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Accelerated Stress Testing and Reliability Conference

An Integrated Reliability Growth Planning in the New Complex Engineering Product

Development

Mohammadsadegh Mobin Zhaojun (Steven) Li

Western New England University [email protected], [email protected]

ASTR 2016, Sep 28- 30, Pensacola Beach, FL www.ieee-astr.org 1

mailto:[email protected]



New Product Development (NDP)

www.ieee-astr.org ASTR 2016, Sep 28- 30, Pensacola Beach, FL 2

Continuous

Product

Development High % of

total

revenue

Consumer needs

changes Marketing

environment

changes

To stay ahead of

Competition

Changing

technology

Not to lose

market share



Planning

Product/Process design & development

Product/Process V&V Production

Prototype/pilot (Build components/

system) Prototype test

(Test components/ System)

Production phase Field

performance

Verification &Validation

(System and process V&V)

Business case (new idea)

Concept design (System requirement

identification)

Detail design (Component requirement

identification)

NPD process



Recent news on NPD delays, cost overruns, and quality issues

NPD programs are often plagued with:

Cost overruns, Schedule delays, and Quality issues.

Product Company Issues Year Source

787 Dreamliner Boeing Co Delay due to a structural flaw 2009 The Wall Street Journal

Chevy Volt General Motors Cost overrun during design 2009 CNN Money

The Honda/GE

HF120 turbofan

engine

Honda

Design issues: An unanticipated test

program glitch. A part of the gearbox

failed during the test. Rebuild the engine

and begin the test again.

2013 Flying

F-35 United Technologies Corp.’s

Pratt and Whitney unit

Delays in delivering engines. Quality flaws

and technical issues. Systemic issues and

manufacturing quality escapes.

2014 Defence-aerospace.com

Bloomberg Business

Sikorsky US Marine Corps' (USMC's)

A failure in the main gear box and need for

redesign of the component. Problems

with wiring and hydraulics systems. Budget

constraints.

2015 HIS Jane’s 360

NPD challenges (1 of 2)



NPD challenges (2 of 2)

NASA’s main projects that faced cost

and time overrun:

The International Space Station.

Prime contract had grown: 25%

(from $783M to $986M, the 3rd increase in 2 years).

The NASA Ares-I launch system.

Cost overrun: 43%

(from $28 billion original estimate to $40 billion)

The Department of Defense (DoD)

The set of 96 major new weapon system

development programs (2000-2010) have:

an average development cost growth of

42%,

an average delay of 22 months.

50% of the DOD’s NPD programs faced cost

overrun.

80% experienced an increase in unit costs from

initial estimates.



The main goal in product reliability improvement:

Reliability Management Process in NPD



Test Time

MT

BF

(D

ays)

1.000

10.00

100.0

1000

Initial MTBF:

3 days

Achieved

MTBF:

70.0 days

Reliability goal = 73

390 Days

A single stage Reliability Growth Plan

One RG plan can be:

390 Days,

Required test units

Required test time

Total cost: $50k

Objective of RGP: To determine the number of test units, test time, and cost to maximize the reliability growth.

Another RGP can be:

490 Days,

Test units and test time

Total cost $70k 490 Days

Reliability growth planning (RGP)



Motivation for multi-stage reliability growth planning

The importance of multi-stage RGP

An example of multi-stage NPD plan



Multi-stage reliability growth planning Challenges for multi-stage RGP in early product development stage:

1. How to allocate test units and time to individual stage.

2. How to determine the proportion of new technology introduction in each stage

The schematic of multi-stage reliability growth planning

A multi-stage reliability growth plan



Duane Model (1964)

• An empirical model, based on the learning curve,

• Also known as power law model

• Duane model in terms of cumulative failure rate:

𝒍𝒏 𝑪 𝒕 = 𝜹 − 𝜶 𝒍𝒏 𝒕

𝐶 𝑡 : The average failure rate 𝐶 𝑡 = 𝑁(𝑡)/𝑡

𝑁 𝑡 : The cumulative number of failures up to time 𝑡 during the reliability growth

testing.

