POKOK BAHASAN Mata Kuliah ini membahas tentang Linear
Programming, Model Transportasi, Model Penugasan, Manajemen Proyek,
Model Antrian, Linear Goal Programming dan Dynamic Programming yang
bermanfaat untuk pengambilan keputusan manajemen.
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LITERATUR Richard, dkk (2000), Pengambilan Keputusan Secara
Kuantitatif, Rajawali Pers Jakarta Subagyo. P, dkk (1983).
Dasar-Dasar Operation Research. BPFE Yogyakarta Supranto. J (1988),
Riset Operasi Untuk Pengambilan Keputusan, UI Press, Jakarta
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What is OPERATION RESEARCH ? Operations research (OR) is a
discipline explicitly devoted to aiding decision makers. This
section reviews the terminology of OR, a process for addressing
practical decision problems and the relation between Excel models
and OR.
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Linear Programming A typical mathematical program consists of a
single objective function, representing either a profit to be
maximized or a cost to be minimized, and a set of constraints that
circumscribe the decision variables. In the case of a linear
program (LP) the objective function and constraints are all linear
functions of the decision variables. At first glance these
restrictions would seem to limit the scope of the LP model, but
this is hardly the case. Because of its simplicity, software has
been developed that is capable of solving problems containing
millions of variables and tens of thousands of constraints.
Countless real-world applications have been successfully modeled
and solved using linear programming techniques.
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Network Flow Programming The term network flow program
describes a type of model that is a special case of the more
general linear program. The class of network flow programs includes
such problems as the transportation problem, the assignment
problem, the shortest path problem, the maximum flow problem, the
pure minimum cost flow problem, and the generalized minimum cost
flow problem. It is an important class because many aspects of
actual situations are readily recognized as networks and the
representation of the model is much more compact than the general
linear program. When a situation can be entirely modeled as a
network, very efficient algorithms exist for the solution of the
optimization problem, many times more efficient than linear
programming in the utilization of computer time and space
resources.
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Integer Programming Integer programming is concerned with
optimization problems in which some of the variables are required
to take on discrete values. Rather than allow a variable to assume
all real values in a given range, only predetermined discrete
values within the range are permitted. In most cases, these values
are the integers, giving rise to the name of this class of models.
Models with integer variables are very useful. Situations that
cannot be modeled by linear programming are easily handled by
integer programming. Primary among these involve binary decisions
such as yes-no, build-no build or invest-not invest. Although one
can model a binary decision in linear programming with a variable
that ranges between 0 and 1, there is nothing that keeps the
solution from obtaining a fractional value such as 0.5, hardly
acceptable to a decision maker. Integer programming requires such a
variable to be either 0 or 1, but not in-between.
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Nonlinear Programming When expressions defining the objective
function or constraints of an optimization model are not linear,
one has a nonlinear programming model. Again, the class of
situations appropriate for nonlinear programming is much larger
than the class for linear programming. Indeed it can be argued that
all linear expressions are really approximations for nonlinear
ones.
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Dynamic Programming Dynamic programming (DP) models are
represented in a different way than other mathematical programming
models. Rather than an objective function and constraints, a DP
model describes a process in terms of states, decisions,
transitions and returns. The process begins in some initial state
where a decision is made. The decision causes a transition to a new
state. Based on the starting state, ending state and decision a
return is realized. The process continues through a sequence of
states until finally a final state is reached. The problem is to
find the sequence that maximizes the total return.
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Stochastic Programming The mathematical programming models,
such as linear programming, network flow programming and integer
programming generally neglect the effects of uncertainty and assume
that the results of decisions are predictable and deterministic.
This abstraction of reality allows large and complex decision
problems to be modeled and solved using powerful computational
methods.
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Combinatorial Optimization The most general type of
optimization problem and one that is applicable to most spreadsheet
models is the combinatorial optimization problem. Many spreadsheet
models contain variables and compute measures of effectiveness. The
spreadsheet user often changes the variables in an unstructured way
to look for the solution that obtains the greatest or least of the
measure. In the words of OR, the analyst is searching for the
solution that optimizes an objective function, the measure of
effectiveness. Combinatorial optimization provides tools for
automating the search for good solutions and can be of great value
for spreadsheet applications.
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Stochastic Processes In many practical situations the
attributes of a system randomly change over time. Examples include
the number of customers in a checkout line, congestion on a
highway, the number of items in a warehouse, and the price of a
financial security, to name a few. When aspects of the process are
governed by probability theory, we have a stochastic process. The
example for this section is an Automated Teller Machine (ATM)
system and the state is the number of customers at or waiting for
the machine
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Discrete Time Markov Chains Say a system is observed at regular
intervals such as every day or every week. Then the stochastic
process can be described by a matrix which gives the probabilities
of moving to each state from every other state in one time
interval. Assuming this matrix is unchanging with time, the process
is called a Discrete Time Markov Chain (DTMC). Computational
techniques are available to compute a variety of system measures
that can be used to analyze and evaluate a DTMC model. This section
illustrates how to construct a model of this type and the measures
that are available.
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Continuous Time Markov Chains Here we consider a continuous
time stochastic process in which the duration of all state changing
activities are exponentially distributed. Time is a continuous
parameter. The process satisfies the Markovian property and is
called a Continuous Time Markov Chain (CTMC). The process is
entirely described by a matrix showing the rate of transition from
each state to every other state. The rates are the parameters of
the associated exponential distributions. The analytical results
are very similar to those of a DTMC. The ATM example is continued
with illustrations of the elements of the model and the statistical
measures that can be obtained from it.
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Terminology OPERATIONS The activities carried out in an
organization related to attaining its goals and objectives.
