National Institute of Technology Rourkela

Production and Operations Management

ME 4232
{ 3-0-0 / 3}

By
Dr. Gaurav Kumar
Assistant Professor
Department of Mechanical Engineering
 COURSE CONTENT
1. Module I (6 h): Cost Accounting and Break-Even Analysis
Fundamentals of Cost Accounting: Introduction to Cost Accounting, Objectives and Advantages of Cost Accounting,
Cost Concepts and Classifications, Cost Components, Calculation of Prime Cost, Factory Cost, and Total Cost, Problem-
Solving on Material, Labour, and Overhead Allocation.
Break-Even Analysis: Introduction, Break-Even Point (BEP), Break-Even Analysis, Sensitivity Analysis, PV chart,
Multi-Product Break-Even Analysis, Decision-Making with Break-Even Models, Applications of Break-Even Analysis in
Different Industries.
2. Module II (6 h): Inventory Management and Control
Inventory Control: Introduction, Role of Inventory in Business Operations, Inventory Cost Components, Economic Order
Quantity (EOQ) Models (Basic EOQ Model, EOQ with Production Quantity, EOQ with Shortages), Quantity Discounts
and Multi-Item Inventory Models, Probabilistic Inventory Models, Inventory Control with Uncertain Demand, Reorder
Levels and Service Levels, Selective Inventory Control Techniques, Information Systems for Inventory Management, Q-
System and P-System Comparison.
3. Module III (6 h): Investment Decisions
Investment Decision: Introduction, Time Value of Money, Cash Flow Diagram, Interest Formulas, present worth method
of comparison future worth method annual equivalent method rate of return method.

NIT Rourkela
2
 COURSE CONTENT
4. Module IV (6 h): Maintenance Management
Replacement and maintenance analysis: Introduction, Types of Maintenance, Types of Replacement Problem,
Determination of Economic Life of an Asset, Replacement of Existing Asset with a New Asset, Capital Recovery with
Return, Concept of Challenger and Defender, Simple Probabilistic Model for Items Which Fail Completely.

5. Module V (6 h): Supply Chain Management

Supply Chain Management: Introduction, Domain Applications, SCM– The Breakthrough Article, Supply Chain
Management, Views on Supply Chain, Bullwhip Effect in SCM, Collaborative Supply Chain, Financial Supply Chain
– A New Revolution within the SCM Fold.

6. Module VI (6 h): Product/ Production Relationships, Manufacturing metrics and economics

Production quantity and product variety, product and part complexity, cycle time and production rate, production
capacity and utilization, manufacturing lead time and work-in-process, cost of equipment usage, cost of
manufacturing part

NIT Rourkela
3
 BOOKS
Textbooks

• Buffa, Sarin, Modern Production/Operations management, John Wiley and Sons.

• Aswathappa, K., Bhat, S., Production and Operations management, Mumbai: Himalaya
Publication House.

SUPPLEMENTARY READING
• Paneerselvam, R., Production and Operations management (3rd ed.)., PHI Leaning.
• Stevenson, W. J., Operations Management (12th ed.), McGraw-Hill.

NIT Rourkela
4
INVENTORY MANAGEMENT
 INVENTORY
• Any kind of resource that has economic value and is maintained to fulfil the present and future needs
of an organization.

• Inventory consists of usable but idle resources.

• Resources: - men, materials, machines.

• Generally, inventory of men or machines is not carried, and management hires machines or consults
experts whenever required, as an economical alternative.

• When the resources involved is materials or goods, it is referred as a stock or simply as inventory.

• Though inventory of materials is an idle resource – it is not meant for immediate use – it is almost
essential to maintain some inventories for the smooth functioning of an enterprise.

6
 INVENTORY
• Inventory held by various organizations: -

Type of Organization Type of Inventories Held

Manufacturer Raw materials; semi-finished goods; finished goods; spare parts, etc.
Hospital Number of beds; stock of drugs; specialized personnel, etc.
Bank Cash reserves; tellers, etc.
Airline company Seating capacity; spare parts; specialized maintenance crew, etc.

• Traditionally, inventory is viewed as a necessary evil – too little of it causes costly interruptions and
too much of it results in idle capital.

• Inventory control aims at maintaining the balance between these two extremes.

7
 INVENTORY CONTROL
Following few basic factors are required to be taken into consideration for an efficient control of
inventory: -

• Items to be stocked
• Inventory level of existing items is kept at reasonable level.

• Unnecessary items are not added to the inventory.

• Items which have not been used for long time are removed from the inventory

• Time to replenish inventory

• Periodic review system.

• Continuous review system or fixed order quantity system or two-bin system.

• Quantity of replenishment order

• Demand pattern
8
• Price of an item, discount options, total budget and warehouse space, etc.

• Lead time
 R E A S O N S F O R C A R RY I N G I N V E N T O RY
1. Service to the customers: To ensure adequate supply of items to customers and avoids the shortages
as far as possible at the minimal cost.

2. Effective use of capital: To make an effective use of capital, i.e. lock-up of capital should be barest
minimum (as low as possible).

3. Economy in purchasing: To take advantages of quantity discount on bulk purchase.

4. To utilize the benefits of price fluctuations.

5. To smoothen production when demand varies seasonally.

6. To carry buffer stocks (reserve stocks) in case of delayed deliveries by the suppliers.

7. To minimize cost of material handling in the plant by bulk movement of materials between
operations.
9
 R E A S O N S F O R C A R RY I N G I N V E N T O RY
8. Minimization of risk obsolescence and deterioration: The possibility of the risk of loss on
account of obsolescence and deterioration should be minimized. In-built checks in the system should
enable the management to weed out obsolete and non-moving items periodically and automatically.

9. Reduction of administrative workload: The administrative workload on the purchasing, receiving,

inspection, stores, accounts and other related departments should be barest minimum.

10. Up-to-date accurate records: In order to enable the company to prepare financial statements,
minimizing of discrepancies between physical stock and book balance, an up-to-date stock records
must be maintained.

10
 R E A S O N S F O R C A R RY I N G I N V E N TO RY

11. Decoupling of operations by making them independent of supply condition

• Breakdown of equipment at one stage will not affect operations at other stations.

• Operations can be scheduled independently.

12. To satisfy other business constraints, such as: (i) Supplier’s condition of minimum quantity (ii)
Government regulations (iii) Seasonal availability

Conclusion
With a good inventory, a firm can make purchases in economic lots,
maintain continuity of operations, avoid small-time consuming
orders and guarantee for the prompt delivery of finished goods.

11
 INVENTORY CLASSIFICATION
1. Transit (or pipeline) inventory: -

• Inventories can’t provide service while in transportation and such inventories are called transit (or
pipeline) inventory.

• The amounts of pipeline inventory depend on inventory supply time (or lead time) and the nature
of the demand.

2. Cycle inventory:-

• It is the inventory necessary to meet the average demand during the successive replenishments.

• The amount of such inventory depends upon the production lot size, economical order quantities,
warehouse space available, lead time, discount, and inventory carrying cost, etc.

12
 INVENTORY CLASSIFICATION
3. Buffer or Safety Stock : -

• The specific level of additional stock of inventory that is maintained for protection against
unexpected demand and the lead time necessary for delivery of goods is called buffer stock.

• Additional stock of inventory items is maintained in addition to the regular stock.

• The level of buffer stock is determined by trade-off between protection against demand and lead
time and the desired level of investment in stock of inventory.

13
 INVENTORY CLASSIFICATION
4. Anticipation (seasonal) inventory: -

• It is the inventory of pre-established or procured stock levels of items like fashion products,
agricultural goods, children's toys, calendars, etc., which are influenced by seasonal demand,
ensuring readiness during peak periods.

• Level of additional stock of inventory items required to meet unexpected demand should be
determined by balancing the holding (or carrying) and shortage costs of seasonal inventories.

14
 INVENTORY CLASSIFICATION
5. Decoupling inventory: -

• Inventory is maintained in the successive operation of manufacturing processes to mitigate the

adverse effects of interdependence, caused by breakdowns or disturbances in one stage, aiming
to enhance organizational productivity by creating an optimal stock level among adjacent stages,
fostering independence.

• The decoupling inventories may be classified into four groups:

a) Raw Materials and Component Parts

b) Work-in-Process Inventory

c) Finished Goods Inventory

d) Spare Parts Inventory

15
 INVENTORY SYSTEM
• Purchase Cost/Unit
System's Performance • Ordering Cost/Unit
(Total Inventory Cost) • Holding Cost/Time/Unit
• Shortage Cost/Unit

Replenishment Pattern Inventory System Demand Pattern

• Instantaneous • Deterministic
• Gradual • Probabilistic
Operating Operating
Constraints Decision Rules
• Warehouse
• Finance
• Number of Units of
• Buffer Stock
Each Item
How Much to Order When to Order • Safety Stock
(Order quantity) (Reorder point) • Reserve Stock
• Customer Service Level
16
FA C T O R S A F F E C T I N G I N V E N T O RY C O N T R O L

1. Demand:

• The number of units required per period is called demand.

