Systems for unified life-cycle mechanical engineering design: Shared ...

8 downloads 7809 Views 1MB Size Report
Individual CAD/CAM/CAE software, developed under various programming ... of current computer hardware, available commercial software and data bases, and ...
Engineering with Computers 6, 211-222 (1990)

Ewngineering computers 9 Springer-Vedag New York Inc. 1990

Systems for Unified Life-Cycle Mechanical Engineering Design: Shared-Tool Architectures Versus Distributed Tool Architectures David A . Hoeltzei and Wei-Hua Chieng Laboratory for Intelligent Design, Department of Mechanical Engineering, Columbia University, New York, NY, USA

Abstract. A unified life-cycle engineering design (ULCED) system embodying conceptual design, detailed design, redesign, design for manufacturing, and design maintainability forms the basis of a new computer-based design discipline for improving the overall quality of manufactured goods. The objective of such a system is to review, evaluate, and analyze the entire life cycle of a product and to incorporate, in an integrated fashion, life-cycle knowledge within the design process. Individual CAD/CAM/ CAE software, developed under various programming environments and operating systems, and on various types of computer hardware, have met with only limited success in achieving total computational integration. At the initial stages of development of such a system exploration of both technologically and economically feasible architectures is of critical importance. Two ULCED system architectures, broadly classified as shared-tool and distributed-tool, are described and compared on the basis of current computer hardware, available commercial software and data bases, and existing computer programming environments. Important evaluation issues to be considered include implementation costs and complexity and the feasibility of system creation.

iterations) contributes only 10 percent of resource costs to the total cost of a manufactured product and little or nothing is contributed (0 percent) from the data dissemination stage (Fig. 1). Based on these statistics, it is evident that the commitment/$cost ratio corresponding to the conceptual design stage of the overall design process is high, equaling 50/3 (or 16.67), such that each 1 percent of resource costs is financed by 16.7 percent of the commitment of the product cost, performance, and manufacturability. The cost-effectiveness of conceptual design in current CAD/CAM environments to life-cycle engineering design is therefore minimal. Conversely, the commitment/$cost ratio corresponding to the redesign and data dissemination stages is much too low, equaling 10/55 + 0/15 (or 0.182), such that each 7 percent of resource costs is financed by only 1 percent of commitment. This clearly demonstrates poor utilization and distribution of resource costs.

1 Existing CAD/CAM Environments Studies [1] have shown that currently existing CAD/CAM environments distribute only 3 percent of total product resource (research and development) costs to the conceptual design stage, 27 percent to the detailed design stage, 55 percent to the redesign stage, and 15 percent to the data dissemination stage throughout the life cycle of a manufactured product, while in theory the total commitment of a product [2], that is, its cost, performance, manufacturability, and so on, are predominantly established during the conceptual design stage (50 percent) and the detailed design stage (40 percent). In contrast, the redesign stage (design revisions and

1O0 [] 75-

$resource cost

(%)

]

commitment (%)

]

comrnitmenU$costratio (~

50--

25-

I

Conceptual

[

I

Detail

Redesign

Design stages

Offprint requests: David A. Hoeltzel, Laboratory for Intelligent Design, Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA.

Fig. 1. Distribution of resource cost in current CAD/CAM environments.

212

D.A. Hoeltzel and W-H. Chieng

,~lmaUan b u e ~k n d y ~ In~

H~mful moblity det~-tion

~ /

lll~mstlo I laab:i C,~lxml,,t comltrailt

Evoluotlm

removal Degr~ of

freldom

Rdldillty

and

ili~ml~r

/vlmwmmu [i.lnliipilol

\q..lh.,.i=\ lollltll

Index

DlmenManal

Off Ibm l~ooemMm

ModlfkmUon

L ~ ,p~rcd k~uhdUm Klnetoetatk= Anal~b

9

PraducUlan )OpUml._]

~

,

],oballxe

Mop

/~ation /

Optlmizatkm

Indtx

0n line l~-ooemms

/ ~m'.dgNon_thread. mid i Jalnhg

Adh~vm

9 obSHy / and . Falur. /

~

.

