An Application of Constructal Theory in the Multi-Objective Design of ...

An Application of Constructal Theory in the Multi-Objective Design of Product Platforms Michael Carone Christopher Williams Janet K. Allen Farrokh Mistree Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia

DETC2003/DTM-48667 Proceedings of DETC’03 ASME 2003 Design Engineering Technical Conferences Chicago, Illinois September 4, 2003

Systems Realization Laboratory http://www.srl.gatech.edu

Research Focus Objective Objective ●● To Todevelop develop formal, formal,mathematically mathematicallycorrect, correct,and and rigorous rigorous principles principles for for designing designing product product architectures architecturesthat thatfacilitate facilitatethe theproduction productionof of customized customized products. products. Relevance Relevance ●● This Thiswould wouldprovide providemanufacturing manufacturingfirms firmsan anefficient efficient (through (through rigorous rigorous and and systematic systematic methodology) methodology) foundation foundation for for realizing realizing customized customized products, products, thus thus enhancing enhancing the the responsiveness responsiveness of of manufacturing manufacturing organizations organizations to to changes changes in in the the market market or or demands demands for forcustomization. customization. 2 Systems Realization Laboratory

Product Platform Design: A Requirements List ●● ●● ●● ●●

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● ●

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Top-Down Top-Down Approach Approach to to Avoid Avoid Redesign Redesign Effective Effective--Systematic Systematicand andRigorous Rigorous Efficient Efficient--Simple Simpleand andPractical Practical Explicit Explicit Consideration Consideration of of other other approaches approachesto to “manage” “manage”product productvariety variety(e.g., (e.g.,modular modulardesign, design, dimensional dimensional customization, customization, adjustable adjustable controls/features, controls/features, etc.) etc.) Explicit ExplicitConsideration Considerationof ofthe the“extent” “extent”of ofthe theproduct product platform platform Applicable Applicableto tofamilies familieswith withvery verylarge largenumber numberof of products products Multiple Multiplecustomizable customizable specifications specifications Multiple Multiple design design objectives objectives Systems Realization Laboratory

Product Platform Constructal Theory Method Developed by Gabriel Hernandez (DETC2002/DAC-34095)

Fundamental FundamentalQuestion: Question: How Howto todetermine determineand andorganize organizemultiple multiple approaches approachesfor foroffering offeringproduct productvariety? variety? Constructal ConstructalTheory Theory –– The Thehierarchic hierarchicstructures structures(tree (treenetworks) networks)that thatwe weobserve observeinin natural naturaland andartificial artificialsystems systemsare arethe the“fingerprint” “fingerprint”ofofthe the minimization minimizationofofflow flowresistance resistancebetween betweenaafinite finitevolume volume and andone onepoint. point. An access problem can be solved through the optimization of the shape of the smallest, inner-most space elements and the hierarchic assembly of these elements into larger “constructs” until covering the entire geometric space. (A. Bejan)

(A. Bejan)

S3

S4

S5

S2

The Smallest Area, S1 y

.

S1 D1 H1

4

E

V0 V1

L1

S6

P(x,y) x

Time, Complexity, Evolution

Systems Realization Laboratory

Product Platform Design as a Problem of Access in a Geometric Space ● ● ●

●

●

●

First Firstestablish establishaageometric geometricspace… space… Want to determine an optimal arrangement of product variety techniques that link all points in the space for customization in order to satisfy any customer demand such that cost is minimized. Modes of managing product variety ● Customization through speed control ● Standardization of motors ● Standardization of blades Must organize mode hierarchy in a way so that each segment of the market space is represented Solution: Optimize the first space element; iterate until entire space is covered

Power [Watts]

Volumetric Flow [L/s]

Synthesis In Time

3rd Space Element

2nd Space Element

1st Space Element 20

Capacity [L/s]

500

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Power [Watts]

Product Platform Constructal Theory Method Step 1: Define the Space of Customization

Volumetric Flow [L/s]

(i.e., (i.e.,minimize minimizecost costper perproduct, product, maximize overall profit, maximize overall profit,etc.) etc.)

