost is unrealised profit per note. That is, C shortage = C = Revenue per
note-cost. If a shortage or stock – out
relates to an items used in production or to a spare parts for a machine, then shortage cost refer
to the actual cost of production. Excess
cost pertains to items left over at the end of the period. In effects, excess cost is the difference between
purchase cost and salvage value. That is
C excess = C2 = Original cost per note – salvage value per note. If there is cost associated with disposing of
excess items, the salvage will be
negative and will therefore increase the excess cost per note. The goal
of the single-period model is to
identify the order quantity or stocking level that will minimize the long-run excess and shortages
costs.
There are two general categories of problem that we will consider;
those for which demand can be
approximated using a continuous distribution (perhaps a theoretical one such as a uniform or normal
distribution) and those for which demand
can be approximated using a discrete distribution (say historical frequencies or a theoretical distribution
such as the Poisson). The kind of
inventory can indicate which types of model might be appropriate. For
example demand for petroleum, liquid and
gases tend to vary over some continuous
scale, thus lending itself to description by a continuous distribution.
Demand for tractors cars and computer is
expressed in terms of the number of notes
demanded and lends itself to description by a discrete distribution.
3.7.1 Continuous
Stocking Levels The concept of
identifying an optimal stocking level is perhaps easiest to visualize when demand is uniform. Choosing
the stocking level is similar to
balancing a seesaw, but instead of a person on each end of the see saw,
we have excess cost per note (Ce) on one
and of the distribution and shortage cost per
note (Cs) on the other. The optioned stocking level is analogous to the
fulcrum of the seesaw; the stocking
level equalizes the cost weights, as illustrated in the figure below.
The service level is the probability that demand will not exceed the
stocking level, and computation of the
service level is the key to determining the optimal stocking level, so Service level = Cs Cs + Ce
Where Cs = shortage cost
per note Ce = Excess cost per note Ce Cs
Service level S0 quantit
Balany c e point So = optimum
stocking quantity If actual demand
exceeds S0 there is a shortage: hence Cs is on the right end of the distribution. When Ce = Cs the optimal
stocking level is half way between the
end points of the distribution. If one cost is greeter than the other, S0 will
be closer to the larger cost. A similar approach applies if demand is
normally distributed.
3.7.2 Discrete Stocking Level.
When stocking level are indiscrete rather than continuous, the
service level computed using the ratio
Cs / (Cs + Ce) usually does not coincide with a
feasible stocking level (e.g. the optimal amount may be between a five
and six notes). The solution is to stock
at the next higher level (e.g. six notes).
In other words, choose the stocking level so that the desire service
level is equalled or exceeded.
3.8 Operation Strategy Inventories are
necessary parts of doing business, but having too much inventory is not good. One reason is that
inventories tend to hide problems: they
make it easier to “live with” problems rather than eliminate them. Another reason is that inventories are costly to
maintain. Consequently, a wise operation
strategy is to work toward cutting back inventories by
(1) reducing lot size
(2) reducing safety
stocks. Japanese manufactures use
smaller lots sizes than their western counterparts because they have a different perspective on
inventory carrying costs. Recall that
carrying costs and ordering costs are equal at the EOQ. A higher carrying cost, results in a steeper carrying-cost
line, and the resulting intersection with
the ordering-cost line at a smaller quantity; hence, a smaller EOQ.
The second factor in the EOQ mode that can contribute to smaller
lot seizes is the set up or ordering
processing cost. Numerous cases can be cited where these costs have been reduced through
research efforts. However while
reduction due to carrying costs stems from a reassessment of those
costs, a reduction due to ordering or
set up cost must come from actually pursuing
improvement. Together, these cost reduction can lead to even smaller
lot seizes.
Additional reductions in
inventory can be achieved by reducing the amount of safety stock carried. Important factor in
safety stock are lead time variability,
reductions of which will result in lower safety stocks. These reductions
can often be realized by working with
supplier, choosing suppliers located close to
the buyer, and shifting to smaller lot sizes. To achieve these reductions, an A-B-C
approach is very beneficial. This means
that all phases of operation should be examined, and those showing the
greatest potential for improvement (A
items) should be attacked first. Last,
it is important to make sure that inventory records be kept accurate and up to date. Estimated of holding costs, setup
costs, and lead time should be reviewed
periodically and updated as necessary.
4.0 CONCLUSION
In this note you have learnt the management of finished goods, raw
materials, purchased parts and retail
items. You have also learnt the different functions of inventories, requirements for effective
inventory management, objective of
inventory control, and the techniques for determining how much to order
and when to order.
5.0 SUMMARY
Good inventory management is often the mark of a well-run
organization. Inventory levels must be
planned carefully in order to balance the cost of holding inventory and the cost of providing
reasonable levels of customer service.
Successful inventory transactions, accurate information about demand and lead times, realistic estimates for
certain inventory-related costs, and a
priority system for classifying the items in inventory and allocating
control efforts.
The models described in this note are relevant for instances where
demand for inventory items is
independent. Four classes of models are described; EOQ, ROP, fixed-interval and the single-period
models. The first three are appropriate
if unused items can be carried over into subsequent periods. The single-period model is appropriate when items
cannot be carried over. EOQ models
address the question of how much to order.
The ROP models address the
question of when to order and are particularly helpful in dealing with situations that include variations in either
demand rate or lead time. ROP models
involve service level and safety stock considerations. When the time between orders is fixed, the F0I model is
useful. The single-period model is used
for items that have a “shelf life” of one period.
Thanks for sharing the useful information. This is very excellent information.
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