inventory mangement pomii class

Rajiv Misra

INVENTORY MANAGEMENT

Rajiv Misra

•Independent Demand

•Deterministic•Stochastic (Newsvendor)

•Dependent Demand


Rajiv Misra

E(1)

Independent Demand (Demand not related to other items or the final end-product)

Dependent Demand

(Derived demand items for

component parts,

subassemblies, raw materials,

etc.)

INDEPENDENT VS DEPENDENT DEMAND

Rajiv Misra

•Periodic Review

•Continuous Review


Rajiv Misra

1. Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously.

2. Delivery is immediate – there is no time lag between production and availability to satisfy demand.

3. Demand is deterministic – there is no uncertainty about the quantity or timing of demand.

4. Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit.

5. A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant.

6. Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products.

EOQ MODEL: ASSUMPTIONS

Rajiv Misra

Basic Fixed-Order Quantity Model and Reorder Point Behavior

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Time

Numberof unitson hand

1. You receive an order quantity Q.

2. Your start using them up over time. 3. When you reach down to

a level of inventory of R, you place your next Q sized order.

4. The cycle then repeats.

EOQ MODEL: MECHANISM

Rajiv Misra

D demand rate (units per year)

c unit production cost, not counting setup or inventory costs

A fixed or setup cost to place an order

h holding cost (Rs per year); if the holding cost is consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate.

Q the unknown size of the order or lot size decision variable

EOQ MODEL: NOTATION

Rajiv Misra

Annual Holding Cost

Annual Setup Costs:

Annual Purchase Cost:

EOQ MODEL: COSTS

Rajiv Misra

EOQ MODEL: AVERAGE INVENTORY

Rajiv Misra

Annual Holding Cost:

Annual Setup Costs:

Annual Purchase Cost: cD per year

Cost Function:

2cost holding annual

2inventory average

hQ

Q

=

=

== QDQAD /(cost setup

cDQDAhQQY ++=

2)(

No of setups/yr)

Holding Setup Production

EOQ MODEL: TOTAL COST

Rajiv Misra

The total cost curve reaches its minimum where the carrying and ordering costs are equal.

QO

Ann

ual C

ost

T C Q H DQ

S= +2

Ordering Costs

Order Quantity (Q)(optimal order quantity)

EOQ MODEL: TOTAL COST

Rajiv Misra

hADQ

QDAh

dQQdY

2

02

)(

*

2

=

=−=

EOQ Square Root Formula

cDQDAhQQY ++=

2)(

EOQ

Rajiv Misra

A grocery store pays Rs 100 for each delivery of milk from the dairy. They sell 200 liters per week. Each liter costs the store Rs2, and they sell it for Rs3. They earn a 15% return on cash that is invested in milk. They have a large refrigerator that holds up to 2,000 liter. It costs them Rs10,000 per year to maintain.

EOQ EXAMPLE

Rajiv Misra

D ~ 200(52) = 10,400 liter per year.h ~ 15%(Rs 2) = $.30 per liter per yearA ~ Rs 100

Q* = 2,633 Roughly one replenishment every 9 weeks?!?!

EOQ EXAMPLE: SOLUTION

A grocery store pays Rs 100 for each delivery of milk from the dairy. They sell 200 liters per week. Each liter costs the store Rs2, and they sell it for Rs3. They earn a 15% return on cash that is invested in milk. They have a large refrigerator that holds up to 2,000 liter. It costs them Rs10,000 per year to maintain.

Rajiv Misra

Suppose that the refrigerator held only 100 liter. How much would it be worth to expand capacity to 200 liter?

EOQ EXAMPLE: STORAGE CAPACITY

Rajiv Misra

Suppose that the refrigerator held only 100 liter. How much would it be worth to expand capacity to 200 liter?

415,102

1003.0100

400,10100)100( RsRsRsTC =+=

230,52

2003.0200

400,10100)200( RsRsRsTC =+=

Rs 5,185 per year in operational savings from doublingfreezer size.

EOQ EXAMPLE: STORAGE CAPACITY

Rajiv Misra

A bank has determined that it costs Rs30 to replenish the cash in one of its suburban ATMs. Customers take cash out of the ATM at a rate of Rs 2000 per day (365 days per year). The bank earns a 10% return on cash that is not sitting in an ATM. Let us define Rs1000 to be a basic unit of cash.

EOQ EXAMPLE: ATM

Rajiv Misra

A bank has determined that it costs Rs30 to replenish the cash in one of its suburban ATMs. Customers take cash out of the ATM at a rate of Rs 2000 per day (365 days per year). The bank earns a 10% return on cash that is not sitting in an ATM. Let us define Rs1000 to be a basic unit of cash.

D ~ Rs 2K(365) = $730K per year.h ~ 10%($1K) = Rs100 per K per yearS ~ Rs30

Q* = Rs 20.9K Roughly one replenishment every 10 days.

EOQ EXAMPLE: ATM

Rajiv Misra

Reorderpoint, R

Q

0

Inve

ntor

y le

vel

LT LTTime

Safety stock

EOQ MODEL: MANAGING UNCERTAINTY

Rajiv Misra

LT

Expected demandduring lead time

Maximum probable demand during lead time

Reorder point R

Qua

ntity

Safety stock


Rajiv Misra

R

Risk of a stock-out

Service level =

probability of no stock-out

Expecteddemand

Safety-stock

0 z

Quantity

z-scale

Q


The value for “z” is found by using the Excel NORMSINV (Service Level) function.