• The demand pattern of a commodity may either be deterministic or probabilistic.

2. Lead time:

• The time gap between placing of an order and its actual arrival in the inventory is known as lead
time.

• Lead time has two components, namely the administrative lead time – from initiation of
procurement action until the placing of an order, and the delivery lead time – from placing of an
order until the delivery of the ordered material.

17
FA C T O R S A F F E C T I N G I N V E N T O RY C O N T R O L

3. Time horizon:

• The time period over which the inventory level will be controlled is called the time horizon.

• This horizon may be finite or infinite depending upon the nature of the demand for the
commodity.

4. Order cycle:

• The time period between placement of two successive orders is referred to as an order cycle.

• The order may be placed on the basis of following two types of inventory review systems:

a) Continuous review

b) Periodic review

18
FA C T O R S A F F E C T I N G I N V E N T O RY C O N T R O L

Continuous review:

• The record of the inventory level is checked continuously until a specified point (reorder point) is
reached where a new order is placed.

• It is often referred to as the two-bin system.

• It divides the inventory into two parts and places it physically, or on paper, in two bins. Items are
drawn from only one bin and when it is empty, a new order is placed.

• Demand is then satisfied from the second bin until the order is received.

• Upon receipt of the order, enough items are placed in the second bin to make up the earlier total.

• The remaining items are placed in the first bin.

• This procedure is then repeated.

19
FA C T O R S A F F E C T I N G I N V E N T O RY C O N T R O L

Periodic review:

• In this system, the inventory levels are viewed at equal time intervals and orders are placed at such
intervals.

• The quantity ordered each time depends on the available inventory level at the time of review.

20
FA C T O R S A F F E C T I N G I N V E N T O RY C O N T R O L

5. Re-order level:

• The level between maximum and minimum stock, at which purchasing (or manufacturing)
activities must start for replenishment, is known as re-order level.

6. Stock replenishment:

• Although an inventory problem may operate with lead time, the actual replenishment of stock may
occur instantaneously or uniformly.

• Instantaneous replenishment occurs in case the stock is purchased from outside sources whereas
uniform replenishment may occur when the product is manufactured by the company.

21
FA C T O R S A F F E C T I N G I N V E N T O RY C O N T R O L

7. Inventory turnover:

• It is defined as the ratio of the annual sales volume of materials to the average investment in
inventories for the same period.

• It is an important index for inventory control.

• The higher the index, the lower the inventory levels and lower the cost of maintaining the
inventories.

22
FA C T O R S A F F E C T I N G I N V E N T O RY C O N T R O L

8. Standardization:

• In inventory control, standardization is the determination of fixed sizes, shapes, quantity and
dimensions of material.

• As it is desirable to encourage the use of standards prescribed for the type of industry in which the
company operates, by keeping the material to standards, a great deal of reductions are possible in
inventory space, obsolescence, handling costs, and inventory as well as improved quality.

23
 C O S T S A S S O C I AT E D W I T H I N V E N TO R I E S

• Purchase cost (C):

• It is the actual price per unit (in Rs) paid for the procurement of items.

• The purchase price will become important when quantity discounts are allowed for purchases
above a certain quantity or when economics of scale suggest that the per unit production cost can
be reduced by a larger production run.

Purchase cost = (Price per unit) × (Demand per unit time) = C × D

24
 C O S T S A S S O C I AT E D W I T H I N V E N TO R I E S

• Carrying (or holding) cost (Ch):

• The inventory cost incurred for carrying (or holding) inventory items in the warehouse is referred
as carrying cost.

• This cost generally includes the costs such as rent for space used for storage, interest on the
money locked-up, inventory handling cost for payment of salaries, insurance cost against fire
or other form of damage , taxes, depreciation of equipment and furniture used etc.

Carrying cost = (Cost of carrying one unit of an item in the inventory for a given length of time,
usually one year) × (Average number of units of an item carried in the inventory for a given length of
time)

25
 C O S T S A S S O C I AT E D W I T H I N V E N TO R I E S

• Carrying (or holding) cost (Ch):

Carrying Costs (Inventory holding costs) consist of expenditure connected with

1. Interest on capital investment

2. Cost of storage facility, up-keep (maintenance) of material, record keeping etc.

3. Cost involving deterioration and obsolescence, and

4. Cost of insurance, property tax etc.

Annual Inventory Carrying costs are almost directly proportional to the order size (order
quantity) or lot size.

26
 C O S T S A S S O C I AT E D W I T H I N V E N TO R I E S

• Ordering or set-up cost (Co or Cs) :

• The inventory cost incurred each time an order is placed for procuring items from the vendors is
referred as ordering cost.

• Advertisements, consumption of stationeries and postage, telephone charges, rent for space used
by the purchase department, travelling expenditures incurred etc. constitute the ordering cost.

Ordering cost = (Cost per order/per set-up) × (Number of orders/set-ups placed in the given period)

• Set-up cost:

• This is a cost associated with the setting up of machinery before starting production.

• Set-up cost is generally assumed to be independent of the quantity produced.

27
 C O S T S A S S O C I AT E D W I T H I N V E N TO R I E S

• Ordering or set-up cost (Co or Cs) :

Ordering Costs (Inventory procurement costs) consist of expenditure connected with

1. Processing purchase requisition.

2. Receiving quotations, following up and expediting purchase order.

3. Cost of services which includes cost of mailing, telephone calls, transportation, and other follow
up actions.

4. Materials handling cost incurred in receiving, sorting, inspecting and storing the items included
in the order.

5. Processing seller’s (vendor’s invoice) , accounting and auditing, etc.

Annual Ordering cost decreases as the order size (order quantity) or lot size increases.
28
 C O S T S A S S O C I AT E D W I T H I N V E N TO R I E S

• Shortage or stock-out or back-order cost (Cb):

• The penalty cost for running out of stock (i.e. when an item cannot be supplied on the customer’s
demand is known as shortage cost.

• This cost includes the loss of potential profit through sales of items and loss of goodwill, in terms
of permanent loss of customers and its associated lost profit in future sales.

Shortage cost = (Cost of being short one unit of an item) × (Average number of units short)

29
 C O S T S A S S O C I AT E D W I T H I N V E N TO R I E S

Total inventory cost

• If price discounts are offered, the purchase cost per unit becomes variable, and depends on the
quantity purchased.

• In such a case, the total inventory cost is calculated as follows:

Total variable inventory cost (TVC) = Purchase cost + Ordering cost + Carrying cost + Shortage cost

• But, if price discounts are not offered, the purchase cost per unit of an item remains constant and is
independent of the quantity purchased, then the total inventory cost is calculated as follows:

Total inventory cost (TIC) = Ordering cost + Carrying cost + Shortage cost

Total Annual Cost (TAC) = Purchase cost + Total inventory cost (TIC)
30
 INVENTORY CONTROL MODEL
Sale
Order Quantity
EOQ Model
Procurement
Company A

Supplier
Market/
Customers
Sale
Company B

EPQ Model

Production
In-house

31 Production lot size

 INVENTORY CONTROL MODEL

Inventory Control Model

EOQ Models without EOQ Models with price EPQ Models

price quantity discount quantity discount

Deterministic Demand: Probabilistic Demand:

EOQ Models EOQ Models

32 EOQ Models without EOQ Models with

Shortage Shortage
 EOQ MODELS

Maximum Inventory Level

Inventory Level

Order
Placed Reorder Level (or Point)
ROL

Order Quantity, Q

LT LT
𝑄 Time (Year)
t=𝐷
Reorder
Cycle

33
LT = Lead Time
 EOQ MODELS
• The concept was first developed by Ford W. Harris in 1913.

• The size of an order(s) affects inventory level to be maintained at various stocking points.

• By the term ‘order quantity’, we mean the quantity procured during one procurement cycle.

• Large order size for an item may reduce (i) the frequency of orders to procure inventory items, and
(ii) the total ordering cost.

• But large order size for an item will, however, increase the cycle stock inventory and carrying cost for
excess inventory.

• Any decision on replenishment order size (or batch size for production) should facilitate economical
trade-off between relevant inventory costs, viz., ordering, carrying and shortage costs.

34
 EOQ MODELS
• Demand is assumed to be fixed and completely pre-determined and items are procured from external
sources.

• When the size of order increases, the ordering costs will decrease whereas the inventory carrying
costs will increase.

• Thus, there are two opposite costs, one encourages the increase in the order size and the other
discourages.

• EOQ is that size of order which minimizes total annual costs of carrying inventory and cost of
ordering.

35
 EOQ MODELS

Cost
Total Cost
∗
𝑇𝑉𝐶

𝐷
Ordering Cost, 𝑄 𝐶𝑜

Purchase Cost, DC
0 𝑄∗ (EOQ) Order Size, Q

Thus, EOQ is the optimal replenishment order size (or lot size) of inventory item (or items) that achieves
the optimum total (or variable) inventory cost during the given period of time.
36
 EPQ MODELS
• Demand is assumed to be fixed and completely pre-determined.

• Items are produced instead of being purchased from outside.

