textbook: pp. 203-254€¦ · • can reduce costs • may result in stockouts (frequent inventory...

Chapter 6: Inventory Control Models

Textbook: pp. 203-254

Learning Objectives

After completing this chapter, students will be able to:

• Understand the importance of inventory control.

• Understand the various types of inventory related

decisions.

• Use the economic order quantity (EOQ) to determine how

much to order.

• Compute the reorder point (ROP) in determining when to

order more inventory.

• Handle inventory problems that allow non-instantaneous

receipt.

• Handle inventory problems that allow quantity discounts.

Learning Objectives

After completing this chapter, students will be able to:

• Understand the use of safety stock.

• Compute single period inventory quantities using marginal

analysis.

• Understand the importance of ABC analysis.

• Describe the use of material requirements planning in solving

dependent-demand inventory problems.

• Discuss just-in-time inventory concepts to reduce

inventory levels and costs.

• Discuss enterprise resource planning systems.

• Inventory is an expensive and important asset

• Any stored resource used to satisfy a current or

future need

o Raw materials

o Work-in-process

o Finished goods

• Balance high and low inventory levels to minimise

Introduction (1 of 3)

• Lower inventory levels

• Can reduce costs

• May result in stockouts (frequent inventory outages)

and dissatisfied customers

• All organisations (banks, hospitals, universities …)

have some type of inventory planning and control

system

• We want to study “how organisations achieve their

objectives by supplying goods/services to their

customers”!

Components of Inventory Planning and Control System

manufactured or purchase

from third party

Planning Phase

Feedback L

Five uses of inventory:

1. The decoupling function

2. Storing resources

3. Irregular supply and demand

4. Quantity discounts

5. Avoiding stockouts and shortages

• Decoupling Function

o Reduces delays and improves efficiency

o A buffer between stages

Importance of Inventory Control (1 of 3)

• Storing resources

o Seasonal products stored to satisfy off-season

demand

o Materials stored as raw materials, work-in-process,

or finished goods

o Labour can be stored as a component of partially

completed subassemblies

• Irregular supply and demand

o Not constant over time

o Inventory used to buffer the variability

• Quantity discounts

o Lower prices may be available for larger orders

o Higher storage and holding costs (spoilage,

damaged stock, theft, insurance…)

o By investing in more inventory, you will have less

cash to invest elsewhere!

• Avoiding stockouts and shortages

o Stockouts may result in lost sales

o Dissatisfied customers may choose to buy from

another supplier

o Loss of goodwill

Two fundamental decisions:

1. How much to order

2. When to order

Major objective is to minimise total inventory costs:

1. Cost of the items (purchase cost or material cost)

2. Cost of ordering (often involves personnel time)

3. Cost of carrying, or holding, inventory (taxes, insurance,

cost of capital…)

4. Cost of stockouts (lost sales and goodwill that result from

not having the items available for the customers)

Inventory DecisionsEconomic Order Quantity (EOQ)

Reorder Point (ROP)

Inventory Cost Factors (1 of 2)

• Ordering costs are generally independent of order

quantity

o Many involve personnel time

o The amount of work is the same no matter the size

of the order

• Holding costs generally vary with the amount of

inventory or order size

o Labour, space, and other costs increase with order

o Cost of items purchased can vary with quantity

discounts

Inventory Cost Factors (2 of 2)

• Economic order quantity (EOQ) model

o One of the oldest and most commonly known

inventory control techniques

o Easy to use

o Several important assumptions limits the

applicability of this model!

• Objective is to minimise total cost of inventory

Economic Order Quantity (EOQ) (1 of 2)

Assumptions:

1. Demand is known and constant over time

2. Lead time* is known and constant

3. Receipt of inventory is instantaneous (arrives in one batch, at one

point in time!)

4. Purchase cost (= cost of the inventory!) per unit is constant

throughout the year (Quantity discounts are not possible!)

5. The only variable costs are ordering cost and holding or

carrying cost, and these are constant throughout the year

6. Orders are placed so that stockouts or shortages are avoided

completely

When these assumptions are not met – adjustments to the EOQ

must be made!

