Inventory control

Krishna Murari

Operations Management

Inventory & inventory system

• Inventory is the set of items that an organization holds for later use by the organization.

• An inventory system is a set of policies that monitors and controls inventory. It determines how much of each item should be kept, when items should be replenished, and how many items should be ordered or made when replenishment is needed.

Inventory & inventory system

• Inventory includes raw materials, semi- finished goods (work in progress) and finished goods

• A Firm can have inventory of personnel, machines and working capital.

• Airlines can have inventory of seats; a modern drugstore- an inventory of medicines and an engineering firm – inventory of engineering talent.

• Basic purpose of inventory analysis is to specify when item should be ordered and how large the order should be.

Basic types of inventory

• independent demand,

• dependent demand, and

• supplies.

Independent and Dependent Demands

• Independent demand items are those items that we sell to customers.

• Dependent demand items are those items whose demand is determined by other items. Demand for a car translates into demand for four tires, one engine, one transmission, and so on. The items used in the production of that car (the independent demand item) are the dependent demand items.

• Supplies are items such as copier paper, cleaning materials, and pens that are not used directly in the production of independent demand items

Purpose For Inventory

1. Smooth Production to take care seasonal demand

2. Better service to customer – even in case of temporary stoppage in production, customer can get the required product.

3. To support strategic plan – Match with aggregate plan of manufacturing taking the inventory in account and allow flexible production schedules;

4. To protect against business uncertainties – helps in taking advantages of speculative and unexpected opportunities like rise in raw material prices

5. To meet variations in demand.

6. As a safeguard against variations in delivery time; and

7. To take advantage of economies of scale: To get materials at lower price.

The cost of Inventory

• Holding costs-it consists of i) storage cost,

ii) capital cost and iii) obsolescence/ shrinkage cost.

Storage cost includes rent, depreciation, insurance, tax, security, personnel, etc; capital cost includes loss of interest, opportunity cost interest paid; Shrinkage cost includes pilferage and breakage.

• Setup or ordering costs – These are fixed costs associated with the production of a lot internally and placing an order externally with a vendor. These are independent of the no. of units ordered. Setup costs includes time for setup of jigs/fixture etc. Ordering cost includes telephone charges, delivery fee, time required for purchase order, expediting cost.

The cost of Inventory

• Purchase costs – These are actual costs of the material purchase. Purchase cost remains constant unless discounts are offered.

• Shortage (or stock out) costs – When the stock

of an item is reduced and a customer orders, stock out occurs. This is sum of lost profit and “ill-will” generated.

Inventory Control

The control of inventories is accomplished The control of inventories is accomplished through a series of inventory records and reports.through a series of inventory records and reports.

These reports provide the following information:These reports provide the following information:– inventory use inventory use – inventory balancesinventory balances– minimum and maximum levels of stockminimum and maximum levels of stock

Inventory System An inventory system provides the organizational An inventory system provides the organizational

structure and operating policies for maintaining structure and operating policies for maintaining and controlling goods to be stocked. and controlling goods to be stocked.

The system is responsible for :The system is responsible for :

i) ordering and receipt of goodsi) ordering and receipt of goods

ii) timing for order placementii) timing for order placement

iii) Keeping track of what has been ordered, how iii) Keeping track of what has been ordered, how much and from whom. much and from whom.

iv) to check that vendor has received the orderiv) to check that vendor has received the order

v) to check that vendor has dispatched the item.v) to check that vendor has dispatched the item.

vi) Procedure for reordering and returning vi) Procedure for reordering and returning undesirable goods.undesirable goods.

Classified Models

i) i) Fixed order quantity models (called as Q systems) Fixed order quantity models (called as Q systems) or Q/R inventory systemor Q/R inventory system

Fixed quantity models are event triggered i.e. Fixed quantity models are event triggered i.e. these initiate an order when the event of reaching these initiate an order when the event of reaching a specified reorder level occurs. It happens any a specified reorder level occurs. It happens any time depend on the demand for the item.time depend on the demand for the item.

ii) ii) Fixed –time period models (called as P systems) Fixed –time period models (called as P systems) or periodic inventory systemor periodic inventory system

These are time triggered. These initiate an order a These are time triggered. These initiate an order a the end of predetermined time period and only the end of predetermined time period and only passage of time triggers the model. passage of time triggers the model.

. .

Difference between Fixed order quantity and Fixed-time period Models

The fixed time period model has a larger average inventory The fixed time period model has a larger average inventory to protect against stock out during the review period. Fixed to protect against stock out during the review period. Fixed quantity model has no review period.quantity model has no review period.

