inventory system - bijay lal pradhan, ph.d. · 10/8/2016 · inventory system introduction...

INVENTORY SYSTEM

Introduction

Inventory is the stock of any item or resources used in an organization. It is idle resource, but usable.

The term stock is generally used to refer to resource in its physical and tangible form, like materials.

Thus stock and inventory are identical, although inventory has wider implications. Managing the

inventory is the task of maintaining an adequate supply of materials to meet the expected demand

pattern for a given financial investment. Thus inventory management has the responsibility to decide

upon the optimal materials under given circumstances for procurement of commodities.

An inventory system is the set of policies and controls that monitor and evaluate the levels of

inventory and then determines that:

What level of inventory should be maintained

When stock should be reordered

How large orders should be placed.

The inventory which contributes to be a part of product or which becomes a product is called

production inventory or manufacturing inventory. It can be classified into 3 parts, viz: raw materials,

semi-finished goods (work-in-progress) and finished goods respectively. Where as in service

organization inventory refers to the tangible goods which are to be served or sold. For example, no.

of beads, drugs, tonics, serums etc… are the inventory of hospitals.

An inventory system is basically a process whereby a business keeps track of the goods and material

it has available. In its simplest sense it can be done manually by a count at the end of each day. In

this way it is possible to keep a record of the goods coming in to the business and goods being sold.

However this is only really appropriate for small businesses that do not have a lot of stock. For larger

business it is more likely that a computerized system will be required.

If the firm is a retail establishment, a customer may look elsewhere to have his or her needs satisfied

if the firm does not have the required item in stock when the customer arrives. If the firm is a

manufacturer, it must maintain some inventory of raw materials and work-in-process in order to

keep the factory running. In addition, it must maintain some supply of finished goods in order to

meet demand.

Sometimes, a firm may keep larger inventory than is necessary to meet demand and keep the factory

running under current conditions of demand. If the firm exists in a volatile environment where

demand is dynamic (i.e., rises and falls quickly), an on-hand inventory could be maintained as a

buffer against unexpected changes in demand. This buffer inventory also can serve to protect the

firm if a supplier fails to deliver at the required time, or if the supplier's quality is found to be

substandard upon inspection, either of which would otherwise leave the firm without the necessary

raw materials. Other reasons for maintaining an unnecessarily large inventory include buying to take

advantage of quantity discounts (i.e., the firm saves by buying in bulk), or ordering more in advance

of an impending price increase.

86 / PRODUCTION AND OPERATIONS MANAGEMENT

Importance of Inventory

Inventory management is an important part of a business because inventories are usually the largest

expense incurred from business operations. Most companies will use an inventory management

system that will track and maintain the inventory required to meet customer demand. It is very

difficult to visualize a real work situation, where the required material will be made available at the

point where it is used, at the very instant, it becomes necessary. That is what makes the maintenance

of inventories a virtual necessity. Viewed that way, inventories could be seen as a 'necessary evil'.

The inventory system is more important due to following reasons:

It ensures smooth flow of work

It ensures proper utilization of labor & machines

It helps to provide & maintain good customer service

It helps to take advantage of economic purchase order

It allows flexibility in production schedule

It increases Return on Investment

It provides the protection against future uncertainties

Reasons for Carrying Inventory

Inventory is undesirable, but necessary, because its aim is to absorb the uncertainties of demand

some major reasons for carrying inventory in an organization may be:

When lead times are longer, stocks have to be maintained for system operation.

Demand and lead times are uncertain; and that makes it necessary to build up a safety stock

(buffer stocks) in order to ensure that various sub-systems succeed in operating in a somewhat

decoupled manner.

Cycle stocks may be a means to avail of economies of scale, which in turn will enable the

reduction of total system cost, due to ordering, carrying inventory, and backlogging.

Technological requirements of batch processing in production, also lead to build up of cycle

stocks.

Stocks may grow on account of pipeline inventory, or work-in-process inventory, due to the

fixed nature of production and transportation rates. This includes materials actually being

worked on, or materials which are moving between different work centers, or being in transit

en route distribution centers and customers.

Inventory Costs

In order to maximize the profit of an organization, the management should focus on minimization of

inventory cost. The total inventory costs can be classified into following four parts. They are as

follows:

1. Ordering cost (setup cost)

The cost which is incurred for placing an order of inventory is called ordering cost. Managerial cost, transportation cost and clerical cost to prepare purchase order may be examples of ordering cost. It is also known as procurement cost. This cost is usually assumed to be independent of the quantity ordered, because the fixed cost component is generally more significant than the variable cost component. Therefore, it is expressed as a fixed amount per order for example Rs. 20 per order.

2. Carrying Cost (Holding cost)

INVENTORY SYSTEM CHAPTER - 8 87

The cost which is incurred for keeping stocks in store is called carrying cost or holding cost. Carrying cost comprises cost elements including, security, heat, light, power taxes, cost of blocking, capital i.e interest rate, cost of insurances, storage cost, cost due to obsolescence, leakage, spoilage, pilferage, deterioration, etc. The attributed carrying cost will be obviously higher for perishable items. It is generally expressed as a fraction of the value of the goods stocked per year. For example, if the fraction of carrying charge is 20% per year, and a material worth Rs. 1,000 is kept in inventory for one year, the unit carrying cost will be Rs 200 per item per year. Alternatively it can be expressed as a rand (i.e. countable) value per item. For example, holding cost per item is Rs. 1.50.

3. Material cost (Purchase price cost)

The price or amount paid to supplier for buying materials of inventory is called material cost. It

inventory is called material cost. It directly affects the cost of production.

4. Stock out cost (Shortage cost)

It is the opportunity cost of not having an item in stock, when it is demanded. These types of costs are only incurred when the demand exceeds the supply. This situation may be the result of lost sales e.g., absence of a product in stock in sufficient quantities to meet demand, on account of the company's earlier decision against the production of the item in sufficient quantities to meet expected demand in the near future, or backlogging. It includes cost of back orders, loss of potential profit, lost demand, penalty cost, emergency replenishment, loss of good will etc.

