IE304_LectureNotes_1

1

Manufacturing Models

Manufacturing can be defined as a series of interrelated activities and

operations involving the design, material selection, planning,

manufacturing process, quality assurance, management and marketing of

the products of the manufacturing industries.

The goal is to produce products that meet customer expectations in terms

of functionality, quality, and reliability BUT at a minimum cost.

The industrial engineer determines how best to utilize the available inputs

of labor, technology, capital, energy, materials and information to achieve

the objectives.

2

Manufacturing Processes

INP

UT

S

Material

Labor

Information

Capital

Energy

Manufacturing Processes:

a) consist of physical

elements interacting

with each other

b) can be monitored via

performance measures

Products

Physical Elements

• Machinery

• Tools

• Computerized Equipment

• People

• Material Handling Equipment

Performance Measures

• Production rate

• % On-Time Delivery

• Defects per Million

• Unit cost

3

Transformation Process at a Food Processor

Inputs:

• Raw vegetables

• Metal Sheets

• Water

• Energy

• Labor

• Building

• Equipment

• Chemicals

• Coloring material

Processes:

• Cleaning

• Making Cans

• Cutting

• Cooking

• Packing

• Labelling

Outputs:

• Canned vegetables

4

Transformation Process at a Hospital

Inputs:

• Doctors

• Nurses

• Other personnel

• Building(s)

• Equipment

• Labs

• Medical supplies

Processes:

• Examination

• Surgery

• Monitoring

• Medication

• Therapy

Outputs:

• Treated patients

5

Products vs Services

Products:

• Tangible

• Can be stocked

• No interaction between customer

and process

Services:

• Intangible

• Cannot be stocked

• Direct interaction between

customer and process

Service industries have been on the rise and constitute most of the economy

in the Western world.

6

Manufacturing Systems

1. Product Design: blueprints, Computer-Aided Design

2. Process Planning:

• Sequence of operations to convert raw material into

finished goods

• Machine selection: Right equipment and tools to be able to process

parts according to design specifications

3. Production/Manufacturing Operations:

• Fabrication: Drilling a hole, plastic injection molding, bending a flange

• Assembly: Combination of separate parts into a more valuable

combined unit

4. Material Flow/Facilities Layout:

• Material Handling: Techniques for transporting parts, tools, scrap in

the facility

• Facility Layout: Placing production facilities, power supply,

compressed air within the facility

5. Production Planning/Control:

Use market demand, consider production capacity and current inventory

levels, determine planned production levels by product family.

7

Manufacturing Facility Layouts

There are four types of layouts:

1. Product Layout

2. Process Layout

3. Group Technology

4. Fixed Position (not to be covered)

The types of products in terms of their volume, variety dictate the layout. In

other words, these 4 layouts are not alternatives to one another.

8

How do We Assess and Characterize the Layout Types?

Throughput Time: The period required for a material, part, or subassembly to

pass through the manufacturing process.

Work in process (WIP): The set of items, parts and material for products not

completed. They are waiting in buffers (areas between workstations) for further

processing.

Skill level of workers

Product variety

Worker and machine utilization

Unit production cost

9

Fixed position (Project/Unit Production)

• “Unique” products with certain complexity (ships, yachts, aircrafts, trains, …)

• The production process is managed like a project

• Makes a one-of-a-kind product

• Uses general purpose equipment

• Tools and equipment come around the product since the product is too big

to move.

10

Product Layout (Mass Production/Flow Lines)

A layout structure designed to make discrete parts. Parts move

through a set of specially designed workstations at a controlled rate.

Characteristics:

1. Makes few products in large volume

2. Uses specialized high-volume equipment

3. Workstations and machines for production are specific for the

product, and cannot be easily adjusted to other products.

4. Short throughput time, low WIP (work-in-process) inventories

11

Process Layout (Batch Production/Job Shop)

• 75% of manufacturing occurs in batches of less than 50 items.

• In this environment, a wide range of products with small volumes have to

be produced.

• General purpose equipment is grouped by the same functionality in the

same departments (e.g., milling machines in one department, punch presses

in another department, etc)

• Since successive batches may require different tooling and setup, highly

skilled operators are typically required.

