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## Inventory Models

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Title: Inventory Models

1
Inventory Models
• 3 Models with Deterministic Demand
• Model 1 Regular EOQ Model
• Model 2 EOQ with Backorder
• Model 3 EPQ with Production
• 2 Models with Stochastic Demand
• Model 5 Models with Discrete Probability
• Model 6 Model with Continuous Probability

2
Deterministic ModelsModel 1 The EOQ Model
Prototype
• In the deterministic models under study, the firm
faces a known linear demand.
• As an example of deterministic models, we will
demonstrate in detail Model 1, the basic EOQ
model, In this model the firm orders the product.
The goal of the firm is to determine Q, the
Economic Order Quantity that minimizes the firms
total cost.

3
• Determining the economic order quantity uniquely
determines the cycles length T.
• The determination depends on the relative cost
of making an order relative to the cost of
holding item in inventory

4
Types of costs
• The firm has three types of costs
• Procurement Cost
• Ordering Cost
• Holding cost

5
Inputs of the EOQ Model
• A Annual number of items demanded (Annual
Demand)
• k Fixed cost per order
• c Unit cost of procuring an item
• h Annual cost per dollar value of . holding
items in inventory.

6
Planning Period and holding items dollars in
Inventory
• Costs are computed for a planning period.
• In the textbook, this planning period is a year.
• Length of use Cycle ,T, (Time between orders) is
then specified in yearly terms. Conversion to
days would require multiplying the result by the
number of days in a year.

7
Holding Cost
• It is important to notice, the primary component
in the holding cost is the cost of holding the
monetary value of the item in inventory.
Typically this cost accounts for about 80 of the
holding cost.
• The holding cost of the monetary value held in
inventory is evaluated based on the opportunity
cost for investment of the value held in
inventory or based on the cost of borrowed funds
needed to hold the item.
• For example, a dollar invested may yield 10
return on investment or may be required borrowed
fund at 10 interest cost. In these examples,
the cost is .10 per dollar.

8
• Other costs associated with holding are also
therefore prorated per dollar value.
• Therefore, in the model inputs we report, h, the
annual cost per dollar value of holding the item
in inventory. Hence, the corresponding holding
cost for the item is hc.

9
Cost Computation is easy.
• For every Cost Type the cost is,
• Cost per unit of measurement times the Number of
units

10
Units of Measurement and Number of Units for Type
of Cost
• Type of cost UOM
Number of Units
• Procurement One procured item
Annual demand
• Holding One item held
• in inventory for a year
Average Inventory
Number of orders in a
year

11
Costs Per Unit
• Type of Cost Cost Per Unit
• Procurment c
• Holing hc
• Ordering k

12
Annual Number of Units per Type of Cost
• Cost Type Number of Units Formula
• Procurement Annual Demand A
• Holding Average Number of Q/2
units in Inventory
• Ordering Annual number of A/Q
Orders

13
Some Outputs
• The ordered (and use) Quantity per cycle,
Q
• Number of Orders, A/Q
• Length of order and use Cycle, Q/A
• Maximum Inventory level, Q
• Average Inventory level, Q/2
• Annual Holding cost per item, hc

14
Total Cost
• Total annual Cost is the sum of procurement cost
ordering costs and holding cost.
• cA k(A/Q) hc(Q/2).
• Since, the procurement cost cA is fixed, and does
not affect optimization. The Relevant Cost for
decision is then,
• TC(Q) k(A/Q) hc(Q/2).

15
Model 2 Inventory with Backorder.
• In Model 2. the firm designs an optimal order
per cycle as well as optimal waiting list.
• As soon as the order of size Q arrives, the firm
supplies the waiting lists demand. The stock,
S, of the remaining units, is left as inventory
to serve next customers not on the waiting list.
• The stock S is depleted till it disappears. The
firm is then beginning to collect orders on a
waiting list, till the optimal level of the
waiting list, Q-S, is reached. At which point a
new order of size Q arrives.

16
Model 2, Inventory with Backorder continues
• Since customers do not like to wait, there is a
shortage penalty per item on the waiting list, p.
• The penalty p may either be given directly, or
imputed from a service level, L, where L is the
proportion of demand met on time or alternatively
the probability of providing an item from
inventory (rather than from the waiting list).
It is straight forward to impute p from L. (pp.
600-01).
• The total relevant cost equals the ordering cost
holding cost shortage cost.