• By the ‘order quantity’, we mean the quantity produced during one production cycle (production
cycle).

• The amount ordered is not delivered all at once, but the ordered quantity is sent or received gradually
over a length of time at a finite rate (i.e. given supply rate) per unit of time.

• Gradual (i.e. non-instantaneous) supply may arise in two cases:

37
 EPQ MODELS
Inventory Production
Level Rate
Amount Produced

Amount Sold during

production time
Maximum Inventory Level
Lot Size, Q
Production

p d
Reorder Level
(p-d) (ROL)

𝑡1 LT Time (Year)
𝑡2 , Inventory
Depletion Time
𝑄
Production cycle time, t = 𝐷
38
LT = Lead Time
 EPQ MODELS
• When the ordered quantity is sent or received at an infinite rate (i.e., given supply rate) per unit
of time i.e. instantaneously

Maximum Inventory Level

Inventory Level

Average Inventory

𝑄 Time (Year)
t=𝐷
Cycle time

39
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

• The objective is to determine an optimal order quantity (EOQ) such that the total inventory cost is
minimized.

Assumptions:

1. Demand is known and uniform.

2. Shortages are not permitted, i.e., as soon as the level of inventory reaches zero, the inventory is
replenished.

3. Supply of commodity is instantaneous (abundant availability).

40
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

• The objective is to determine an optimal order quantity (EOQ) such that the total inventory cost is
minimized.

Assumptions:

4. Ordering cost per placement of an order is constant.

5. Holding cost per unit in inventory is constant.

6. Unit purchase price is also constant.

41
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

NOTATIONS:

• D = Annual demand

• Q = Quantity procured per order (Order quantity)

• C = Price per unit item (Rs per unit)

• C0 = Ordering cost per order (Rs per order)

• Ch = Inventory carrying cost per unit (Rs per item per unit time)

• Ch = I.C; where, C = Unit cost, I is called inventory carrying charge expressed as % of the value of
unit price of inventory.
42
• d = Demand rate (quantity per unit time)
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

NOTATIONS:

• ROL = Reorder level (or point) at which an order is placed

• LT = Replenishment lead time (delivery time or period)

• n = Number of orders per time period

• t = Time interval between successive orders to replenish inventory stock.

43
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

Maximum Inventory Level

Inventory Level

Order
Placed Reorder Level (or Point)
ROL

Order Quantity, Q

LT LT
𝑄 Time (Year)
t=𝐷
Reorder Cycle

LT = Lead Time
44
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

• This fundamental situation can be shown on an inventory-time diagram, with Q on the vertical axis
and time on the horizontal axis.

• The total time period (one year) is divided into parts of equal length t.

• Assume, n is the total number of orders placed per year.

1 = 𝑛𝑡
D = 𝑛𝑄
𝐷
That means, n =
𝑄

• Order quantity replenished in one inventory cycle,

45 𝑄 = 𝐷. 𝑡 SAW-teeth diagram.
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

Total inventory cost, TIC = Annual inventory carrying cost + Annual ordering cost

Average inventory level Number of orders placed per

= +
× ( Carrying cost/unit/year ) year × ( Ordering cost/order )

𝐼max + 𝐼min 𝐷 𝑄 𝐷
= ⋅ 𝐶ℎ + ⋅ 𝐶0 = 𝐶ℎ + ⋅ 𝐶0
2 𝑄 2 𝑄

46
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

Cost
Total Cost
∗
𝑇𝑉𝐶

𝐷
Ordering Cost, 𝑄 𝐶𝑜

Purchase Cost, DC
0 𝑄∗ (EOQ) Order Size, Q
The total inventory cost is minimum at a value of 𝑄, which appears to be at the point where

47
inventory carrying and ordering costs are equal.
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

𝐷 𝑄 2𝐷𝐶0
⋅ 𝐶0 = ⋅ 𝐶ℎ or 𝑄2 =
𝑄 2 𝐶ℎ

2𝐷𝐶0 2 × Annual demand × Ordering cost

Economic Order Quantity EOQ , 𝑄∗ = =
𝐶ℎ Carrying cost

This formula for Q* is also known as the Wilson or Harris lot size formula.

48
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

1. Optimal interval, 𝒕∗ between the successive orders

𝑄∗ = Annual demand × Reorder cycle time = D. 𝑡

or

𝑄∗ 1 2𝐷𝐶0 2𝐶0
𝑡∗ = = × =
𝐷 𝐷 𝐶ℎ 𝐷𝐶ℎ

2. Optimal number of orders 𝒏∗ to be placed in the given time period (assumed as

one year)

Annual demand 𝐷 1 𝐷𝐶ℎ

𝑛∗ = = ∗=𝐷× =
Optimal order quantity 𝑄 2𝐷𝐶0 /𝐶ℎ 2𝐶0
49
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1: The Fundamental Problem of EOQ

3. Optimal (minimum) total inventory cost (TIC*)

𝐷 𝑄 ∗
TIC∗ = ∗ 𝐶0 + 𝐶ℎ
𝑄 2
1 𝐶ℎ
TIC∗ = 𝐷 ⋅ 𝐶0 × + × 2𝐷 𝐶0 Τ𝐶ℎ = 2𝐷𝐶0 𝐶ℎ
2𝐷𝐶0 Τ𝐶ℎ 2

TIC∗ = 2𝐷𝐶0 𝐶ℎ

4. Optimal total annual cost

TAC ∗ = Fixed purchase cost + Total inventory cost

= 𝐷. 𝐶 + TIC∗
50
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Example 1

Maximum inventory level is 2000 units, and it is achieved with infinite rate of replenishment. Inventory
becomes zero in one and half month due to consumption at a uniform rate and same cycle continues
throughout a year if ordering cost = Rs 800 per order and inventory carrying cost, Ch is Rs 5 per unit per
month. Find EOQ and TIC*.
Solution 1
Time interval between successive orders, t = 1.5 months
12
Total number of orders placed per year, n = =8
1.5

Annual Demand (𝐷) = 2,000 × 8 = 16,000

Ordering cost, C0 = 800

51
Inventory carrying cost, Ch = Rs 5 per unit per month = 5*12 = Rs 60 per unit per year
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Solution 1

2𝐷𝐶0 2×16000×800
Economic Order Quantity EOQ , 𝑄 ∗ = = 60
= 653.197 ≅ 654
𝐶ℎ

TIC∗ = 2𝐷𝐶0 𝐶ℎ = 2 × 16000 × 800 × 60 ≅ Rs 39, 192

or
𝐷 𝑄∗ 16000 654
TIC ∗ = 𝑄∗ 𝐶0 + 𝐶ℎ = × 800 + × 60 ≅ Rs 39, 192
2 654 2

𝐷 16000
Optimal number of orders 𝑛∗ = = = 24.46 ≅ 25
𝑄∗ 654

52
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Example 2

Annual Demand (𝐷) = 18,000 units per year

Price per unit item, C = Rs 4 per unit

Ordering cost, C0 = Rs 120 per order

Inventory carrying cost = 12% of C

Lead time, LT = 8 days

Working day per year = 300 days

Determine (i) EOQ (ii) Optimum number of order per year (iii) Length of inventory cycle (iv) TIC* (v)
ROL (vi) Amount of saving with EOQ against the earlier practice of 4 order per year (vii) increase in
total cost associated with ordering (a) 25% > EOQ (b) 40% < EOQ.
53
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Solution 2

2𝐷𝐶0 2×18000×120
(i) Economic Order Quantity EOQ , 𝑄 ∗ = = 0.12∗4
= 3000 units
𝐶ℎ

𝐷 18000
(ii) Optimal number of orders 𝑛∗ = 𝑄∗ = =6
3000

1 1 1
(iii) Length of inventory cycle = = years = * 300 = 50 days
𝑛∗ 6 6

(iv) TIC∗ = 2𝐷𝐶0 𝐶ℎ = 2 × 18000 × 120 × 0.48 = Rs 1440

𝐷 18000
(v) d = = = 60 units per day
𝑁𝑜. 𝑜𝑓 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 300
54
ROL = 8*60 = 480 units
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Solution 2

(vii) n = 4 orders

18000
𝑄= = 4500 units
4

𝐷 𝑄 18000 4500
TIC = 𝑄 𝐶0 + 2 𝐶ℎ = × 120 + × 0.12 × 4 = Rs 1560
4500 2

Savings = Rs 1560 – Rs 1440 = Rs 120

(viii) (a) 25% > EOQ

𝑄 = 𝑄 ∗ ×1.25 = 3000 units ×1.25 = 3750 units

𝐷 18000
Number of orders n = = = 4.8 ≅ 5
𝑄 3750
𝐷 𝑄 18000 3750
55 TIC = 𝐶0 + 𝐶ℎ = × 120 + × 0.12 × 4 = Rs 1476
𝑄 2 3750 2

Increase in total cost = Rs 1476 – Rs 1440 = Rs 36

 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Solution 2

(viii) (b) 40% < EOQ

𝑄 = 𝑄 ∗ ×0.6 = 3000 units ×0.6 = 1800 units

𝐷 18000
Number of orders n = 𝑄 = = 10
1800

𝐷 𝑄 18000 1800
TIC = 𝑄 𝐶0 + 2 𝐶ℎ = 1800
× 120 + 2
× 0.12 × 4 = Rs 1632

Increase in total cost = Rs 1632 – Rs 1440 = Rs 192

56
 PROBLEM
Example 3

A Company operating 50 weeks in a year is concerned about its stocks of copper cable. This costs Rs.
240/- a meter and there is a demand for 8000 meters a week. Each replenishment costs Rs. 1050/- for
administration and Rs. 1650/- for delivery, while holding costs are estimated at 25% of value held a year.
Assuming no shortages are allowed, what is the optimal inventory policy for the company?

