Transcript Slide 1

OPTIMASI
Bahan kajian pada MK. Metode Penelitian Kajian Lingkungan
Disarikan oleh:
Prof Dr Ir Soemarno MS
PMPSLP PPSUB OKTOBER 2010
https://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 PROGRAMMING
The 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
http://en.wikipedia.org/wiki/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.
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 function
Consider 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