Types of Inventory Systems

There are types of commonly used inventory system. They are as follows:-

1. Continuous Inventory System (Q/R model)

Continuous inventory system is the perpetual inventory system which describes the systems

of inventory where information on inventory quantity and availability is updated on a

continuous basis as a function of doing business. Generally this is accomplished by connecting

the inventory system with order entry and in retail the point of sale system. In this case, book

inventory would be exactly the same as, or almost the same, as the real inventory.

The basic feature of this inventory system is to manage and maintain balance the use rate of inventory. Under this system, re-ordering level (R) and fixed order quantity (Q) are already pre-determined. Whenever the quantity of stock reaches at re-ordering level, then new order for purchasing is placed with fixed order quantity (Q). The order can be placed any time and it depends upon the demand of stock. Therefore withdrawals and additions of stock should be recorded and monitored regularly to make balance of inventory and ensure the re-ordering level. The graphical presentation of continuous inventory system can be seen below:

Q1 Q2 Q3

Time

Safety Stock

LT LT LT

ROL

Current Inventory

Quantity

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Assumptions of continuous inventory system

Constant and certain demand over the period.

Constant and known lead time (Delivery time)

Constant purchase price of material.

Constant and independent carrying cost and ordering cost.

No stock out cost.

2. Periodic Inventory System

Periodic inventory system is an operating doctrine on which ordering level and order quantity

are not predetermined; it depends upon the actual time period. It is a system of inventory in

which updates are made on a periodic basis. This differs from perpetual inventory systems,

where updates are made as seen fit. In a periodic inventory system no effort is made to keep up-

to-date records of either the inventory or the cost of goods sold. Instead, these amounts are

determined only periodically - usually at the end of each year. This physical count determines

the amount of inventory appearing in the balance sheet. The cost of goods sold for the entire

year then is determined by a short computation. Under this system, inventory is counted in a

fixed time period to determine the quantity of inventory for placing an order of inventory. It is

period based or time based model, on which it requires higher level of safety stock rather than

Q/R model. It is also known as fixed period inventory system. The graphical presentation of

periodic inventory system can be seen below:

Figure: Periodical Inventory System.

Economic Order Quantity (EOQ)

The quantity for ordering goods which minimizes the total cost of inventory is called economic order

quantity (EOQ). It is the optimal lot size or optimal orders. It is one of the major tools for controlling

inventory. Under this, total carrying cost and total ordering cost of inventory remains equal.

EOQ only applies where the demand for a product is constant over the year and that each new order

is delivered in full when the inventory reaches zero. There is a fixed cost charged for each order

placed, regardless of the number of units ordered. There is also a holding or storage cost for each

unit held in storage (sometimes expressed as a percentage of the purchase cost of the item).

We want to determine the optimal number of units of the product to order so that we minimize the

total cost associated with the purchase, delivery and storage of the product. The optimal number of

units (i.e. EOQ) depends upon the total demand for the year, ordering cost and carrying cost. So, the

required parameters to the solution of EOQ are the total demand for the year, the purchase cost for

each item, the fixed cost to place the order and the storage cost for each item per year. Note that the

number of times an order is placed will also affect the total cost, however, this number can be

determined from the other parameters.

Q1

Q2

Stock Out

Safety Stock

LT LT LT

Current Inventory

Quantity

● ●

●

Q2

Q3

Place

Order

Place

Order

Place

Order

ROL

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Assumptions of EOQ

a. Constant and certain demand over the period.

b. Constant and known lead time (Delivery time)

c. Constant purchase price of material.

d. Constant and independent carrying cost and ordering cost.

e. No stock out cost.

Graphical approach of EOQ

Since EOQ is the quantity of materials to be ordered economically. This level of quality total ordering

cost & total currying cost becomes equal. So EOQ can be determined graphically at that paint, one

ordering cost and currying cost interested each other.

Figure: Economic order quantity.

This figure shows costs on the vertical axis or Y axis and the order quantity on the horizontal or X

axis. The straight line which commences at the origin is the carrying cost curve, the total cost of

carrying units of inventory. As expected, as we order more on the X axis, the carrying cost line

increases in a proportionate manner. The downward sloping curve which commences high on the Y

axis and decreases as it approaches the X axis and moves to the right is the ordering cost curve. This

curve represents the total ordering cost depending on the size of the order quantity. Obviously the

ordering cost will decrease as the order quantity is increased thereby causing there to be fewer

orders which need to be made in any particular period of time.

The point at which these two curves intersect is the same point which is the minimum of the curve

which represents the total cost for the inventory system. Thus the sum of the carrying cost curve and

the ordering cost curve is represented by the total cost curve and the minimum point of the total cost

curve corresponds to the same point where the carrying cost curve and the ordering cost curve

intersect.

ABC Inventory System

ABC inventory system refers to the priority system of inventory on which total inventories are

categorized into three parts. It is based upon the assumption that all inventories are not always

equally important for any organization. So some discrimination should be made on total inventory

according to their relative importance or priorities.

0

Y

X

Carrying cost(QC/2)

Total cost

Ordering cost(AO/Q)

EOQ

Co

st

Quantity


Large manufacturing companies should maintain large numbers of several types of inventories and

it is not possible to control all inventories with equal attention. Therefore, the firm should pay more

attention to those items which have highest value. In order to pay the optimum level of attention for

inventories, they (i.e. inventories) should be prioritized.

High investment is made on inventory for accurate production and supply of the product to the

consumer. ABC inventory system enables management to reduce the investment to the minimum

possible level because a reasonable quantity of 'A' items representing a significant portion of the

material cost is purchased. To reduce investment in materials, close control of 'A' items contributes

much more than close control of 'C' items. On the other hand, no other inventory control system

provides such technique.

Management time is saved by this system because this technique requires minimum attention to be

paid to some items only. But, other inventory system do not facilitates such separation and in case of

any misguidance about the inventory system arises this burden goes to the management and have to

spend lots of time on such deviation.

ABC inventory system stands for Always Better Control in inventory. Following 3 classifications are

made under this inventory system.

Group "A" – It includes few items with large value.

Group "B" – It includes moderate items with moderate value.

Group "C" – It includes more items with small value.

Great effort is made on category A items; a small percentage reduction in their inventory costs has a

large impact on the organization. In contrast, we often use simpler & lower cost methods to control

the inventories of category C. Substantial reduction in inventory cost may produce only small overall

saving under this category.