17
Model 3 Inventory with Production
• In this model the firm produces the item for its
own use. The firm wishes to optimize the
quantity to be produced and used in every cycle,
Q and correspondingly the optimal length of the
production and use cycles. (T1 and T
respectively,).
• First the firm sets up the machines, produces the
product and uses it. (production rate B must be
greater than use rate A.). Since the production
is on-going, inventory is gradually built up to
an optimal maximal level.
• Once the maximum inventory level is reached the
firm stops production. For the rest of the use
cycle, the firm only uses the already produced
inventory in stock. The process is repeated.

18
Model 3, Inventory with Production, Continued
• In this model set up costs replaces ordering
cost. Mathematically, it has the same formula,
• The quantity Q stands for the quantity to use and
produce in the cycle.
• Average inventory is half the maximum inventory,
as in Model 1. However it is lesser than half of
Q, the quantity used in a cycle, as the inventory
• Total Relevant Cost equals Set Up Cost Holding
Costs.

19
An interesting feature of the Deterministic Models
• In the Regular EOQ Model, at optimum,
• Ordering Cost Holding Cost
• In other models, similar observation is made
• In the Inventory with Back Order Case
• Ordering Cost Holding Cost Shortage Cost
• In the inventory with Production case,
• Set Up Cost Holding Cost

20
Models with Stochastic Demand The Newsvendor
Problem
• In this case we are dealing with stochastic
demand and given short cycles.
• Unlike the deterministic models of chapter 15 we
are not determining the cycles length There is
some given cycle and we are provided with the
holding cost of left over item at the end of the
cycle.

21
Properties of the Newsvendor Problem Continued.
• The firm faces possibilities of either shortage
and surplus penalties
• We wish to find Q, the optimal order quantity to
be stocked at the beginning of a cycle, expected
surplus and surplus cost, and expected shortage
and Shortage costs, and total expected costs.

22
Inputs for Model 4 An Inventory Model with
Discrete Demand Distribution
• C Procurement Cost
• hE Additional cost of each item held at the end
of the inventory cycle.
• pS Penalty for each item short (loss of
goodwill)
• pR Selling Price
• Discrete Distribution of the quantity demand for
the item in a cycle.

23
Per Unit Penalties for left over Item and for
Shortage
• The penalty for a left over item hE C, the
surplus penalty per item, is the item cost c and
the residual cost associated with the left over
item. If the left over item has some residual
value hE will be negative and would reduce the
surplus penalty per item.
• Shortage penalty per item, pS (pR C) . It has
two components. The first is the good will loss,
pS. The second is the opportunity loss
encountered. If the firm had the item it could
have made a profit pR C by selling it.

24
The Critical Share and Finding the optimal Order
Quantity
• The critical share is the share of the per item
shortage penalty in the sum of the per units
shortage and surplus penalties,

25
The critical share formula
• The critical share is,
• pS (pR c)
• ____________________________
• pS (pR c) (hEc)

26
The critical share and optimal Q
• Marginal considerations, based on calculus led to
the result that the optimal quantity is to be
this where the cumulative probability of the
demand equal the critical share.
• In the discrete case we look at the level of the
cumulative distribution just surpass the critical
share. Mean demand for a cycle is labeled µ.

27
Total Expected Costs
• Total expected cost is equal to the expected
procurement cost the expected shortage cost
the expected surplus cost.
• Expected shortage cost equals the penalty per
unit of shortage times the expected shortage,
B(Q).
• Expected surplus cost equals the penalty per unit
of surplus times the expected surplus.

28
Model 5. Inventory with Normally Distributed
Demand
• In this case the mean and standard deviation of
the demand are known and the determination of the
ordered quantity, Q, based on the critical share
may be very precisely determined. The rest of
the inputs are the same as in Model 4.
• The following computations of the expected
shortage and surplus using the Normal Loss
Function L(x), and the computations of the
associated expected costs depend on the whether
Q is smaller or larger than the mean demand µ.
See page 630.