Models of Inventory• There are different models of inventory. The

inventory models can be classified into deterministic and probabilistic models. The various deterministic models are as given below

– Purchase model with instantaneous replenishment and without shortages

– Manufacturing model without shortages– Purchase model with instantaneous replenishment and

with shortages– Manufacturing model with shortages

Purchase model with instantaneous replenishment and without shortages

In this model of inventory, 1. Orders of equal size are placed at periodical

intervals. 2. The items against an order are replenished

instantaneously3. The items are consumed at a constant time4. The purchase price per unit is the same

irrespective of order size.

• Let D be the annual demand in units• Co be the ordering cost / order

• Cc be the carrying cost / order• P be the purchase price per unit• Q be the order size

Time

Units

Q

t

• The Number of orders/ Year = D/Q

• Average inventory = Q/2

• Cost of ordering / year = ( D/Q) x Co

2)(

dQd

Yields Q w.r.t atingDifferenti

P D2Q

QD

yearTC)/ (cost inventory totalTheP D year cost/ Purchase

2Q /year carrying ofCost

2c

o

co

c

CCQDTC

CC

C

D*Q ordersbetween Time

*QD Orders of No

2*

2

02

2

2

c

o

c

o

co

CDCQ

CDCQ

CCQD

• Example: Alpha industry estimates that it will sell 12000 units of its products for the forthcoming year. The ordering cost is Rs. 100 per order and the carrying cost per unit per year is 20% of the purchase price per unit. The purchase price per unit is Rs. 50.

• Find – a) Economic order qty– b) No of orders per year– c) Time between successive orders

• a) D = 12000 units / year

• Co = Rs. 100/ order

• Cc = Rs. 50 X 0.2• = Rs. 10/ unit /year• Solution:

year

Aproxunits

CDCQc

o

12000490

D*Q ordersbetween Time

49.24490

12000*Q

D Orders of No

)(49010

120001002

2*

• Manufacturing Model without Shortages– If a company manufactures its components

which is required for its main product then the corresponding model of inventory is called Manufacturing model.

– This model will be with shortages or without shortages.

– The rate of consumption of items is uniform throughout the year.

– The cost of production per unit is same irrespective of production lot size.

• Let• R be the annual demand of an item• k be the production rate of the item (No of

Units produced per year)• Co be the ordering cost / order• Cc be the carrying cost / order• P be the production per unit• EBQ be economic batch qty• The operation of the manufacturing model

without shortages is show in figure

• Let• R be the annual demand of an item• k be the production rate of the item (No of

Units produced per year)• Co be the ordering cost / order• Cc be the carrying cost / order• P be the production per unit• EBQ be economic batch qty• The operation of the manufacturing model

without shortages is shown in the foll. figure

• Let• R be the annual demand of an item• k be the production rate of the item (No of

Units produced per year)• Co be the ordering cost / order• Cc be the carrying cost / order• P be the production per unit• EBQ be economic batch qty• The operation of the manufacturing model

without shortages is show in figure

Manufacturing Model without stock out

Manufacturing Model with out stock.

Units

Timet1 t2

K-r r K-r r

• During the period t1,the item is produced at the rate of k units per period

• simultaneously it is consumed at the rate of r units per period.

• So during this period, the inventory is built at the rate of k-r units per period.

• During the period t2, the production of the item is discontinued but the consumption of that item is continued. Hence the inventory is decreased at the rate of r units per period during this period. The various formulas for this situation are given here

21

2

1

t timeCycle

]/1[*

*

)/1(2

tr

krQt

kQt

krCrCEBQ

c

o

• If a product is to be manufactured within the company, the details are as follows

• r = 24000 units/year• K= 48000 units/year• Co = Rs. 200per set up

• Cc = Rs. 20/unit/year

Find the EBQ and Cycle time.

montht

monthr

krQt

monthyearkQt

krCrCEBQ

c

o

48.0 t timeCycle

24.048000240001

24000980]/1[*

24.002.048000/980*Approx) ( Units980

48000/240001(20240002002

)/1(2

21

2

1

Purchase model with shortages• In this model, the items on order will be received

instantaneously and they are consumed at a constant rate.