Timely inventory records and accuracy are the essential components of inventory management

system. It is obvious that actual number of inventory may differ from the number of units reported

by the inventory records. In order to understand the theme of ABC system, we need to take an

example. For this we assume gold, kerosene and coal in group A, B and C respectively. The

procedures of controlling the stock, assuming gold (A), kerosene (B) and coal (C) are given below:

Group Percentage of items Percentage of cost

A (Gold)

B (Kerosene)

C (Coal)

10%

20%

70%

70%

20%

10%

Let, a store has 2,000 items of consumption and a monthly expenditure in consumption of Rs.

10,00,000. According to above report, 200 items will have a consumption of Rs. 7,00,000 on gold and

400 items ill account for Rs. 2,00,000 on kerosene and finally 1,400 items consume material worth Rs.

1,00,000 only on coal.

The significance of this analysis is that a very close control is exercised over the items of gold, which

account for a high percentage of costs and very little control would be enough for category coal.

A tentative weight given to items and value under each group of ABC inventory system can be

shown in the figure below:


Here in the figure above upper triangle PQR refers to the value of inventory and lower triangle

OPR refers to the item of inventory.

Again,

Where,

Value = Periodic consumption × unit cost.

Benefit of ABC inventory system is listed below:

Inventories are stored to a minimum level.

Materials are acquired or produced when needed.

Production scheduling and material tracking e.g. how much in hand, the status of outstanding

orders, and the schedule for production.

Expand short-term capacity or reschedule production and how to reallocate among time

periods to stay within capacity limits.

Identification of production bottlenecks and capacity problems.

Reduction in selling price.

Quality service to the customers.

Cummulative % of

value

100%

90%

70%

Cummulative %

of Item

100%

30%

10%

Q P

20%

10%

20%

70%

70% 10%

A

B

C

O 100 80 60 40 20

Cu

m %

of

valu

e

20

40

60

80

100

X

Cumulative % of items

A B C

Y


Example 1 From the information given below, draw a ABC inventory charts.

Items 1 2 3 4 5 6 7 8 9 10

Annual usage in units 1500 5000 1800 2300 800 7000 400 10000 8000 1500

Unit cost (Rs) 5 4 3 1 5 10 8 6 7 3

Solution:

Calculation of sales value and assignment of ranks

Items Annual sales

(in units)

Unit cost (in Rs.)

Sales value (in Rs.)

= units × rate

Rank

(as per sales value)

1

2

3

4

5

6

7

8

9

10

1500

5000

1800

2300

800

7000

400

10000

8000

1500

5

4

3

1

5

10

8

6

7

3

7500

20000

5400

2300

6000

70000

3200

60000

56000

4500

5

4

7

10

6

1

9

2

3

8

Again,

Table showing classification of inventory(i.e. product) as per the percentage of sales value and items.

Rank Sales value

Prob. of sales

Group probability

Items

Percent value of each item

=100% ÷ 10

Group probability

Group

1

2

3

70000

60000

56000

0.298

0.255

0.238

0.791

6

8

9

0.10

0.10

0.10

0.30

A

4

5

6

20000

7500

6000

0.086

0.032

0.026

0.144

2

1

5

0.10

0.10

0.10

0.40

B

7

8

9

10

5400

4500

3200

2300

0.023

0.019

0.09

0.010

0.065

3

10

7

4

0.10

0.10

0.10

0.10

0.30

C

Total 234900 1.000 1.000 - 1.00 1.00

Now

Table showing cumulative percentage value of sales and items.


Again, plotting the above cumulative percentage of probability of sales value and items into graph.

Example 2 From the following information, construct the ABC diagram.

Items Units Unit Price

(Rs) Items Units

Unit Price (Rs)

1

2

3

4

5

6

700

2,400

150

60

3,800

4,000

5.00

3.00

10.00

22.00

1.50

0.50

7

8

9

10

11

12

6,000

300

30

2,900

1,150

410

0.20

3.50

8.00

0.40

7.10

6.20

Solution

Working out usage (consumption) value and ranking the items in descending order of consumption value.

Items Units Unit Price (Rs)

Items Units Unit Price (Rs)

1

2

3

700

2,400

150

5.00

3.00

10.00

3,500

7,200

1,500

9.8

20.2

4.2

4

2

7

Cumulative % of value

Cumulative % of

Item

100%

60%

30%

C

B

A

R

Q P

O

30%

30%

40% 6.5%

14.4%

79.1%

100%

93.5%

79.1%

100 80 60 40 20

Cu

mu

lati

ve %

of

valu

e

20

40

60

80

100

X

Cumulative % of items

A B C

Y


4

5

6

7

8

9

10

11

12

60

3,800

4,000

6,000

300

30

2,900

1,150

410

22.00

1.50

0.50

0.20

3.50

8.00

0.40

7.10

6.20

1,320

5,700

2,000

1,200

1,050

240

1,165

8,165

2,542

3.7

16.0

5.6

3.4

3.0

0.7

3.3

23.0

7.1

8

3

6

9

11

12

10

1

5

35,577 100

Cumulative Data

Ranks Item No. Usage percentage

Cumulative Value

% Number Cumulative Percentage Quantity

1

2

3

4

5

6

7

8

9

10

11

12

11

2

5

1

12

6

3

4

7

10

8

9

23

20.2

16

9.8

7.1

5.6

4.2

3.7

3.4

3.3

3.3

0.7

23

43.2

59.2

67

76.1

81.7

85.9

89.6

93.0

96.3

99.3

100

–

10.9

17.4

5.2

18.2

0.7

0.3

27.4

13.3

1.3

0.1

–

5.3

16.2

33.6

36.8

38.7

56.9

57.6

57.9

85.3

98.6

99.9

100

The curve can be drawn as follows:

Hence items 2, 11 are classified category A items requiring outmost attention and items 1, 5 and 12 as

category B items for moderate control. Other items fall under C category.