The fixed time period model is preferred when several The fixed time period model is preferred when several different items are purchased from same vendor as it different items are purchased from same vendor as it results in savings of ordering same time.results in savings of ordering same time.

The fixed order quantity model is useful for more The fixed order quantity model is useful for more expensive items as average inventory is low.expensive items as average inventory is low.

The fixed order quantity model is more appropriate for The fixed order quantity model is more appropriate for important items as it has close monitoring.important items as it has close monitoring.

The fixed –order quantity model requires more time and The fixed –order quantity model requires more time and resources to maintain as every addition or withdrawal is resources to maintain as every addition or withdrawal is recorded. recorded.

Basic Fixed –Order Quantity Model (Deterministic Model)

Assumption- all aspects of the situation are known with certainty. E.g. demand, Setup cost and holding cost are known precisely.

It tries to identify the specific point R at which order is to be placed for quantity Q. R is always a specified number of units actually in inventory.

Basic Fixed –Order Quantity Model

R

NO

. O

F U

NIT

S O

N

HA

ND

Time L

Q

Saw tooth relationship shows that when inventory drops to point R, reorder takes place at end of time period L (lead time)

Basic Fixed –Order Quantity Model

Optimal order quantity is derived based on Optimal order quantity is derived based on assumption:assumption:

• Demand is known, constant and uniform Demand is known, constant and uniform throughout the periodthroughout the period

• Lead time is constantLead time is constant• Price per unit is constantPrice per unit is constant• Ordering or setup cost is constantOrdering or setup cost is constant• All demands for the product will be satisfied All demands for the product will be satisfied

(no back orders are allowed)(no back orders are allowed)• There is no interaction with other product. There is no interaction with other product.

Basic Fixed –Order Quantity Model Functional RelationshipFunctional Relationship

Total annual cost = annual purchase cost + annual Total annual cost = annual purchase cost + annual ordering cost + annual holding costordering cost + annual holding cost

TC=DC+ (D/Q)S + (Q/2) HTC=DC+ (D/Q)S + (Q/2) H

Where Where

TC = total costTC = total cost

D= Demand (annual)D= Demand (annual)

C= Cost per unitC= Cost per unit

Q= Quantity to be orderedQ= Quantity to be ordered

S= setup cost or cost of placing orderS= setup cost or cost of placing order

R= Reorder point R= Reorder point

L= lead timeL= lead time

H= Annual holding cost and storage cost per unit of H= Annual holding cost and storage cost per unit of

average inventoryaverage inventory

D/Q = no. of orders placed

Q/2 = average inventory

Basic Fixed –Order Quantity Model

lead time is the time between placing an order and the receipt of the goods.

inventory usage rate is the quantity of the goods used over a period of time.

Reorder point

It is the level of inventory at which it is desirable to order or produce additional items to avoid an out-of-stock condition.Reorder point= Lead time in days × Average

inventory usage rate/dayor R= dL = average demand/time period x no. of time periods Economic order quantity (EOQ)

It is the order quantity that minimizes total inventory order costs and inventory carrying costs.

Basic Fixed –Order Quantity Modelc

ost

Order Quantity Size

TC=Total Cost

Q/2 x H= Annual Holding cost

DC=Annual purchasecost of items

D/Q x S= Annual ordering cost

EOQ

Basic Fixed –Order Quantity Model

EOQ = economic order quantityEOQ = economic order quantity D = Demand (annual)D = Demand (annual) S = Set up cost or cost of placing order S = Set up cost or cost of placing order H = inventory carrying cost per unit H = inventory carrying cost per unit

(insurance, taxes, interest, storage costs)(insurance, taxes, interest, storage costs)

EOQ =EOQ = 2 2 ×× D D ×× S S HH

In 1913 F.W. Harris developed the Economical Order Quantity model to determine optimum order quantity

Basic Fixed –Order Quantity Model

If the carrying cost is expressed as a fraction If the carrying cost is expressed as a fraction of the inventory value then, H can be of the inventory value then, H can be replaced in the EOQ formula by s.f where s replaced in the EOQ formula by s.f where s is the prices per unit and f is the fraction of is the prices per unit and f is the fraction of inventory valueinventory value

H = s.f H = s.f

EOQ =EOQ = 2 2 ×× D D ×× S S s.f s.f

Basic Fixed –Order Quantity Model

If EOQ is expressed in terms of rupees and If EOQ is expressed in terms of rupees and not in terms of items, the classical formula not in terms of items, the classical formula for EOQ is modified as follows for EOQ is modified as follows

EOQ =EOQ = 2 2 ×× D D ×× S S f f

f = fraction (carrying cost expressed as fraction f = fraction (carrying cost expressed as fraction of the inventory value)of the inventory value)

Note : D also to be in Rupees i.e. total worth of Note : D also to be in Rupees i.e. total worth of the items.the items.