I ~ \ . . ~--~---"~Sequmoln~ md

\

~t,rnauv,

machining

mMhod,

Pmd~tla~ IIModiflcoUon

Fig. 2. Processes required for total life-cycle evaluation of engineered mechanical components within the ULCED system9

A ULCED system would utilize computer-aided engineering (CAE) techniques to normalize the distribution of resource costs throughout all design levels (concept, detail, and redesign) by applying concept engineering tools (concept geometry, early design review, graph enumeration) to strengthen the capability of the conceptual design process, using relational and unified knowledge representation to expedite setup procedures for detailed design and employing a centralized control strategy to guide optimization of the redesign stage of a manufactured product design.

2

The ULCED System Environment

A ULCED system encompasses an integrated understanding of, and detailed knowledge about, de-

sign engineering, manufacturing and production engineering (including optimal scheduling), and software engineering. We specifically identify such a system to be one that is capable of including information (knowledge) about all aspects of the life cycle of a component, from its initial conception (geometry, features, materials) to its strength and durability (fatigue, wear), to its manufacture (processes, assemblability), and to the final evaluation of its performance (maintainability as well as causes and rates of failure) and economics during in-field use, in a parallel, closed-loop configuration. Our concept of the life cycle of mechanical components has been succinctly described in Fig. 2. The ULCED system, for the near optimal design of mechanical components, has been modeled as an integrated three-phase program that includes the following design phases:

Systems for Unified Life-Cycle Mechanical Engineering Design

Subetructure ~ / / ~ Supemtructure(supportbase)

Head

Posi~donlng

(Arm swing llrniter)

Fig. 3. Three subclasses of a mechanical design in the ULCED system: Example of a hard disk drive.

1. Conceptual design and design synthesis 2. Dimensional design and design analysis 3. Economics-of-quality design and production analysis Three subclasses of a mechanical design (design categories) are included in the ULCED system; they are superstructures (or foundations), substructures, and mechanisms (Fig. 3).

3

Complexity Involved in the Implementation of an ULCED System

The system diagram illustrated in Fig. 2 includes the following four primary features required for the development of an ULCED system:

1. Multidisciplinary, multiple-dimensional. The ULCED system employs knowledge that covers mechanical engineering, computer science, material science, surface engineering, industrial engineering, and so on. Optimality of the designed product must be evaluated from a multiple-viewpoint perspective. 2. Optimality-oriented. Other than constraint satisfaction, optimization of an overall design, at all levels of the design process, must be achieved. The ULCED system is concerned not only with search through the design space but also with realistic evaluation of optimality criteria. The optimality criteria must include a knowledge base of nonnumerically based information, in addition to traditional mathematical optimality criteria, such as the Karush-Kuhn-Tucker (KKT) necessary conditions or monotonicity analysis [3]. 3. Hybrid (symbolic-numeric) programming. The ULCED system requires tools that include both

213

knowledge-based and numerical programming capabilities. For example, knowledge-based programming strategies and techniques have the potential for effective application to the automation of the mechanical assembly process (shape and feature-based descriptions) and the selection of engineering materials. In addition, various types of numerically based analysis programs can be used to perform stress analyses and geometric modeling, and can be effective for predicting numerically based effects of design variations. 4. Distributed and parallel tasking. Some design and production processes are independently separable. For example, material selection, mechanical assembly processes, and kinematic analysis represent a set of independently separable design processes. The control of coupled, nonseparable, ULCED processes must occur in parallel to ensure simultaneous information control and guidance for the implementation of distributed tasks. The total integration of design, manufacturing, and in-field evaluation of mechanical components would permit design, manufacturing, production, and quality control engineers to work in a more harmonious "team-oriented" manner to realize the design of mechanical components that are of higher quality and reliability, lower cost, and superior maintainability and modifiability. 4 Tools Incorporated Within the ULCED Conceptual Design Phase The elements of conceptual design include discovery, analogy [4], identification, and enumeration [5]. Effective use of these elements or design "skills" commonly requires the accumulation of 20 to 30 years of industrial experience. Existing CAD/ CAM environments neither supply nor store the knowledge required to assist engineers during the conceptual design stage. Technological evidence has demonstrated that implanting design concepts within computer programs through the use of classification and coding schemes can produce effective results [5]:

1. Classification. Categorization of the types of elements required for implementation of conceptual design 2. Encoding. The storage of existing design concepts through efficient and effective coding schemes Classification enables systematic enumeration of design alternatives through the application of gener-

214

D.A. Hoeltzel and W-H. Chieng

I I

Disk locking control

I L _ _ _

] Head location I [ control

A

I I IIi

P. . . .

] I I. . . .