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

(16) (39)

A B C

(43)

(76)

Percentage of Total Number of Items

Per

cen

tag

e o

f T

ota

l In

ven

tory

Val

ue


Inventory Models

Organizations spend lot of money in materials. Material cost represent 20 to 60 percent of the cost of

production, even a small saving in material will reflect in profit. Idle scarce material resource is

called inventory. Since we invest lot of money in materials and if materials are idle for long time, it is

not good for the health of the organization. So it is must to exercise a control over idle scares

resources, otherwise lot of money is tied in inventory. Inventory models help to find out the order

quantity which minimizes the total costs. Inventory models are classified into following four parts as

shown below:

Basic inventory model

Production run model

A Inventory Model when Shortages are permitted

Inventory Model with Price Discounts

MODEL 1:

Basic inventory model (Economic order quantity with instantaneous stock replenishment):

The purpose of the study of this model is to find an optimum order quantity (EOQ) denoted by Q* so

that the total inventory cost is minimized. It shows the behavior of an inventory system which

operates some common assumptions. At the beginning of the cycle time we start with a maximum

amount of inventory equal to the order quantity Q. As this amount is used up, the level of inventory

drops at a fixed rate equal to the demand rate A. When it reaches a reorder level ROL, enough

inventory is available to cover expected demand during the lead time LT. At this, an order is placed

equal to Q which arrives at the end of lead time, when the inventory level reaches zero. This amount

is placed in stock all at once and the inventory level goes up to its maximum value.

This inventory system is characterized in terms of the following assumptions:

Assumptions

1. Annual demand (A) rate is constant and known.

2. Production rate is infinite.

3. Lead time (LT) is constant and known with certainty and independent of demand.

4. Cost per unit (P) is constant.

5. Unit costs of carrying inventory (C) and ordering (O) are known and constant (inventory has

one stock points).

Graphical representation of the model is shown in figure below.

Max

. In

ven

tory

Time

Consumption rate

Average inventory

Quality


Figure: Basic inventory model.

Let D be the annual demand (units per year)

C0 = Ordering costs per order)

Ch = Inventory carrying costs per unit per unit year

Cp = Price per unit

Q = Order quantity

Q* = Economic order quantity

N = Number of orders placed per annum

Tc = Total cost per annum

Here,

Annual ordering cost = Number of orders × Ordering cost per order

= Annual demandOrder quantity × Ordering cost per order

= DQ × C0 … (i)

Annual inventory carrying cost = Average inventory investment × Inventory carrying cost

=

Max. inventory – Minimum inventory

2 × Inventory

carrying cost

= QCh

2 …….… (ii)

Annual total cost (TC) = Annual ordering cost + Annual total cost

= DC0

Q + QCh

2 …….…(iii)

To determine economic order quantity (EOQ),

Differentiating equation (iii) w.r. t. Q, we get

dTCdQ =

ddQ

DC0

Q + d

dQ

QCh

2

0 = DC0

d

dQ

1

Q + Ch

2

d

dQ Q (setting equals to zero)

0 = – DC0

1

Q2 + Ch

2

DC0

Q2 = Ch

2

Q* = 2DC0

Ch

Q* is economic order quantity.

Note: If the inventory carrying cost is expressed as a percentage of annual average inventory investment, then

Q* = 2DC0

CpI

where, it is expressed as a percentage of annual inventory investment.

The optimal number of orders placed per annum.


n* = Annual demand

Economic order quantity = DQ*

Optimal time interval between two orders

T* = Number of working days in a year

N*

Minimum total yearly inventory cost is given by

Tcm = DC0

Q0 +

Q*Ch

2 ………………..(iv)

Substituting for Q* from Eqn. (iv)

Tcm = DC0

2DC0

Ch

+ 2DC0

Ch ×

Ch

2

Tcm = 2 DC0 Ch

Example 3 XYZ Company requires 12,000 units of materials annually. If ordering costs are Rs 250

per order, expected lead time is 5 days, unit cost is Rs 25 per unit and annual inventory holding costs

are charged at 20% and the company operates 250 days a year, compute EOQ and ROL.

Solution

Given,

Annual demand (D) = 12,000 units

Ordering costs (O) = Rs 250 per order

Lead time (L) = 5 days

Annual holding cost per unit = 20% of unit cost

= 20100 × 25 = Rs 5

Number of working days in a year = 250 days

We have,

EOQ = 2DO

C

= 2 × 12‚000 × 250

5 = 1095.5 units per order

ROL = Daily demand × Lead time + Safety stock

= Annual demand

Number of working days × Lead time + Safety stock

= 12‚000

250 × 5 + 0 = 240 units

MODEL-II

Production run model (economic order quantity when stock replenishment is non-

instantaneous)

This model is applicable when inventory continuously builds up over a period of time after placing

an order or when the units are manufactured and used (or sold) at a constant rate. This model is also

based on the assumptions in previous model except that of instantaneous replenishment (supply).

This is because of the fact that in many situations, the amount ordered is not delivered all at once,


but available at a finite rate (i.e. given supply rate) per unit of time, i.e. order is either produced

internally or will be supplied at some uniform rate, p. This Situation can arise in case of an order

being filled by a machine having finite production rate. Graphically it can be illustrated below:

Assumptions

1. Annual demand rate (A) is constant and known.

2. The production rate (p) per unit time is greater than demand rate (d).

3. Lead time (LT) is constant and known with certainty and independent of demand.

4. Cost per unit (P) is constant.

5. Unit costs of carrying inventory (C) and ordering (O) are known and constant.

6. Supply is continuous and constant until Q units are supplied to stock, then it stops.

7. Production begins immediately after production set-up.

Inventory under this situation, builds at the rate of (p - d) units and inventory is maximum at the end

of production period tp.

Maximum inventory at the end of production run = (p - d) × tp

Average inventory = (p - d) tp

2

Now, the quantity produced during production period'(Q) = P × tp

tp = Qp

Substituting for tp,

Average inventory = (P – D) Q2P

= Q2

1 –

DP

Annual inventory carrying cost = Average inventory × Inventory carrying cost

= Q2

1 –

DP Ch

Annual set-up cost = Number of set-up × Set-up cost per set-up

= DQ × C0

Total annual cost (TC) = DC0

Q + Q2

1 –

DP Ch ……………………(i)

Sto

ck l

evel

Time

Max. inventory level Quality

T

tp tc


To determine, the economic batch (Lot) size differentiate Eqn. (i) w.r.t. Q, be the first derivative is

equals to zero.

dTcdQ =

12

1 –

DP Ch –

DC0

Q2 = 0

Q* = 2DC0

(1 – D/P) Ch

where Q* is economic batch quantity.