How would this analysis differ if the company wanted to maximize profit rather than minimize cost?
What is the gross profit if the company sell cable for Rs. 360/- a meter?

• Profit = Revenue – Cost

𝐷 1
• Profit = 𝐷 × 𝑆𝑃 − 𝐷 × 𝐶 + 𝐶 + 𝑄𝐶ℎ
𝑄 𝑜 2

57
 PROBLEM
Solution 3
we have
Annual Demand (𝐷) = 8,000 × 50 = 4,00,000 meters a year
Purchase cost (𝐶) = Rs. 240 a meter
Ordering cost 𝐶𝑜 = 1,050 + 1,650 = Rs. 2,700
Holding cost (𝐶ℎ ) = 0.25 × 240 = Rs. 60 a meter a year
2×4,00,000×2,700
Optimal order quantity, 𝑄∗ = 2𝐷𝐶0 /𝐶ℎ = = 6,000 meters,
60

Total inventory cost, TIC = Annual inventory carrying cost + Annual ordering cost
𝑄∗ 𝐷
= 𝐶ℎ + ∗ ⋅ 𝐶0
2 𝑄
Total annual cost, TAC = Fixed purchase cost + Total inventory cost
= 𝐷. 𝐶 + TIC∗
58
𝑄∗ 𝐷
= 𝐷. 𝐶 + 𝐶ℎ + ∗ ⋅ 𝐶0
2 𝑄
 PROBLEM
Solution 3
𝑄∗ 𝐷
Total annual cost, TAC = 𝐷. 𝐶 + 𝐶ℎ + ∗ ⋅ 𝐶0
2 𝑄
1 4,00,000
= 4,00,000 × Rs. 240 + × 6,000 × Rs. 60 + × Rs. 2,700
2 6,000
= Rs. 9,60,00,000 + Rs. 3,60,000 = Rs. 9,63,60,000.

𝐷 1
Profit = Revenue − Cost = 𝐷 × 𝑆𝑃 − 𝐷 × 𝐶 + 𝐶0 + 𝑄𝐶1
𝑄 2
= 𝐷 × 𝑆𝑃 − Rs. 9,63,60,000.

If the company sells the cable for Rs. 360 a metre,

its revenue is Rs. 360 × 4,00,000 = Rs. 14,40,00,000 per year.

∴ Gross profit = Rs. 14,40,00,000 − Rs. 9,63,60,000 = Rs. 4,76,40,000 a year.
59
 PROBLEM
Example 4

A factory follows an economic order quantity system for maintaining stocks of one of its component
requirements. The annual demand is for 24,000 units, the cost of placing an order is Rs. 300/-, and the
component cost is Rs. 60/- per unit. The factory has imputed 24% as the inventory carrying rate.

a) Find the optimal interval for placing orders, assuming a year is equivalent to 360 days.

b) If it is decided to place only one order per month (1 month ~ 30 days), how much extra cost does
the factory incur per year as a consequence of this decision?

Solution 4

60
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1(a): EOQ Model with Different Rates of Demand

Maximum Inventory Level

Inventory Level

1
Q𝑡
2 1

𝑡1 𝑡2 𝑡3 𝑡𝑛 Time (Years)

• Demand is constant and varies from period to period.

• Objective: - To determine the order size (or production quantity) in each reorder cycle (or
61
period) that will minimize the total inventory cost.
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1(a): EOQ Model with Different Rates of Demand

• If t1, t2, . . ., tn denotes time for successive replenishment and D1, D2, . . ., Dn are the demand
rates at these cycles, respectively, then the total period T is given by:
T = t1 + t2 + . . . + tn.
• Suppose each time a fixed quantity, Q is ordered, then the number of orders in total time period
T will be:
n = D/Q, where D is the total demand in time period T.
• Thus, the inventory carrying cost for the time-period, T will be
1 1 1 1
Carrying cost = Q𝑡1 × 𝐶ℎ + 2 Q𝑡2 × 𝐶ℎ + 2 Q𝑡3 × 𝐶ℎ +………+ 2 Q𝑡𝑛 × 𝐶ℎ
2

1 1
= Q × 𝐶ℎ × (𝑡1 + 𝑡2 + 𝑡3 +………+ 𝑡𝑛 ) = Q × 𝐶ℎ × 𝑇
2 2

62
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 1(a): EOQ Model with Different Rates of Demand
𝐷
Annual ordering cost = ⋅ 𝐶0
𝑄

Total inventory cost, TIC = Annual inventory carrying cost + Annual ordering cost

𝑄 𝐷
= 𝐶ℎ 𝑇 + ⋅ 𝐶0
2 𝑄
For optimum value of Q that minimizes TIC
𝐷 𝑄 2
2𝐷𝐶0
⋅𝐶 = ⋅ 𝐶ℎ ⋅ 𝑇 or 𝑄 =
𝑄 0 2 𝑇𝐶ℎ

2𝐷𝐶0
Economic Order Quantity EOQ , 𝑄 ∗ =
𝑇𝐶ℎ

63
TIC∗ = 2𝐶0 𝐶ℎ (𝐷 Τ𝑇)
 EPQ MODELS
Case 2: The Fundamental Problem of EPQ

The objective is to determine an optimal production quantity (EPQ) such that the total inventory cost is
minimized.

Assumptions:

• Demand is assumed to be fixed and completely pre-determined.

• Demand is continuous and at a constant rate.

• Shortages are not permitted, i.e., as soon as the level of inventory reaches zero, the inventory is
replenished.

• Production of commodity is instantaneous (abundant availability of raw material).

64
 EPQ MODELS
Case 2: The Fundamental Problem of EPQ

The objective is to determine an optimal production quantity (EPQ) such that the total inventory cost is
minimized.

Assumptions:

• Set-up cost per production run is constant.

• Holding cost per unit in inventory is constant.

• Unit production price is also constant.

• Single type of item is involved.

65
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 2: The Fundamental Problem of EPQ

NOTATIONS:

• D = Annual demand

• Q = Quantity produced per production run (production lot size)

• CP = Production cost per unit item (Rs per unit)

• Cs = Set-up cost per setup (Rs per setup)

• Ch = Inventory carrying cost per unit (Rs per item per unit time)

• Ch = I. CP; where I is called inventory carrying charge expressed as % of the value of unit price of
inventory.
66
• d = Demand rate (quantity per unit time)
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 2: The Fundamental Problem of EPQ

NOTATIONS:

• ROL = Reorder level (or point) at which an order is placed

• LT = Replenishment lead time (delivery time or period)

• n = Number of orders per time period

• t = Time interval between successive orders to replenish inventory stock.

• t1 = Production period (time)

• p = Production rate (quantity per unit time) at which quantity Q is added to inventory

67
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 2: The Fundamental Problem of EPQ

• This fundamental situation can be shown on an inventory-time diagram, with Q on the vertical axis
and time on the horizontal axis.

• The total time period (one year) is divided into parts of equal length t.

• Assume, n is the total number of orders placed per year.

1 = 𝑛𝑡
D = 𝑛𝑄
𝐷
That means, n =
𝑄

68
SAW-teeth diagram.
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 2: The Fundamental Problem of EPQ

Total inventory cost, TIC = Annual Set-up Cost + Inventory Carrying Cost

𝑄
= 𝑛 ⋅ 𝐶𝑠 + 𝐶ℎ
2

𝐷 𝑄
⋅𝐶 + 𝐶
𝑄 s 2 ℎ

2𝐷𝐶s
Economic Production Quantity EPQ , 𝑄∗ =
𝐶ℎ

69
 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Case 2: The Fundamental Problem of EPQ

1. Optimal interval, 𝒕∗ between the successive orders

𝑄∗ = Annual demand × Reorder cycle time = D. 𝑡

or

𝑄∗ 1 2𝐷𝐶s 2𝐶s
𝑡∗ = = × =
𝐷 𝐷 𝐶ℎ 𝐷𝐶ℎ

2. Optimal number of production runs executed per year

Annual demand 𝐷 1 𝐷𝐶ℎ

𝑛∗ = = ∗=𝐷× =
Optimal production quantity 𝑄 2𝐷𝐶s /𝐶ℎ 2𝐶s
70
 EPQ MODELS
Case 3: Problem of EPQ with Finite Production (Replenishment)

• In this problem, all the assumptions are same as in case of simple EPQ model, except that of
instantaneous replenishment. Assume that each production run of length 𝑡 consists of two parts, say
𝑡1 𝑎𝑛𝑑 𝑡2 , such that

Assumptions:

• Items are produced instead of being purchased from outside.