Economic Order Quantity (EOQ) (2 of 2)

*Lead time = the time between the placement of the order and

the receipt of the order!

Inventory Usage Over Time

e.g. 1,000 bananas

If this amount is 1,000 bananas, all 1,000 bananas arrive at one time when

an order is received. The inventory level jumps from 0 to 1,000 bananas (Q).

Demand is constant over time

inventory drops at a uniform rate!

A new order is placed so that when the inventory level reaches 0, the new order

is received and the inventory level again jumps to Q units (vertical line)!

• Annual ordering cost is number of orders per year times cost of

placing each order

• Annual carrying cost is the average inventory times carrying cost

per unit per year

Computing Average Inventory:

Inventory Costs in the EOQ Situation

Blank Blank INVENTORY LEVEL Blank

DAY BEGINNING ENDING AVERAGE

April 1 (order received) 10 8 9

April 2 8 6 7

April 3 6 4 5

April 4 4 2 3

April 5 2 0 1

Maximum level April 1 = 10 units Total of daily averages = 9 + 7 + 5 + 3 + 1 = 25

Number of days = 5 Average inventory level = 25÷5 = 5 units

Average Inventory Level =𝑄

Based on our assumptions - if we minimise the sum of the

ordering and carrying costs we are minimising the total costs!

Inventory Costs in the EOQ Situation (1 of 3)

Q = number of pieces to orderEOQ = Q* = optimal number of pieces to orderD = annual demand in units for the inventory itemCo = ordering cost of each orderCh = holding or carrying cost per unit per year

Number ofOrdering cost

orders placedper order

cost per year

Ordering costAnnual demand

Number of units per order

in each order

Annual

Ordering

Q = number of pieces to orderEOQ = Q* = optimal number of pieces to orderD = annual demand in units for the inventory itemCo = ordering cost of each orderCh = holding or carrying cost per unit per year

Carrying costAverage

holding per unitinventory

cost per yea

Annual

Order quantity (Carrying cost per unit per year)

Total Cost as a Function of Order Quantity:

• When the EOQ assumptions are met, total cost is

minimised when:

Annual ordering cost = Annual holding cost

Solving for Q:

Finding the EOQ (1 of 2)

D QC C

Q C DC

Equation summary:

Finding the EOQ (2 of 2)

• Sells pump housings to other companies

• Reduce inventory costs by finding optimal order quantity

Annual demand = 1,000 units

Ordering cost = $10 per order

Average carrying cost per unit per year = $0.50

Sumco Pump Company (1 of 5)

* 2 2(1,000)(10)40,000 200 units

Q = number of pieces to order; D = annual demand in units for theinventory item; Co = ordering cost of each order; Ch = holding or carryingcost per unit per year

optimal number of

units per order

The total annual inventory cost is the sum of the

ordering costs and the carrying costs:

Number of orders per year = (D÷Q) = 5

Average inventory (Q÷2) = 100

1,000 200(10) (0.5)

$50 + $50 $100

D QTC C C

Q = number of pieces to order; D = annual demand in units for theinventory item; Co = ordering cost of each order; Ch = holding or carryingcost per unit per year

• The total inventory cost can be written to include the

cost of purchased items

o Annual purchase cost is constant at D × C no matter

the order policy, where

C is the purchase cost per unit

D is the annual demand in units

• The average dollar level of inventory

Purchase Cost of Inventory Items (1 of 2)

( )Average dollar level

• Inventory carrying cost is often expressed as an

annual percentage of the unit cost or price of the

inventory

• When this is the case, a new variable is introduced:

I = (Annual inventory holding charge as a percentage of

unit price or cost)

Cost of storing inventory for one year = Ch = IC

Purchase Cost of Inventory Items (2 of 2)

* 2 oDCQ

D = annual demand in units for the inventory itemCo = ordering cost of each orderC = unit price or cost of an inventory item

• The EOQ model assumes all input values are know

with certainty and fixed over time!