J

I

,

I

Hord dink drive

Decoder

[ L.

F: Fastening J: Joining R: Revolute Joint AG: Air gop DF: Dlrectlon al fastening

Disk locking control

Power

Slgno[ out

B GT classlficotion code: 11481074

II

t ~ C e n t e r hole: "~ru & "l~r~d

~___.y.~..~oit c i r o l ,

C

.,~

il

Grooves: Surfoce groove | I MAX O.D.: obout 1.5 In " ~ J l MAX Length: about 3 In

Fig. 4. (A) Functional representation: Example of a hard disk drive. (B) Graph representation: Example of a hard disk drive assembly. (C) Group technology (GT) classification and coding system: Example of a rotor in the hard disk drive assembly.

alization and specialization processes [6], whereas encoding enables design through analogy by retrieving similar, existing designs from a concept design data base [5]. Graph theory [7] can be applied to the mechanical design process in order to encode the structure, and potentially the function of designs. In this way designs can be identified and grouped together to take advantage of their similarities. Methods for the classification and encoding of mechanisms began 20 years ago and continue today [6,8]. The processes of classification and encoding the structures of mechanisms is concerned with identifying similarities among mechanisms through the use of structural attributes (types of links and joints and the manner in which these elements are connected) and formally representing their similarities in an encoded format. Some important mechanism classification schemes include the link adjacency matrix (LIAM) approach, the loop adjacency matrix (LOAM) ap-

proach, and the stratified representation of mechanisms (SRM) approach [6]. As an example, the LIAM stores information about how adjacent link pairs in a mechanism are connected and includes information concerning the types of joints that connect the links. When a new mechanism design is required within the concept design phase, an engineer can devote a few minutes to determining its minimum required connectivity. Existing designs that match its encoded structure can then be retrieved to see if one of them is capable of satisfying the desired functional (design) requirements (Fig. 4A). An example illustrating the LIAM classification scheme is shown in Fig. 4B. Group technology (GT) [9] is utilized in manufacturing to identify and group together mechanical parts to take advantage of the similarities among their features. In this situation part classification and encoding is concerned with identifying the similarities among parts, including both design attributes (such as geometric shape and size) and manufacturing attributes (the type and sequence of processing steps required to produce the parts), and relating these similarities to a coding system. Several of the more important classification and coding systems include the Brisch System, Optiz, CODE, CUTPLAN, DCLASS, and Multiclass [5]. For example, the Optiz system uses the first five digits of a numerical coding scheme to represent information about the class of a part, its external and internal shapes, planar surface finish, and the type of auxiliary holes or gear teeth present. When a new part design is required during the concept design phase, the engineer can devote a few minutes to determine the code corresponding to the required part. Then existing part designs that may match the code can be retrieved to see if one of them will satisfy the desired functional requirements. An example illustrating the GT classification scheme is shown in Fig. 4C.

5

Utility Software for the ULCED Detailed Design Phase

The detailed design phase represents the most developed phase in existing CAD/CAM/CAE environments that can be directly utilized in an ULCED system. Individual software modules that satisfy the needs of the proposed U L C E D system are listed here. (While only a single software program has been listed under each category, this does not necessarily imply that the quality of the indicated soft-

Systems for Unified Life-Cycle Mechanical Engineering Design

T = Tool

IULC'ED] environment

215

Aries, and I D E A S TM by General Electric's Calma Division. Most of the existing packages are oriented toward (1) kinematics, (2) dynamics, and (3) stress analysis, for which the underlying theories have been well developed. Little or nothing has been included to evaluate critical design processes that would elevate the "quality" of a design during the conceptual design phase, the design for manufacturing phase, and the in-field analysis phase of a product design.

Fig. 5. A shared-tool ULCED architecture.