Note: If the inventory carrying cost is expressed as a percentage of annual inventory investment, then

Q* = 2DC0

(1 – D/P) Ch I where / is expressed in %.

Optimal total cost is given by

TC = 2D C0 Ch (1 - D/P)

Optimal number of production runs

N* = Annual demand

Economic batch quantity (EBQ) = DQ*

Example 4 The annual demand of a product is 100,000 units. The rate of production is 200,000

units per year. The set up cost per production run is Rs 500 and the variable production cost of each

item is Rs 10. The annual holding cost per unit is 20% of its value.

Find the optimum production lot size and the length of the production run.

Solution

Given,

D = 100,000 P = 200,000 Ch = 0.2 and C0 = 500

Optimum production lot size is given by

Q(EOQ) = 2DC0

Ch

P

P – D

= 2 × 10‚000 × 500

0.2

200‚000

200‚000 – 100‚000

= 10,000 units

Length of the production runs (PN) = QP =

10‚000200‚000 = 0.05 year

MODEL III:

Inventory Model with Price Discounts

Then items are bought in large quantities, the supplier often gives discounts. However, if the

material is purchased to take advantage of discount, the average inventory level and so the inventory

carrying costs ill increase.

Benefits for the purchaser from large orders are lower cost per unit, lower shipping and

transportation cost, reduced handling cost and reduction in ordering costs due to less number of

orders. These benefits are to be compared with the increase in carrying costs. As the order size or

increases, more space should be provided to stock the items.

As with other inventory models discussed so far, the overall objective will be to minimize the total

variable cost. As the discount quantity increases the material cost decreases, but the carrying cost


increases because the order are large. Thus, the major trade-off when considering quantity discount

is between the reduced material cost and the increased carrying cost. When we include the cost of the

material, the equation for the total annual inventory cost becomes:

Total variable cost = Material cost + Ordering cost + Carrying cost

Total variable cost = DC + AQ C0 +

Q2 Ch

A decision is, therefore, to be taken whether the buyer should stick to economic order quantity or

increase the same to take advantage that, at large quantities, the production costs per price are lower

(economies of scale) and, hence, part of the savings can be passed on to the customer.

Figure: Quantity discount.

Example 6 An aircraft company uses rivets at a constant rate of 25,000 per year. Each unit cost Rs

30 and the company personnel estimate that is costs 130 to place an order, and that carrying cost of

inventory is 10 percent per year. How frequently should orders be placed? Also determine the

optimum size of each order.

Solution

Given,

Aircraft company uses rivets at a constant rate (D) = 2500 per year

Each unit cost (C*) = Rs 30

Ordering cost per order (C0) = 130

Carrying cost of inventory is 10% per year

Carrying cost (Ch) = 30 × 0.10 = Rs 3

EOQ = 2DC0

Ch =

2 × 25000 × 1303 = 465 units

Number of order (N) = 5 times

Periodic Model

In a fixed–time period system, inventory is counted only at particular times, such as every week or every month. Counting inventory and placing orders periodically is desirable in situations such as when vendors make routine visits to customers and take orders for their complete line of products, or when buyers want to combine orders to save transportation costs. Other firms

Q(d1) Qoptimal Q(d2) Quantity discount

Inv

ento

ry c

ost

(R

s)

Ordering cost

Carrying cost

TC

TC

TC

O


operate on a fixed time period to facilitate planning their inventory count; for example, Distributor X calls every two weeks and employees know that all Distributor X’s product must be counted.

Fixed–time period models generate order quantities that vary from period to period, depending on the usage rates. These generally require a higher level of safety stock than a fixed–order quantity system. The fixed–order quantity system assumes continual tracking of inventory on hand, with an order immediately placed when the reorder point is reached.

In contrast, the standard fixed–time period models assume that inventory is counted only at the time specified for review. It is possible that some large demand will draw the stock down to zero right after an order is placed. This condition could go unnoticed until the next review period. Then the new order, when placed, still takes time to arrive. Thus, it is possible to be out of stock throughout the entire review period, T, and order lead time, L. Safety stock, therefore, must protect against stockouts during the review period itself as well as during the lead time from order placement to order receipt.

Fixed–Time Period Model with Safety Stock

In a fixed–time period system, reorders are placed at the time of review ( T ), and the safety stock that must be reordered is

Safety stock = z σ T + L

above shows a fixed–time period system with a review cycle of T and a constant lead time of L. In this case, demand is randomly distributed about a mean d . The quantity to order, q, is

where

q = Quantity to be ordered

T = The number of days between reviews

L = Lead time in days (time between placing an order and receiving it)

d = Forecast average daily demand

z = Number of standard deviations for a specified service probability

σ T + L = Standard deviation of demand over the review and lead time

I = Current inventory level (includes items on order)

Note: The demand, lead time, review period, and so forth can be any time units such as days, weeks, or years so long as they are consistent throughout the equation.


Note: The demand, lead time, review period, and so forth can be any time units such as days, weeks, or years so long as they are consistent throughout the equation. In this model, demand ( d ) can be forecast and revised each review period if desired or the yearly average may be used if appropriate. We assume that demand is normally distributed. The value of z is dependent on the probability of stocking out and can be found using table value of Z attached in the appendix.

Example

Daily demand for a product is 10 units with a standard deviation of 3 units. The review period is 30 days, and lead time is 14 days. Management has set a policy of satisfying 98 percent of demand from items in stock. At the beginning of this review period, there are 150 units in inventory. How many units should be ordered?

SOLUTION

The quantity to order is q = d ( T + L ) + z σT + L

– I = 10(30+ 14) + z σT + L

+ L − 150

Before we can complete the solution, we need to find σT + L

and z.

To find σT + L

we use the notion, as before, that the standard deviation of a sequence of independent random variables equals the square root of the sum of the variances. Therefore, the standard deviation during the period T + L is the square root of the sum of the variances for each day:

= = 19.9

The z value for P = .98 is 2.05.

The quantity to order, then, is

q = d ( T + L ) + z σ

T + L − I = 10(30+ 14) + 2.05(19.90) − 150 = 331 units

To ensure a 98 percent probability of not stocking out, order 331 units at this review period

WORKED OUT EXAMPLES

Example 7 The Prajwal Company requires 1000 units per month through the year at constant rate. If ordering costs are Rs 250 per order, unit cost of item is Rs 25 and annual inventory holding cost are charged at 20%. Then, determine EOQ for the time.