• The amount ordered is not delivered all at once, but the produced or ordered quantity is sent or
received gradually over a length of time at a finite rate (i.e. given supply rate) per unit of time.

• The inventory is building up at a constant rate of (𝑝 − 𝑑) units per unit time during 𝑡1 , 𝑝 > 𝑑;

71
• There is no replenishment (production) during time 𝑡2 and the inventory is decreasing at the rate of 𝑑
per unit of time.
 EPQ MODELS
Inventory Production
Level Rate
Amount Produced

Amount Sold during

production time
Maximum Inventory Level
Lot Size, Q
Production

p d
Reorder Level
(p-d) (ROL)

𝑡1 LT Time (Year)
𝑡2 , Inventory
Depletion Time
𝑄
Production cycle time, t = 𝐷
72
LT = Lead Time
 EPQ MODELS
Case 3: Problem of EPQ with Finite Production (Replenishment)

• Here, the total (production) quantity Q is produced over a period 𝑡1 , which is defined by the
production rate 𝑝.

• Since the inventory does not pile up in one shot but rather continuously over a time period and is also
consumed simultaneously, the average inventory level would be determined not only by the lot size
Q, but also be affected by the production rate 𝑝 and depletion (demand) rate 𝑑.

73
 EPQ MODELS
Case 3: Problem of EPQ with Finite Production (Replenishment)

To determine the average inventory, we proceed as follows:

𝑄
• Since 𝑡1 is the time required to produce Q at a rate 𝑝, we shall have 𝑄 = 𝑝𝑡1 or 𝑡1 =
𝑝

• During production period 𝑡1 , inventory is increasing at the rate of 𝑝 and simultaneously decreasing at
the rate of 𝑑.

• Thus, inventory accumulates at the rate of (𝑝 − 𝑑) units.

• Therefore, the maximum inventory level shall be equal to 𝑡1 (𝑝 − 𝑑).

1 1 𝑑 𝑄
• Hence, average inventory = 𝑡 𝑝−𝑑 = 𝑄(1 − ) Since 𝑡1 =
2 1 2 𝑝 𝑝
74
 EPQ MODELS
Case 3: Problem of EPQ with Finite Production (Replenishment)

𝐷 1 𝑑
• Hence, annual inventory cost = 𝐶𝑠 + 𝑄 1 − 𝐶ℎ
𝑄 2 𝑝

2𝐷𝐶s
Economic Production Quantity EPQ , 𝑄∗ = Economic Batch Quantity
𝑑
𝐶ℎ 1−
𝑝

𝐷 𝐷𝐶ℎ 𝑑
Optimal number of production runs per year = 𝑛∗ = ∗= 1−
𝑄 2𝐶𝑠 𝑝

75
 EPQ MODELS
Case 3: Problem of EPQ with Finite Production (Replenishment)

∗
𝑄 2𝐷𝐶𝑠
Optimal length of each lot size production run = 𝑡1∗ = =
𝑝 𝐶ℎ 𝑝 𝑝 − 𝑑

∗
𝑑
Total minimal inventory cost (annual), TIC = 2𝐷𝐶𝑠 𝐶ℎ 1−
𝑝

76
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Example 1

A contractor has to supply 10,000 bearings per day to an automobile manufacturer. He finds that when he
starts production run, he can produce 25,000 bearings per day. The cost of holding a bearing in stock for
a year is Rs 2 and the set-up cost of a production run is Rs 1,800. How frequently should production run
be made? Assume 300 working days in the year.
Solution 1
we have
Cs = Rs 1,800 per production run;
Ch = Rs 2 per bearing per year
p = Rs 25,000 bearings per day
d = 10,000 bearing per day

77 D = 10,000 × 300 = 30,00,000 units/year

 D E T E R M I N I S T I C I N V E N T O RY P R O B L E M S
W I T H N O S H O RTA G E S
Solution 1
Economic batch quantity for each production run is given by

2𝐷𝐶s 2×30 00 000×1800

𝑄∗ = 𝑑 = 10000 ≅ 94868
𝐶ℎ 1−𝑝 2× 1−
25000

𝑄∗ 94868
Frequency of inventory cycle 𝑡 ∗ = = 10000 = 9.49 days
𝑑

∗ 𝑄∗ 94868
Length of production cycle 𝑡1 = = 25000 = 4 days (Approx.)
𝑝

Thus, the production cycle starts at an interval of 9.49 days and production continues for 4 days so
78 that in each cycle a batch of 94,868 bearings is produced.
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Example 2

A manufacturing company needs 2500 units of a particular component every year. The company buys it
at the rate of Rs. 30/- per unit. The order processing cost for this part is estimated at Rs. 15/- and the cost
of carrying a part in stock comes to about Rs. 4/- per year.

The company can manufacture this part internally. In that case, it saves 20% of the price of the product.
However, it estimates a set-up cost of Rs. 250/- per production run. The annual production rate would be
4800 units. However, the inventory holding costs remain unchanged.

[1] Determine the EOQ and the optimal number of orders placed in a year.

[2] Determine the optimal production lot size and the average duration of the production run.

[3] Should the company manufacture the component internally or continue to purchase it from the
79
supplier?
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)

Solution 2
we have
D = 2500 units/year
Co = Rs 15 per order
Ch = Rs 4 per unit per year

2𝐷𝐶o 2×2500×15
Economic order quantity, 𝑄 ∗ = = ≅ 137 units
𝐶ℎ 4

𝐷 2500
Optimal number of orders, 𝑛∗ = = = 18
𝑄∗ 137

80
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Solution 2
we have
D = 2500 units/year
Cs = Rs 250 per production run;
Ch = Rs 4 per unit per year
p = 4800 units/year
d = 2500 units/year

2𝐷𝐶s 2×2500×250
Economic lot size, 𝑄 ∗ = 𝑑 = 2500 ≅ 808 units
𝐶ℎ 1−𝑝 4× 1−
4800

∗ 𝑄∗ 808
Average duration of production run, 𝑡1 = = year = 0.17 year
𝑝 4800
𝑄∗ 808
81 Frequency of production run 𝑡∗ = = = 0.32 year
𝑑 2500
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Solution 2
When item is purchased from outside:

Total minimal annual cost, TAC∗ = 𝐷. 𝐶 + TIC∗ = 𝐷. 𝐶 + 2𝐷𝐶𝑜 𝐶ℎ

= 2500 × 30 + 2 × 2500 × 15 × 4
= Rs. 75547.72 ≅ Rs. 75548
When item is produced internally:
𝑑
Total minimal annual cost, TAC∗ =𝐷. 𝐶 + TIC∗ = 𝐷. 𝐶 + 2𝐷𝐶𝑠 𝐶ℎ 1 −
𝑝

2500
= 2500 × 24 + 2 × 2500 × 250 × 4 × 1 −
4800

= Rs. 61547.85 ≅ Rs. 61548

82
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Example 3

(a) At present a company purchases an item X from outside suppliers. The consumption of this item is
10,000 units/year. The cost of the item is Rs 5 per unit and the ordering cost is estimated to be Rs 100 per
order. The cost of carrying inventory is 25 per cent. If the consumption rate is uniform, determine the
economic purchasing quantity.

(b) In the above problem assume that company is going to manufacture the item with the equipment that
is estimated to produce 100 units per day. The cost of the unit thus produced is Rs 3.50 per unit. The set-
up cost is Rs 150 per set-up and the inventory carrying charge is 25 per cent. How has your answer
changed?
Solution 3
(a) Economic order quantity = 1265 units
83
(b) Economic batch quantity = 2391 units
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Example 4

Company A wants to know what production cost its major competitor, company B, has assigned to
product item 𝑝7. After a bit of investigation, company A has collected the following data about company
B’s production of item 𝑝7.

Production lot size = 2600 units; Set-up cost = Rs. 135/-; Annual demand = 30,000 units;

Daily demand = 100 units; Production rate = 200 units per day;

Inventory holding costs = 28% of the average value per year

Company A has further learnt that company B produces according to ‘economic lot size’ model. What is
the company B’s cost of producing product item 𝑝7 ?
Solution 4
84
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Example 5

The details of a part to be machined are as follows:

Annual requirement = 2400 pieces

Machine rate = 10 pieces/shift

Number of working days in the year = 320 shifts

Cost of machining a component = Rs. 100/- per piece

Inventory carrying cost per annum = 12% of value

Set-up cost per production run = Rs. 400/-

Find the optimal run size for machining.

Solution 5
85 Optimal run size for machining = 800 pieces
 EPQ WITH FINITE PRODUCTION
(REPLENISHMENT)
Example 6

A Company requires 12000 units of a Component X in a year. Component X is made in 30 batches of

400 units each on a machine that produces 8 units per hour. The company operates for 4000 hour per
year, and it costs Rs. 500 to setup the machine. Holding Cost, Ch is Rs 10/unit/year. Find out whether the
existing production plan is optimum or not. If not, suggest a new production plan and amount of saving
possible. Also, determine production cycle time, maximum inventory level and cycle time corresponding
to optimum cycle.