• Values are estimated or may change

• Sensitivity analysis determines the effects of these

changes

• Because the EOQ is a square root, changes in the

inputs (D; Co; Ch) result in relatively minor changes

in the order quantity

Sensitivity Analysis with the EOQ Model (1 of 2)

* 2EOQ o

Sumco Pump example:

If we increase Co from $10 to $40

In general, the EOQ changes by the square root of the

change to any of the inputs!

Sensitivity Analysis with the EOQ Model (2 of 2)

2(1,000)(10)EOQ 200 units

2(1,000)(40)EOQ 400 units

* 2EOQ o

• Next decision is “When to order”

• The time between placing an order and its receipt is

called the lead time (L) or delivery time

• “On hand” and “on order” inventory must be

available to meet demand during lead the total of

these is called “inventory position”

• Generally the “when to order” decision is usually

expressed in terms of a reorder point (ROP; inventory

position at which an order should be placed)

Reorder Point: Determining When To Order

ROP (Demand per day) (Lead time for a new order in days)

D = Annual demand = 8,000

d = Daily demand = 40 units

L = Delivery in 3 working days

An order for the EOQ (400) is placed when the inventory

reaches 120 units

The order arrives 3 days later just as the inventory is

depleted to 0!

Procomp’s Computer Chips (1 of 2)

ROP 40 units per day 3 days

120 units

d = Demand per day

L = Lead time for a new order in days

Reorder Point Graphs

Time between placing an order and its receipt is called lead time (L)

Q = 400

d = 40 units

ROP = d * L = 120

ROP < Q 120 < 400

3 days

120 = 120 + 0

New order placed

when inventory = 120

Annual demand = 8,000

Daily demand = 40 units

Delivery in three twelve working days

Suppose the lead time for Procomp Computer Chips was

12 days instead of 3 days. How would this affect the

“reorder point”?

Annual demand = 8,000

Daily demand = 40 units

Delivery in three twelve working days

New order placed when inventory = 80 and one order is

in transit!

Q = 400

d = 40 units

ROP = d * L = 480

ROP > Q 480 > 400

12 days

480 = 80 + 400

New order placed

when inventory = 80

and one order is in

transit!

Q = 400

d = 40 units

ROP = d * L = 480

ROP < Q 480 < 400

Q = 400

d = 40 units

ROP = d * L = 120

ROP < Q 120 < 400

3 days

12 days

• When a firm receives its inventory over a period of

time, a new model is needed that does not require the

“instantaneous inventory receipt assumption”!

• This new model is applicable when inventory

continuously flows or builds up over a period of

time after an order has been placed or when units

are produced and sold simultaneously.

Under these circumstances, the daily demand rate (d)

must be taken into account.

EOQ Without Instantaneous Receipt

• Production run model: This graph shows inventory

levels as a function of time. Model especially suited to

the production environment!

Inventory Control and the Production Process:

EOQ Without Instantaneous Receipt

Units are produced and sold simultaneously!

• In the production process, instead of having an

ordering cost, there will be a setup cost

Cost of setting up the production facility to manufacture

the desired product!

• Salaries and wages of employees responsible for

setting up the equipment

• Engineering and design costs of making the setup

• Paperwork, supplies, utilities …

Annual Carrying Cost for Production Run

Model (1 of 3)

• So how can we derive the optimal production quantity?

• We need to set our “setup costs” equal to “carrying

costs” and then solve for the “order quantity”.

• Let’s start by developing the expression for carrying

Model (1 of 3)

• As with the EOQ model, the carrying costs of the

production run model are based on the average

inventory, and the average inventory is one-half the

maximum inventory level.