Solution

Given,

Monthly requirements (d) = 1000 units per month

Annual requirement of company (D) = 1000 × 12 = 12,000

Ordering cost per order (C0) = Rs 250

Carrying or holding cost (Ch) = Rs 25 × 0.20 = 5

EOQ = 2DC0

Ch =

2 × 12‚000 × 2505 = 1049.45 units


Price or unit costs are denoted by P or C.

Example 8 XYZ company busy in lots of 500 boxes which a 3 months supply. The cost per box is Rs. 125 and the ordering cost is Rs 150. The inventory cost is estimated at 25% of unit value.

a. What is the annual cost of he existing inventory policy?

b. How much money could be saved by employing the economic order quantity?

Solution

Given,

Ordering cost (C0) = Rs. 150

Quantity of one order = 500

Annual demand (D) = 500 × 12/3 = 2,000

Carrying Cost (Cc) = Rs. 125 × 25% = Rs. 25

a. Total annual cost of existing inventory = DQ C0 +

Q2 Ch

= 2‚000500 × 150 +

500200 × 25 = Rs 6,800

b. Economic order quantity = 2DC0

Ch

or Q(EOQ) = 2 × 2‚000 × 150

25 = 1500

Minimal annual cost = 2D C0 Ch

= 2 × 2‚000 × 150 × 25 = 3‚873

Hence saving = 6,850 – 3,873 = Rs 2,977.

Example 9 Mohan is a confectioner. He buys plastic boxes in bulk and uses them to pack his chocolates. His annual requirement of boxes is 12000, and each box costs him Rs. 30. He has estimated that his ordering costs are m Xs. 10 per order and his carrying costs are 20%. How many boxes should he order at a time so as to minimise his expenses?

Solution

Given, D = 12000 boxes; Ch = Rs. 6 (Rs. 30 x 0.2); C0 = Rs. 10

Now,

EOQ = 2DC0

Ch =

2 × 12‚000 × 1030 × 0.2 = 63 boxes

Example 10 If the supplier sells the boxes only in lots of 25, should we buy 50 or 75?

Solution

In order to make a decision on this problem, we should calculate the total in both cases. The cost of the inventory can be ignored as it remains the same.

If Q = 50

Total annual cost (TC) = DQ C0 +

Q2 Ch

= 12‚000

50 × 10 + 502 × 6 = Rs 390

If Q = 75

Total annual cost (TC) = DQ C0 +

Q2 Ch


= 12‚000

75 × 10 + 752 × 6 = Rs 385

As the total cost of buying lots of 75 is lesser, we should buy in lots of 75.

Example 11 A contractor has to supply 10,000 paper cones per day to a textile unit. He finds that

when he starts a production run, he can produce 25,000 paper cone per day. The cost of holding a

paper cone in stock for one year is 2 paisa and set up cost of production run is Rs 18. How frequently

should the production run be made?

Solution

Given,

Let us assume that the production runs on 300 days in a year

D = 10,000

C0 = Rs 18

Ch = 0.02 per year

P = 25,000

d = 10,000

Q(EOQ) = 2DC0

Ch

P

P – D

= 2 × 30‚000 × 18

0.02

25‚000

25‚000 – 10‚000 = 95,000 units

Frequency of production = 95‚00025‚000 = 4 days

Example 12 Yamaha Motors Ltd. has been offered a particular component part it uses according

to the following discount pricing schedule provided by the supplier.

Order size Unit cost

1 – 199

200 – 599

600 +

65

59

56

The company uses 700 of the components annually, the annual carrying cost is Rs 14 per unit, and the ordering cost is Rs 275. Determine the amount the firm should order.

Solution

Given,

Annual demand (D) = 700 components

Ordering cost (C0) = Rs 275

Carrying cost (Ch) = Rs 14

Discount category Order size Discount Unit cost

1

2

3

1 – 199

200 – 599

600 +

0

9

14

65

59

56

We have,

EOQ1 = 2DC0

Cn =

2 × 700 × 27514 = 165.83


EOQ2 = 2DC0

Cn =

2 × 700 × 27514 = 165.83

EOQ3 = 2DC0

Cn =

2 × 700 × 27514 = 165.83

Here,

EOQ1 = 165.83 belongs to order size 1 – 199, therefore, optimal quantity is 165.83. On the other

hand, EOQ2 and EOQ3 are also 165.83 respectively which does not belongs to order size 200 – 599

and 600 +, so both are disqualified. Hence, 200 and 600 is considered as optimal quantity for second

and third category respectively.

Thus,

EOQ1 = 164.83

EOQ2 = 200

EOQ3 = 600

Now,

Computation of total cost for all categories

For, EOQ1 = 165.836 and discount price = Rs 65

TC1 = DQ1

C0 + Q1

× Ch + Cp × D

= 700

165.83 × 275 + 165.83

× 14 + 65 × 700

= Rs 47,821

For, EOQ2 = 200 and discount price = Rs 59

TC2 = DQ2

C0 + Q2

× Ch + Cp × D

= 700200 × 275 +

200

× 14 + 59 × 700

= Rs 43,662.50

For, EOQ3 = 600 and discount price = Rs 56

TC3 = DQ3

C0 + Q3

× Ch + Cp × D

= 700600 × 275 +

600

× 14 + 59 × 700 = Rs 43,720.83

Decision: Since the minimum total annual cost is Rs 43,622.5 for second category where optimal

quantity is 200 units. Therefore, the firm should order 200 quantities for optimality.

Some operating parameters to control inventory System

1. Optimum number of orders to be placed in one year (N)

N = D

EOQ

2. Cycle Time between two order (CT)

CT = No. of days in a year

No. of orders =

365D

EOQ = 365A EOQ


3. ROL (Re-ordering level) = (Daily Average Consumption Average LT) + SS

Or,

Maximum Consumption Maximum LT

Where,

LT = Lead Time

SS = Safety Stock or Buffer Stock

4. Average stock level = EOQ

2 + SS Or ½ (min. stock + Max. stock)

5. Maximum stock level (M) = SS + ordering quantity

Example 13 A company uses annually 50,000 units of an item costing Rs. 1.20. It operates 250 days in a year and procurement times is 10 days. Each order costs Rs. 45 and inventory carrying cost is 15% of the annual average inventory value. If the safety stock is 500 units, find EOQ, Re-order level, minimum, maximum and average inventory levels.