Example 1: Basic Fixed order Quantity model

Find the economical order quantity and Find the economical order quantity and reorder point, given the following data :reorder point, given the following data :

Annual demand (D)= 1,000 unitsAnnual demand (D)= 1,000 units

Ordering cost (S) = Rs 1000 per orderOrdering cost (S) = Rs 1000 per order

Holding Cost (H) = Rs. 200 per unitHolding Cost (H) = Rs. 200 per unit

Cost per unit ( C) = Rs. 5000 Cost per unit ( C) = Rs. 5000

Lead time (L) = 7 daysLead time (L) = 7 days

Average daily demand = 1,000 / 365 Average daily demand = 1,000 / 365

Also calculate total annual cost. Also calculate total annual cost.

Example1 : Basic Fixed order Quantity model

EOQ = EOQ = 2 2 ×× D D ×× S S = 2 x 1000 x 1000 = 2 x 1000 x 1000 H 200H 200 = 100= 100Reorder Point R = dL = (1000/365) x 7 = 19.17 =19Reorder Point R = dL = (1000/365) x 7 = 19.17 =19

When no. of unit drops to 19, order for 100 more is When no. of unit drops to 19, order for 100 more is placed.placed.

Total Annual cost TC = DC + (D/Q) S + (Q/2) HTotal Annual cost TC = DC + (D/Q) S + (Q/2) H = 1000 x 5000 + (1000/100) 1000= 1000 x 5000 + (1000/100) 1000

+ (1000/2) 200+ (1000/2) 200 = Rs 51,10,000= Rs 51,10,000

Fixed order Quantity model with Usage -Gradual Replacement Model (Deterministic Model)

In some situation item does not come in lot rather In some situation item does not come in lot rather production of inventory item and its usage takes place production of inventory item and its usage takes place simultaneously. Also if a company asks for staggered simultaneously. Also if a company asks for staggered quantity supply. Where vendor makes delivery more quantity supply. Where vendor makes delivery more frequently. frequently.

If r = demand rate and p = production rate If r = demand rate and p = production rate

TC= DC +(D/Q)S + (Imax /2) HTC= DC +(D/Q)S + (Imax /2) H

Where, I max = (p-r)(Q/p) as Q will not be maximum Where, I max = (p-r)(Q/p) as Q will not be maximum inventoryinventory

p-r = amount of inventory accumulate each time, and p-r = amount of inventory accumulate each time, and Q/p= no. of time periods required to fill the orderQ/p= no. of time periods required to fill the order

Fixed order Quantity model with Usage TC= DC +(D/Q)S + (p-d)(Q/2p) HTC= DC +(D/Q)S + (p-d)(Q/2p) H

EOQ = 2DS x pEOQ = 2DS x p

H x(p-d)H x(p-d)

R

NO

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S O

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Time

I maxBuild up = production rate- usage rate=p-d

Usage rate d

L

Q

Example 2

TMC shoes Ltd is a manufacturing company which produces

variety of shoes. It has a retail shop attached to it. The

production manager wants to ascertain the optimal number of

shoes to produced with each production run. Following data

has been collected by him.

Annual demand for shoes = 12,000 pairs

Days/ year when retail shop opens = 240

Daily production rate = 200 pairs

Set up cost to start shoe production = Rs 8000

Annual storage cost per pair of shoes = Rs. 200

What should be optimum production lot size.

Example 2

D= Annual demand = 12000

d= daily demand = 12000/ 240 = 50

p = daily production = 200 pairs

S= set up cost = Rs 8000

Annual storage cost H= Rs 200

EOQ = 2DS x pEOQ = 2DS x p H x(p-d)H x(p-d)

= 1131.4 = 1131= 1131.4 = 1131

=2 x 12000 x 8000 x 200

200 x (200-50)

Example2 : Basic Fixed order Quantity model

Bharati Fibers Production Limited produces a Bharati Fibers Production Limited produces a special fibre at the rate of 5000 meters per special fibre at the rate of 5000 meters per hour. The fibre is used in the other products hour. The fibre is used in the other products made at Bharati, at the rate of 20,000 meters made at Bharati, at the rate of 20,000 meters per day ( in the 8 hour day). The cost of fiber is per day ( in the 8 hour day). The cost of fiber is Rs. 5 per meter. The inventory carrying cost is Rs. 5 per meter. The inventory carrying cost is 25 percent and the set-up cost are Rs. 4050 per 25 percent and the set-up cost are Rs. 4050 per set-up. Compute the optimum number of set-up. Compute the optimum number of cycles required in a year for the manufacture of cycles required in a year for the manufacture of this special fibre. this special fibre.