Model variables:

Q = number of pieces per order (production run)

Cs = setup cost

Ch = holding or carrying cost per unit per year

p = daily production rate

d = daily demand rate

t = length of production run in days

Model (1 of 3)

• Maximum inventory level =

(Total produced during the production run)

− (Total used during production run) =

= (Daily production rate)(Number of days production)

− (Daily demand)(Number of days production) =

= (pt) − (dt)

Since Total produced = Q = pt and

Maximum inventory level =

Model (2 of 3)

– – 1–Q Q d

pt dt p d Qp p p

p = daily production rated = daily demand ratet = length of production run in days

Average inventory is one-half the maximum:

Model (3 of 3)

Average inventory 1–2

Annual holding cost 1–2

When a product is produced over time, setup cost

replaces ordering cost:

Both of these are independent of the size of the order

and the size of the production run! This cost is simply

the number of orders (or production runs) times the

ordering cost (setup cost).

Annual Setup Cost for Production Run Model

Q = number of orders or production run; Cs = setup cost; D = annual demand rate

• Set setup costs equal to holding costs and solve for

the optimal order quantity

o Annual holding cost = Annual setup cost

Solving for Q, we get the optimal production quantity Q*

Determining the Optimal Production Quantity

Equation summary:

Production Run Model

Annual holding cost 1 – 2

Annual setup cost

2Optimal production quantity

Brown Manufacturing produces commercial refrigeration units in

batches. The firm’s estimated demand for the year is 10,000

units. It costs about $100 to set up the manufacturing process,

and the carrying cost is about 50 cents per unit per year. When

the production process has been set up, 80 refrigeration units

can be manufactured daily. The demand during the production

period has traditionally been 60 units each day. Brown operates

its refrigeration unit production area 167 days per year.

How many units should Brown produce in each batch?

How long should the production part of the cycle last?

Brown Manufacturing (1 of 4)*

Annual setup cost

Produces commercial refrigeration units in batches:

Annual demand = D = 10,000 units

Setup cost = Cs = $100

Carrying cost = Ch = $0.50 per unit per year

Daily production rate = p = 80 units daily

Daily demand rate = d = 60 units daily

1. How many units should Brown produce in each batch?

2. How long should the production part of the cycle last?

Brown Manufacturing (1 of 4)*

Annual setup cost

Brown Manufacturing (2 of 4)

2 10,000 100

600.5 1 –

2,000,00016,000,000

4,000 units

Annual demand = D = 10,000 units

Setup cost = Cs = $100

Carrying cost = Ch = $0.50 per unit per year

Daily production rate = p = 80 units daily

Daily demand rate = d = 60 units daily

2 10,000 100

600.5 1 –

2,000,00016,000,000

4,000 units

Quantity produced

in each batch

Daily production

If Q* = 4,000 units and we know that 80 units can be produced

daily, the length of each production cycle will be days. When

Brown decides to produce refrigeration units, the equipment will

be set up to manufacture the units for a 50-day time span.

The number of production runs per year will be

D/Q = 10,000/4,000 = 2.5.

The average number of production runs per year is 2.5.

There will be 3 production runs in one year with some inventory

carried to the next year

Therefore only 2 production runs are needed in the second year.

In developing the EOQ model, we assumed that quantity

discounts were not available.

However, many companies do offer quantity discounts.

If such a discount is possible, but all of the other EOQ

assumptions are met, it is possible to find the quantity

that minimises the total inventory cost by using the EOQ

model and making some adjustments!

Quantity Discount Models (1 of 7)

• When quantity discounts are available the basic EOQ

model is adjusted by adding in the “purchase or

materials cost” (Logic they change based on order

quantity!):

Total cost = Material cost + Ordering cost + Holding cost

Where D = annual demand in units

Co = ordering cost of each order

C = cost per unit

Ch = holding or carrying cost per unit per year

Total cost + + 2

D QDC C C

Q = number of orders or production run

Holding cost per unit is based on cost per unit (C), so

Ch = IC

Where I = holding cost as a percentage of the unit cost (C)

You are now offered following discounts by your supplier:

Will buying at the lowest unit cost

result in lowest total cost?

Overall our objective is to minimise the total cost!

• Overall our objective is to minimise the total cost.