𝛿, 𝛼 > 0 , 𝛼 is known as growth rate

Duane reliability growth model



RGP Literature Review

Our contribution: Multi-objective & multi-stage RGP

Author

Early

design

stage

Field (test)

stage

Single

objective

Multi

objective

Single -

stage

Multi-

stage

Duane (1964) [1] No Yes Yes No Yes No Crow (1974) [2] No Yes Yes No Yes No Lloyd (1986) [3] No Yes Yes No Yes No Robinson and Dietrich (1987) [4] No Yes Yes No Yes No Coit (1998) [5] No Yes Yes No Yes No Walls & Quigley (1999) [6] Yes No Yes No Yes No Walls & Quigley (2001) [7] No Yes Yes No Yes No Quigley and Walls (2003) [8] No Yes Yes No Yes No Krasich et al. (2004) [9] Yes No Yes No Yes No Johnston et al. (2006) [10] Yes No Yes No Yes No Jin and Wang. (2009) [11] No Yes No Yes Yes No Jin et al. (2010) [12] No Yes Yes No Yes No Jin et al. (2013) [13] No Yes Yes No Yes No Jin and Li (2016) [14] Yes Yes Yes No Yes No Jackson (2016) [15] No Yes Yes No Yes No Li et al (2016) [16] Yes No No Yes No Yes



Objectives : 1. Minimize failure rate at the final stage

2. Minimize total development time

3. Minimize total test cost

Decision

variables:

1- Number of test units for each subsystem in each stage (discrete)

2- Test time for each subsystem in each stage (continuous)

3- Percentage of introduced new contents of subsystem in each stage (continuous)

1- Total product development time

2- Number of available test units in each development stage

3- The lower and upper bounds on the test time of each subsystem in each stage

4- The lower and upper bounds on the number of test units for each subsystem in each stage

5- The lower and upper bounds on the percentage of subsystem added/introduced to the system

in each stage

Constraints:

Multi-Objective Multi-Stage Reliability Growth Planning

with continuous and discrete decision variables

Proposed RGP model



Method 1: Creating a weighted composite objective function Shortcomings:

1. Difficulties in determining appropriate utility functions (weights).

2. Objectives have different scale and cannot easily be added up.

Method 2: Consider one as main objective function and others as constraints Shortcomings:

1. Difficulties in determining boundary values.

2. Defining boundaries may reduce the solution space.

Method 3: Multi-objective Evolutionary algorithms (MOEAs)

e.g., MOPSO, MOACA, NSGA, etc.

1. Simultaneously optimizing two or three (or more) conflicting objectives.

2. Effective methods in exploring feasible solutions and providing a population of approximately optimal solutions (Pareto-

optimal frontier).

3. Apply evolutionary operators, e.g., crossover and mutation to generate variety of new solutions.

Solution methodology (1 of 3)



Based on theory of natural selection.

Mimics genetic operations in developing new potential solutions.

Terminology of GA:

• Population: a group of potential solutions.

• Chromosome: a solution.

• Gene: a component of the solution.

• Fitness function: how desirable a solution is in terms of surviving into

the next generation.

Results: A population of Pareto-optimal solutions.

Challenge: Selection of the best solution:

Application of Multiple Criteria Decision Making (MCDM)

approaches to rank the optimal solutions.

Genetic algorithm

Generate initial population of solutions

Evaluate fitness of each solution

Selection of solutions for reproduction

Mutation and crossover

New population generated and fitness evaluated

Sufficient solution quality or max iteration reached

YES

NO

End

Start




A powerful quantitative, analytical tool for ranking the DMUs

• DMUs convert multiple inputs (negative criteria ) into multiple outputs (positive criteria).

• MCDM compares DMUs based on their distance from the best DMU.

MCDM tools can be used to compare and rank the Pareto-optimal solutions in terms of their

objective values (criteria).

Criteria: Cost type criteria (minimization objectives)

Benefit type criteria (maximization objectives)

Results:

A set of Pareto-optimal high-ranked solutions in a manageable size.

