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Dr Valentina Plekhanova
CISM02: Decision Support for Management
Unit 8
Inventory Systems
Unit 8: Learning Outcomes
2CISM02 Decision Support for Management Unit 8
1. To understand the inventory problems
2. To be able to distinguish between a variety of the major stock costs
3. To understand the nature of demands
4. To understand the simple Economic Order Quantity (EOQ) model
5. To construct an algebraic model for a simple inventory system
6. To be able to apply inventory models to the practical problems
7. To be able to explain the limitations of inventory control models
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Inventory
3
Inventory: Stock of items for internal or external demand.
Tools, machinery , equipment
Raw materials
Working capital
Finished goods
Thepurpose of inventory managementis to determine the amount
of inventory to keep in stock.
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Inventory Management
4
The word inventory was first recorded in 1601. The French term inventaire, or
"detailed list of goods," dates back to 1415.
Inventory management is primarily about specifying the size and placement of
stocked goods.
Inventory management is required at different locations within a facility or
within multiple locations of a supply network to protect the regular and planned
course of production against the random disturbance of running out of
materials or goods.
The scope of inventory managementalso concerns the fine lines between
replenishment lead time, carrying costs of inventory, asset management,
inventory forecasting, inventory valuation, inventory visibility, future inventory
price forecasting, physical inventory, available physical space for inventory,
quality management, replenishment, returns and defective goods and demand
forecasting.
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Basic Inventory Problem
5
Any inventory problem gives an answer to the following questions:
How much to order?
When to order?
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Inventory Control
6
The problem of inventory exists in all companies whether big or small.
Exceeding inventory has contributed to the failure of many companies.
If stock is taken regularly to honour a demand at any time, a company
will never be out of stock and thereby lose the sales and goodwill of its
customers. But the cost of investment and keeping up the inventory
(holding cost) will increase.
Less inventory leads to frequent orders and stock out and costs
increase as a result.
The ideal inventory policy is to find out the break-even of these costs.
Also the policy should be adjustable to suit the existing system.
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Buffer Stocks
7
Many organisations keep an additional amount of inventory called
safetyorbuffer stocksfor the following reasons:
Variations in demand
High seasonal demand
Variations in supplier deliveries
Volume discounts
High ordering price
etc.
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Inventory Costs
8
Inventory must be sufficient to provide high quality customer service.
Inventory Costs:
Carrying costs/holding costsare the costs of holding an item in
inventory (rent, heating, cooling, lighting, security, refrigeration,
record keeping, deterioration, taxes, etc.)Ordering costsincrease/decrease with the number/size of orders
Shortage costsinsufficient inventory for customer demand
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Inventory Models:Economic Parameters
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Set-up cost: This involves the fixed charge associated with theplacement of an order or with the initial preparation of a production
facility. The set-up cost is usually assumed independent of the quantity
ordered or produced.
Purchase price or production cost: This parameter is of special interest
when quantity discounts or price breaks can be secured or when large
production runs may result in a decrease in the production cost. Under
these conditions, the order quantity must be adjusted to take advantage
of these price breaks.
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Inventory Models:Economic Parameters
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Holding cost: This represents the cost of carrying inventory in storage.
It includes the interest on invested capital, storage costs, handling
costs, depreciation costs, etc. Holding costs usually are assumed to
vary directly with the level of inventory as well as the length of time
the item is held in stock. As the inventory increases, this cost
increases.
Shortage cost or penalty cost: These are the penalty costs incurred asa result of running out of stock when the commodity is needed (i.e.
when the demand is fulfilled). They generally include costs due to
loss in customersgoodwill and potential loss in income.
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Inventory Models: Terminology
11
Replenishment: An inventory system may operate with delivery lags, the
actual replenishment of stock may occur instantaneously or uniformly.
Lead Time or time lag or delivery lag: The time between placing an
order and its arrival in stock. If the lead time is known and is not equal to
zero, and if demand is known, all that one needs to do is to order in
advance by an amount of time equal to the lead time. If, however, it is a
variable that is known only probabilistically, the question of when to
order is more difficult one. If either the demand or the lead time is known
only probabilistically, the amount and timing of replenishment if found by
considering expected costs of holding and storage over the lead time
period.
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Water Tank Analogy for Inventory
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Supply Rate
Inventory Level
Demand Rate
Inventory LevelBuffers Demand Ratefrom Supply Rate
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Two Forms of Demand
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Dependent:
items used to produce final products (table top, legs, hardware, paint, etc.)Demand determined once we know the type and number of final products
Independent:items demanded by external customers (Kitchen Tables; Cars,appliances, computers, and houses are examples of independent demand
inventory)
Independent demand
Uncertain / forecasted
Continuous Review / Periodic Review
Dependent demand
Requirements/ planned
Materials Requirements Planning / Just in Time
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Nature of Demand
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Sometimes, the demand is known with certainty with its rate remaining the
same or not in relation to time. If demand is certain it is called a
deterministic demand.
