OPTIMASI Bahan kajian pada MK. Metode Penelitian Kajian Lingkungan

Disarikan oleh:Prof Dr Ir Soemarno MS

PMPSLP PPSUB OKTOBER 2010https://marno.lecture.ub.ac.id

DEFINITION OF OPTIMUM

1. The amount or degree of something that is most favorable to some end; especially : the most favorable condition for the growth and reproduction of an organism.

2. Greatest degree attained or attainable under implied or specified conditions .

DEFINITION OF OPTIMIZE

Optimize : to make as perfect, effective, or functional as possible

Examples of OPTIMIZE

The new system will optimize the efficiency with which water is used.

DEFINITION OF OPTIMIZATION

Optimization: an act, process, or methodology of making something (as a design, system, or decision) as

fully perfect, functional, or effective as possible;

Specifically : the mathematical procedures (as finding the maximum of a function) involved in this process.

OPTIMIZATION(Stephen J. Wright)

Optimization, also known as mathematical programming, collection of mathematical principles and methods used for solving quantitative problems in many disciplines, including

physics, biology, engineering, economics, and business.

The subject grew from a realization that quantitative problems in manifestly different disciplines have important mathematical

elements in common.

Because of this commonality, many problems can be formulated and solved by using the unified set of ideas and methods that

make up the field of optimization.

MATHEMATICAL PROGRAMMINGThe historic term mathematical programming, broadly synonymous

with optimization, was coined in the 1940s before programming became equated with computer programming.

Mathematical programming includes the study of the mathematical structure of optimization problems, the invention of methods for

solving these problems, the study of the mathematical properties of these methods, and the implementation of these methods on

computers. Faster computers have greatly expanded the size and complexity of optimization problems that can be solved.

The development of optimization techniques has paralleled advances not only in computer science but also in operations research,

numerical analysis, game theory, mathematical economics, control theory, and combinatorics.

OPTIMIZATION PROBLEMS

Optimization problems typically have three fundamental elements.

The first is a single numerical quantity, or objective function, that is to be maximized or minimized.

The objective may be the expected return on a stock portfolio, a company’s production costs or profits, the time of arrival of a vehicle at a specified destination, or the vote share of a political candidate.

OPTIMIZATION PROBLEMS

Optimization problems typically have three fundamental elements.

The second element is a collection of variables, which are quantities whose values can be manipulated in order

to optimize the objective.

Examples include the quantities of stock to be bought or sold, the amounts of various resources to be allocated to different production

activities, the route to be followed by a vehicle through a traffic network, or the policies to be advocated by a candidate.

OPTIMIZATION PROBLEMS

Optimization problems typically have three fundamental elements.

The third element of an optimization problem is a set of constraints, which are restrictions on the values that the variables

can take.

For instance, a manufacturing process cannot require more resources than are available, nor can it employ less than zero resources. Within

this broad framework, optimization problems can have different mathematical properties.

Problems in which the variables are continuous quantities (as in the resource allocation example) require a different approach from problems in which the variables are discrete or combinatorial quantities (as in the selection of a vehicle route from among a

predefined set of possibilities).

LINEAR PROGRAMMING

An important class of optimization is known as linear programming.

Linear indicates that no variables are raised to higher powers, such as squares.

For this class, the problems involve minimizing (or maximizing) a linear objective function whose variables are real numbers that

are constrained to satisfy a system of linear equalities and inequalities.

NONLINEAR PROGRAMMING

In nonlinear programming the variables are real numbers, and the objective or some of the constraints

are nonlinear functions (possibly involving squares, square roots, trigonometric functions, or products of

the variables).

FUNGSI TUJUAN Dan/atau FUNGSI KENDALANYA…… NON LINEAR

MATHEMATICAL OPTIMIZATION

Mathematical optimization (optimization or mathematical programming) refers to the

selection of a best element from some set of available alternatives.

The optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the

value of the function. The generalization of optimization theory and techniques to other formulations comprises a

large area of applied mathematics. More generally, optimization includes finding

"best available" values of some objective function given a defined domain including a

variety of different types of objective functions and different types of domains.

http://en.wikipedia.org/wiki/Mathematical_optimization

OPTIMIZATION PROBLEM

An optimization problem can be represented in the following way

Given: a function f : A R from some set A to the real numbers

Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A ("minimization") or such that f(x0) ≥ f(x) for all x in A ("maximization").

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for

example in linear programming).

Many real-world and theoretical problems may be modeled in this general framework. Problems formulated using this technique in the fields of physics and computer vision

may refer to the technique as energy minimization, speaking of the value of the function f as representing the energy of the system being modeled.

Typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the

members of A have to satisfy.

The domain A of f is called the search space or the choice set, while the elements of A are called candidate solutions or feasible

solutions.

The function f is called, variously, an objective function, cost function (minimization), utility function (maximization), or, in

certain fields, energy function, or energy functional.

A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

By convention, the standard form of an optimization problem is stated in terms of minimization.

Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be

several local minima, where a local minimum x* is defined as a point for which there exists some δ > 0 so that for all x such that

|X - X*| ≤ δ

the expression: F(x*) ≤ f(X)

holds; that is to say, on some region around x* all of the function values are greater than or equal to the value at that point. Local maxima are defined similarly.

Optimization problems are often expressed with special notation. Here are some examples.

Minimum and maximum value of a functionConsider the following notation:

This denotes the minimum value of the objective function x2 + 1, when choosing x from the set of real numbers . The minimum value in this case is 1,

occurring at x = 0.

Similarly, the notation

asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is

unbounded, so the answer is "infinity" or "undefined".

MULTI-OBJECTIVE OPTIMIZATION

Adding more than one objective to an optimization problem adds complexity.

For example, to optimize a structural design, one would want a design that is both light and rigid. Because these two objectives conflict, a trade-off exists. There will be one lightest design, one stiffest design, and an infinite number of designs that are some

compromise of weight and stiffness. The set of trade-off designs that cannot be improved upon

according to one criterion without hurting another criterion is known as the Pareto set. The curve created plotting weight against

stiffness of the best designs is known as the Pareto frontier.

A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other

design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto

optimal.

MULTI-MODAL OPTIMIZATION

Optimization problems are often multi-modal; that is they possess multiple good solutions. They could all be globally good (same

cost function value) or there could be a mix of globally good and locally good solutions.

Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.

Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different

solutions will be obtained even with different starting points in multiple runs of the algorithm.

Evolutionary Algorithms are however a very popular approach to obtain multiple solutions in a multi-modal optimization task.

FEASIBILITY PROBLEM

The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without

regard to objective value. This can be regarded as the special case of mathematical

optimization where the objective value is the same for every solution, and thus any solution is optimal.

Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to relax the feasibility conditions

using a slack variable; with enough slack, any starting point is feasible. Then, minimize that slack variable until slack is null or

negative.

1. Konsep Program Linier :

Merupakan model umum yang dapat digunakan dalam pemecahan masalah pengalokasian sumber-sumber yang terbatas agar bisa digunakan secara optimal

Merupakan teknik matematik tertentu untuk mendapatkan kemungkinan pemecahan masalah terbaik atas suatu persoalan yang melibatkan sumber-sumber organisasi yang terbatas

Metode matematis yang dapat digunakan sebagai alat bantu pengambilan keputusan bagi seorang manajer berkaitan dengan masalah maksimisasi atau minimisasi

Prosedur Penyelesaian LP:

Pembuatan Model Matematis (Logika Matematis), merupakan faktor kunci/utama dalam permasalahan linier programming

Perhitungan bisa diselesaikan dengan cara manual (metode grafik, metode simplex, konsep dualitas) maupun dengan Komputer.

Analisis hasil hitungan, sebagai salah satu alat alternatif keputusan dan pengambilan keputusan.

Tahapan Pembuatan Model Matematis

Identifikasi Masalah : Masalah Maksimisasi (berkaitan dengan Profit/Revenue) atau Masalah Minimisasi (berkaitan dengan dengan Cost/biaya)

Penentuan Variabel Masalah :

1) Peubah Keputusan (Variabel yang menyebabkan tujuan maksimal atau minimal)

2) Fungsi Tujuan (Objective Function) Z maks. atau min.

3) Fungsi Kendala (Constraint Function) Identifikasi dan merumuskan fungsi kendala yang ada

Program Linear adalah bagian ilmu

matematika terapan yang digunakan untuk

memecahkan masalah optimasi

(pemaksimalan atau peminimalan suatu

tujuan) yang dapat digunakan untuk mencari

keuntungan maksimum seperti dalam bidang

perdagangan, penjualan dsb

MULTI-OBJECTIVE OPTIMIZATION

Multi-objective optimization (or multi-objective programming), also known as multi-criteria or multi-attribute optimization, is the process of simultaneously optimizing two or more conflicting

objectives subject to certain constraints.

Multiobjective optimization problems can be found in various fields: product and process design, finance, aircraft design, the oil and gas industry, automobile design, or wherever optimal decisions need to

be taken in the presence of trade-offs between two or more conflicting objectives.

Maximizing profit and minimizing the cost of a product; maximizing performance and minimizing fuel consumption of a vehicle; and minimizing weight while maximizing the strength of a particular

component are examples of multi-objective optimization problems.

In mathematical terms, the multiobjective problem can be written as:

where μi is the i-th objective function, g and h are the inequality and equality constraints, respectively, and x is the vector of optimization or decision

variables. The solution to the above problem is a set of Pareto points. Thus, instead of

being a unique solution to the problem, the solution to a multiobjective problem is a possibly infinite set of Pareto points.

PARETO EFFICIENCY

Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after

Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.

Given an initial allocation of goods among a set of individuals, a change to a different allocation that makes at least one individual better off without making any other individual worse off is called a

Pareto improvement.

An allocation is defined as "Pareto efficient" or "Pareto optimal" when no further Pareto improvements can be made.

Pareto efficiency is a minimal notion of efficiency and does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a

society