Example3 : Basic Fixed order Quantity model

Solution: Solution: Rate or production , p = 5000 m/ hrRate or production , p = 5000 m/ hrRate of usage, r = 20,000 m per day = 2500 m / hr.Rate of usage, r = 20,000 m per day = 2500 m / hr.s = Rs 5 per meter, f = 0.25 s = Rs 5 per meter, f = 0.25 set up cost, S= Rs 4050 per set upset up cost, S= Rs 4050 per set upD = annual demand =20,000 x 365 per year D = annual demand =20,000 x 365 per year

EBQ = Qopt = √ 2DS / [s.f ( p-r)/p ]EBQ = Qopt = √ 2DS / [s.f ( p-r)/p ]

= √ 2x4050x20,000 x365 / [ 5 x 0.25 (5000-2500)/5000= √ 2x4050x20,000 x365 / [ 5 x 0.25 (5000-2500)/5000

= 3,07,584 m= 3,07,584 m

Optimum number of cycles = D/ EBQ = 20,000 x 365 / 3,07,584Optimum number of cycles = D/ EBQ = 20,000 x 365 / 3,07,584 =23.7 ≈ 24 cycles.=23.7 ≈ 24 cycles.

Multiple Products by Common facilities

• Optimum number of joint cycle for multiple products from common facilities

k

∑ Cci Ai ( 1- ri / pi)

n opt = i=1

k

2 ∑ Coi

i=1

Cci= carrying cost for ith item Coi = order/set up cost for ith item

Ai = annual requirements for ith item

ri = usage rate of ith item per day = annual demand / no. of days

pi = production rate of ith item

EOQ of ith item = annual demand of ith item / n opt

Example 4 : Multiple Products by Common facilities

• Elisa watches assembles four different type of watches on their assembly line. The assembly is done in batches. Given the following data, which batch sizes should each type of watches be produced.? The company has 300 working days.

Watches Annaul demand

Ai

Set up cost Rs.

Coi

Carrying cost Rs.

CCi

Assembly rate /day

pi

Poise 9000 1000 40 100

Elegant 5000 1000 50 120

Dainty 10000 1000 120 100

Intellectual 3000 1000 30 90

Example 4 : Multiple Products by Common facilities

Ai ri =Ai/300

pi 1-ri/pi Cci CciAi (1-r1/pi)

9000 30 100 0.700 40 25200

5000 16.66 120 0.861 50 51660

10000 33.33 100 0.667 120 66700

3000 10 90 0.889 30 240030

total 16,75,630

Example 4 : Multiple Products by Common facilities

• Optimum number of joint cycle for multiple products from common facilities

k

∑ Cci Ai ( 1- ri / pi)

n opt = i=1 16,75,630

k =

2 ∑ Coi 2 x4000

i=1

= 14.47

Batch quantity of Poise = 9000/ 14.47 = 621.97 = 622

Batch quantity of Elegant = 5000/ 14.47 = 345.54 = 346

Batch quantity of Dainty = 10,000/ 14.47 = 691.08 = 691

Batch quantity of Intellectual = 3000/ 14.47 = 207.32 = 207

EOQ under inflation

• If an item is going to cost more in future, one tends to procure more of items now than in future. EOQ formula is modified to take care of inflation

EOQ = 2 Co A ( 1 + i/2) s(f-i)

S = purchase price at beginning of yearF = inventory carrying cost as fraction of inventory

valuesi= inflation rate per year, expressed as fraction

Example 5

Ravi foods buys 1000 tonnes of rice from market every year. Each order costs Rs. 10,000. Cost of holding inventory is 30 percent. Present price of wheat per tones is Rs. 10,000. Inflation rate is 15 percent. What should be optimum order quantity for wheat ?

EOQ = 2 x 10,000 x 1000 ( 1+ 0.15/2)

10000 x (0.30-01.5)

= 119.72 = 120 tonnes approx.

EOQ with Quantity Discount

When the items are bought in bulk, the supplier often gives a price discount. But the inventory level and carrying cost increases. The situation may be :

Quantity Price per unit Less than b s1B or more than B s2The total relevant costs areWhen Q<b TC = Co.D/Q+Q/2.s1.f +D.s1 where Co –ordering cost ,f is inventory carrying cost in fraction of inventory value When Q>or = b TC = Co.D/Q+Q/2.s2.f +D.s2

contd..