6 A Shared-Tool ULCED System Architecture ware is better than others that have not been included in the same category.): 9 Concept design a n d manufacturing. Initial com-

mercial versions exist, as well as academic prototypes (e.g., The Concept Modeller by Wisdom Systems Div. of McDermott International, Inc., and DOMINIC by the University of Massachusetts). 9 Knowledge-based system shell. KnowledgeCraft by the Carnegie Group Inc. 9 Geometric modeling. Such as C A T I A TM by Dassault Systems, USA. 9 Mechanism and machine design, synthesis, and analysis. Lincages T~ by Minnesota Technology Transfer (MINTT). 9 Automatic tolerance and sensitivity analysis. Such as Complete Engineering M o d e l ~Mby Cognition Inc. 9 Engineering data bases. Material property data bases such as that produced by the NBS (National Bureau of Standards, Washington, D.C.). TM

TM

TM

9 Manufacturing m a n a g e m e n t . X e r o x Cost Planning and Control ~Mby Xerox Computer Services,

Inc. 9 Product variation analysis. Variation Simulation Analysis TM by Applied Computer Solutions. 9 Design f o r machining. A u t o S p o t TM by IBM Inc. 9 Design f o r assembly. No commercial versions

exist, but handbooks do exist. 9 Stress analysis. A N S Y S ~ , by Swanson Analysis Systems, Inc. 9 D y n a m i c analysis. A D A M S ~ by Mechanical Dynamics, Inc. 9 In-field analysis. No formal commercial versions are known to exist. Literature describing these topics can be found respectively in Refs. [10-16]. Existing commercially available software providing a limited portion of a ULCED system capabilities include the Mechanical A d v a n t a g e TM by Cognition Inc., M C A D ~ by

A single mechanical design engineer assisted by a ULCED system environment could, in theory, undertake all required design processes since the utility software programs incorporated within it can be equally shared by different, concurrent system users. We define such a system to be a "shared-tool system," as shown in Fig. 5. A shared-tool system can be considered to be a fully automated implementation of the ULCED process. The scope of a true shared-tool system includes all aspects of the life cycle of a design, from its initial conception to its strength and durability, to its manufacture, and to the final evaluation of its performance and economics during in-field use in a parallel, closed-loop configuration. An automated shared-tool ULCED system must contain the capability for automatic data transfer and results interpretation. It transfers design data among different design processes as well as interprets the design results for the designer. More specifically, the main functions that this system must include are: 1. A unified knowledge representation scheme 2. A user-friendly man-machine interface 3. Knowledge-based, object-oriented data base utilities A unified knowledge representation scheme and associated language, which is both hierarchical and relational, serves as the underlying basis for the entire ULCED system. The feature-based, knowledge representation scheme for a mechanical component includes design features (detailed design knowledge) and manufacturing features (manufacturing knowledge). The design attributes include basic external shape, basic internal shape, length-diameter ratio, material type, part function, major dimensions, minor dimensions, tolerance, part loading, and surface finish. The manufacturing attributes include major manufacturing processes, minor manu-

216

facturing operations, machine tool types, machine tool operation sequence, production time, batch size, fixtures needed, and types of cutting tools. The hierarchically structured knowledge representation scheme must permit the design function (design idea) performed by a set of structures (conceptual design knowledge) and the specification of the structure that integrates (jointing, joining, fastening) a set of mechanical components to be represented. For example, a linear motor performs a design function. It is comprised of a set of structures each of which contains a set of components. The concept of the functionality of a design is a highlevel abstraction. There are currently no existing generalized, comprehensive mapping schemes that exist to establish relationships between structure and function in mechanical design. Further research is required to establish such a mapping scheme. The relationally based knowledge representation scheme must allow relations to be declared among different components, hence any variations made to a component will automatically be propagated to the other components in the knowledge hierarchy to maintain the consistency of the design. The knowledge representation scheme for completeness must also include knowledge about geometric tolerances. In other words, the specification of design parameters must contain not only a preferred or "nominal" value but also upper and lower bounds that establish allowable variations on the nominal dimensions [17]. This property is of importance when performing variational geometric analysis during the optimization stage [18,19]. The man-machine interface (intelligent front end) is aimed at providing guidance and smooth access to the individual design software modules within the U L C E D system for its users (designers and engineers). Front ends for numerical analysis programs are intended to help the user to select and use numerical analysis processes and to assist the user in interpreting the results generated by such programs [20]. Front ends for symbolic processes are intended to help the user to select, interpret, and evaluate different types of symbolically represented objects and to provide assistance in evaluating candidate objects, occurring within the design process, that have received preliminary acceptance. Engineering data base utilities [21] are important for selecting engineering materials, manufacturing processes, or standardized components. An intelligent (knowledge-based) data base system must assist the user in setting up property requirements (upper or lower bounds), establishing preferences for properties (ranges of allowable materials), and

D.A. Hoeltzel and W-H. Chieng

Ultility e 9 e 9

Programs:

Concept design tool Expert system shell Solid Modeler Kinematic and d ~ a m l c

/~k /

analysis 9 Assembly process s[mula+.ion / . . 9 ~ t r e. s s . onolysls i/

9 O,~olitycontrol ~ 9

Manufacturing

9,mu,o~,o~

/.