Depending upon whether the rate of demand remains the same or not
with respect to time it is called staticor dynamicdemand.
Instead of a deterministic demand, sometimes a demand is expressed in
terms of probability distribution in which case it is termed a probabilistic
demand. In this case also the demand may change in relation to time.
If it does not change it is called stationary, and if it does as non-
stationary.
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Reasons to Hold Inventory (1)
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Meet variations in customer demand:Meet unexpected demandSmooth seasonal or cyclical demand
Pricing related:Temporary price discountsHedge against price increasesTake advantage of quantity discounts
Process & supply surprisesInternalupsets in parts of or our own processesExternaldelays in incoming goods
Transit
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Reasons to Hold Inventory (2)
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Meet unexpected demand
Smooth seasonal or cyclical demand
Meet variations in customer demand
Take advantage of price discounts
Hedge against price increases
Quantity discounts
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Reasons to Not Hold Inventory
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Carrying cost
Financially calculable
Takes up valuable factory space
Especially for in-process inventory
Inventory covers up problems
That are best exposed and solved
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Inventory Control Systems
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Functions of Inventory Management
Track inventory
How much to order
When to order
Prioritization (e.g. ABC classification/prioritization)
Inventory Management ApproachEOQ
Continuous / Periodic
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Inventory Control Objectives
19
Inventory Control Objectives Twofold:
1. Maximize customer service
2. Minimize costs
Cost Objective: Minimize sum of relevant costs
Service Objective: desired customer service levels significantly impact
inventory levels.
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Inventory Control Models
20
Service level may be defined in a number of ways, such as:
Inventory Control Models
Remindertwo important issues in inventory control:
order quantity and order timing.
Two general classes of models:
1. continuous review (fixed order quantity)
2. periodic review (fixed order period)
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Inventory Control Systems:
Continuous & Periodic
21
In a continuous system, an order is placed for the same constant amountwhenever the inventory on hand decreases to a certain level, whereas in
a periodic system, an order is placed for a variable amount after specific
regular intervals.
Fixed-order-quantity system ( or Continuous Review or Q-systems) -
constant amount ordered when inventory declines to predetermined level
Fixed-time-period system (or Periodic review or P-systems) - order
placed for variable amount after fixed passage of time
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Inventory Control Systems:Continuous
22
In a continuous inventory system (also referred to as a perpetual system
and a fixed-order-quantity system), a continual record of the inventory
level for every item is maintained. Whenever the inventory on hand
decreases to a predetermined level, referred to as the reorder point,a
new order is placed to replenish the stock of inventory.
The order that is placed is for a fixed amount that minimizes the total
inventory costs. This amount, called the economic order quantity.
A positive feature of a continuous system is that the inventory level is
continuously monitored, so management always knows the inventory
status. This is advantageous for critical items such as replacement parts
or raw materials and supplies. However, maintaining a continual record of
the amount of inventory on hand can also be costly.
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Inventory Control Systems:
Continuous
23
Continuous Review or Fixed Order Quantity Systems (Q-systems)
Multi period models
1. Fixed order quantity, variable time between orders (EOQ, EPQ,
and Quantity Discount)
2. On-hand inventory balance serves as order trigger (R)
3. Perpetual inventory count
4. 2-bin system
Single period Model
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Inventory Control Systems: Periodic
24
In a periodic inventory system (also referred to as a fixed-time-period
systemor a periodic review system), the inventory on hand is counted at
specific time intervals; for example, every week or at the end of each
month.
After the inventory in stock is determined, an order is placed for an
amount that will bring inventory back up to a desired level. In this system
the inventory level is not monitored at all during the time interval between
orders, so it has the advantage of little or no required record keeping.
The disadvantage is less direct control. This typically results in larger
inventory levels for a periodic inventory system than in a continuous
system to guard against unexpected stockouts early in the fixed period.
Such a system also requires that a new order quantity be determined
each time a periodic order is made.
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Inventory Control Systems: Periodic
25
Periodic Review or Fixed Order Period Systems (P-systems)
1. Variable order quantity, fixed time between orders
2. Time serves as order trigger
3. Periodic count
4. Process: When a predetermined amount of time has elapsed, a
physical inventory count is taken. Based upon the number of units in
stock at that time, OH, and a target inventory of TI units, an order is
placed for Q = (TI-OH) units
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Single Item Static Model
26
The simplest type of inventory model occurs when demand is constant
over time with instantaneous replenishment and no shortages.
