Engineering Optimization

Second EditionAuthors: A. Rabindran, K. M. Ragsdell,

and G. V. ReklaitisChapter-1 (Introduction)

Presenter: Pulak ChowdhuryJune 4, 2010

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Life is about Optimization

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Introduction to Optimization

Finding the best candidate from a set of alternatives without explicitly enumerate and evaluate all possible alternatives

Root of Engineering - design new, better, more efficient, and less expensive systems

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Requirements to Apply OptimizationClearly delineate the boundary of the engineering system

Define quantitative criterion - objective

Select system variables

Define a model for expressing relation of variables

Problem Formulation

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Defining System Boundaries (1)

System - A restricted portion of the Universe

System boundary - separate the system from the universe

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Defining System Boundaries (2)Initial choice may be too restrictive

Expand system boundary - increase in size and complexity

Example: A manufacturing operation has a paint shop in which finished parts are

mounted on an assembly line and painted in different colors. Optimize batch size or

color sequence.

Considering paint shop in isolation or not?

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Performance Criterion

Criteria on the basis of which the performance or design of the system can be evaluated

Examples: Total capital cost, annual cost, return on investment, cost-benefit ratio, production time, maximum production rate, etc.

Maximum/mimimum

Only one criterion

Shop foreman - design with minimum color and part changes

Sales department - design with maximized inventory of parts

Financial Officer - design that minimizes inventories

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Independent Variables

Variables adequate to characterize possible candidate designs or operating conditions of the system

Distinguishing Variables

Including all variables

To what detail?

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System Model

Describes the relations among problem variables

Simplified mathematical representation of real system

Collection of equations

Real system too expensive

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Optimization Problem

Performance Measure - Objective

A set of independent variables

A model relating the variables

Defined boundary - Constraints

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Engineering Design Process

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Application of OptimizationDesign of components or entire systems

Design of a welded beam, Design of an oxygen supply system

Planning and analysis of existing operationsRefinery production planning

Engineering analysis and data reductionNonlinear curve fitting

Control of dynamic systems

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Structure of Optimization Problems

Minimize

Subject to Constraints

Objective

Constrained Optimization ProblemProblem with no constraints, that is, Unconstrained Optimization Problem. Here,

and

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Classification of Optimization Problems

Objective

Constraints

Variables Program

Linear Linear Continuous Linear Program

Linear Linear IntegerInteger Linear

Program

nonlinear

linear N/A

Linearly Constrained Nonlinear Program

Unconstrained problems in which is a one-component vector are called single-

variable problems

Quadratic Program

Fractional Linear Program

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Refinery Production Planning: DescriptionA refinery processes crude oils to produce a number of

raw gasoline intermediates that must subsequently be blended to make two grades of motor fuel, regular and premium. Each raw gasoline has a known perform- ance rating, a maximum availability, and a fixed unit cost. The two motor fuels have a specified minimum performance rating and selling price, and their blending is achieved at a known unit cost. Contractual obligations im- pose minimum production requirements of both fuels. However, all excess fuel production or unused raw gasoline amounts can be sold in the open market at known prices. The optimal refinery production plan is to be deter- mined over the next specified planning period.

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Refinery Production Planning: Schematic

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Refinery Production Planning: Formulation

System: Raw gasoline intermediates, blending operation, fluid motor fuels

Performance Index: Net profit over planning period

Motor fuel and intermediate sales - blending costs - cost of intermediates

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Refinery Production Planning: Variables

Flows depicted as directed arcs in

the figureEach intermediate - one variable

for each intermediate = amount used for regular

= amount used for premium= amount sold directly

for each product = amount allocated to contracts= amount sold in open market

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Refinery Production Planning: Model

Material balance equations

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Refinery Production Planning: ObjectivePerformance Criterion - Net profit

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Questions?