Solution 6

86
 EOQ

Example 7

Neon lights of A campus are replaced at the rate of 100 units per day. The physical plant orders the neon
lights periodically. It costs $100 to initiate a purchase order. A neon light kept in storage is estimated to
cost about $.02 per day. The lead time between placing and receiving an order is 12 days. Determine the
ROL.

a) 1200 b) 1000 c) 200 d) 100

Solution 7

c) 200

87
 Case 4: Deterministic Inventory Problem with Shortages

Q = Q1 + Q 2
88
t = t1 + t 2
 Case 4: Deterministic Inventory Problem with Shortages

• Shortages are permitted.

• As soon as the desired units of a certain commodity arrive in inventory, the back orders are satisfied.

• Let 𝐶2 be the shortage cost per unit of time per unit quantity.

• Here, the total time period is one year and is divided into equal parts, say of interval 𝑡. Further, this
time interval 𝑡 is further divided into two parts 𝑡1 and 𝑡2 , such that 𝑡 = 𝑡1 + 𝑡2 .

89
 Case 4: Deterministic Inventory Problem with Shortages

• During the interval 𝑡1 , the items are drawn from the inventory as needed and during 𝑡2 , orders for the
item are being accumulated but not filled.

• Then at the end of the interval 𝑡 an amount 𝑄 is procured.

• The amount 𝑄 is has been divided into 𝑄1 and 𝑄2 such that 𝑄 = 𝑄1 + 𝑄2 , where 𝑄1 denotes the
amount which goes into inventory, and 𝑄2 denotes the amount which is immediately taken to satisfy
past orders or unfilled demand.

90
 Case 4: Deterministic Inventory Problem with Shortages

• The problem now is concerned with the areas of triangles above the time axis (representing items in
inventory) and below the same axis (representing items in shortage).

1
• Now, total inventory over the time period 𝑡 = 𝑄1 𝑡1
2

1
𝑄 𝑡
2 1 1
• Average inventory at any time =
𝑡

1
𝑄 𝑡
2 1 1
• Annual holding cost = 𝐶ℎ
𝑡

91
 Case 4: Deterministic Inventory Problem with Shortages

1
• Similarly, total amount of shortage over time period, 𝑡 = 𝑄 𝑡
2 2 2

1
𝑄 𝑡
2 2 2
• Average shortage at any time =
𝑡

1
𝑄 𝑡
2 2 2
• Annual shortage cost = 𝐶𝑏
𝑡

92
 Case 4: Deterministic Inventory Problem with Shortages

• Total (annual) inventory cost = Ordering cost + Holding cost + Shortage cost

1 1
𝐷 𝑄 𝑡
2 1 1
𝑄 𝑡
2 2 2
= 𝐶 + 𝐶ℎ + 𝐶𝑏
𝑄 𝑜 𝑡 𝑡

• Now, using the relationship for similar triangles, we have

𝑡1 𝑄1 𝑄1
• = => 𝑡1 = 𝑡 ……………………….. (1)
𝑡 𝑄 𝑄

𝑡2 𝑄2 𝑄2 𝑄−𝑄1
• = => 𝑡2 = 𝑡 => 𝑡2 = 𝑡 ………………… (2)
𝑡 𝑄 𝑄 𝑄

93
 Case 4: Deterministic Inventory Problem with Shortages

𝐷 1 𝑄1 2 1 𝑄−𝑄1 2
• Annual inventory cost = 𝐶 + 𝐶 + 𝐶
𝑄 𝑜 2 ℎ 𝑄 2 𝑏 𝑄

𝜕(𝐴𝑛𝑛𝑢𝑎𝑙 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑐𝑜𝑠𝑡)

• = 0 …………………… (3)
𝜕𝑄1

𝜕(𝐴𝑛𝑛𝑢𝑎𝑙 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑐𝑜𝑠𝑡)

• = 0 …………………… (4)
𝜕𝑄

94
 Case 4: Deterministic Inventory Problem with Shortages

𝑄𝐶𝑏
• From Eq. (3) we get, Q∗1 =
𝐶ℎ +𝐶𝑏

2𝐶0 𝐷+𝐶ℎ 𝑄12

• From Eq. (4) we get, 𝑄∗ = + 𝑄12
𝐶𝑏

• Thus, the optimum quantities are given by (on simplification)

2𝐷𝐶𝑜 𝐶ℎ +𝐶𝑏
• 𝑄∗ =
𝐶ℎ 𝐶𝑏

𝐶𝑏 2𝐷𝐶𝑜
• 𝑄1∗ = (optimal stock level)
𝐶ℎ +𝐶𝑏 𝐶ℎ

95
 Case 4: Deterministic Inventory Problem with Shortages

A dealer supplies you the following information with regard to a product dealt in by him:

Annual demand: 10,000 units; Ordering cost: Rs. 10/- per order; Price: Rs. 20/- per unit; Inventory
carrying cost: 20% of the value of inventory per year.

The dealer is considering the possibility of allowing some back-order (stock-out) to occur. He has
estimated that the annual cost of back-ordering will be 25% of the value of inventory.

[1] What should be the optimal number of units (product) he should by in one lot?

[2] What quantity of the product should be allowed to be back-ordered, if any?

[3] What would the maximum inventory level?

[4] Would you recommend to allow back-ordering? If so, what would be the annual cost saving/ loss by
96

adopting the policy of back-ordering?

 Case 4: Deterministic Inventory Problem with Shortages
Solution

In the usual notations, we are given:

𝐷 = 10,000 units, 𝐶o = Rs. 10 per order, 𝐶h = 20% of Rs. 20 = Rs. 4 per unit per year, and 𝐶b =
25% of Rs. 20 = Rs. 5 per unit per year.
(i) (a) When stock-outs are not permitted:

2𝐷𝐶o 2 × 10,000 × 10
𝑄∗ = = = 223.6 units
𝐶h 4

(b) When back-ordering is permitted:

2𝐷𝐶0 𝐶ℎ + 𝐶b 2 × 10,000 × 10 (4 + 5)
𝑄∗ = × = × = 300 units
𝐶ℎ 𝐶b 4 5
97
 Case 4: Deterministic Inventory Problem with Shortages
Solution
(ii) Optimum quantity of the product to be backordered is given by:

∗
4
𝑄2 = 300 × = 133 units (approx.)
4+5

(iii) Maximum inventory level = 300 − 133 = 167 units.

(iv) Minimum total variable inventory cost in the cases (a) and (b) are:

𝑇𝐼𝐶(223.6) = 2 × 10,000 × 10 × 4 = Rs. 894.43

5
𝑇𝐼𝐶(300) = 2 × 10,000 × 10 × 4 × = Rs. 666.67
(4 + 5)

98 Since, 𝑇𝐼𝐶(223.6) > 𝑇𝐼𝐶(666.67), the dealer should accept the proposal for back-ordering
as this will result in a saving of (894.43 − 666.67) = Rs. 227.76 per year.
 Case 5: Problem of EOQ with Price Breaks

• In the real world, it is not always true that the unit cost of an item is independent of the quantity
produced.

• Often, discounts are offered for the purchase of large quantities. These discounts take the form of
price breaks.

• Let us now consider a firm which is encountered with a problem of determining an optimal order
quantity for each procurement run and an optimal interval between successive runs.

• The following conditions are assumed to hold:

(i) Demand is known and uniform.

(ii) Shortages are not allowed.

99
(iii) Supply of commodities is instantaneous.
 Case 5: Problem of EOQ with Price Breaks

𝐷 1
𝑇𝐴𝐶 = 𝐷𝑘1 + 𝐶𝑜 + 𝑄𝐶ℎ
𝑄 2

𝐷 1
𝑇𝐴𝐶 = 𝐷𝑘1 + 𝐶𝑜 + 𝑄 𝐼𝑘1
𝑄 2

∗
2𝐷𝐶𝑜
𝑄 =
𝐼𝑘1

100
 Case 5: Problem of EOQ with One Price Breaks

• When there is only one price break (one quantity discount), the situation may be illustrated as
follows:
𝑅𝑎𝑛𝑔𝑒 𝑜𝑓 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡

0 ≤ 𝑄1 < 𝑏1 𝑘11

𝑏1 ≤ 𝑄2 ≤ 𝑏2 𝑘12

where 𝑏1 is that quantity at and beyond which the quantity discount applies and 𝑘12 < 𝑘11 .

• The procedure for obtaining EOQ may be summarized in the following steps:

Step 1: Compute 𝑄2∗ , i.e., optimum order quantity for the lowest price (highest discount), i.e., 𝑘12 and
compare it with the quantity 𝑏1 .

101
 Case 5: Problem of EOQ with One Price Breaks

Step 2: If 𝑄2∗ > 𝑏1 , then optimum order quantity will be 𝑄2∗ , i.e., 𝑄 ∗ = 𝑄2∗ .