Multiple Criteria Decision Making (MCDM) approach

Application of MCDM in MOPs




Overview of the proposed solution methodology

Mathematical model:

• Objective functions

• Constraints

• Decision variables

A set of Pareto-

optimal solutions

Negative criteria

(Minimization objective functions)

Positive Criteria

(Maximization objective functions)

MCDM tools

Multiple

Objectives

Evolutionary

Algorithm

Optimal

high-ranked

solutions

Proposed solution model



Notations 𝑚𝑖 Total number of subsystems in stage 𝑖 ( 𝑗 = 1, 2,… ,𝑚𝑖)

𝑛 Total number of stage (𝑖 = 1, 2,… ,𝑛)

𝑘 Index for the unique subsystem in the system (𝑘 = 1,2,… ,𝐾)

𝛼𝑖𝑗 The growth rate in stage 𝑖 for subsystem 𝑗

𝛼𝑖 The total growth rate in stage 𝑖 𝑛𝑖𝑗 Number of test units for subsystem 𝑗 in stage 𝑖 [𝑛𝑙 (𝑖𝑗 )

,𝑛𝑢 (𝑖𝑗 )] Lower and upper bounds on the number of test units for subsystem 𝑗 in stage 𝑖

𝑁𝑖 Total number of test units in stage 𝑖 [𝑁𝑙(𝑖),𝑁𝑢(𝑖)] Lower and upper bounds for the total number of test units in stage 𝑖

𝑁 Total number of test units (𝑁 = 𝑁𝑖𝑛𝑖=1 )

𝑡𝑖𝑗 Planned testing time for each subsystem 𝑗 in stage 𝑖 [𝑡𝑙 (𝑖𝑗 )

, 𝑡𝑢 (𝑖𝑗 )] Bounds on the test time for subsystem 𝑗 in stage 𝑖

𝑇𝑖 Total planned test time for stage 𝑖 (𝑇𝑖 = 𝑡𝑖𝑗𝑚 𝑖𝑗=1 𝑛𝑖𝑗 ,∀𝑖 = 1, 2,… ,𝑛)

𝑇 Total planned test time for all stages (𝑇 = 𝑇𝑖𝑛𝑖=1 )

𝜏𝑢 Upper bound of total development time

𝜏𝑖 Development time for stage 𝑖 𝜏 Total development time, (𝜏 = 𝜏𝑖

𝑛𝑖=1 ), (0 ≤ 𝜏 ≤ 𝜏𝑢 )

𝑐𝑖𝑗 Fixed cost of subsystem 𝑗 in stage 𝑖 Cv(i) Variable cost which is a function of total test time for stage 𝑖 ( 𝑇𝑖 )

𝐶𝑖 Total test cost of stage 𝑖, including fixed cost and variable cost

𝐶 Total test cost (𝐶 = 𝐶𝑖𝑛𝑖=1 , 𝑖 = 1,… ,𝑛)

𝜃𝑖𝑗𝑘 Percentage of introduced new contents of subsystem 𝑗 in stage 𝑖, [𝜃𝑙 (𝑖𝑗𝑘 )

,𝜃𝑢 (𝑖𝑗𝑘 )] Bounds on the percentage of introduced technology in stage 𝑖 for subsystem 𝑗

𝑊 The effective working hours in each year.

𝑉 The unit variable cost for each hour of testing

𝜆𝑖𝑗 Failure rate of subsystem 𝑗 in stage 𝑖 𝜆 𝑛(𝑖)

Failure rate of all new subsystems in stage 𝑖

𝜆 𝐼(𝑖) Initial failure rate in stage 𝑖,

𝜆𝑖 Total failure rate at the end of stage 𝑖 𝑀𝐼(𝑖) Initial MTBF in stage 𝑖

𝑀(𝑇𝑖 ) MTBF at the end of stage 𝑖



MS MO RGP mathematical modeling (1 of 2)

Min: 𝜆𝑖=𝑛 = 𝑓 𝜆 𝑖−1, 𝜆 𝑛(𝑖)

,𝛼𝑖 ,𝑇𝑖 (1)

Min: 𝜏 = 𝜏𝑖𝑛𝑖=1 , 𝑖 = 1,… ,𝑛 (2)

Min: 𝐶 = 𝐶𝑖𝑛𝑖=1 , 𝑖 = 1,… ,𝑛 (3)

s.t. 0 ≤ 𝜏 ≤ 𝜏𝑢 (4)

𝑁𝑙(𝑖) ≤ 𝑁𝑖 ≤ 𝑁𝑢(𝑖),∀𝑖 = 1, 2,… ,𝑛 (5)