RESEARCH The process of observation and testing characterized by
the scientific method. The steps of the process include observing
the situation and formulating a problem statement, constructing a
mathematical model, hypothesizing that the model represents the
important aspects of the situation, and validating the model
through experimentation.
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ORGANIZATION The society in which the problem arises or for
which the solution is important. The organization may be a
corporation, a branch of government, a department within a firm, a
group of employees, or perhaps even a household or individual.
DECISION MAKER An individual or group in the organization capable
of proposing and implementing necessary actions.
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ANALYST An individual called upon to aid the decision maker in
the problem solving process. The analyst typically has special
skills in modeling, mathematics, data gathering, and computer
implementation. TEAM A group of individuals bringing various skills
and viewpoints to a problem. Historically, operations research has
used the team approach in order that the solution not be limited by
past experience or too narrow a focus. A team also provides the
collection of specialized skills that are rarely found in a single
individual. MODEL An abstract representation of reality. As used
here, a representation of a decision problem related to the
operations of the organization. The model is usually presented in
mathematical terms and includes a statement of the assumptions used
in the functional relationships. Models can also be physical,
narrative, or a set of rules embodied in a computer program.
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SYSTEMS APPROACH An approach to analysis that attempts to
ascertain and include the broad implications of decisions for the
organization. Both quantitative and qualitative factors are
included in the analysis. OPTIMAL SOLUTION A solution to the model
that optimizes (maximizes or minimizes) some objective measure of
merit over all feasible solutions -- the best solution amongst all
alternatives given the organizational, physical and technological
constraints. OPERATIONS RESEARCH TECHNIQUES A collection of general
mathematical models, analytical procedures, and optimization
algorithms that have been found useful in quantitative studies.
These include linear programming, integer programming, network
programming, nonlinear programming, dynamic programming,
statistical analysis, probability theory, queuing theory,
stochastic processes, simulation, inventory theory, reliability,
decision analysis, and others. Operations research professionals
have created some of these fields while others derive from allied
disciplines.
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The Operations Research Process Recognize the Problem Formulate
the Problem Construct a Model Find a Solution Establish the
Procedure Implement the Solution The OR Process
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Recognize the Problem Decision making begins with a situation
in which a problem is recognized. The problem may be actual or
abstract, it may involve current operations or proposed expansions
or contractions due to expected market shifts, it may become
apparent through consumer complaints or through employee
suggestions, it may be a conscious effort to improve efficiency or
a response to an unexpected crisis. It is impossible to
circumscribe the breadth of circumstances that might be appropriate
for this discussion, for indeed problem situations that are
amenable to objective analysis arise in every area of human
activity
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Formulate the Problem The first analytical step of the solution
process is to formulate the problem in more precise terms. At the
formulation stage, statements of objectives, constraints on
solutions, appropriate assumptions, descriptions of processes, data
requirements, alternatives for action and metrics for measuring
progress are introduced. Because of the ambiguity of the perceived
situation, the process of formulating the problem is extremely
important. The analyst is usually not the decision maker and may
not be part of the organization, so care must be taken to get
agreement on the exact character of the problem to be solved from
those who perceive it. There is little value to either a poor
solution to a correctly formulated problem or a good solution to
one that has been incorrectly formulated.
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Construct a Model A mathematical model is a collection of
functional relationships by which allowable actions are delimited
and evaluated. Although the analyst would hope to study the broad
implications of the problem using a systems approach, a model
cannot include every aspect of a situation. A model is always an
abstraction that is, by necessity, simpler than the reality.
Elements that are irrelevant or unimportant to the problem are to
be ignored, hopefully leaving sufficient detail so that the
solution obtained with the model has value with regard to the
original problem
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Find a Solution The next step in the process is to solve the
model to obtain a solution to the problem. It is generally true
that the most powerful solution methods can be applied to the
simplest, or most abstract, model. Here tools available to the
analyst are used to obtain a solution to the mathematical model.
Some methods can prescribe optimal solutions while other only
evaluate candidates, thus requiring a trial and error approach to
finding an acceptable course of action. To carry out this task the
analyst must have a broad knowledge of available solution
methodologies.
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Establish the Procedure Once a solution is accepted a procedure
must be designed to retain control of the implementation effort.
Problems are usually ongoing rather than unique. Solutions are
implemented as procedures to be used repeatedly in an almost
automatic fashion under perhaps changing conditions. Control may be
achieved with a set of operating rules, a job description, laws or
regulations promulgated by a government body, or computer programs
that accept current data and prescribe actions.
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Implement the Solution A solution to a problem usually implies
changes for some individuals in the organization. Because
resistance to change is common, the implementation of solutions is
perhaps the most difficult part of a problem solving exercise. Some
say it is the most important part. Although not strictly the
responsibility of the analyst, the solution process itself can be
designed to smooth the way for implementation. The persons who are
likely to be affected by the changes brought about by a solution
should take part, or at least be consulted, during the various
stages involving problem formulation, solution testing, and the
establishment of the procedure
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The OR Process
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Combining the steps we obtain the complete OR process. In
practice, the process may not be well defined and the steps may not
be executed in a strict order. Rather there are many loops in the
process, with experimentation and observation at each step
suggesting modifications to decisions made earlier. The process
rarely terminates with all the loose ends tied up. Work continues
after a solution is proposed and implemented. Parameters and
conditions change over time requiring a constant review of the
solution and a continuing repetition of portions of the process. It
is particularly important to test the validity of the model and the
solution obtained. Are the computations being performed correctly?
Does the model have relevance to the original problem? Do the
assumptions used to obtain a tractable model render the solution
useless? These questions must be answered before the solution is
implemented in the field.