/ ~/

/ /"

/

I\

l~tt~t Bases"

J \ '\ ~' / \ \ / \ ~\ i \

\ ~\ \ ~ \

~'~/2"/

\~.\

9 Material 9 dornts and Bearings 9 Machining processes

* Jo~i~g o.d Vostsnl~g o Other=

. r--,

Designer

Fig. 6. Overview of a shared-tool ULCED system.

eliminating those materials and properties deemed to be unimportant or undesirable. An intelligent data base system must be able to perform a minimally satisfying search (in accordance with property requirements) as well as an optimal search (in accordance with preferred properties). A shared-tool U L C E D system possesses the potential to allow for fast design and manufacturing setup and is especially useful for developing innovative designs. New designs and plans for their manufacturing processes can be systematically enumerated and simulated within the ULCED system. Since design trade-offs (optimization) can be judged at an early stage in the design process, less resource costs are required during the redesign and manufacturing stages, thereby fulfilling, more completely, the need for normalization of resource costs throughout the entire life cycle of a manufactured product. The attempt at fully automating, through integration, the U L C E D system (Fig. 6), presently stands as a future goal rather than a currently achievable reality, due to a number of ULCED system bottlenecks:

1. System. Existing design processes, such as automatic finite element mesh generation, automated mechanical assembly, and machining process selection, have not yet reached a fully automated stage. 2. Application. Unsuccessful planning of system implementation may result in an U L C E D system whose scope is either too broad, which can unpredictably increase the complexity and cost of system implementation, or too narrow, which

Systems for Unified Life-Cycle Mechanical Engineering Design

M =

Man

T =

Tool

The requirements that the knowledge representation be feature-based, hierarchical, and relational are the same as those required for the shared-tool system. ~__~D]

".'5"] i {ULgED] v~onrnent

Fig. 7. A distributed-tool ULCED system architecture.

can reduce user satisfaction and system applicability. 3. People. In real life, designers develop different talents for different portions of the design process. The fate and quality of a design will ultimately be determined by human factors. Designers must be retrained to follow, in a cooperative manner, the design steps established by the U L C E D system design philosophy.

7

217

A Distributed-Tool ULCED System Architecture

A distributed-tool U L C E D system is shown in Fig. 7. The control facilities have been uncoupled from software utility tools. Under this distributed-tool system architecture, each designer handles only a few design processes. This represents an important difference from the shared-toll architecture previously described. For example, the stress analyst handles only shape optimization, whereas the manufacturing engineer makes decisions germane only to the design and sequencing of machining processes. This system aims at providing guidance to a engineering design team (group design). The core of a distributed-tool U L C E D system is based on a central control and designer communication network. This network communicates different U L C E D design processes undertaken by different process designers and supervises the overall progress of the design team. Ultimately, this central control scheme must be independent of the specific software and hardware used in various design processes. The main functions required of the central control program include: 1. A unified knowledge representation 2. A relational map with constraint propagation 3. A parallel design optimization and planning control

8

Establishing a Relational Map for a Distributed Tool ULCED System

A relational map may be thought of as a networking mechanism that establishes a communication link among the various types of design information residing within the distributed-tool U L C E D environment. The influence that design variations play in any or all of the design processes will automatically be propagated to the other relevant design processes. These design variations include component connectivity variations (fastening, joining, jointing, and other methods of assembly), design attributes associated with individual components (major shape, minor shape, internal shape, dimensions, surface finish, etc.), manufacturing attributes (major process, minor operation, selection and sequencing of machine tools, etc.). This knowledge about design variations, and their relationships with the various design processes, can be represented in a coded form and stored in a relational map. The relational map is initially hard-coded with knowledge acquired from experienced designers. A schematic representation of this map is shown in Fig. 8 in a ROM (read only memory)-like configuration, where the inputs (outputs) are rows (columns). Each input corresponds to a series of codes (bits representing design attributes and bits representing manufacturing attributes) as previously described. Each active (relevant) node connecting an input to an output establishes a set of design rules (ruleset) relating the corresponding design processes. The designer may select nodes through the top-level supervising program (Fig. 10). For example, the ruleset for the node that connects the mechanical assembly process to the stress analysis process may be represented as follows:

Ruleset: Assembly-to-Stress Rule: Minor-shape-modification: :contactingsurface Rule: Minor- shape-modification:: shape-edges Rule: Minor- shape-modification:: drill-holes Rule: Major-shape-modification: :removematerial Rule: Major-shape-modification: :creategroove

218

D.A. Hoeltzel and W-H. Chieng

Joinlng Fastonlnc. Joints

Camp. Conn.