• Long throughput times, high WIP

12

• When there is a high variety of low demand products, if similar parts can be

grouped together in sufficient quantity, they are processed in a cell

• Different machines are placed in the same cell for similar parts

• Thus, scheduling and material handling are streamlined, low WIP and short

throughput time

Group Technology/Cellular Manufacturing

13

Comparison of Process Layout and Group Technology

14

Measuring Process Performance

• Productivity: Ratio of output to input

• Utilization: Ratio of the time that a resource is actually activated relative to

the time that it is available for use.

• Cycle Time: Average time between the completion of successive units.

• Run Time: Time required to produce a batch of parts.

• Setup Time: Time required to prepare a machine to make a particular item

• Operation Time: Sum of setup and run time.

• Throughput Rate: Number of parts processed per unit time.

• Value Added Time: Time that useful work is actually done

Assembly Lines

• Assembly operation: joins two or more components to create a

new entity, which is called an assembly, or subassembly.

• Assembly line: A production line consisting of a sequence of

workstations where assembly tasks are performed by human

workers or machines as the product moves along the line.

• Organized to produce a single product or a limited range of

products

– Each product consists of multiple components joined together by

various assembly work elements

15

16

• Products are assembled as they move along the line • At each station a portion of the total work content is performed

on each unit

• Base parts are launched onto the beginning of the line at

regular intervals (cycle time) • Workers add components to progressively build the product

• A mechanized material transport system is typically used to

move the base parts along the line.

Assembly Lines

Assembly Lines

• Factors favoring the use of assembly lines:

– High or medium demand for product

– Identical or similar products

– Total work content can be divided into work elements

• Why assembly lines are so productive?

– Specialization (Division) of labor

• A large job is divided into small tasks and each task is assigned to one

worker

– Interchangeable parts

• Each component is manufactured to sufficiently close tolerances

– Work flow principle

• The work is moved to the worker

– Line pacing

• Workers are required to complete their assigned tasks on each unit within a

certain cycle time

• A specified production rate is maintained

17

Advantages

• Assembly lines reduced production cost and increased

production volume

• Keeps direct labor or automated machines busy doing

productive work

• Minimal setup times since the tasks are repeated

• Assembly lines do not require large queues, thus

• reduced WIP and lower inventory holding cost

• reduced space requirements

• shorter throughput time

18

Assembly Lines

Assembly Lines

• Most consumer products are assembled on assembly

lines

Automobiles Personal computers

Cooking ranges Power tools

Dishwashers Refrigerators

Dryers Telephones

Furniture Toasters

Lamps Trucks

Luggage Video DVD players

Microwave ovens Washing machines

19

Assembly Line Types

• Paced lines vs. Unpaced lines

Paced lines

o Each workstation is given exactly the same amount of time

(C, cycle time) to operate on a unit of product.

o At the end of C time units, the handling system

automatically indexes each unit to the next station.

o Encourages the workers to maintain the proper pace

o Randomness of performance may cause some items not to

be completed

o Extra time may be allowed

o Small buffers can be used to prevent starvation.

Unpaced lines do not have such restrictions.

New unit is removed from the handling system when the

previous one is completed

20

• Single product vs. Mixed lines

Mixed lines

o Used when single item types do not have sufficient demand

to justify an assembly line

o Several products are produced simultaneously

o Different workstations may process different productions at

the same time.

o Problems

• Scheduling the sequence of different products

• More complex line balancing problem

• Logistics - get the right parts for the models currently

processed in each workstation

21

Assembly Line Types

Fundamentals of Assembly Lines

• Assembly workstations

• Work Transport Systems

– Manual transport systems

– Mechanized transport systems

22

23

Workstation: A designated location along the work flow path at which one or more work elements are performed by one or more workers

Typical operations performed at assembly stations

Adhesive application

Sealant application

Arc welding

Spot welding

Electrical connections

Component insertion

Press fitting

Riveting

Snap fitting

Soldering

Stitching/stapling

Threaded fasteners

Assembly Workstations

• Manual transport methods

– Work units are moved between stations by the workers

without the aid of a powered conveyor

• Types:

– Work units moved in batches

– Work units moved one at a time

• Problems:

o Starving of stations – worker is available for the next unit, but

the unit has not yet arrived

o Blocking of stations – worker cannot pass the unit to the next

station since that worker is not ready yet

o No pacing – production rates tend to be lower

24

Work Transport Systems

• Mechanized transport methods

– Work units are moved by powered conveyor or other

types of mechanized material handling equipment

– Types

• Continuous transport

– Conveyor moves at constant speed

• Synchronous transport

– Work units are moved simultaneously with stop-and-go

(intermittent) motion to next stations

• Asynchronous transport

– Work units are moved independently between workstations

– Queues of work units can form in front of each station

25

Work Transport Systems

26

Continuously moving conveyor operates at constant velocity

Can be implemented in two ways:

(1) work units are fixed to the conveyor

(2) work units are removable from the conveyor

Continuous Transport

27

All work units are moved simultaneously to their respective next

workstations with quick, discontinuous motion

The task must be completed within a certain time limit

Ideal for automated production lines

Synchronous Transport

28

Work units move independently, not simultaneously.