𝑡𝑙 (𝑖𝑗 )≤ 𝑡𝑖𝑗 ≤ 𝑡𝑢(𝑖𝑗 ), 𝑖 = 1, 2,… ,𝑛 𝑎𝑛𝑑 𝑗 = 1, 2,… ,𝑚𝑖 (6)

𝑛𝑙 (𝑖𝑗 )≤ 𝑛𝑖𝑗 ≤ 𝑛(𝑖𝑗 ), 𝑖 = 1, 2,… , 𝑛 𝑎𝑛𝑑 𝑗 = 1, 2,… ,𝑚𝑖 (7)

𝜃𝑙(𝑖𝑗𝑘 ) ≤ 𝜃𝑖𝑗𝑘 ≤ 𝜃𝑢(𝑖𝑗𝑘 ), 𝑖 = 1, 2,… ,𝑛 , 𝑗 = 1, 2,… ,𝑚𝑖 ,∀𝑘 = 1,2,… ,𝐾 (8)

𝜃𝑖𝑗𝑘

𝑛

𝑖=1

= 100% , 𝑖 = 1, 2,… , 𝑛 , 𝑗 = 1, 2,… ,𝑚𝑖 ,∀𝑘 = 1,2,… ,𝐾 (9)



MS MO RGP mathematical modeling (2 of 2)

𝜏𝑖 = 𝑚𝑎𝑥𝑗 {𝑡𝑖𝑗 } , 𝑗 = 1,… ,𝑚𝑖 (10)

𝜏 = 𝜏𝑖𝑛𝑖=1 (11)

𝑇𝑖 = 𝑡𝑖𝑗𝑚 𝑖𝑗=1 𝑛𝑖𝑗 ,∀𝑖 = 1, 2,… ,𝑛 (12)

𝑇 = 𝑇𝑖𝑛𝑖=1 (13)

𝐶𝑖 = (𝜃𝑖𝑗𝑘 ∗ 𝑐𝑖𝑗𝑚 𝑖𝑗=1 ∗ 𝑛𝑖𝑗 ) + Cv(i) , ∀𝑖 = 1, 2,… ,𝑛 (14)

𝐶𝑣(𝑖) = 𝑓(𝑇𝑖 ) (15)

𝐶 = 𝐶𝑖𝑛𝑖=1 (16)

𝑁𝑖 = 𝑛𝑖𝑗𝑚 𝑖𝑗=1 ,∀𝑖 = 1, 2,… ,𝑛 (17)

𝑁 = 𝑁𝑖𝑛𝑖=1 (18)

𝑙𝑛 𝜆𝑖 = 𝑙𝑛 𝜆 𝐼(𝑖)− 𝛼𝑖 𝑙𝑛 𝑇𝑖 ,∀𝑖 = 1, 2,… ,𝑛 (19)

𝛼𝑖 = 𝛼𝑖𝑗𝑚 𝑖𝑗=1

𝑚 𝑖,∀𝑖 = 1, 2,… ,𝑛

(20)

𝜆𝐼(𝑖) = 𝜆 𝑛(𝑖)+ 𝜆 𝑖−1

,∀𝑖 = 1,… ,𝑛 (21)

𝜆 𝑛(𝑖)= (𝜆𝑖𝑗 ∗ 𝜃𝑖𝑗𝑘 )

𝑚 𝑖 𝑗=1 ,∀𝑖 = 1, 2,… ,𝑛 (22)

𝑀𝐼(𝑖) = 1

𝜆𝐼(𝑖) ,∀𝑖 = 1,… ,𝑛 (23)

𝑙𝑛 𝜆𝑖 = 𝑙𝑛 𝜆 𝐼(𝑖)− 𝛼𝑖 𝑙𝑛 𝑇𝑖 ,∀𝑖 = 1,… ,𝑛 (24)

𝜏𝑖 = 𝑚𝑎𝑥𝑗 {𝑡𝑖𝑗 } , 𝑗 = 1,… ,𝑚𝑖 (10)

𝜏 = 𝜏𝑖𝑛𝑖=1 (11)

𝑇𝑖 = 𝑡𝑖𝑗𝑚 𝑖𝑗=1 𝑛𝑖𝑗 ,∀𝑖 = 1, 2,… ,𝑛 (12)