Mech. Camp.

Klne. Material AnallPais Selection

Stress Llfe--r Analysim VeHf.

In--field Evol.

Prodn. Variation

Auern. Machn. Process Procen

jFastenlm o•ghg I, 1111=r-Jationol map

C~'np.

~

co~, ~

2. 9

0

0

9

9

0

~..o.o0 9

~

0

0

Strm

9

9

9

9

Klne.

,.~,. 0 Matedal

..~,. 9 I.]fe-c~.

~,.

9

(~

9

0 0

9

~

0

9

means that the column the row component are moans that the column the row component are

0 Prodn,

~o.ot,o.

0

@

0

9

9

0

.. . . .

@

9

9

9

0

@

is s~metricol.

means that the column component and the row component are highly con-elated.

@l:x 0

Rule: Major-shape-modification: :drill-holes Rule: Part-loading-modification: :add-

attachment Rule: Part-loading-modification: :merge-parts Rule: Length/diameter-modification: :reducethe-contacting-surface M o r e rules . . .

When the relational map undergoes on-line execution, new design rules can be added and existing design rules can be modified or deleted in accordance with specific design purposes. This knowledge acquisition mechanism, which provides the capability for manipulating the design rules in the relational map, is not a novel concept, and has been successfully adopted in the development of many expert systems. MYCIN represents one example of this [22].

9 Design Optimization in a Distributed-Tool ULCED System Since design processes occur in parallel in a distributed-tool U L C E D system, the corresponding design optimization process must be dynamic. This differs from the traditional approach to CAD optimization that employs an iterative decision-making process. Based on the requirement for a dynamic optimizer, a three-level, parallel architecture optimization processor, which proceeds from local design optimization to global design optimization, is proposed:

component and

partially corse;areal. component and independent.

-.... o

o

Fig. 8. A prototype relational map for the ULCED distributedtool system.

1. Constraint satisfaction. The process of design

modification, or redesign, is driven by the necessity to reconcile constraint violations; that is, a design is fulfilled when there are no remaining violated design constraints. These design constraints originate from (1) resource-cost distribution (design and manufacturing costs), (2) reliability requirements, (3) functional requirements (input and output specifications), and (4) existing design achievements (a least-commitment design philosophy). 2. L o c a l p a r a m e t r i c optimization. Based on the requirement of constraint satisfaction, parametric optimization [3] allows for the minimization (or maximization) of mathematically defined local design objective functions, including cost indices (for design and manufacturing) and reliability indices, within each of the design processes. 3. Heuristically based global design optimization (design planning). A combination of success-

fully achieved locally optimal designs does not necessarily imply the existence of an overall or global optimum. Therefore, we advocate parallel optimization control, fabricated in accordance with design process planning (scheduling and sequencing). Appropriately designed sequencing and scheduling can avoid combinatorial explosion that can easily occur when global design optimization is attempted. A potential problem associated with design optimization in a distributed-tool U L C E D system is convergence of the final design result. The satisfaction of a globally optimal design relies heavily on

S y s t e m s for Unified Life-Cycle Mechanical Engineering Design

219

local design optima (i.e., optimum solutions to subproblems that taken together comprise the solution to the overall optimization problem). Azarm and Li [23] provide an excellent discussion of optimization problem decomposition strategies. As a simple example, consider a parametrically formulated design optimization problem containing four subproblems, where the overall objective function is the weighted sum of the subproblem objective functions: Global

objective

= W l C 1 @ w 2 C 2 -}- w 3 C 3 -]- w 4 C 4

It is required to minimize the "hidden" (none of the local design processes, that is, subproblems, explicitly contains the global objective) design cost, where the individual subproblems or processes can be expressed as Process

1=

z C 1 - (x 2 -}- 1)

Process 2 = C2 = y - 5z 3 In(x) Process 3 = C3 - (y2 + 1) Process 4 = C4 = 4x - y whereO_