# OPERATION RESEARCH

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OPERATION RESEARCH

DR.BAMBANG SUDARYANA MSI DEAOPERATION RESEARCH1POKOK BAHASANMata 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.

LITERATURRichard, dkk (2000), Pengambilan Keputusan Secara Kuantitatif, Rajawali Pers JakartaSubagyo. P, dkk (1983). Dasar-Dasar Operation Research. BPFE YogyakartaSupranto. J (1988), Riset Operasi Untuk Pengambilan Keputusan, UI Press, Jakarta

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.Linear ProgrammingA 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.Network Flow ProgrammingThe 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.Integer ProgrammingInteger 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.Nonlinear ProgrammingWhen 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.

Dynamic ProgrammingDynamic 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.

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.

Combinatorial OptimizationThe 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. 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 machineDiscrete Time Markov ChainsSay 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.Continuous Time Markov ChainsHere 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.TerminologyOPERATIONS 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.

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.

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.

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.

The Operations Research ProcessRecognize the ProblemFormulate the ProblemConstruct a ModelFind a SolutionEstablish the Procedure Implement the Solution The OR Process

Recognize the ProblemDecision 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

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.

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

Find a SolutionThe 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.

Establish the ProcedureOnce 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.

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

25The OR Process

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.