𝑇 = 𝑇𝑖𝑛𝑖=1 (13)

𝐶𝑖 = (𝜃𝑖𝑗𝑘 ∗ 𝑐𝑖𝑗𝑚 𝑖𝑗=1 ∗ 𝑛𝑖𝑗 ) + Cv(i) , ∀𝑖 = 1, 2,… ,𝑛 (14)

𝐶𝑣(𝑖) = 𝑓(𝑇𝑖 ) (15)

𝐶 = 𝐶𝑖𝑛𝑖=1 (16)

𝑁𝑖 = 𝑛𝑖𝑗𝑚 𝑖𝑗=1 ,∀𝑖 = 1, 2,… ,𝑛 (17)

𝑁 = 𝑁𝑖𝑛𝑖=1 (18)

𝑙𝑛 𝜆𝑖 = 𝑙𝑛 𝜆 𝐼(𝑖)− 𝛼𝑖 𝑙𝑛 𝑇𝑖 ,∀𝑖 = 1, 2,… ,𝑛 (19)

𝛼𝑖 = 𝛼𝑖𝑗𝑚 𝑖𝑗=1

𝑚 𝑖,∀𝑖 = 1, 2,… ,𝑛

(20)

𝜆𝐼(𝑖) = 𝜆 𝑛(𝑖)+ 𝜆 𝑖−1

,∀𝑖 = 1,… ,𝑛 (21)

𝜆 𝑛(𝑖)= (𝜆𝑖𝑗 ∗ 𝜃𝑖𝑗𝑘 )

𝑚 𝑖 𝑗=1 ,∀𝑖 = 1, 2,… ,𝑛 (22)

𝑀𝐼(𝑖) = 1

𝜆𝐼(𝑖) ,∀𝑖 = 1,… ,𝑛 (23)

𝑙𝑛 𝜆𝑖 = 𝑙𝑛 𝜆 𝐼(𝑖)− 𝛼𝑖 𝑙𝑛 𝑇𝑖 ,∀𝑖 = 1,… ,𝑛 (24)



Step 1

𝒇𝟏 𝒇𝟐

𝒇𝟑

𝒇𝟏 𝒇𝟐

𝒇𝟑

To obtain final Pareto optimal solutions

through iteratively generating and assessing

the intermediate solutions in the Pareto

frontiers.

Fast Non-dominated sorting

genetic algorithm (NSGA-II)

Step 2 Multiple Criteria Decision

Making (MCDM)

To compare the optimal solutions in terms

of their objective values and as a result,

reduce the number of optimal solutions into

a workable size.

Considering:

all feasible solutions,

quality and diversity of solutions,

all objective functions simultaneously,

constraints when generating solutions.

Considering: each solution as a decision making unit (DMU).

cost type criteria as input for a DMU.

benefit type criteria as output for a DMU.

Proposed optimization method



Fast non-dominated sorting genetic algorithm (NSGA-II) Step 1: Create a random parent population P0 of size N. Set t = 0.

Step 2: Apply crossover and mutation to 𝑃0 to create offspring population 𝑄0 of size 𝑁.

Step 3: If the stopping criterion is satisfied, stop and return to Pt. Step 4: Set 𝑅𝑡= 𝑃𝑡 ∪ 𝑄𝑡.

Step 5: Identify the non-dominated fronts F1, F2, … , Fk in Rt (using FNDS)

Step 6: For 𝑖 = 1,… , 𝑘 do following steps:

Step 6.1: Calculate crowding distance of the solutions in 𝐹𝑖

Step 6.2: Create 𝑃𝑡+1 as follows:

Case 1: If 𝑃𝑡+1 + 𝐹𝑖 ≤ 𝑁, then set 𝑃𝑡+1 = 𝑃𝑡+1 ∪ 𝐹𝑖 ;

Case 2: If 𝑃𝑡+1 + 𝐹𝑖 > 𝑁, then add the least crowded 𝑁 − 𝑃𝑡+1 solutions from 𝐹𝑖 to 𝑃𝑡+1 Step 7: Use binary tournament selection based on the crowding distance to select parents from Pt+1.

Step 8: Apply crossover and mutation to Pt+1 to create offspring population Qt+1of size N.

Step 9: Set t = t + 1, and go to step 3.