Step 2: Formulate an Objective Function Step 3: Identify the Modes for Managing Product Variety

(i.e., (i.e.,modular modulardesign, design,standardization, standardization, dimensional dimensionalcustomization, customization,etc.) etc.) 1st Space Element

2nd Space Element

r Market Space

1

Step 4: Identify the Stages and Define a Baseline Decision for Each Stage

Mode 1 Mode 2 r 2

Step 5: Formulate the Multistage Design Problem Step 6: Solve the Multistage Design Problem

r

r

1

1

Mode 3

S 1

3rd Space Element

S 2

S 1 S 1S

S 3

1

r 2

Mode 4

r 2

Optimal Objective Functions Stage 1 (1st Space Element)

Stage 2 (2rd Space Element)

Stage 3 (3rd Space Element)

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Product Platform Design: A Requirements List ●● ●● ●● ●●

●●

●●

● ●

7

●●

Top-Down Top-Down Approach Approach to to Avoid Avoid Redesign Redesign Effective Effective--Systematic Systematicand andRigorous Rigorous Efficient Efficient--Simple Simpleand andPractical Practical Explicit Explicit Consideration Consideration of of other other approaches approachesto to “manage” “manage”product productvariety variety(e.g., (e.g.,modular modulardesign, design, dimensional dimensional customization, customization, adjustable adjustable controls/features, controls/features, etc.) etc.) Explicit ExplicitConsideration Considerationof ofthe the“extent” “extent”of ofthe theproduct product platform platform Applicable Applicableto tofamilies familieswith withvery verylarge largenumber numberof of products products Multiple Multiplecustomizable customizable specifications specifications Multiple Multiple design design objectives objectives Systems Realization Laboratory

Utility-Based Compromise Decision Support Problem AAdecision decisionsupport supportconstruct constructbased basedon onutility utilitytheory theorythat thatpermits permits mathematically mathematicallyrigorous rigorousmodeling modelingof ofdesign designpreferences preferencessuch suchthat that decisions decisionscan canbe beguided guidedby byexpected expectedutility utilityof ofdecision decisionoutcome outcome Given: Given: AAfeasible feasiblealternative, alternative,assumptions, assumptions,parameter parametervalues, values,and andgoals goals Find: Values Find: Valuesofofdesign designand anddeviation deviationvariables variables Satisfy: Satisfy: System Systemconstraints, constraints,system systemgoals, goals,and andbounds boundson onvariables variables Minimize: Minimize:Deviation Deviationfunction functionthat thatmeasures measuresdistance distancebetween betweengoal goaltargets targetsand and design points design points Step 1: Assess Utility Functions for Each Goal

Step 2: Combine into a MultiAttribute Utility Function

1

n

0.9

U = ∑ ki ui ( Ai )

0.8

0.7

Utility

0.6

0.5

0.4

0.3 2

y = -0.0101x - 0.0069x + 1.0609 0.2

0.1

0 0

2

4

6 Deflection (in.)

8

10

12

i =i

Step 3: Formulate System Goals

E[ui ( Ai ( X ))] − i

+ i

+d − d = 1

Step 4: Formulate the Deviation Function

Z = 1 − E[U ( X )] n

= ∑ ki (di− − di+ ) i =1

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Augmented Product Platform Constructal Theory Method Step 2: Formulate an Objective Function

Step 1: Assess Utility Functions for Each Goal 1

0. 9

0. 8

0. 7

0. 6 Utility

Step 1: Define the Space of Customization

0. 5

0. 4

0. 3 y = -0. 0101x 2 - 0 .00 69x + 1 .06 09 0. 2

0. 1

0 0

2

4

6

8

10

12

Defle ctio n (i n. )

Step 3: Identify the Modes for Managing Product Variety

Step 2: Combine into a Multi-Attribute Utility Function U =

Step 4: Identify the Stages and Define a Baseline Decision for Each Stage

n

∑ k u (A ) i=i

i

i

i

Step 3: Formulate System Goals E [ u i ( Ai ( X ) ) ]

Step 1: Define the Space of Customization Step 2: Formulate the Objective Functions Step 3: Identify the Modes for Managing Product Variety Step 4: Identify the Decision Stages