A work unit departs a given station when the worker releases it.

Small queues of parts are permitted to form at each station.

Forgiving of variations in worker task times.

Asynchronous Transport

29

Determination of the Cycle Time

• Production rate = 2200 units / week

• Number of working days / week = 5 days

• Number of shifts = 2 / day

• Number of hours / shift = 4 hours

• Breaks = 2*10 min / shift

• Net minutes per shift = ?

• Net minutes per week = ?

• C = ?

30

Cycle Time

Bottleneck station establishes the cycle time for the entire line

• Given:

– Total work content consists of many distinct work

activities

– The sequence in which the activities can be

performed is restricted

– The line must operate at a specified cycle time

• Problem:

– Assign tasks to the minimum number of stations

such that the workload assigned to each station does

not exceed the cycle time and the idle time is

minimized.

31

Line Balancing Problem

• C: cycle time

• n (possible) workcenters, m tasks

• ti : time to perform task i, i = 1,…,m

• Assume

– 𝐶 ≥ max𝑖{𝑡𝑖}

– 𝑛 ≥ 𝑡𝑖𝑚𝑖=1

𝐶

• 𝑐𝑖𝑗: cost of assigning task i to station j, i=1,…,m, j = 1,…,n

– To minimize the idle time force tasks into lowest numbered stations.

Assume 𝑚 𝑐𝑖𝑗 < 𝑐𝑖,𝑗+1, 𝑗 = 1,… , 𝑛 − 1

• Precedence constraints: IP

• Zone constraints: ZS and ZD

32

Line Balancing Problem - Formulation

necessary?

33

• Restrictions on the order in which work elements can be

performed

• IP = {(u,v): task u is an immediate predecessor of task v }

IP = ?

Precedence Constraints

Precedence

diagram

Zone Constraints

• Limitations on the grouping of tasks and/or their

allocation to workstations

– ZS: Positive zoning constraints

• Tasks should be grouped at same station

• Example: spray painting elements

ZS = {(u,v) | u and v must be assigned to the same station}

– ZD: Negative zoning constraints

• Elements that might interfere with each other

• Ex: Separate delicate adjustments from loud noises

ZD = {(u,v) | u and v cannot be assigned to the same station}

34

Problem Formulation

35

36

m

i

n

j

jiji xc1 1

,,min

m

i

jii njCxt1

, ,...,1 ,

Sum of task times of tasks assigned to each station cannot exceed cycle time

,...,m ixn

j

i,j 1 ,11

Each task must be assigned to exactly one station

h

j

juhv IPu,v ,n ,h,xx1

,, )(and 1

If task v is assigned to station h its immediate predecessor(s) u must be assigned to

some station between 1 and h

n

j

jvju ZSu,v,xx1

,, )( 1

u and v must be assigned to the same station

ZDu,v ,...,n j,xx jvju )(and 1 1,,

u and v cannot be assigned to the same station

Problem Formulation

,...,n j,x ji 1 and m,1,..., i }1,0{, xij is a binary variable

(LB - I)

37

Assign tasks to a fixed number of stations n such that the cycle time, C,

is minimized. This also maximizes the output rate.

m

i

jiij

xt1

,maxmin

s.t. ,...,m ix

n

j

i,j 1 ,11

h

j


,, )(and 1

Assembly Line Balancing – Different Objective

n

j

jvju ZSu,v,xx1

,, )( 1

,...,n j,x ji 1 and m1,..., i }1,0{,


38

We formulate the problem as follows:

min

njCxtm

i

jii ,...,1 1

,

s.t.

