02b Deterministic Inventory Models

8/9/2019 02b Deterministic Inventory Models

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Deterministic Inventory Models

(PartII)

1Sasadhar Bera, IIM Ranchi


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Outline


Inventory Models

Continuous Review Single Item Models

Basic EOQ model

Economic production quantity model

Quantity discount model

Determining Safety Stock and Reorder Point

For variable demand and constant lead time

For variable demand and variable lead time


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Outline (Contd.)


Impact of Service Level on Safety Stock

Determining Order Quantity for Periodic Review Models

For variable demand and constant lead time

For variable demand and variable lead time

Visual System

Two-bin system

One-bin system


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Introduction


The objective of inventory management is to provide the

required level of customer service and to reduce the sum of

all costs involved. To achieve these objectives, two basic

questions must be answered:

How much should be ordered at one time?

When should an ordered be placed?

Management must establish decision rules to answer these

question so that inventory management personnel know

when to order and how much.


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Inventory Models



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Inventory Models


Inventory control problem is closely related to the demand

pattern. Demand is the principle factor in the design of

inventory model.

i. Deterministic

a) Static demand: constant over time

b) Dynamic demand: Demand is known withcertainty but varies from one time period to

other.

ii. Probabilistic

a) Stationary demand: Demand probability densityfunction remains unchanged over time.

b) Non-stationary demand: Demand probability

density function changes with time.


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Inventory Models (Contd.)


Other classification of an inventory problem is

i. Single item inventory: Handles only one item or SKU.

ii. Multiple item inventory: When the inventories consists

of many items. Such inventory problems may have

different types of limitations such as finance, coststructure, space etc. As the number of restrictions

increases, the inventory problem becomes more

complicated.


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Continuous Review Single Item Models

Basic EOQ ModelProduction Quantity Model

Quantity Discount Model



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EOQ Model


Economic Order Quantity(EOQ) - The order size for which

total inventory cost is minimum known as EOQ. Acalculation is made which considers demand rate, ordering

cost, holding cost, lead time, and service level.

The function of the EOQ model is to determine the optimal

order size that minimizes total inventory cost. There are

different variations of EOQ model, depending on the

assumptions made about the inventory system. We will

discuss two basic versions:

i. Basic EOQ model

ii. Production quantity model


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Basic EOQ Model


The basic EOQ model determines optimal order size that

minimizes the sum of carrying costs and ordering costs. The

assumptions considered are:

i. Demand is known with certainty and is constant over

time

ii. The lead time is also known with certainty and constant

over time

iii. No shortages are allowed

iv. The order quantity is received all at once (i. e.instantaneous replenishment).

v. Decisions for one item can be made independently of

decisions for other items


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Basic EOQ Model (Contd.)


Inventory Pattern in Basic EOQ model


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To determine optimal order size two types of cost are

considered viz. ordering cost and inventory carrying cost.

These two costs react inversely to each other. As order size

increases fewer orders are required, causing the ordering

cost to decline, whereas the average amount of inventory

on hand will increase, resulting in an increase in inventory

carrying cost. Thus in effect, the optimal order quantityrepresents a compromise between these two inversely

related costs.

D = Annual demandC0= ordering cost per order

Ch= Inventory carrying cost (or holding cost) per unit item

Q = Economic order quantity


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The figure below shows the inverse relationship between

ordering cost and inventory carrying cost.


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Inventory carrying cost increases linearly with order size Q,

while annual ordering cost decreases exponentially with

order-size Q.

The optimal order quantity occurs at the point where the

total cost is minimum, which coincide exactly with the point

where the inventory carrying cost curve intersects the

ordering cost curve. The optimal value of Q can be

determined by differentiating the total cost curve with

respect to Q.

Annual ordering cost = C0 * (D/Q)

Average inventory level = (0 + Q)/2 = Q/2

Annual inventory holding cost or carrying cost = Ch* (Q/2)

Total inventory cost (TC) = C0*(D/Q) + Ch*(Q/2)


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To determine optimal order,

= 0

Optimal order size = Qopt

=

Optimal number of order per year = Nopt =

Order cycle time =

Q

)TC(

h

0

C

DC2

OptQ

D

OptNdaysworkingofNumber


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Production Quantity Model


In production quantity model, the order quantity is received

gradually over time and the inventory level is depleted atthe same time it is being replenished.

