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QUANTITATIVE METHODS FOR MANAGEMENT

Lecture 10Inventory Control

April 2008

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Learning Objectives

When you complete this chapter, you should

be able to explain:

• the purpose of inventory

• the costs involved in inventory

• independent and dependent demand

• EOQ Model

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Outline• Functions of inventory

- Types of inventory• Inventory Models

- Independent versus dependent demand

- Holding, ordering, set up costs..• Inventory Models for independent demand

- Basic Economic Order Quantity (EOQ) Model

- Minimizing costs

- Reorder points (ROP)

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What is Inventory?

• Inventory is one of the most expensive and important assets to many companies, representing 50% of total invested capital. Managers have long recognised that good inventory control is crucial.

• Definition of inventory

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Types of Inventory

• Typical items carried in inventory can include:

»raw materials

»purchased parts

»components

»Sub-assemblies

»work-in-progess

»finished goods and supplies

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Basic components of inventory

• Planning phase

• Forecasting

• Controlling of the inventory levels

• Feedback measurements

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Functions of Inventory

• To provide a stock of goods that will provide a ‘selection’ for customers

• To take advantage of quantitative discounts

• To care for against inflation and upward price changes

• Seasonal inventory

• Safety inventory

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Where do we keep inventory• Retail• Wholesale/distributor• Warehouse• Producer• Input supplier

- kept closer to the customer if quick response to demand for finished goods is needed

- kept further down the chain if more customisation is needed

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Inventory decisions

Two fundamental decisions that have to be made

when controlling inventory are:

i. How much to order

ii. When to order

The purpose of all inventory models and

techniques is to determine rationally how much to

order and when to order.

What is the objective of inventory?

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Inventory Costs• Purchasing costs – cost incurred when purchasing a

unit of an item.

• Holding costs – associated with holding or ‘carrying’ inventory over time.

• Ordering costs – associated with costs of placing order and receiving goods.

• Setup costs – cost to prepare a machine or process

• Stockouts costs – penalty costs for running out of stock.

• Safety stock – extra stock kept at hand in order to avoid stockouts.

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Disadvantages of Inventory• BUT maintaining inventory incurs costs such as:

» Ordering costs (setup cost – flat charge for delivery and admin)

» Unit costs (charge per item)» Holding costs (insurance, interest lost on

capital, cost of storage)» Shortage costs (cost of placing an order

for immediate delivery, cost of lost trade, loss of goodwill)

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Independent v/s Dependent Demand

• Independent demand – demand for item is independent of demand for any other item

• Dependent demand – demand for item is dependent upon the demand for some other items

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Inventory Management for Independent Demand

• Answers two basic questions:

1. how much should we order?

2. when should we order?

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Inventory Models

Help answer the inventory planning questions.

• Fixed order-quantity models- Economic Order Quantity (EOQ): one of the oldest and most commonly known inventory control techniques.- Quantity discount: a reduced cost for an item when it is purchased in larger quantities.

• Probabilistic models

• Fixed order-period models

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Economic Order Quantity (EOQ)Assumptions• Known and constant demand.• Known and constant lead time (time between the

placement of the order and the receipt of the order)• Instantaneous receipt of material: the inventory from

an order arrives in one batch, at one point in time.• No quantity discounts.• The only variable costs are the cost of placing an

order (ordering cost) and the cost of holding (holding or carrying cost)

• No stockouts: if orders are placed at the right time, shortages can be avoided completely.

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EOQ Objective

• To determine the quantity to order which will minimise the total annual inventory cost

Total annual inventory cost = annual inventory

holding costs + annual ordering/setup costs

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Inventory Usage Over Time

Time

Inve

ntor

y Le

vel

AverageInventory

(Q*/2)

0

Minimum inventory

Order quantity = Q (maximum inventory level)

Usage Rate

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EOQ Model (How Much to Order?)