Step 3: If 𝑄2∗ < 𝑏1 , we cannot place order at the reduced price 𝑘12 . Therefore, in order to obtain the
optimal order quantity, we need to compare the total (annual) cost for 𝑄 = 𝑄1∗ (for price 𝑘11 ) with
𝑄 = 𝑏1 .

102
 Case 5: Problem of EOQ with One Price Breaks

• The values of 𝑇𝐴𝐶 𝑄1∗ and 𝑇𝐴𝐶 𝑏1 may be determined as follows:

𝐷 1 ∗
𝑇𝐴𝐶 𝑄1∗ = 𝐷𝑘11 + ∗ 𝐶𝑜 + 𝑄1 𝑘11 𝐼
𝑄1 2

𝐷 1
𝑇𝐴𝐶 𝑏1 = 𝐷𝑘12 + 𝐶𝑜 + 𝑏1 𝑘12 𝐼
𝑏1 2

If 𝑇𝐴𝐶 𝑄1∗ > 𝑇𝐴𝐶 𝑏1 , then 𝑄 ∗ = 𝑏1 , Otherwise, 𝑄∗ = 𝑄1∗ .

103
 Case 5: Problem of EOQ with One Price Breaks

Problem:

• Find the optimal order quantity for a product for which the price breaks are as follows:
𝑅𝑎𝑛𝑔𝑒 𝑜𝑓 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡

0 ≤ 𝑄1 < 800 𝑘11 (Rs. 1/-)

800 ≤ 𝑄2 𝑘12 (Rs. 0.98/-)

The yearly demand for the product is 1600 units, cost of placing an order is Rs. 5/-, the cost of
storage is 10% per year.

Solution: EOQ = 800 units

104
 Case 5: Problem of EOQ with One Price Breaks

Problem:

A company uses 8000 units of a product as raw material, costing Rs. 10/- per unit. The administrative
cost per purchase is Rs. 40/-. The holding costs are 28% of the average inventory. The company is
following an optimal purchase policy and places orders according to EOQ. It has been offered a quantity
discount of 1% if it purchases its entire requirement only 4 times a year.

Should the company accept the offer of quantity discount of 1%? If not, what minimum discount should
the company demand?

Solution: EOQ = 478 units; TAC = Rs 81338.66; Discount = 2%

105
 Case 5: Problem of EOQ with more than One Price Breaks

A manufacturing firm requires 2000 units of a material per year. The ordering costs are Rs. 10/- per
order. While carrying costs is 16% per unit of average inventory, and the purchase price is Rs. 1/- per
unit. Find the economic order quantity and total (annual) cost. If a discount of 5% is available for orders
of 1000 units or more but less than 2000 units, should the manufacturer accept this offer? Also, if it
purchases a single lot of 2000 units or more, it has to pay Rs. 0.93/- per unit. What purchase quantity
would you recommend?
𝑅𝑎𝑛𝑔𝑒 𝑜𝑓 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡

0 < 𝑄1 < 1000 𝑘11 (Rs. 1/-)

𝟏𝟎𝟎𝟎 ≤ 𝑸𝟐 < 𝟐𝟎𝟎𝟎 𝒌𝟏𝟐 (Rs. 0.95/-)

2000 ≤ 𝑄3
106
𝑘13 (Rs. 0.93/-)
 Case 5: Problem of EOQ with more than One Price Breaks

Find the optimal order quantity for a product where the annual demand for the product is 500 units, the
cost of storage per unit per year is 10% of the unit cost and ordering cost per order is Rs. 180/-. The unit
costs are given below:
𝑅𝑎𝑛𝑔𝑒 𝑜𝑓 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡

0 < 𝑄1 < 500 𝑘11 (Rs. 25/-)

500 ≤ 𝑄2 < 1500 𝑘12 (Rs. 24.80/-)

1500 ≤ 𝑄3 < 3000 𝑘13 (Rs. 24.60/-)

3000 ≤ 𝑄4 𝑘14 (Rs. 24.00/-)

107
 Case 5: Problem of EOQ with Price Breaks

108
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• When the inventories consist of several items under some limitations such as the availability of
warehouse space, total investment in inventories, or the total number of orders to be placed per year
for all items, number of deliveries which can be accepted, size of delivery which can be handled, etc.,
then it is not possible to consider each item separately, since there exists a relation among the items.

• Then the technique Lagrange’s multipliers is used to handle simple cases.

• In all such problems, first of all, we solve the problem ignoring the limitations and then consider the
effect of limitations.

• Consider the problem of having 𝑛 items (types) in inventory, with instantaneous procurement and no
lead time.

109
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• Assume that the demand is known and is uniform at a rate 𝐷𝑖 per unit time for the 𝑖 𝑡ℎ item
𝑖 = 1,2,3, … … , 𝑛 .

• Let 𝐶ℎ𝑖 be the carrying cost or inventory holding cost per unit of the quantity of the 𝑖 𝑡ℎ item and 𝐶𝑜𝑖
be the ordering cost per order for the 𝑖 𝑡ℎ item.

• If shortages are not allowed, then the total inventory cost of 𝑖𝑡ℎ item per annum is given by:

𝐷𝑖 1
𝑇𝐼𝐶 𝑖 = 𝐶𝑜𝑖 + 𝑄𝑖 𝐶ℎ𝑖
𝑄𝑖 2

where 𝑄𝑖 is the quantity of 𝑖 𝑡ℎ item in inventory at the beginning of the period.

110
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• Thus, the total inventory cost is given by:

𝐷𝑖 1
TIC = σ𝑛𝑖=1 𝐶 + 𝑄𝑖 𝐶ℎ𝑖
𝑄𝑖 𝑜𝑖 2

• Assuming the storage space required per unit item of 𝑖 𝑡ℎ type is 𝑓𝑖 , then total floor area (or volume)
required by all inventory items must be less than or equal to the total floor area (or volume) of the
warehouse.

• Constraint (Warehouse capacity)

σ𝑛𝑖=1 𝑓𝑖 𝑄𝑖 ≤ 𝑊 and 𝑄𝑖 ≥ 0 for 𝑖 = 1,2, … , 𝑛

111
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• This constraint indicates that even if all items reach their maximum inventory levels simultaneously,
the warehouse will accommodate (house) them all.

• Here, it is assumed that all the items are received together.

• Thus, the problem becomes to minimize the total variable inventory cost for each item together under
warehouse capacity constraint. Thus, we have to

𝐷𝑖 1
• Minimize TC = σ𝑛𝑖=1 𝐶 + 𝑄𝑖 𝐶ℎ𝑖 subject to the constraints:
𝑄𝑖 𝑜𝑖 2

σ𝑛𝑖=1 𝑓𝑖 𝑄𝑖 ≤ 𝑊 and 𝑄𝑖 ≥ 0 for 𝑖 = 1,2, … , 𝑛

• In order to find the optimal value for 𝑄𝑖 𝑖 = 1,2, … , 𝑛 so as to minimize (TC), we use the technique
of Lagrange multiplier.
112
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• We write the Lagrangian function,

𝐷𝑖 1
𝐿 𝑄𝑖 , 𝜃 = σ𝑛𝑖=1 𝐶 + 𝑄𝑖 𝐶ℎ𝑖 + 𝜃 σ𝑛𝑖=1 𝑄𝑖 𝑓𝑖 − 𝑊 ,
𝑄𝑖 𝑜𝑖 2

where, 𝜃 is non-negative Lagrange multiplier.

• The optimum values of 𝜃 and 𝑄𝑖 𝑖 = 1,2, … , 𝑛 are obtained by setting

𝜕𝐿 𝜕𝐿
= 0 and =0
𝜕𝜃 𝜕𝑄𝑖

• Thus, σ𝑛𝑖=1 𝑄𝑖 𝑓𝑖 − W = 0

or σ𝑛𝑖=1 𝑄𝑖 𝑓𝑖 = 𝑊, since 𝜃 ≥ 0; …………………………………………….(1)

113
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

1 𝐷𝑖
• 𝐶 − 𝐶 + 𝜃𝑓𝑖 = 0
2 ℎ𝑖 𝑄𝑖2 𝑜𝑖

2𝐷𝑖 𝐶𝑜𝑖
• or 𝑄𝑖∗ = ………………………………………………………………..(2)
𝐶ℎ𝑖 +2𝜃𝑓𝑖

Here, 𝜃 indicates an additional cost related to the storage area used by each unit of the item.

114
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• The procedure to find suitable value of 𝜃 so that it may satisfy the optimum values is given in
the following steps:

• Step 1: For 𝜃=0, compute EOQ for each item separately.

2𝐷𝑖 𝐶𝑜𝑖
• 𝑄𝑖∗ =
𝐶ℎ𝑖 +2𝜃𝑓𝑖

• Step 2: If 𝑄𝑖∗ 𝑖 = 1,2, … , 𝑛 satisfy the storage space constraint, then stop. Otherwise go to the next
step.