Solution

Given

Co = Rs.45; Ch = 0.15 of 1.20 = Rs. 0.18; D = 50,000.

(i) Economic order quantity = 2 × 45 × 50‚000

0.18 = 5,000 units

(ii) Average daily demand = 50‚000

250 = 200 units per day (for 250 days annual working)

Lead time = 10 days

Average DDLF = 200 x 10 = 2,000 units

Safety stock = 500 units

ROL = Safety stock + Average DDLF

= 500 + 2,000 = 2,500 units

Maximum Inventory = 500 + 5,000 = 5,500 units

Minimum Inventory = Safety stock = 500 units

Average Inventory = 500 + 1/2 EOQ = 3,000 units.

EXERCISE – 8

Theoretical Questions

1. What do you mean by inventory? What are the reasons for keeping inventory?

2. Explain the inventory system.

3. Discuss about the different types of inventory cost.

4. What is ABC inventory system? Describe the benefits of ABC system.

5. What is material handling? Explain its basic principles.

6. Write short notes on:

a. Economic order quantity

b. Basic inventory model

c. Production run inventory model

d. Inventory model with price discount

Practical Problems

1. What is the percentage change in EOQ when the annual demand is doubled?

2. What will be the percentage change in EOQ if 50% decreases in the ordering cost and the

Holding cost?

3. What will be the % change in EOQ if 40% decrease in the Holding cost and 60% decreases

in the ordering cost?

4. What will be the % change in EOQ if 40% decrease in the Holding cost and 60% increase

in the ordering cost?

5. Compute total annual cost of inventory if number of order is 5 @Rs. 20 per order and

optimum order quantity is 400 units with holding cost Rs. 2 per unit.

6. A manufactures has to supply his customers with 2400 units of his product per year.

Shortages are not allowed and storage cost amounts to 1.20 Rs. per unit per year. The

setup cost per run is Rs. 160. Find the optimal order quantity, optimum number of orders

in a year, and cycle time in days.

7. A manufacturing company with stable demand for its product is managing its inventory

system for EOQ 320 units. No of order per year are 12 and set up or order preparation

cost is Rs. 50 per order. Find what value of the holding cost adjusts this system. Also find

the Reorder Level, if lead time is 10 days.

8. A firm requires 18,000 units of raw materials in a period of 6 months. The carrying cost is

20% of cost. The set up cost per set up is Rs. 10. The cost per unit is Rs. 19. The average

lead time is 5 days (Assuming 360 days in a year.)

Required

(a) Economic order quantity

(b) Total annual cost

(c) No of days between orders (i.e. cycle time)

(d) Reorder level

9. A computer shop stocks and sells a mercantile brand of personal computer. It costs the

store Rs. 450 each time it Places an order with the manufacturer for the personal

computers. The annual cost of carrying the PCs in inventory is Rs. 170. The store manager

estimates that annual demand for the PCs will be 1200 units. Lead time is given as 10

days and assuming working days per annum is 300.

Determine:

(a) Economic order quantity

(b) Re-order point

(c) Number of orders

10. A firm consumes 50,000 units of an item per annum, each costing Rs. 10. The order costs

are expected to be Rs.40 and inventory carrying costs 10% of the annual average

inventory value. Find EOQ. If the company operates 250 days per year, the procurement

time is 10 days and the buffer stock is 500 units, find the reorder level and average

inventory level.


11. A company demands 50,000 units per quarter. The ordering cost is Rs.40 per order and

carrying cost is 10%. The cost price of inventory is Rs. 10 per unit. The lead time for

placing order is 5 days. Required:

(a) Economic order quantity (EOQ)

(b) No of orders

(c) Cycle time

(d) Total cost of inventory.

(e) Reorder level (assuming 300 working days per annum)

12. S and P company requires 1000 units of materials monthly. If ordering costs are Rs. 250

per order, expected lead time is 5 days, unit cost is Rs. 25 per unit and annual inventory

holding costs are charged at 20%, and the company operates 250 days a year: compute

EOQ and ROL.

13. The production department of a food and beverage company requires 1800 units of raw

materials for manufacturing a special food semi-annually. It has been estimated that the

cost of placing an order is Rs. 36 and the cost of carrying inventory is 12.5% of the

investment in the inventories. The price is Rs. 20 per unit. Purchase manager wishes to

determine an ordering policy for the raw material. Recommend the manager in the

following aspects:

(a) The optimal lot size

(b) The optimal reorder time.

14. Star electronics is a dealer of the Sony mobile sets. It has observed that the annual

demand is about 768 sets and the annual cost of the holding a mobile set in stock is Rs. 30

where as an order placed for the sets costs Rs.20. Using the information below find:

(a) Optimum order quantity.

(b) Optimum number of the orders to be placed in a year's time.

(c) Total minimum variable cost.

15. What are the main objectives of holding inventory in an organization? What are the major

costs involved in holding inventory? The ABC requires 100 units per months though out

the year at constant rate. If ordering cost are Rs. 250 per order, unit cost of he item is Rs.

25 and annual inventory holding cost are charged at 30%, then determine the EOQ for the

item.

16. Nepal soft drink Co. has a soft drink product which has constant annual demand rate of

3000 cases and cost Rs. 200/case. If ordering cost are Rs. 20 and inventory holding cost

are charged at 25%, what is the EOQ for this product? Also determine the cycle time

(in days).

17. A manufacturer has to supply his customers with 600 units of his product per year.

Shortages are not allowed and the storage cost amounts to Rs. 0.6 per unit per year. The

set up cost per run is Rs. 80. Find the optimum run size and the minimum average yearly

cost.

18. A manufacturing organization experienced constant annual demand of 10,000 unit of an

item which cost Rs. 40 per unit. If the order cost is Rs. 20 per order and the holding cost is


25% of the item costs determine the economic order quantity and the reorder level with

no lead-time known.