C

Assembly Line Balancing – Different Objective

,...,m ixn

j

i,j 1 ,11

h

j


,, )(and 1

n

j

jvju ZSu,v,xx1

,, )( 1

,...,n j,x ji 1 and m1,..., i }1,0{,


(LB - II)

• Three heuristics:

– Largest Candidate Rule

– Kilbridge and Wester Method

– Ranked Positional Weights Method

• Assume there is no zone constraints

• Assume only one worker will be assigned to

each station

39

Line Balancing Algorithms

(0) Arrange tasks in descending order according to their

processing times ti, consider the first workstation

(1) Assign tasks to the workstation by starting at the top of

the list and selecting the first task that

– satisfies precedence requirements and

– does not cause the total workload of the station to exceed C

When a task is assigned to the station, start from the

top of the list

(2) When no more task can be assigned to the station,

proceed to the next station

(3) Repeat steps 1 and 2 until all tasks have been assigned

40

Largest Candidate Rule

41


• Production rate = 2200 units / week

• Number of working days / week = 5 days

• Number of shifts = 2 / day

• Number of hours / shift = 4 hours

• Breaks = 2*10 min / shift

Cycle time C = 1 min

Number of workstations ≥ 4

1 = 4

42


Task

(i)

ti Preceded

by

3 0.7 1

8 0.6 3, 4

11 0.5 9, 10

2 0.4 -

10 0.38 5, 8

7 0.32 3

5 0.3 2

9 0.27 6, 7, 8

1 0.2 -

12 0.12 11

6 0.11 3

4 0.1 1, 2

Tasks are arranged

43


Station Task ti Station Time

1

2 0.4

5 0.3

1 0.2

4 0.1 1.0

2 3 0.7

6 0.11 0.81

3 8 0.6

10 0.38 0.98

4 7 0.32

9 0.27 0.59

5 11 0.5

12 0.12 0.62

Tasks are assigned to stations

Task

(i)

ti Preceded

by

3 0.7 1

8 0.6 3, 4

11 0.5 9, 10

2 0.4 -

10 0.38 5, 8

7 0.32 3

5 0.3 2

9 0.27 6, 7, 8

1 0.2 -

12 0.12 11

6 0.11 3

4 0.1 1, 2

Tasks are arranged

44


Assignment of tasks to workstations

Physical sequence of stations with assigned tasks

• Select tasks for assignment to stations

according to their position in the precedence

diagram

• Tasks in the precedence diagram are arranged

into columns

• Tasks are listed according to their column and

processing times

– If a task can be located in more than one column, all

the columns for that task should be listed

• Apply the three-step procedure of the Largest

Candidate Rule to this list

45

Kilbridge and Wester Method

46


Task Column ti Preceded

by

2 I 0.4 -

1 I 0.2 -

3 II 0.7 1

5 II,III 0.3 2

4 II 0.1 1, 2

8 III 0.6 3, 4

7 III 0.32 3

6 III 0.11 3

10 IV 0.38 5, 8

9 IV 0.27 6, 7, 8

11 V 0.5 9, 10

12 VI 0.12 11

Tasks are listed according to their columns

47


Station Task Column ti Station

Time

1

2 I 0.4

1 I 0.2

5 II 0.3

4 II 0.1 1.0

2 3 II 0.7

6 III 0.11 0.81

3 8 III 0.6

7 III 0.32 0.92

4 10 IV 0.38

9 IV 0.27 0.65

5 11 V 0.5

12 VI 0.12 0.62

• Ranked positional weight of task i (RPWi):

𝑅𝑃𝑊𝑖 = 𝑡𝑗𝑗∈𝑉(𝑖)

𝑉(𝑖) is the set of all successors of node i (including i) in the

precedence diagram

• Compute the ranked positional weight of each task

• Order tasks according to their RPW value

• Apply the three-step procedure of the largest candidate

rule to this list

48

Ranked Positional Weights Method

49

Ranked Positional Weights Method

Task RPWi ti Preceded

by

1 3.30 0.2 -

3 3.00 0.7 1

2 2.67 0.4 -

4 1.97 0.1 1, 2

8 1.87 0.6 3, 4

5 1.30 0.3 2

7 1.21 0.32 3

6 1.00 0.11 3

10 1.00 0.38 5, 8

9 0.89 0.27 6, 7, 8

11 0.62 0.5 9, 10

12 0.12 0.12 11

Station Task ti Station

Time

1 1 0.2

3 0.7 0.9

2

2 0.4

4 0.1

5 0.3

6 0.11 0.91

3 8 0.6

7 0.32 0.92

4 10 0.38

9 0.27 0.65

5 11 0.5

12 0.12 0.62

Tasks are ordered according to their RPW Tasks are assigned to stations

• When only a small number of tasks are assigned to each station,

the idle time may be very high.