This situation is commonly found when the inventory user is

also producer, as in a manufacturing operation where parts

are produced by other section to use in assembly operation.

This situation is also can occur when orders are delivered

continuously over time or when a retailer is also the

producer.


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Production Quantity Model (Contd.)


In order to determine the average inventory level, we

define the following parameters:

Q = order quantity per order

p = daily rate at which the order is received over time, also

known as the production rate

d = the daily rate at which the inventory is depleted

In this model, we are still assuming that no shortages occur.

Hence demand rate cannot exceed production rate i. e.

d p . If d = p, items are consumed as fast as they are

produced, there is no order size.


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Inventory pattern of production quantity model


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D = Annual demand

C0= Setup cost per production run (ordering cost)

Ch= Inventory carrying cost (or holding cost) per unit item

Q = Economic order quantity known as lot size

Time required to finish receiving an order = t/ = Q/p

The amount of inventory that will be depleted or used upduring this time period = t/d = (Q/p)d

As a result, the maximum inventory on hand = Q - (Q/p)d

= Q(1- d/p)

Average inventory level = (Q/2) (1- d/p)

Inventory carrying cost = Ch(Q/2) (1- d/p)


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To determine optimal order,

= 0

Optimum lot size = Qopt=

Length of production run = (Qopt/p)

Number of production runs = Nopt =

Q

)TC(

)p

d

1(C

DC2

h

0

optQ

D


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Quantity Discount Model


When items are purchase, suppliers often give a discount in

price over a certain volume or quantity. This is done

because of larger orders reduce the suppliers ordering

costs and cheaper transportation.

Many manufacturing companies receive price discounts for

ordering in high volume from supplier. Retail stores receive

price discounts for ordering merchandise in large

quantities.

A quantity discount is a price discount on an item ifpredetermined numbers of units are ordered.


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Quantity Discount Model (Contd.)


The basic EOQ model can be used to determine the optimal

order size with quantity discounts. The purchase price of

the item being ordered:

D = Annual demand

C0= ordering cost per order

Ch= Inventory carrying cost (or holding cost) per unit itemQ = Economic order quantity

P = per unit price of item

Total cost (TC) =

The basic assumptions considered are instantaneous

replenishment, and no shortages.

PD2

QC

Q

DC

h0


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Total cost curve shows step function behaviour


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Safety stock and reorder point calculation depends on the

following factors:

i. Demand type: Fixed demand or probabilistic demand

ii. Service level: Allowable % of stock out

iii. Lead time: Constant lead time or variable lead time

iv. Frequency of reorder: One prefers replenishment to

take place on a continuous basis. Japanese companies

have managed to get their replenishment from suppliers

three to four times a day in trips involving multiple

deliveries or pickups called milk runs.


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Safety Stock


Inventory level with safety stock

Safety stock: It is an additional inventory that is kept to take

care of demand and supply uncertainty. Safety stock reduces

chances of stockout situation.

Reorder Point

Safety Stock

Inventory

Time

Lead time

Cycle Stock

Avg. inventory


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Safety Stock (Contd.)


The average inventory for a period (considering safety stock) is

equal to average of opening inventory and ending inventory.

Opening inventory = order quantity (Q) + safety stock (SS)

Ending inventory = safety stock (SS)

Average inventory = Q/2 + SS


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Reorder Point for Basic EOQ Model


In basic EOQ model we assumed that demand is constant i. e.

rate of demand is same. We also consider that lead time is

constant i. e. we get order quantity in exactly specified timeperiod.

Reorder point: The level of inventory at which a new order is

placed. Determining the reorder point depends on thedemand during the lead time and the safety stock required. If

demand during the lead time is greater than expected, there

will be a stockout unless sufficient safety stock is available.

Let d = demand rate during lead time

L = lead time

Reorder point (R) = d*L


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Capturing Uncertainty in Demand and Supply


Uncertainty is captured by one of the following measures:

range, standard deviation, and coefficient of variation.