Rajiv Misra


0 1 2 3 4 5 6 7 8

Average Demand each week = 100Standard Deviation = 4Lead Time = 2Service Level = 95%

Find out the Safety Stock?

Rajiv Misra


0 1 2 3 4 5 6 7 8

Average Demand during two weeks = 200

Rajiv Misra

R

Risk of a stock-out

Service level =

probability of no stock-out

Expecteddemand

Safety-stock

0 z

Quantity

z-scale

Q


The value for “z” is found by using the Excel NORMSINV (Service Level) function.

200 ?

Rajiv Misra


0 1 2 3 4 5 6 7 8

Average Demand Lead Time = 200Standard Deviation =

Rajiv Misra


0 1 2 3 4 5 6 7 8

Average Demand Lead Time = 200Standard Deviation =

SD 4 4

( )( ) =42 2

Rajiv Misra

( )( )

z Stock Safety

5.656854 =42 =

L

2L

σ

σ

=

The value for “z” is found by using the Excel NORMSINV function.

So, if we want to be 95 percent sure of not stocking out, z = 1.64, we need to carry 10 units of safety stock (1.64 x 5.6569 = 9.3).


Rajiv Misra

( )

2wL

w

L

1i

2wL

(L) =

constant, is σ andt independen iseach week Since

=

σσ

σσ ∑=

The standard deviation of a sequence of random events equals thesquare root of the sum of the variances.

SAFETY STOCK CALCULATION

Rajiv Misra

DEVELOPING INVENTORY INTUITION

Rajiv Misra

Consider a situation with an item that we wish to stock on an ongoing basis. The demand for the item averages 100 units per week with a standard deviation of 30 units. Lead time is 3 weeks and the EOQ is 400 units. Probability of not stocking out is 97%.

What is the reorder point?


Rajiv Misra

Reorder Point = Expected Demand During Lead Time + Safety Stock = 300 + 98 = 398

Expected Demand During Lead Time = 100 x 3 = 300 unitsSafety Stock

• Standard Deviation of demand during lead time = sqrt( 3 x 302) = 51.96• NORMSINV(.97) = 1.88• Safety Stock = 1.88 x 51.96 = 97.688 (98 units)


Rajiv Misra

How often do you expect the item to be ordered?

What is the expected average inventory for the item.


Rajiv Misra

Expect to order every 4 weeks (400/100 = 4).

Average Inventory = Q/2 + Safety Stock = 298


Rajiv Misra

In general, what percentage of the yearly demand do you expect to meet? Would it equal 97%? Be greater than 97%? Or less than 97%? Why?


Rajiv Misra

More that 97%.

Why?? Only taking the risk over the lead time.


Rajiv Misra

Currently there are 278 units on hand. The firm has just placed an order for the item. What it the probability they will stock out before the order arrives?


Rajiv Misra

Expected demand over the 3 weeks is 300 units. Therefore, there are -22 units of safety stock. The z-score associated with

-22 units = -22/51.96 = -.4234. This corresponds to a probability of not stocking out of .336 (NORMSDIST(-.4234)). Probability of stocking out .664 (66 percent).


Rajiv Misra

Dependent vs Independent DemandFixed Order Quantity Model

• Reorder Point, Order Quantity

Safety Stock – Demand Uncertainty

SUMMARY

FIXED TIME PERIOD REVIEW

Rajiv Misra

Periodic Review System

Time

Inve

ntor

y Le

vel

T T

T – Time between ordersThis is fixed.

Max Level – At the beginning of each order cycle order up tothis level.

If the lead time (L) were zero,what should the max level be?

Q = Max – Current Inv. Position

Max = DemandT + SST

Q


Rajiv Misra

Periodic Review System

Time

Inve

ntor

y Le

vel

T T

T – Time between ordersThis is fixed.

Max Level – At the beginning of each order cycle order up tothis level.

What if the lead time (L) were not zero, what should the max level be?

Q = Max – Current Inv. Position

Max = DemandT+L + SST+L

Q

L L


order)on items (includes levelinventory current = I timelead and review over the demand ofdeviation standard =

yprobabilit service specified afor deviations standard ofnumber the= zdemanddaily averageforecast = d

daysin timelead = Lreviewsbetween days ofnumber the= T

ordered be toquantitiy = q:Where

I - Z+ L)+(Td = q

L+T

L+T

σ

σ

q = Average demand + Safety stock – Inventory currently on handq = Average demand + Safety stock – Inventory currently on hand


Multi-Period Models: Fixed-Time Period Model: Determining the Value of σT+L

( )σ σ

σ

σ σ

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

=∑( )σ σ

σ

σ σ

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

=∑

The standard deviation of a sequence of random events equals the square root of the sum of the variances


Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The daily demand standard deviation is 4 units.

Given the information below, how many units should be ordered?Given the information below, how many units should be ordered?


( )( )σ σT+ L d2 2 = (T + L) = 30 + 10 4 = 25.298

The value for “z” is found by using the Excel NORMSINV function, The value for “z” is found by using the Excel NORMSINV function,

z = 1.75z = 1.75


or 644.272, = 200 - 44.272 800 = q

200- 298)(1.75)(25. + 10)+20(30 = q

I - Z+ L)+(Td = q L+T

units 645+

σ

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period


inventory mangement pomii class

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