Typical situations to which this model may apply are
Use of light bulbs in a building
Use of clerical suppliers, such as paper, pads, and pencils, in an office
Use of certain industrial suppliers such as blots and nuts.
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Single Item Static Model:
Economic Order Quantity (EOQ) Model (1)
27
This model known as the economic order quantitymodel is the oldest
and best known inventory model which originated as early as 1915 with
Wilson and so the formula derived under this model to decide the
ordering quantity is called the Wilson formula (i.e. Wilsons economic
lot size).
It basically answers the question how much to order. Out of the four
components of inventory cost, here we are considering only two of
them, namely, theset-up costand theholding cost.
The other two costsshortage cost and purchasing cost, we adjust in
such a way that shortage is not allowed and the holding cost takes care
of purchasing cost which is possible to some extent with theassumption that there is no price break possible even if we order
quantity in big lots.
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Single Item Static Model:Economic Order Quantity (EOQ) Model (2)
28
Our aim is to reduce thetotal costper unit(TCU) time. This is expressed
it terms of inventory as we are finding out the optimal inventory level
which reduces TCU.
With our assumptions TCU can be expressed as
TCU= set-up cost per unit time+holding cost per unit time
Let us denote
B = rate of demand or consumption
Y= quantity to be ordered
H= holding cost per unit per item
K = set-up cost per order
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Single Item Static Model:Economic Order Quantity (EOQ) Model
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Figure: Variation of the Inventory Level
Points in time at which orders are placed
Inventory Level
Time
SS
Average Inventory:Y/2
Y
t0=Y/B
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Single Item Static Model:Economic Order Quantity (EOQ) Model
30
The smaller the order quantity Y, the more frequent will be the placement of new orders. However, the
average level of inventory held in stock will be reduced. On the other hand, larger order quantities
indicate larger inventory level but less frequent placement of orders.
Inventory Level
TimeSS
Low ordering frequency
High ordering frequency
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Single Item Static Model:Economic Order Quantity (EOQ) Model (3)
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Holding cost per unit time= Average inventory x Holding cost per unit per item ==Y x H / 2
Y/2becoming average inventory.
The cycle time is denoted byt0 and defined as
Set-up cost per unit time= K x Number of orders per unit time =K x B/Y =
= K/ (Y/B)
Reminder:TCU= set-up cost per unit time+holding cost per unit time
or TCU = K x B/Y + (Y/2) H
0
0
2( )
YK H t
TCU Y
t
0
Yt
B
CISM02 Decision Support for Management Unit 8
Single Item Static Model:Economic Order Quantity (EOQ) Model (4)
32
Using the calculus method to find the optimal inventory we have to differentiate
TCUwith respect toY (assuming Y is a continuous variable)and equate to zero:
This will give us the value ofY as
(2 ) /Y KB H
2
( )0
2
dTCU Y KB H
dY Y
The above order quantity is usually referred as Wilsons economic lot size.
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Single Item Static Model:an Example
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The daily demand for a commodity is approximately100 units. Every time anorder is placed, a fixed cost of100is incurred. The daily holding cost per unit
inventory is0.02. If the lead time is12days, determine the economic lot size
and reorder point.
From the above formulas, the economic lot size is:
* 2 2 100 1001000 ( )
0.02
KBY units
H
Since the lead time is12 daysand the cycle length is10 days, reordering
occurs when the level of inventory is sufficient to satisfy the demand for two
(12-10)days. Thus, the quantityY*=1000is ordered when the level of inventory
reaches2 x 100 = 200 units. And the optimum costTCUis defined as
100 1000( ) 0.02 20 ( per day)
10002 2
100
K YTCU Y H
Y
B
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Single Item Static Model:Remarks
34
The assumptions of the above model are usually rarely satisfied for real
situations, mainly because demand is probabilistic.
Nevertheless, a crudeprocedure has evolved among practitioners
which, while retaining the simplicity of applying the economic lot size
model, does not completely ignore the effect of probabilistic demand.
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Quantity Discount Model
35
Form of an economic order quantity (EOQ) model that takes into accountquantity discounts. Quantity discountsare price reductions designed to induce
large orders.