• Balance delay is a performance measure that represents the

proportion of idle time

• Example: C = 100 min, 3 tasks with task times 75, 50, and 70

min

– Assign each task to one station (n = 3)

– Balance delay is D = 0.35

Some Practical Issues (I)

Cn

tCn

D

m

i

i

1

50

• Consider 2 parallel lines (n=2) and assign a cross trained worker to

each station

• We can produce a unit within 100 mins with one less workstation

• Cycle time is 200 min for each station, but the balance delay is

D = 0.025

• In general, with K parallel stations ( or number of workers that can

be grouped) with KxC as the cycle time for each station, excessive

idle time can be shortened

51

Some Practical Issues (II)

• If demand is uncertain and there is no idle time in the current line,

then we might need overtime or another shift.

• This is not a big problem for labor intensive lines.

• If, on the other hand, we have fixed equipment on the line, then a

different approach might be needed.

• Random processing times, i.e., ’s are the mean task times:

• Let

– 𝑆𝑗 be the set of tasks assigned to station j.

– 𝑠𝑗 be a random variable for the station time, i.e., the sum of task times

of all tasks assigned to station j.

• If task times are independent

52

Some Practical Issues (III)

𝑡𝑖

jSi

ij tsE

jSi

ijsVar 2

variance of task time i

• If each task time is normally distributed or we can invoke the

Central Limit Theorem, then 𝒔𝒋 is also normally distributed.

• If we require, e.g., 99% of the time to complete the assigned tasks in

each cycle at station j, then the following should hold:

𝐸 𝑠𝑗 + 2.33 𝑉𝑎𝑟[𝑠𝑗] ≤ 𝐶

• If all stations are created under this rule, the probability that ALL n

stations complete their tasks within C is 0.99𝑛

• In the case of random process times, we can assign utility workers

to help assembly workers in case of difficulty or provide for a rework

area where they can complete the unfinished tasks.

53

Some Practical Issues (IV)

54

Unpaced lines

In a paced line, each item spends nC time units in the system.

In an unpaced line, the slowest station 𝑛𝑏 is the bottleneck station, which

blocks the preceeding stations and starves the succeeding stations.

Then, flow time of a product in an unpaced line is

n

nj

jnb

b

bssn1

Positioning the bottleneck closest to the beginning of the line (preferably as

the first station), therefore, decreases the flow time and WIP although

the output rate will be constant as

bns

1

55

• Several different product types can be assembled simultaneously and

continuously (not in batches).

•Stations are sufficiently flexible that they can perform their respective tasks

on different models.

•Mixed lines typically used to accomplish final assembly of automobiles,

small and large trucks, major and small appliances.

•The amount of work required to assemble the units can vary from model to

model, creating an uneven flow of work along the line.

• Problems in designing and operating a mixed assembly line:

•Line balancing - same basic problem as in single product assembly line

except differences in work elements among models must be considered

•Model launching - determining the sequence in which different models will be

launched onto the line

Mixed Lines

56

Mixed Line Balancing

• There are P different products to be produced.

• 𝑅𝑗 is the production rate for product product j = 1,…,P

• 𝑡𝑖𝑗 is the necessary time for performing task i for product j,

i=1,…,m, j = 1,…,P

• Spread the workload amongst stations as evenly as possible.

• Compute the total time to perform each task

𝑇𝑇𝑖 = 𝑅𝑗𝑡𝑖𝑗𝑃𝑗=1 for 𝑖 = 1, … ,𝑚

• Assign the tasks to stations by using one of the line balancing

algorithms (Largest candidate rule, Kilbridge and Wester method,

Ranked positional weights method)

57


Task 𝑡𝑖𝑨 Preceded by 𝑡𝑖𝑩 Preceded by

1 3 - 3 -

2 4 1 4 1

3 2 1 3 1

4 6 1 5 1

5 3 2 - -

6 4 3 2 3

7 - - 4 4

8 5 5, 6 4 7

Construct the precedence diagram for each model and for both of the models

combined into one diagram.

Suppose that the line is up and operating 94% of the time. Use the Kilbridge

and Wester method to solve the line balancing problem.

Two models A and B with hourly production rates 4 units/hr and 6 units/hr,

respectively.