𝑃

𝑄

Parent population

Offspring population

Crossover &

Mutation 𝑅

𝐹1 𝐹2

𝐹3

𝐹𝑘

Fast Non-Dominated Sorting

Population in next

generation 𝐹3

Crowding Distance Sorting

𝐹1 𝐹2

𝐹3 𝑃

Proposed

Optimization Method

(Step 1)

[16-18]



Proposed

Optimization Method

(Step 2)

Multiple Criteria Decision Making tools: VIKOR

• The name VIKOR is from Serbian:

VIseKriterijumska Optimizacija I Kompromisno Resenje,

• That means:

Multi-criteria Optimization and Compromise Solution.

The basic mechanism: • Calculate the “distance” from each alternative (optimal solution) to a “Positive Ideal

Solution” (PIS) and a “Negative Ideal Solution” (NIS)

• PIS and NIS are defined in “n-dimensional” space, where n represent the number of criterion

(the objective functions of the optimal solutions) in the decision problem.

• The chosen alternative should have the smallest vector distance from the PIS and the greatest

from the NIS.

[19]



Case study Application of MS MO RGP for next generation dual engine development process

𝜏𝑢: 3.5 years

The effective work hours in each year: 2000 hours

The variable cost per hour: $2000

Decision variables:

• 𝑛𝑖𝑗 (discrete)

• 𝑡𝑖𝑗 (continuous)

• 𝜃𝑖𝑗𝑘 (continuous)

𝑖

Subsystem 𝑘 𝑗 𝜆𝑖𝑗 𝑐𝑖𝑗

($1000)

𝛼𝑖 𝑗 𝑚𝑖 Cv(i)

($1000)

𝑁𝑙(𝑖) 𝑁𝑢(𝑖)

1 Engine Block 𝑘 = 1 𝑗 = 1 1.38 600 𝛼11 = 0.40

2 2 ∗

(𝑇1*2000) 4 8

Turbocharger 𝑘 = 2 𝑗 = 2 0.03 45 𝛼12 = 0.35

2

Engine control 𝑘 = 3 𝑗 = 1 0.30 62.5 𝛼21 = 0.32

4 2 ∗ (𝑇2*

2000) 8 16

Cooling

System 𝑘 = 4 𝑗 = 2 0.02 30 𝛼22 = 0.45

Fuel System 𝑘 = 5 𝑗 = 3 0.25 50 𝛼23 = 0.38 Lubricating

system 𝑘 = 6 𝑗 = 4 0.06 25 𝛼24 = 0.30

3

Engine control 𝑘 = 3 𝑗 = 1 0.30 62.5 𝛼31 = 0.25

3 2 ∗

(𝑇3*2000) 6 20

Fuel system 𝑘 = 5 𝑗 = 2 0.25 50 𝛼32 = 0.20 Lubricating

system 𝑘 = 6 𝑗 = 3

0.06 25 𝛼33 = 0.20



Mathematical modeling

Min: 𝜆𝑖=3 = 𝑓 𝜆 𝐼(𝑖−1), 𝜆 𝑛(𝑖)

,𝛼𝑖 ,𝑇𝑖

Min: 𝐶𝑖 = 𝐶1 + 𝐶2𝑛𝑖=1 + 𝐶3

Min: τ = τini=1 = τ1 + τ2 + τ3

0 ≤ 𝜏 ≤ 𝜏𝑢 ⟹ 0 ≤ 𝜏1 + 𝜏2 + 𝜏3 ≤ 3.5 (note:3.5 year)

𝑁𝑙(1) ≤ 𝑁1 ≤ 𝑁𝑢(1) ⟹ 4 ≤ 𝑛11 + 𝑛12 ≤ 8

𝑁𝑙(2) ≤ 𝑁2 ≤ 𝑁𝑢(2) ⟹ 8 ≤ 𝑛21 + 𝑛22 + 𝑛23 + 𝑛24 ≤ 16

𝑁𝑙(3) ≤ 𝑁3 ≤ 𝑁𝑢(3) ⟹ 6 ≤ 𝑛31 + 𝑛32 + 𝑛33 ≤ 20

𝑛𝑖𝑗 [min max] 𝑡𝑖𝑗 [min max] 𝜃𝑖𝑗𝑘 (%)