+ d i− − d i+ = 1

Step 5: Formulate the Multistage Design Problem

Step 4: Formulate the Deviation Function

Step 5: Formulate a Multistage u-cDSP

Z = 1 − E[U ( X )] n

Step 6: Solve the Multistage Design Problem

= ∑ ki (d i− − d i+ ) i =1

Step 6: Solve the Multistage u-cDSP

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Example: Development of a Product Platform for a Line of Customizable Pressure Vessels Th

Ts R

  P Th ≥  R  2σ − 0.2 P   y 

L ● ●

● ●

Manufacturer Manufacturerwants wantstotooffer offeraacustomized customizedline lineofofpressure pressurevessels vesselsthat thatreach reach aaspecified range of pressure and volume specified range of pressure and volume Design DesignVariables: Variables: –– –– ––

● ●

  P Ts ≥  R  σ − 0.6 P   y 

Length Length(L) (L) Radius Radius(R) (R) Thickness Thicknessofofhead headand andshell shell(T (Th, ,TTs) )

Objective: Objective: –– ––

h

s

Minimize Minimizeaverage averagecost costofofproduct productfamily family Maximize Maximizetotal totalnumber numberofofproduct productvariants variants

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Step 1: Define the Space of Customization 30

P (MPa) 10 10

V (m3)

30

Step 2: Define the Objective Functions Cost Cost Material: Material:sheet sheetmetal, metal,weld weldmaterial material Waste: Waste:LL00--LL Manufacturing ManufacturingEquipment: Equipment:$(150,000 $(150,000++30,000R) 30,000R)

Variety Variety 11

Vmax − Vmin   Pmax − Pmin  Variety =    V P ∆ ∆    1 1

Systems Realization Laboratory

Step 3: Identify the Modes for Managing Product Variety Mode ModeV1: V1:Customization Customization of of Shell Shell Length Length Mode ModeV2: V2:Modular Modular Combination Combination of ofVessels Vessels

L1

L2

Mode ModeP1: P1:Standardization Standardization of of Plate PlateThickness Thickness Mode ModeP2: P2:Commonalization Commonalization of of Vessel VesselRadius Radius 12 Systems Realization Laboratory

Step 4: Identify the Number of Decision Stages L2

L1

One Vessels

Two (V2,Vessels P2)

P

Radius 1 (R1)

(Lo,1)

Radius 3 (R3)

R2 (V1, P1)

V

Mode P2: Common Raw Length Radius R 1

∆P1

Radius 2 (R2)

R1 Head/Shell ∆V Thickness2 1 (Th1, Ts1)

Mode V2

∆V1

S1

S

Mode V1: Cutting Length L (R constant)

Head/Shell Thickness 2 (V22, P 22) (Th2, Ts2)

Mode P2 Mode P1: Standard (Common) R3 Thicknesses Ts and Th

Head/Shell Thickness 3 (Th3, Ts3)

S1

∆P

Raw Length2 2 (Lo,2)

Raw Length 3 P (L ) o,3

S3

Head/Shell Thickness 4 (Th4, Ts4)

Mode P1

S1

S2

Raw Length 4 (Lo,4)

Raw Length 5 (Lo,5)

Mode V1

S2 Vmin

V’

(V21, P21)

V 13

V

Systems Realization Laboratory

Step 5: Formulate a Multistage u-cDSP For Foreach eachStage Stagei i N Given: Given:The TheN-Dimensional N-Dimensionalmarket marketspace spaceM MN={(r ={(r1,,rr2,,… …rrN)})} 1

2

N

Find: Find:The Thevalue valueof ofdecision decisionvariable variable x(i) x(i)==[∆ [∆rr11(i), (i),∆∆rr22(i), (i),… …∆∆rrNN(i)] (i)] − + d d The deviation variables and x , i x The deviation variables and ,i Satisfy: Satisfy: Bounds Bounds Constraints Constraints