Order quantity

Annual Cost

Holding Cost CurveTotal Cost Curve

Order (Setup) Cost Curve

Optimal Order Quantity (Q*)

Minimum total cost

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EOQ Model (When To Order)

Reorder Point

(ROP)

Time

Inventory Level Average

Inventory (Q*/2)

Lead Time

Optimal Order

Quantity(Q*)

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Optimal Order Quantity

Expected Number of Orders

Expected Time Between Orders Working Days / Year

Working Days / Year

= =× ×

= =

= =

=

= ×

Q*D Co

Ch

NDQ*

TN

dD

ROP d L

2

D = Demand per year

Co = Setup (order) cost per order

Ch = Holding (carrying) cost

d = Demand per day

L = Lead time in days

EOQ Model Equations

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Finding the EOQ

Four steps for finding the optimum inventory:

• Develop an expression for the ordering cost.

• Develop an expression for the holding cost.

• Set the ordering cost equal to the carrying cost.

• Solve this equation for the optimum desired.

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Example 1

A company that sells pump housings to other

manufacturers, would like to reduce its inventory

cost by determining the optimal number of pump

housings to obtain per order. The annual demand is

1,000 units, the ordering cost is £10 per order and

the average holding cost per unit per year is £0.50.

i. Calculate the optimal number of units per order.

ii. Evaluate the total inventory cost.

iii. The expected number of orders.

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• Answers how much to order• Orders placed at fixed intervals

– Inventory brought up to target amount– Amount ordered varies

• No continuous inventory count– Possibility of stockout between intervals

• Useful when vendors visit routinely– Example: P&G representative calls every 2

weeks

Fixed Period Model

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Example 2

A wholesaler has a steady demand for 50 items of a

given product each month. The purchase cost of each

item is $6 and the holding cost for this item is

estimated to be 20% of the stock value per annum.

Every order placed by the wholesaler costs $10 in

administration charges regardless of the number

ordered.

Given this information, you are required to evaluate all

the relevant costs and attempt to determine the most

economic order quantity for this item of goods.

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Example 3

The Managing Director at Littlewoods, has asked you to devise ordering policies at the central warehouse for all the major drug items required by their stores. Consider a single item of stock: the Becotide inhaler for asthma sufferers. Assume that there is a constant demand for this item of 400 per week. Each inhaler costs $3 to purchase. It is estimated that it costs an average of $2 per week to store 100 inhalers in the warehouse. Each order placed to the suppliers costs $12 in administration fees. Using this information, what is the optimum order quantity and order frequency for this product? Sketch a graph indicating the total costs, holding costs and order costs.

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Reorder point: Determining when to order

• In most simple, inventory models, it is assumed that receipt of an order is instantaneous. That is, we assume that a firm waits until its inventory level for a particular item reaches 0, places an order and receives the items in stock immediately.

• However, the time between the placing and receipt of an order , called the lead time or delivery time, is often a few days or even a few weeks.

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Reorder point: Determining when to order

• Thus, the when to order decision is usually expressed in terms of e reorder point (ROP), the inventory level at which an order should be placed. The reorder point, ROP, is given as

• ROP = (demand per day) * (lead time for a new order in days)

= d * L

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• The figure illustrates the situation where reordering occurs L time units before delivery is expected.

LL

Reorder points

Inventor

y

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Procomp’s demand for computer chips is

8,000 yearly. The firm has a weekly demand

of 280 units. On the average, delivery of an

order takes three working days.

Calculate the ROP?

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• Answer how much & when to order

• Allow demand to vary– Follows normal distribution

– Other EOQ assumptions apply

• Consider service level & safety stock– Service level = 1 - Probability of stockout

– Higher service level means more safety stock• More safety stock means higher ROP

Probabilistic Models

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Safety Stock• Safety stock is additional stock that is kept on

hand.

• If, for example, safety stock for an item is 50 units, you are carrying an average of 50 units more of inventory during the year.

• When demand is unusually high, you dip into the safety stock instead of encountering a stockout.

• Thus, the main purpose of safety stock is to avoid stockouts when the demand is higher than expected.

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Safety Stock• The Hinsdale Company carries an inventory item

that has a normally distributed demand during the reorder point. The mean demand is 350 units and the standard deviation is 10. Hinsdale follow a policy that results in stockouts occurring only 5% of the time.

How much safety stock should be maintained?