If quantity discounts are offered, the buyer must weigh the potential benefits of
reduced purchase price and fewer orders against the increase in carrying costs
caused by higher average inventories. Hence, the buyer's goal in this case is to
select the order quantity that will minimize total costs, where total cost is the sum
of carrying cost, ordering cost, andpurchase cost:
Total Cost = Carrying Cost + Ordering Cost + Purchase Cost =
C x Q/2 + O x (D/Q) + P x D
where C= carrying cost per unit, O= ordering cost per order, D= annual
demand, P= unit price, and Q= order quantity.
CISM02 Decision Support for Management Unit 8
Inventory Models
36
More models:
Single Item Static Model withSimultaneous Production and Sales
Single Item Static Model with Simultaneous Production and Sales
whereShortage is allowed
Single Item Static Modelwith Time LagSingle Item Static Model withPrice Breaks
Multiple-Item Static Model with Storage LimitationSingle-Item N-Period Dynamic Model
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Prioritization (e.g. ABC classification/prioritization)
37
ABC Inventory Classification
Inventory control based on a form of Pareto analysis. The inventory
items are divided into three categories (A, B, and C), according to a
criterion such as revenue generation, turnover, or value.
Typically, Aitems represent 20 percent in terms of quantity and 75 to
80 percent in terms of the value.
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ABC Analysis
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ABC Analysis:Analysis of a range of items, from inventory levels to
customers and sales territories, into three groups:
A= very important
B= important
C= marginal significance.
The goal is to categorize items which would be prioritized, managed, orcontrolled in different ways.
ABC analysis is also called usage-value analysis.
Details of ABC classifications will be discussed in Unit 11.
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ABC classification:
Applications in Inventory Control
39
ABC classification could be used to determine the importance rating fordifferent items and help the Stores in better inventory control. We could
have the following objectives for classifications:
Purchasing and Finance may want to classify the items in three
classes, i.e.A, Band Cbased upon last years consumption value.
Stores may want to classify the items in class High, Medium, and
Lowbased upon on-hand quantities for the purpose of doing a better
store space planning.
Inventory Controller may want to classify the items inHigh
movement, Medium movements and Low movements for
analysing the Min-Max Levels of different items.
CISM02 Decision Support for Management Unit 8
Inventory Control: an Example (1)
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A product is manufactured to meet the demand over the nextn periods.
At periodi, the demandri (units) may be filled from the productionxi
(units) in this period and/or the inventory carried forward from previous
periods. This means it is possible to produce more thanriunits in period
i, with the surplus being used to fill (some of) the demand for one or
more succeeding periods.
No shortages are allowed in any period so that the demand for all
periods must be filled. If the level of production in periodi is increased
over that in period (i-1), that is,xi> xi-1, a costaipounds per excess unitis incurred.
On the other hand, ifxi< xi-1, a costbipounds per decreased unit is
added for decreasing the level of production.
An inventory holding costci pounds is incurred for every unit held over to
the next period.
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Inventory Control: an Example (2)
41
The objective of the modelis to determine the optimal production
schedule such that the total cost of the production-inventory system isminimised.
The balance equation at periodi is:
Ii+1= xi+ Iiri, I = 1,2,,n
whereI1is the initial inventory at the beginning of period1and In+1is
the final inventory at the end of periodn; the demandri (units); the
productionxi (units).
Generally,I1>=0, whileIn+1must be zero in any optimal solution. In
order to guarantee that the demand is satisfied in every period (no
shortage), Ii>=0, I = 1,2,, n-1.
Naturally, production quantities must be nonnegative so that
xi>= 0, i=1,2,,n
CISM02 Decision Support for Management Unit 8
Inventory Control: an Example (3)
1 1
1 1
1
( ), if
( ), if
0, if =
i i i i i
i i i i i i
i i
a x x x x
y b x x x x
x x
42
The above restrictions define all the constraints of the problem. The
objective function includes three components:cost of holding inventory,
cost of increasing production, andcost of decreasing production.
The holding cost is the simplest, and it is given directly for period i by
ci Ii+1(notice the inventory holding cost is charged on units left at the
end of the period). The cost of increasing and decreasing production in
periodi is given by
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Inventory Control: an Example (4)
1 1max{ ( ), ( )}i i i i i i iy a x x b x x
43
Equivalently,yimay be written as
This follows since if production is increased,xi-1xi< 0so that
bi(xi-1xi)0. Thusbi (xi-1xi)
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Inventory Control: an Example (6)
1
1
1
0
0
, , 0, 1,2,...,i
i i i i i
i i i i i
i i i i
i i
a x a x y
b x b x y
x I I r
I x y i n
0 1
1
minimise ( )n
i i i
i
x c I y
45
The linear programming model of the production-inventory problem maybe summarised as
subject to
whereIn+1=0. The variableyiis nonnegative because it represents thecost of increasing or reducing production.
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