∆rj ,min (i ) ≤ ∆rj (i ) ≤ ∆rj ,max

∆rj (i ) ≥ ∆rj (i − 1) d x−,i , d x+,i ≥ 0

d x−,i id x+,i = 0 Goals Goals

E[u (ox ,i )] + d x−,i + d x+,i = 1 y

Minimize: Minimize: Z i = 1 − U i = ∑ k x (d 14

x =1

− x ,i

y

+ d ) ;;where where U i = ∑ k x ,i u (ox ,i ) + x ,i

x =i

Systems Realization Laboratory

Step 6: Solve the Multistage u-cDSP 3

U3 For

Volume (m ) Pressure (MPa) Length (m) Tshell (m) Thead (m) Radius (m) 11.25∆ V ∆V 2 1 12.50 13.75 2 2 1 1 30 15.00 ∆P 0.0127 S16.25 1 1 0.00635 17.50 3rd Space Level 18.75 rd each 3 Space 20.00 10.00 7.00 0.637 21.25 Select 22.50 V’ 2 23.75 25.00 0.0191 ∆P 0.01270 S226.25 S3 20 2 nd 27.50 2 Space Level 28.75 For each 2nd Space 30.00 2 Select Common 11.25 1 12.50 R 0.00635 0.0127 13.75 1 15.00 16.25 1st Space Level 17.50 10 18.75 0.0191 20.00 1 st Space For each 10.896 11.25 3.27 21.25 Select Common 22.50 0.01270 23.75 Lo, Ts30 , Th 10 25.00 20 26.25 0.0254 27.50 V [m 3 ] 28.75 30.00 Systems Realization Laboratory

∆V , ∆P , ∆P , ∆V

∆P

P [ MPa ]

U

∆V , ∆P

U

15

Closing Comments ●●

Infusion Infusion of of utility utility theory theory aided aided with with multimultiobjective objective decision decision making making –– Use Useof ofu-cDSP u-cDSPcaptured capturedand andquantified quantifieddesigners’ designers’ preferences preferences –– Adequately Adequately handled handled tradeoffs tradeoffs between between objectives objectives

Strengthening the PPCTM ●● ●●

●● ●●

Handling Handling non-uniform non-uniform demand demand Development Development of of platforms platforms for for manufacturing manufacturing processes processes Handling Handling risk risk and and uncertainty uncertainty Use Use of of robust robust solution solution techniques techniques

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Acknowledgements ●● ●●

●●

Gabriel Gabriel Hernandez, Hernandez, Siemens Siemens Westinghouse Westinghouse Christopher Christopher Williams Williams and and Michael Michael Carone Carone acknowledge acknowledge the the support support of of NSF NSF Grant Grant DMI DMI #0085136 #0085136 Christopher Christopher Williams Williams also also acknowledges acknowledges the the support support of of aa Georgia Georgia Tech Tech Presidential Presidential Fellowship Fellowship

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Thank you!

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Supplemental Slides: Utility Based Compromise Decision Support Problem Working WorkingAssumptions: Assumptions: •• Designer Designerisis rational rational (preferences (preferences are areconsistent) consistent) •• Designer Designer isis risk risk averse: averse: •• prefer preferalternatives alternativesthat thatoffer offeron-target on-targetoutcomes outcomesto tothose thosethat that have haveconsiderable considerablechances chancesof ofyielding yieldingundesirable undesirableresults results

•• Multi-attribute Multi-attributeutility utilityfunction functionisisformulated formulatedas as aa weighted weighted sum: sum: •• Utility Utilityindependence: independence:risk riskaversion aversionfor foraagoal goalisisindependent independent of ofthe thevalues valuesof ofthe theother othergoals. goals. •• Additive Additiveindependence: independence:no nointeraction interactionbetween betweenpreferences preferences for fordifferent differentgoals goals 19 Systems Realization Laboratory

Establishing Context: Design for Mass Customization ●●

Mass Mass customization customization –– Want Wantto tooffer offervariety varietyeffectively effectively –– Want Wantto todo doititthrough throughproduct productdesign design

●●

Product Product platforms platforms –– Modularity Modularity –– Dimensional Dimensionalscaling scaling –– Standardized Standardizedinterfaces, interfaces,etc. etc.

●●

Need Needaamethodical methodicalway wayto todesign designfor for –– Multiple Multiplemodes modesof ofmanaging managingvariety variety –– Can Canhandle handlecomplexity complexityof ofCustomized Customizedparts parts –– Can Canhandle handlemultiple multipleobjectives objectives

20 Systems Realization Laboratory