𝑛11 [2 4] 𝑡11 [0 0.5] 𝜃111 = 100

𝑛12 [2 4] 𝑡12 [0 0.5] 𝜃122 = 100

𝑛21 [4 8] 𝑡21 [0 1.5] 50 ≤ 𝜃213 ≤ 100

𝑛22 [4 8] 𝑡22 [0 1.5] 𝜃224 = 100

𝑛23 [4 8] 𝑡23 [0 1.5] 30 ≤ 𝜃235 ≤ 100

𝑛24 [4 8] 𝑡24 [0 1.5] 60 ≤ 𝜃246 ≤ 100

𝑛31 [3 15] 𝑡31 [0 1.5] 0 ≤ 𝜃313 ≤ 60

𝑛32 [3 15] 𝑡32 [0 1.5] 0 ≤ 𝜃325 ≤ 50

𝑛33 [3 15] 𝑡33 [0 1.5] 0 ≤ 𝜃336 ≤ 60

Decision variables and the defined interval for each

Objectives: Constraints:



Results: RGP Optimal solutions

Obtaining Pareto optimal frontier for RGP using NDSGA-II



One example set

of Pareto optimal

solutions for the

case study

problem

notations

Solution

1

Solution

2

Solution

3

Solution

4

Solution

5

Solution

6

Solution

7

Solution

8

Solution

9

Solution

10

Decision

variables

𝑛11 1 3 2 2 2 1 3 3 1 3

𝑛12 2 3 3 3 3 3 2 3 1 2

𝑛21 3 3 3 3 3 3 3 3 3 3

𝑛22 4 3 3 3 3 4 3 3 3 3

𝑛23 3 4 4 4 4 3 4 4 4 4

𝑛24 4 4 4 4 4 4 4 4 4 4

𝑛31 5 5 5 5 5 5 5 5 5 5

𝑛32 5 5 5 5 5 5 5 5 5 6

𝑛33 6 6 6 6 6 6 6 6 6 5

𝑡11 0.0100 0.1177 0.2416 0.1913 0.2846 0.0475 0.1193 0.3493 0.0514 0.9900

𝑡12 0.1677 0.1641 0.1768 0.3478 0.1721 0.9900 0.1715 0.2711 0.0139 0.3337

𝑡13 0.0075 0.0698 0.2006 0.3511 0.0767 0.9900 0.1710 0.3948 0.0045 0.9900

𝑡22 0.0100 0.2824 0.1100 0.3458 0.1111 0.0688 0.0603 0.1610 0.0132 0.9900

𝑡23 0.0068 0.2882 0.1615 0.2415 0.1601 0.6900 0.2377 0.4542 0.0090 0.9900

𝑡24 0.0280 0.1288 0.0100 0.0079 0.0102 0.0171 0.0190 0.7703 0.0461 0.8812

𝑡31 0.0151 0.0226 0.0668 0.0763 0.0449 0.4037 0.0225 0.0348 0.0449 0.9752

𝑡32 0.0053 0.0100 0.0127 0.0100 0.1877 0.0116 0.2491 0.1635 0.0080 0.9460

𝑡33 0.0037 0.2561 0.1979 0.2920 0.1502 0.4995 0.1641 0.3383 0.0097 1.3017

𝜃213 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

𝜃235 1.0000 0.4971 0.5551 0.5642 0.3494 0.9177 0.8647 0.9046 1.0000 0.9723

𝜃246 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

𝜃313 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

𝜃325 0.0000 0.5029 0.4449 0.4358 0.6506 0.0823 0.1353 0.0954 0.0000 0.0277

𝜃336 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Objective

functions

𝜆3 7.758 1.355 1.561 1.077 1.587 0.603 1.536 0.704 9.305 0.318

𝜏 0.211 0.708 0.640 0.991 0.632 2.479 0.658 1.458 0.143 3.282

𝐶 1250.159 2558.724 1951.614 1961.082 1962.610 1354.633 2493.321 2563.059 1225.147 2606.756

Other

parameters

𝜆1 2.101 1.502 1.403 1.234 1.367 0.932 1.611 1.117 3.923 0.869

𝜆2 4.938 1.395 1.614 1.166 1.695 0.837 1.813 0.871 7.284 0.582

𝐶1 691.382 1938.382 1339.055 1340.703 1339.342 747.070 1892.804 1942.445 645.261 1904.550

𝐶2 558.280 487.818 494.980 502.688 452.357 566.683 557.323 584.685 578.594 625.669

𝐶3 0.497 132.524 117.578 117.690 170.911 40.880 43.194 35.929 1.292 76.537

𝑇1 0.345 0.845 1.014 1.426 1.086 3.018 0.701 1.861 0.065 3.637

𝑇2 0.195 2.725 1.618 3.088 1.245 5.384 1.720 6.565 0.274 13.425

𝑇3 0.124 1.699 1.585 2.184 2.064 5.073 2.343 3.021 0.323 17.060

Results: RGP Optimal solutions



𝜆3 𝜏 𝐶

Rank 1 20.459 0.030 1223.8

Rank 2 20.459 0.030 1223.8

Rank 3 0.431 2.984 1393.6

Rank 4 0.457 2.970 1307.5

Rank 5 17.211 0.040 1268.8

Rank 6 0.501 2.752 1302.4

Rank 7 0.445 2.684 1389.9

Rank 8 16.214 0.030 1313.9