• σ𝑛𝑖=1 𝑄𝑖 𝑓𝑖 = 𝑊

• Step 3: For different values of 𝜃>0, compute EOQ for each item by systematic trial-and-error
method until the additional constraint on storage space is satisfied.
115
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

PROBLEM

A machine shop produces three products (types) 1, 2, 3 in lots. The shop has a warehouse whose
total floor area is 4000 sq. meters. The relevant data for three items is given below. The inventory
carrying charges for the shop are 20% of the average inventory valuation per annum of each item.
If no stock-outs are allowed and at no time can the warehouse capacity be exceeded, determine the
optimal lot size for each item.
Item 1 2 3
Annual demand (Units/year) 500 400 600
Setup cost per lot (Rs.) 800 600 1000
Production cost per unit (Rs.) 30 20 70
Floor area required (sq. meters) 5 4 10
𝑛
116 2𝐷𝑖 𝐶𝑠𝑖
𝑄𝑖∗ = ෍ 𝑄𝑖 𝑓𝑖 = 𝑊
𝐶ℎ𝑖 + 2𝜃𝑓𝑖
𝑖=1
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• Problem of EOQ with Average Inventory Level Constraint

𝑄𝑖
• Since, the average number of units in the inventory of an item 𝑖 is , and it is required that the total
2

average number of units of all items held in the inventory should not exceed to number 𝐾.

1 𝑛
• Therefore, we must have σ 𝑄 ≤𝐾
2 𝑖=1 𝑖

• The problem of minimizing the total variable inventory cost under the above inventory constraint,
then is to

𝐷𝑖 1
Minimize TC = σ𝑛𝑖=1 𝐶 + 𝑄𝑖 𝐶ℎ𝑖 subject to the constraints:
𝑄𝑖 𝑜𝑖 2

1 𝑛
σ 𝑄 ≤ 𝐾 and 𝑄𝑖 ≥ 0 for 𝑖 = 1,2, … , 𝑛
2 𝑖=1 𝑖

117
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• Problem of EOQ with Average Inventory Level Constraint

• We write the Lagrangian function,

𝐷𝑖 1 1 𝑛
• 𝐿 𝑄𝑖 , 𝜃 = σ𝑛𝑖=1 𝐶 + 𝑄𝑖 𝐶ℎ𝑖 + 𝜃 σ 𝑄 −𝐾 ,
𝑄𝑖 𝑜𝑖 2 2 𝑖=1 𝑖

where, 𝜃 is non-negative Lagrange multiplier.

• The optimum values of 𝜃 and 𝑄𝑖 𝑖 = 1,2, … , 𝑛 are obtained by setting

𝜕𝐿 𝜕𝐿
= 0 and =0
𝜕𝜃 𝜕𝑄𝑖
𝑛
• Thus, we get 2𝐷𝑖 𝐶𝑜𝑖 1
𝑄𝑖∗ = ෍ 𝑄𝑖 = 𝐾
𝐶ℎ𝑖 + 𝜃 2
𝑖=1

118
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• Problem of EOQ with Investment Constraint

• Inventory represents a substantial investment capital for many firms.

• Thus, the decision-maker places a limit on the amount of inventory to be carried.

• The inventory control policy must then be adjusted to meet this objective, if total investment exceeds
the limit.

• If a monetary limit is placed on all items carried, then the constraint on investment can be expressed
as:

σ𝑛𝑖=1 𝑘𝑖 𝑄𝑖 ≤ 𝐹; Here, 𝑘𝑖 is the cost per unit item of 𝑖 𝑡ℎ type for 𝑖 = 1,2, … , 𝑛 .

119
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• Problem of EOQ with Investment Constraint

• The problem of minimizing the total variable inventory cost under the above inventory constraint,
then is to

𝐷𝑖 1
Minimize TC = σ𝑛𝑖=1 𝐶 + 𝑄𝑖 𝐶ℎ𝑖 subject to the constraints:
𝑄𝑖 𝑜𝑖 2

σ𝑛𝑖=1 𝑘𝑖 𝑄𝑖 ≤ 𝐹 and 𝑄𝑖 ≥ 0

120
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

• Problem of EOQ with Investment Constraint

• We write the Lagrangian function,

𝐷𝑖 1
𝐿 𝑄𝑖 , 𝜃 = σ𝑛𝑖=1 𝐶 + 𝑄𝑖 𝐶ℎ𝑖 + 𝜃 𝑘𝑖 𝑄𝑖 − 𝐹 ,
𝑄𝑖 𝑜𝑖 2

where, 𝜃 is non-negative Lagrange multiplier.

• The optimum values of 𝜃 and 𝑄𝑖 𝑖 = 1,2, … , 𝑛 are obtained by setting

𝜕𝐿 𝜕𝐿
= 0 and =0
𝜕𝜃 𝜕𝑄𝑖

• Thus, we get 𝑛
2𝐷𝑖 𝐶𝑜𝑖
𝑄𝑖∗ = ෍ 𝑘𝑖 𝑄𝑖 = 𝐹
𝐶ℎ𝑖 + 2𝜃𝑘𝑖 𝑖=1

121
 M U LT I - I T E M D E T E R M I N I S T I C P R O B L E M S

PROBLEM

• A small shop produces three machine parts I, II, and III in lots. The demand rate for each item
(type) is constant and can be assumed to be deterministic. No back orders are to be allowed.
The pertinent data for the items is given in the following table.

Item 1 2 3
Annual demand (Units/year) 10,000 12,000 7,500
Setup cost per lot (Rs.) 50 40 60
Production cost per unit (Rs.) 6 7 5
Holding cost (Rs.) @unit 20 20 20

Determine approximately the EOQs when the total value of average inventory levels of these
items does not exceed Rs. 1,000/-.
𝑛 Try for 𝛉 = 4.7
122 2𝐷𝑖 𝐶𝑠𝑖
𝑄𝑖∗ = ෍ 𝑘𝑖 𝑄𝑖 = 𝐹
𝐶ℎ𝑖 + 2𝜃𝑘𝑖 𝑖=1
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
• The inventory of an/a enterprise/firm generally comprises of large number of items (types).

• All items are not of equal importance to the firm in such terms as sales, profits, availability etc.

• The firm, therefore, should pay more attention and care to those items whose usage value is high and
less attention to those whose usage and consumption value is low.

• One way of exercising proper degree of control over all and various types of items held in inventory
is to classify them into groups (classes) on the basis of degree of control.

• By selectively applying inventory control policies to these different groups, inventory objectives can
be achieved with lower inventory levels than with a single policy applied to all items.

• These techniques are also known as selective multi-item inventory control techniques.

123
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES

124
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
ABC Analysis (Always, Better, Control)

• The ABC analysis consists of separating the inventory items into three groups: A, B and C, according
to their annual cost volume consumption (unit cost × annual consumption).

• The items are divided into three categories as follows:

A – High consumption value items

B – Moderate consumption value items; and

C – Low consumption value items

• The division of the items into various categories is accomplished by plotting the usage value of the
items to obtain the ABC distribution curve given below.

125
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
ABC Analysis (Always, Better, Control)

• Although the break points between these groups vary according to individual business conditions, a
common breakdown might be as follows:

“Principle of law of Vital Few and Trival Many”

• Carrying out the ABC analysis of the store items helps in identifying the few items that are vital from
the financial point of view and require careful watch, scrutiny and follow-up.
126
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
Procedure of ABC analysis is summarized in the following steps:

Step 1: Obtain data on the annual usage (or consumption) in units and unit cost of each inventory item.

Multiply the annual usage in units and the value of each item to get annual value for each of these items:

Annual value = Unit cost × Annual consumption

Step 2: Arrange these inventory items in a decreasing order of their value computed in Step 1.

Step 3: Express the annual value of each item as percentage of the total value of all items. Also compute

the cumulative percentage of annual consumption rupees spent.

Step 4: Obtain the percentage value for each of the items. That is, if there are 50 items involved in

classification, then each item would represent 100/50 = 2 per cent of the total items. Also cumulate these
127
percentage values.
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
Procedure of ABC analysis is summarized in the following steps:

Step 5: Draw a graph between cumulative percentage of items (on x-axis) and cumulative annual

percentage of usage value (on y-axis), and mark cut-off points where the graph changes slope as shown

in Fig. below.

128
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
Example

A company produces a mix of high technology products for use in hospitals. The annual sales data are as
follows:

For inventory control reasons, the company wants to classify these items into three groups A, B and

129 C, on the basis of annual sales value of each item. You are assigned the task of helping the company.
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
Solution

The annual sales volume (in Rs) for each product and the item ranking on the basis of this volume is
shown in Table

130
 S E L E C T I V E I N V E N T O RY C O N T R O L
TECHNIQUES
Solution

The annual sales volume (in Rs) for each product and the item ranking on the basis of this volume is
shown in Table

131
 THANK YOU

CONTACT DETAILS
ROOM NO.:- ME-240
MECHANICAL ENGINEERING DEPARTMENT
KUMARG@NITRKL.AC.IN | 8791675692