• Because the unit cost (4.75) for the third discount is

lowest, we might be tempted to order 2,000 units or more

to take advantage of the lower material cost.

• Placing an order for that quantity with the greatest discount

cost might not minimise the total inventory cost.

• As the discount quantity goes up, the material cost goes

down, but the carrying cost increases because the orders

are large.

• Key trade-off when considering quantity discounts is

between the reduced material cost and the increased

carrying cost!

Total Cost (TC) Curve for the Quantity Discount Model

Steps in the process:

1. For each discount price (C), compute

2. If EOQ < Minimum for discount, adjust the quantity to Q

= Minimum for discount

3. For each EOQ or adjusted Q, compute

4. Choose the lowest-cost quantity

2EOQ oDC

Total cost + + 2

D QDC C C

Brass Department Store stocks toy race cars. Recently,

the store was given a quantity discount schedule for the

cars. The ordering cost is $49 per order, the annual

demand is 5,000 race cars, and the inventory carrying

charge as a percentage of cost, I, is 20%.

Discount schedule:

What order quantity will minimise the total inventory cost?

Brass Department Store (1 of 6)

Toy race cars:

Quantity discounts available

Step 1 – Compute EOQs for each discount

Total Cost (TC) Curve for the Quantity Discount Model

Step 2 – If EOQ < Minimum for discount, adjust the

quantity to Q = Minimum for discount

2EOQ oDC

Step 2 – If EOQ < Minimum for discount, adjust the

quantity to Q = Minimum for discount

• The EOQ for discount 1 is allowable

• The EOQs for discounts 2 and 3 are outside the

allowable range, adjust to the possible quantity closest

to the EOQ

Q1 = 700

Q2 = 1,000

Q3 = 2,000

Step 3 – Compute total cost for each quantity

Total Cost Computations for Brass Department Store:

Step 4 – Choose the alternative with the lowest total cost

Step 3 – Compute total cost for each quantity

Total Cost Computations for Brass Department Store:

Step 4 – Choose the alternative with the lowest total cost

3 days

12 days

• When the EOQ assumptions

are met, it is possible to

schedule orders to arrive so

that stockouts are completely

avoided!

• If the demand or the lead

time is uncertain, the exact

demand during the lead time

(ROP in the EOQ situation)

will not be known with

certainty!

• Therefore, to prevent

stockouts, it is necessary to

carry additional inventory

called safety stock.

• If demand or the lead time are uncertain, the exact

ROP will not be known with certainty!

• To prevent stockouts, it is necessary to carry additional

inventory called safety stock.

• Can be implemented by adjusting the Reorder Point

ROP = (Average demand during lead time) + Safety stock

ROP = (Average demand during lead time) + SS

where SS = safety stock

Use of Safety Stock (1 of 4)

• When demand is

unusually high during

the lead time, you dip into

the safety stock instead of

encountering a stockout!

• The main purpose of

safety stock is to avoid

stockouts when the

demand is higher than

expected.

• Objective is to choose a safety stock amount the

minimises total holding and stockout costs

• If variation in demand and holding and stockout costs

are known, payoff/cost tables could be used to

determine safety stock

• More general approach is to choose a desired service

level based on satisfying customer demand

• Set safety stock to achieve a desired service level

Service level = 1 − Probability of a stockout

Probability of a stockout = 1 − Service level

ROP = (Average demand during lead time) + ZσdLT

Z = number of standard deviations for a given service level

σdLT = standard deviation of demand during the lead time

Thus Safety stock = ZσdLT

Safety Stock with the Normal Distribution

• Item A3378 has normally distributed demand during

lead time

Mean = 350 units, standard deviation = 10

• Stockouts should occur only 5% of the time

Hinsdale Company (1 of 8) --- Safety Stock and

the Normal Distribution

μ = Mean demand = 350

σdLT = Standard deviation = 10

X= Mean demand + Safety stock

SS= Safety stock = X − μ = Zσ

From Appendix A we find Z = 1.65

ROP = (Average demand during lead time) + ZsdLT

= 350 + 1.65(10)