58

4

2

1

5

8 6 3

2

3

4

4 4

6 4

4

3

1

7

8 6 3

2

3

5

4

4

4 2

7

AB

4

1

5

8 6 3

2

AB

AB

AB

AB

A

B

AB

for model A

Precedence diagrams

for model B

for both models


59


Task RA tiA RB tiB TTi

1 12 18 30

2 16 24 40

3 8 18 26

4 24 30 54

5 12 0 12

6 16 12 28

7 0 24 24

8 20 24 44

7

AB

4

1

5

8 6 3

2

AB

AB

AB

AB

A

B

AB

I II III IV

Task Column TTi Proceeded by

1 I 30 -

4 II 54 1

2 II 40 1

3 II 26 1

6 III 28 3

7 III 24 4

5 III 12 2

8 IV 44 5, 6, 7

Compute task durations

Arrange tasks

according to the

columns

• Suppose that the line is up and operating 94% of the

time. Thus, the available time in one hour is

60min x 0.94 = 56.4 min

60


Station Task TTi Station Time

1 1 30

3 26 56

2 4 54 54

3 2 40

5 12 52

4 6 28

7 24 52

5 8 44 44

• Determine the sequence of models and the time

difference between successive launches

• Two alternatives

– Variable-rate launching

• Time interval between successive launches is set equal to

the cycle time of that model

• The models can be launched in any sequence

• Causes logistical problems (supply of the correct

components to individual stations)

– Fixed-rate launching

• Time interval between two consecutive launches is constant

• The time interval depends on the product mix and production

rates of models

• Models must be launched in a specific sequence

61

Model Launching in Mixed Lines

Fixed-rate launching time interval is determined as

𝑇 = 1

𝑅 𝑅𝑗𝑇𝑗𝑃𝑗=1

𝑛

where

𝑅 = 𝑅𝑗𝑃𝑗=1 is the total production rate for all models

𝑇𝑗 = 𝑡𝑖𝑗𝑚𝑖=1 is the total time necessary for producing one unit of model j

𝑛 is the number of stations (or workers)

Let 𝐶𝑗 = 𝑇𝑗

𝑛 for j = 1,…,P

If model j is launched in position r, let 𝐶 𝑟 = 𝐶𝑖

For each launch position r, select j so as to minimize

𝐶 ℎ + 𝐶𝑗 − 𝑟𝑇

𝑟−1

ℎ=1

2

62


63


Task 𝑡𝑖𝐴 Preceded

by

𝑡𝑖𝑩 Preceded

by

1 3 - 3 -

2 4 1 4 1

3 2 1 3 1

4 6 1 5 1

5 3 2 - -

6 4 3 2 3

7 - - 4 4

8 5 5, 6 4 7

𝑇𝑗 27 25

𝐶𝐴 =27

5= 5.4 𝑚𝑖𝑛

Station Task TTi Station

Time

1 1 30

3 26 56

2 4 54 54

3 2 40

5 12 52

4 6 28

7 24 52

5 8 44 44

Two models A and B with 𝑅𝐴 = 4 units/hr and 𝑅𝐴 = 6 units/hr, respectively.

So, 𝑅 = 10 units/hr

𝑛 = 5

𝐶𝐵 =25

5= 5 𝑚𝑖𝑛

𝑇 =

110 (4 × 27 + 6 × 25)

5= 5.16 𝑚𝑖𝑛

Select the first launch

For model A, 5.4 − 1 × 5.16 2 = 0.0576

For model B, 5 − 1 × 5.16 2 = 0.0256

Select the second launch

For model A, 5 + 5.4 − 2 × 5.16 2 = 0.0064

For model B, 5 + 5 − 2 × 5.16 2 = 0.1024

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Model B will be launched first

Set 𝐶 1 = 𝐶𝐵 = 5

Model A will be launched second

Set 𝐶 2 = 𝐶𝐴 = 5.4

Launch

(r) 𝐶 ℎ + 𝐶𝐴 − 𝑟𝑇

𝑟−1

ℎ=1

2

𝐶 ℎ + 𝐶𝐵 − 𝑟𝑇

𝑟−1

ℎ=1

2

Model

1 0.0576 0.0256 B

2 0.0064 0.1024 A

3 .1024 0.0064 B

4 0.0256 0.0576 A

5 0.16 0 B

6 0.0576 0.0256 B

7 0.0064 0.1024 A

8 0.1024 0.0064 B

9 0.0256 0.0576 A

10 0.16 0 B

The sequence is

B-A-B-A-B-B-A-B-A-B

IE304_LectureNotes_1

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