Demand distribution is captured by two parameters: mean

demand and standard deviation of demand. Standard

deviation of demand is essentially captures uncertainty indemand.

Similarly, supply uncertainty has two parameters, average

lead time and standard deviation of lead time.


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Reorder Point with Variable Demand



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Reorder Point with Variable Demand (Contd.)



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Demand distribution during lead time


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The only time a stockout is possible is during the lead time.

Safety stock is needed to cover only that period in which the

demand during lead time is greater than the average.

If z = 1, it provides service level 84% . It indicates (100 -84)=

16% chance of stockout situations. Similarly, z = 1.65,

provides 95% service level. z = 3, provides 99.83% service

level.

Excel function:

For service factor z = 2, service level = NORMDIST(2, 0,1,1) =

0.977 = 97.7%

For 90% service level, service factor = z = NORMSINV(0.90) =

1.28

d i f i bl d d


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Reorder Point for Variable Demand and

Variable Lead Time


In practice both demand and lead time are variable. In suchsituation, first we have to determine demand distribution

and lead time distribution in same unit. Second we consider

demand and lead time are independent.

To determine the reorder point we have to determine mean

and standard deviation of demand during lead time, which

is also a random variable. Therefore we have to use

conditional distribution to find out standard deviation of

demand.

R d P i t f V i bl D d d


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Reorder Point for Variable Demand and

Variable Lead Time (Contd.)



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f i l f k


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Safety stock level= service factor (z) * Standard deviation of

demand during lead time

If we improves service factor, amount of safety stock

increases. Service factor determine service level. If service

factor improves from 0 to 1, service level increases by 34%and further improve by 1 unit would improves service level

by 14%. At some point improving service factor service level

improves marginally. Thus relationship between safety stock

level and service factor is non-linear.

f S i l S f S k


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Impact of Service Level on Safety Stock (Contd.)


For 90% service level, service factor = z = NORMSINV(0.90) =

1.28

I f S i L l S f S k


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Impact of Service Level on Safety Stock (Contd.)


d f k


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How to Reduce Safety Stock


Managerial decisions to reduce safety stock

Reduction in demand uncertainty: Minimize the deviation

between actual demand and forecast value by using better

forecasting method or by making contracts with known

customers to generate stable demand.

Reduction in supply uncertainty: work with the suppliers to

reduce lead time, reduce suppliers internal processing

time, and faster mode of transport. Otherwise, select

suppliers based on on-time delivery, flexibility, and quality.


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Sasadhar Bera, IIM Ranchi

Determining Order Quantity in Periodic

Review Model

P i di R i S t


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Periodic Review System

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In a continuous review system, the inventory position is

monitored continuously so that an order can be placedwhenever the reorder point is reached.

With a periodic review system, the inventory is checked and

reordering is done only at specified time interval. Forexample, inventory may be checked and orders placed on a

weekly, biweekly, monthly, or some other time interval.

When a firm or business handles multiple products, the

shipping and receiving of orders are easily coordinated

under periodic review system.

P i di R i S t


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Periodic Review System (Contd.)

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Firms running their MRP (material requirement planning)

system place order once in a week, essentially follow theperiodic review system. Similarly, if a retailer places an

order at the time of a salesmansvisit, the periodic review

system is being followed.

While reviewing inventory position, we know that the next

opportunity for ordering will come only after a time interval

(P) and subsequently replenishment will take place after the

lead time (L). In case of continuous review system

vulnerable period is lead time. In case of periodic review

system vulnerable period is review period plus lead time.

Periodic Review System


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Periodic Review System (Contd.)

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Once the inventory in stock is determined after specified time

interval, an order is placed for an amount that will bring back to a

desired inventory level. Hence for periodic review system orderquantity (Q) is variable amount.

Periodic Review: Order Quantity with Variable


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Demand

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Demand and Variable Lead Time

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Visual System


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Visual System

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Bin 2 Bin 1

When bin 1 is empty,use bin 2 as backup and

place an order

Pick parts

as required

Two-bin system

(Q system)

Visual System


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Visual System (Contd.)

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Single-bin system: When inventory drops to level of Red tag,

place new order

ROP

02b Deterministic Inventory Models

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