= 350 + 16.5 = 366.5 units (or about 367 units)

Hinsdale Company (2 of 8)

Three situations to consider:

o Demand is variable but lead time is constant

o Demand is constant but lead time is variable

o Both demand and lead time are variable

Calculating Lead Time Demand and Standard

Deviation (1 of 4)

1. Demand is variable but lead time is constant

Deviation (2 of 4)

ROP ddL Z L

average daily demand

standard deviation of daily demand

lead time in days

2. Demand is constant but lead time is variable

Deviation (3 of 4)

ROP LdL Z d

average lead time

standard deviation of lead time

daily demand

3. Both demand and lead time are variable

• The most general case

• Can be simplified to the earlier equations

Deviation (4 of 4)

2 2 2ROP d LdL Z L d

• Determine safety stock for three other items

• For SKU F5402,

• Desired service level = 97%

o For a 97% service level, Z = 1.88

= 15, = 3, = 4dd L

15(4) + 1.88(3 4) 15(4) + 1.88(6)

60 + 11.28 71.28

ddL Z L

• For SKU B7319,

= 25, = 6, = 3Ld L

25(6) + 2.05(25)(3) 150 + 2.05(75)

150 + 153.75 303.75

LdL Z d

• For SKU F9004,

= 20, = 4, = 5, = 2d Ld L

(20)(5) + 1.55 5(4) + (20) (2)

100 + 1.55 1680

100 + 1.55(40.99) 100 + 63.53 163.53

d LdL Z L d

• As service levels increase

o Safety stock increases at an increasing rate

• As safety stock increases

o Annual holding costs increase

Service Levels, Safety Stock, and Holding

Safety Stock for SKU A3378 at Different Service Levels:

Under standard assumptions of EOQ

o Average inventory = Q÷2

o Annual holding cost = (Q÷2)Ch

With safety stock

Calculating Annual Holding Cost with Safety

Holding cost ofTotal annual Holding cost of

regularholding cost safety stock

inventory

THC (SS)2

QC C where THC = total annual holding cost

Q = order quantity

Ch = holding cost per unit per year

SS = safety stock

Service Level Versus Annual Carrying Costs:

• The purpose is to divide the inventory into three

groups (A, B, and C) based on the overall inventory

value of the items

• Group A items account for the major portion of

inventory costs (high price!)

o Typically 70% of the dollar value but only 10% of the

quantity of items

o Great care should be taken in forecasting the demand

and developing good inventory management policies for

this group!

o Mistakes can be expensive!

ABC Analysis (1 of 5)

A prudent manager should spend more time managing those items

representing the greatest dollar inventory cost because this is where

the greatest potential savings are.

• Group B items are more moderately priced

o Items represent about 20% of the company’s

business in dollars, and about 20% of the items in

inventory

o Not appropriate to spend as much time developing

optimal inventory policies for this group as with the A

group since inventory costs are much lower!

o Moderate levels of control!

• Group C items are very low cost but high volume

o These items may constitute only 10% of the company’s

business in dollars, but they may consist of 70% of the

items in inventory.

o It is not cost effective to spend a lot of time managing

these items!

o For the group C items, the company should develop a

very simple inventory policy, and this may include a

relatively large safety stock.

o Since the items cost very little, the holding cost

associated with a large safety stock will also be very

• More care should be taken in determining the safety

stock with the higher priced group B items.

• For the very expensive group A items, the cost of

carrying the inventory is so high, it is beneficial to

carefully analyse the demand for these so that safety

stock is at an appropriate level!

• Otherwise, the company may have exceedingly high

holding costs for the group A items.

Summary of ABC Analysis:

• End of chapter self-test 1-14

(pp. 243-244)

Compile all answers into one

document and submit at the

beginning of the next lecture!

On the top of the document,

write your Pinyin-Name and

Student ID.

• Please read Chapter 11!

Homework --- Chapter 6

textbook: pp. 203-254€¦ · • can reduce costs • may result in stockouts (frequent inventory...

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