Rank 9 15.812 0.060 1224.0

Rank 10 0.495 2.591 1379.3

Rank 11 11.783 0.045 2514.3

Rank 12 10.531 0.059 2494.6

Rank 13 10.236 0.074 2514.4

Rank 14 0.317 3.388 2582.9

Rank 15 0.318 3.282 2606.8

Rank 16 5.753 0.160 2471.2

Rank 17 0.318 2.970 2631.2

Rank 18 0.317 2.970 2635.1

Rank 19 0.369 2.749 2609.5

Rank 20 0.528 2.263 2580.1

Rank 21 2.549 0.419 1874.78

Rank 22 2.102 0.508 1924.20

Rank 23 1.587 0.632 1962.61

Rank 24 1.561 0.640 1951.61

Rank 25 2.079 0.655 1924.47

High- ranked optimal solutions

Results : MCDM application


Conclusions


Formulate and investigate a multi-stage reliability growth planning model.

Reliability growth planning model is formulated considering multiple objectives.

Considering the percentage of introduced new technology, number of test units, and test time

as decision variables.

Integrate subsystem growth rates within an individual stage into the model.

Applying a multi-objective evolutionary algorithm, called NSGA-II, to find the Pareto

optimal solutions of developed Multi-Objective-Multi-Stage-RGP (MO MS RGP).

Rank the Pareto optimal solutions for solution reduction using an MCDM approach, called

VIKOR.

Future Research:

Considering uncertainty in the parameters of the model

Application and comparison of other MOEAs and MCDM tools


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(RAMS), 424-30, 2004.

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References



Biography


Mohammadsadegh Mobin is a doctoral student in Industrial Engineering and Engineering

Management at Western New England University, MA, USA. He holds a Master degree in Operations

Research (OR) (2011) and a bachelor degree in Industrial Engineering (IE) (2009). He served as a

quality engineer (2006-2012) in different industries. He has published in different journals including:

Expert Systems with Applications and IEEE Transaction on Reliability. He has been the instructor of

“Design and Analysis of Experiments” and “Probability and Statistics”. His research interests lie in

the area of different applications of operations research tools in quality and reliability engineering.

Zhaojun “Steven” Li is an Assistant Professor at the Department of Industrial Engineering and

Engineering Management, Western New England University in Springfield, MA. Dr. Li’s research

interests focus on Reliability, Quality, and Safety Engineering in Product Design, Systems

Engineering and Its Applications in New Product Development, Diagnostics and Prognostics of

Complex Engineered Systems, and Engineering Management. He earned his doctorate in Industrial

Engineering from the University of Washington in 2011. He is an ASQ certified Reliability Engineer,

and Caterpillar Six Sigma Black Belt. Dr. Li’s most recent industry position was a reliability team

lead with Caterpillar Rail Division to support the company’s Tier 4 Locomotive New Four Stroke

Engine and Gas-Diesel Dual Fuel Engine Development.


THANK YOU


Mohammadsadegh Mobin, Zhaojun (Steven) Li

[email protected], [email protected]



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