19. The local supermarket buys cabbage each day to ensure really fresh produce. Each

morning any cabbage that is left from the previous day is sold to a dealer that results it

to farmers who use it to feed their animals. This week the supermarket can buy fresh

cabbage for Rs 40 per box. The cabbage is sold for rs 100 a box and the dealer that sells

old cabbage is willing to pay Rs 15 a box. Past history says that tomorrow’s demand for

cabbage averages 250 boxes with a standard deviation of 34 boxes. How many boxes of

cabbage should the supermarket purchase tomorrow?

20. Next week, super discount airlines has a flight from Kathmandu to New Delhi that will

be booked to capacity. The airline knows from past history that an average of 25

customers (with standard deviation of 15) cancel their reservation or do not show for the

flight. Revenue from a ticket on the flight is Rs 1250. If the flight is overbooked, the

airline has a policy of getting the customer on the next available flight and giving the

person a free round trip ticket on a future flight. The cost of this free round trip ticket

averages Rs 2500. Super discount consider the cost of flying the plane from Kathmandu

to New Delhi a sunk cost. By how many seats should super discount overbook the

flight?

21. KVS Pharmacy fills prescriptions for a popular children’s antibiotic, amoxycilin. The

daily demand for amoxycilin is normally distributed with a mean of 200 ounces and a

standard deviation of 80 ounces. The vendor for the pharmaceutical firm that supplies

the drug calls the drug-store’s pharmacist every 30 days and checks the inventory of

amoxycilin. During a call the druggist indicated the store had 60 ounces of the antibiotic

in stock. The lead time to receive an order is four days. Determine the order size that will

enables the drugstore to maintain a 99% service level.

22. Food place market stocks frozen pizzas in a refrigerated display case. The average daily

demand for the pizzas is normally distributed, with a mean of 8 pizzas and a standard

deviation of 2.5 pizzas. A vendor for a packaged food distributor checks the market’s

inventory of frozen foods every 10 days; during a particular visit there were no pizzas in

stock. The lead time to receive an order is three days. Determine the order size for this

order period that will result in a 98% service level. During the vendor’s following visit

there were 5 frozen pizzas in stock. What is the order size for the next order period?

23. The Mediterranean Restaurant stocks a red Chilean table wine it purchase from a wine

merchant in a nearby city. The daily demand for the wine at the restaurant is normally

distributed, with a mean of 18 bottles and a standard deviation of 4 bottles. The wine

merchant sends a representative to check the restaurant’s wine cellar every 30 days, and

during a recent visit there were 25 bottles in stock. The lead time to receive an order is

two days. The restaurant manager has requested an order size that will enable him to

limit the probability of a stockout to 5%.

24. Daily demand for a product is 10 units with standard deviation of 3 units. The review

period is 30 days and lead time is 14 days. Management has set a policy of satisfying 98

percentage of demand from item in stock. At beginning of this review period, there are

150 units in inventory. How many units should be ordered?


25. Wal-Mart would like to develop an inventory ordering policy of a 95% probability of not

stocking out. To illustrative your recommended procedure, use as an example the

ordering policy for white percale sheet. Demand during white percale sheets is 5000 per

year. The store is open 365 days per year. Every two weeks (14 days) inventory is

counted and a new order is placed. It takes 10 days for the sheets to be delivered.

Standard deviation of demand for the sheets is five per day. There are currently 150

sheets on hand. How many sheets should you order?

26. Acer company is planning to make his monthly (every 30 days) trek to Banglore to pick

up supply chips. The trip will take to chip about two days. Before he leaves, he calls for

the order to the supply store. He uses chip at an average rate of five per day ( seven days

per week) with a standard deviation of one per day. He needs a 98% service probability.

If he currently has 35 chip in inventory, how many should he order? What should be the

most he will ever have to order?

27. Daily demand for a product is 100 units, with standard deviation of 25 units. The review

period is 10 days and the lead time is 6 days. At the time of review there are 50 units in

stock. If 98% service probability is desired, how many units should be ordered?

28. University drug pharmaceuticals order its antibiotics every two weeks (14 days) when a

sales person visit from one of the pharmaceutical companies. Tetracycline is one of its

most prescribed antibiotics, with average daily demand of 2000 capsules. The standard

deviation of daily demand was derived from examining prescriptions filled over past

three months and was found to be 800 capsules. It takes five days for the order arrives.

University drug would like to satisfy 99 percentages of prescriptions. The salesperson

just arrived and there are currently 25000 capsules in stock. How many capsules should

be ordered?

29. After graduation, you decide to go into a partnership in an office supply store that has

existed for a number of years. Walking through the store and stock room, you find a great discrepancy in service levels. Some spaces and bins for items are completely empty; others have supplies that are covered with dust and have obviously been there a long time. You decided to take on the project establishing consistent levels of inventory to meet customer demands. Most of your supplies are purchased from just a few distributors that call on your store once every two weeks. You choose , as your first item for study, computer printer paper. You examine the sales record and purchase order and find the demand for the past 12 months was 5000 boxes. Using your calculator you sample some days demand and estimate that the standard deviation of daily demand is 10 boxes. You also search out the figures. Cost per box of paper is Rs 110, Desired service probability of 98 percentage, store is open every day, sales person visits every two weeks and Delivery time following visit is three days. Using your procedure, how many boxes of paper would be ordered if, on the day the salesperson call, 60 boxes on hand?

30. Dave’s auto supply custom mixes paint for its customers. The shop performs a weekly inventor count of the main colors that are used for mixing paint. Determine the amount of white paint that should be ordered using the following information. Average weekly demand: 20 gallons, standard deviation of demand 5 gallons/week, desired service probability 98%, current inventory 25 gallons and lead time 1 week.


ANSWERS

2. No change 3. 18.35% decrement

4. 63.3% increment 6. Q = 730.30; N = 4(approx); CT = 0.304

7. Rs. 3.75 and 106.67 8. (a) 706.13; (b) Rs. 685851.47; (c) 0.0196; (d) 500

9. (a) 80; (b) 40; (c) 15

11. (a) 4000; (b) 50; (c) 2.02; (d) Rs. 2004000; (e) 3333.33

12. 1095.45 and 240 13. 322 units (b) 32days

15. 1095.45units 16. 49 cases and 6 days

17. 400 units and Rs. 240 18. 283 units, 167 units.

inventory system - bijay lal pradhan, ph.d. · 10/8/2016 · inventory system introduction...

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