CAD-CAM

Oleh :

Haris Setiawan

Pengantar; Optimasi Satu Variabel;Optimasi Banyak Variabel

Optimasi Numerik

INTRODUCTORY EXAMPLE

Persoalan pemilihan diameter pipa untuk mengangkutfluida dari satu proses ke proses yang lain

Diameter pipa optimum, berdasarkan:Biaya investasi, dan biaya operasi

Diameter pipamanayang akan Andapilih?

Definisi dan Jenis Optimasi

Optimasi merupakan suatu proses untuk mencarikondisi yang optimum, dalam arti paling menguntungkan.

Optimasi bisa berupa maksimasi atau minimasi

Jika berkaitan dengan masalah keuntungan, maka keadaan optimum adalah keadaan yang memberikan

keuntungan maksimum (maksimasi).

Jika berkaitan dengan masalahpengeluaran/pengorbanan, maka keadaan optimum adalah keadaan yang memberikanpengeluaran/pengorbanan minimum (minimasi).

Introduction

Numerical optimization is one of the tools needed to produce the optimum (PRODUCT) design.

In studying design optimization, it is important to distinguish between analysis and design.

Analysis is the process of determining the response of a specified system to its environment.Ex. stress calculation in a structure due to applied loads.

Design is the actual process of defining the system.Ex. defining sizes and locations of members necessary to support a prescribed set of loads in actual structural design.

Analysis is a sub-problem of the design process

Fungsi Objektif

Secara umum, fungsi yang akan dimaksimumkan ataudiminimumkan disebut fungsi objektif (objective function), sedangkan harga-harga yang berpengaruhdan bisa dipilih disebut variabel (perubah) ataudecision variable.

Secara analitik, nilai maksimum atau minimum dari suatupersamaan: y = f(x)

dapat diperoleh pada harga x yang memenuhi:y = f(x) = 0Untuk fungsi yang sulit untuk diturunkan atau mempunyaiturunan yang sulit dicari akarnya, proses optimasi dapatdilakukan secara numerik.

Contoh Persoalan Optimasi dalamBidang Engineering

Design pump and heat transfer equipment for maximum efficiency

Design waste water treatment system to meet water-quality standards of least cost

Optimal planning and scheduling Optimal pipeline network Inventory control Maintenance planning to minimize cost etc.

Contoh Constraints yang MenyertaiPersoalan Optimasi

Maximum process temperature Maximum flow rate limitation Maximum conversion limitation Product purity Strength of materials Environmental factor Safety consideration Availability of utilities Corrosion considerations Availability and characteristics of feed stocks Market demand for the product Space limitation An upper limit on the capital investment

Example Unconstrained function minimization

From above we can estimate that the minimum value of F (X) will atx1

* = 1 and x2*= 1

Notes: In this example, the optimum can be found both graphicallyand analytically. This example has a little engineering value, except fordemonstration purpose

Example 3-2 Constrained function minimization

The column must be designed so that the magnitude of theapplied stress is less than the minimum of allowable stress,Euler-buckling stress, and shell-buckling stress, so

In addition to stress limitations, the design must satisfy the geometric conditions as follows

The design problem can now be stated compactly as

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