• The purchase price per unit remains same irrespective of order size.

• If there is no stock at the time of receiving a request for the items, it is assumed that it will be satisfied at a later date with a penalty. This is called backordering

Purchase model of Inventory with stock out• Q- EOQ; Q1-Max inventory;Q2-Max stock out

t1

t2

Q1

Q2

Q

Time

DQt

DtDQt

Q

CCCDC

CCCCD

csc

s

sc

cs

/*

/*Q*/**

*-Q*Q*

)(C2*Q

)(2CEOQQ*

2*2

11

12

o1

o

• Example: The annual demand for an automobile component is 24000 units. The carrying cost is Re 0.40/unit/year, the ordering cost is Rs. 20.00 per order and the shortage cost is Rs. 10.0/ unit/year. Find the optimal values of the following:

• Economic order qty• Maximum Inventory• Maximum Shortage qty• Cycle Time• Inventory period ( t1)• Shortage period ( t2).

• D = 24000 units/ year• Cc = Rs0.40/unit/year

• Co= Rs. 20.00/ order

• Cs =Rs.10.00/unit/year• Solution:

dayDQt

Dt

daysDQt

unitsQCCC

DC

CCCCD

csc

s

sc

cs

12324/*

days 23365 24000/1520/*Q*

24365240001580/**

60*-Q*Q*

1520)(

C2*Q

1580)(2CEOQQ*

2*2

11

12

o1

o

Manufacturing Model with Shortages

• In this model, the items are produced and consumed simultaneously for a portion of the cycle time.

• The rate of consumption of items is uniform throughout the year.

• The cost of production per unit is the same irrespective of production lot size.

• In this model stock out is permitted. It is assumed that the stock out units will be satisfied from the units which will be produced at a later date with a penalty. This is called backordering

r Be the annual demand of an item

k be the production rate of the item

Co The cost / set up

Cc Be the carrying cost/unit/period

Cs Be the shortage cost/unit/period

p Be the cost of production/ unit

• Manufacturing model of Inventory with stock out

t1 t2

K-r

r

t3 t4

Q1

Q2

• In the above model• Q-Economic Batch Qty• Q1- Maximum Inventory• Q2- Maximum Stockout

*2

*1

o*2

o*1

o*

*

)CC(C)(C2C

)CC(CC)(2C

C)(C)CC(2C Q

QQkrkQ

krkrQ

krkrQ

rkkrEBQ

scs

c

scc

s

sc

sc

)/(

/

/

)/(

/

*2

*4

*2

*3

*1

*2

*1

*1

**

rkQt

rQt

rQt

rkQt

rQt

• The demand for an item is 18000 per year. Its production rate is 3000 per month. The carrying cost is Re 0.15/unit/month and the setup cost is Rs. 500 per set up. The shortage cost is Rs20.00 per unit per year. Find various parameters of the inventory system

• r = 18000, units/year• k= 3000 x 12 = 36000units / yr• Co = Rs. 500.00/ Set up• Cc = Rs. (0.15 X 12) = 1.80/ yr• Cs = Rs 20.00 unit/ year

• Solution:

Units466920)1800036000(1.80

)2080.1(18000360005002

C)(C)CC(2C Q o*

sc

sc

rkkrEBQ

Units2142

193466936000

1800036000*

19336000)208.1(20

)1800036000(180000052)CC(C

)(C2C

*2

*1

o*2

QQkrkQ

UnitskrkrQ

scs

c

daysrkQt

daysrQt

daysrQt

daysrkQt

daysrQt

4)1800036000/(193)/(

418000/193/

5.4318000/2142/

5.43)1800036000/(2142)/(

9518000/4669/

*2

*4

*2

*3

*1

*2

*1

*1

**

A plant manager of a chemical plant must determine the lot size for a particular chemical that has a steady demand of 30 barrels per day. The production rate is 190 barrels per day, annual demand is 10,500 barrels, setup cost is $200, annual holding cost is $0.21 per barrel and the plant operates 350 days per year.Determine the economic production lot sizeDetermine the total annual setup and inventory holding

cost for this itemDetermine the cycle length for the ELS(Economic lot size)Determine the production time per lot