EOQ with Quantity Discount –one price break point

s1s2

b

TC

Size of order

EOQ with Quantity Discount EOQ is given by

Q1 opt = 2 Co D for price s1

S1.f

Q2 opt = 2 Co D for price s2 S2.f

The nature of Total Cost curve depends on where the price break b is situated.= vis-à-vis the minimum for two different prices. Q1opt and Q2opt may not fall in its real region. For instance, Q2 opt may be less than b.

Hence for optimization the costs at Q1opt, Q2opt and cost at b are compared .

contd..

EOQ with Quantity Discount

The procedure for one price break is

i) Calculate Q2opt

ii) If Q2 opt is in its range then it is minimum, if it is not, then

iii)Compare TC at Q1opt and b

iv)If TC at Q1opt is less than it is optimum otherwise optimum order quantity is b

EOQ with Quantity Discount – two price break points

s1s2

b1

TC

Size of order

b2

s3

EOQ with Quantity Discount The procedure for two price breaks is

i) Calculate Q3opt

ii) If Q3 opt is in its range (i.e. if Q3opt > or = b2 ) then it is minimum, if it is not, then go to next step

iii)If Q3opt <b2 compute Q2opt

iv)Compare Q2opt with b1 and if Q2opt > or = b1 and Q2opt is < b2 then compare TC(Q2opt) with TC(b2)

if TC(Q2opt ) < TC(b2) then EOQ = Q2opt or otherwise EOQ = b2

v) If Q2opt is less than b1 then compare TC(Q1opt) and TC(b2) and TC(b1). Which ever is minimum that will be the EOQ

Example 6:EOQ with Quantity Discount A book vendor purchase note books from a stockiest the

price are given below.

Quantity, nos. Price per unit. Rs.

1 to 99 50.00

100 to 499 45.00

500 and above 40.00

His annual requirement is 2000 notebooks. The ordering cost to be Rs. 25 per order and the inventory carrying cost is 30 percent. What is the optimum quantity to be ordered by the vendor.

Example 6 :EOQ with Quantity Discount Q3opt = √ 2x2000x25/ 40 x 0.3 = 91.2

It is not in its range of 500 and above, it is in range of Q1opt

Q1 opt = √ 2 x 2000x 25 / 50 x 0.3 = 81.65

So we will compare TC(Q1opt) TC (b2) and TC(b1)

TC = Co. D/Q + Q/2.d.f +D.s

TC(Q1opt) = (25 x 2000/81.65) + (81.65/2 x 50 x 0.3 ) +

2000 x 50 = Rs.101,224.75

TC(b1) = (25 x 2000/100) + (100/2 x 45 x 0.3) + 2000x45

= Rs. 91,250

TC(b2) = (25 x 2000/500) + (500/2 x 40x0.3) + (2000x40)

= Rs.83,100 hence EOQ =b2 = 500

Fixed time Period Model

In this model, inventory is counted at fixed interval such as every week or every month.

This is useful when vendors make routine visits to customers and takes orders for their complete line of products or when buyers want to combine order to save transportation cost. Also some firms operate on fixed time period to facilitate their inventory counts.

The time interval is chosen as convenient. If only one item is involved it can be estimated by using EOQ formula. And no. of order and time period by dividing total demand by EOQ

Fixed time Period Model With a fixed –time period model, there is a ceiling or “par” inventory that is determined for each item. The difference between the par value and the quantity in hand when the count is taken is the amount is ordered. This varies from period to period depending on usage.

Thus in this model, the time period remains constant but quantity varies.

In this system, the uncertainty i.e. enhanced usage rate, can start immediately after an order is placed. But the remedial action can be taken only on next review. Hence a safety stock is required for a period of review plus lead time.

Fixed time Period Model In

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Safety

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Stock

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T T

Place order

Place order

Place order

Q1Q2

Q3

Fixed time Period Model

The two quantities which need to be determined in a P-system are :

1. Desired order quantity

2. Target level i.e. maximum inventory on hand plus on order.

Desirable order cycle is determined as per the organisation's or the suppliers convenience.

Fixed time Period Model

Application:

1. There are so many small items, that it is not advisable to monitor inventory levels, continuously.

2. In order to make a production programme, many items are required at one time.

3. Grouping of orders from the same supplier is necessary in order to effect savings in cost e.g. shipping costs or discounts.

4. The supplier desires that order be placed only at fixed periodic intervals.

Fixed time Period Model

Optimum review period is calculated as

Popt = 1 / Nopt = Qopt / D

Where , Popt = optimum value of review period in years

N opt = optimum number of orders in a year

D = annual demand

Qopt = EOQ