Mathematical theory of administration. operations research

The TGA received many contributions from mathematics in the form of mathematical models in order to provide solutions to business problems.

The mathematical theory applied to the solution of administrative problems is known as Operations Research (IO). The universally established IO designation is genetic and uncertain. Mathematical theory is not properly a school, like the theory of human relations, but rather a current found in several authors who emphasize the decision process and treat it in a logical and rational way through a quantitative, deterministic approach. and logical.

The main topics of operations management are:

1. Operations.- It focuses on productive processes and productivity, especially when globalization imposes world products.

   2. services.- These are service operations systems. Quality.- Involves the statistical treatment of quality, continuous improvement, total quality programs and ISO certification. Operations strategy.- Defines the strategic alignment and nature strategic management of operations.Technology.- The use of the computer in the administration of operations.

ORIGINS OF MATHEMATICAL THEORY IN ADMINISTRATION

Mathematical theory EMERGED in administrative theory from five causes:

1. The classical work on Game Theory by Von Neumann and Morgesnstem. (1947) and Wald (1954) and Savage (1954) for statistical decision theory The study of the decision process by Herbert Simon then a behaviorist author, and the emergence of Decision theories highlighted a greater importance to the decision that to the action derived from it in the organizational dynamics The existence of programmable decisions.- Simon had defined qualitative decisions (not programmable and made by man) and quantitative decisions (programmable and programmed by man) and qualitative decisions (not programmable and programmed for the machine). The computer provided means for the application and development of more complex and sophisticated mathematical techniques.Mathematical theory emerged with the use of operational research (OR) in the course of the Second World War.

The mathematical theory tried to create a science of the administration with logical and mathematical bases.

DECISION PROCESS

Mathematical theory dislocates the emphasis on action to locate it in the preceding decision. The decision process is your basic foundation. It constitutes the field of study of the theory of the decision that is considered here a doubling of the mathematical theory.

Decision making is studied from two perspectives, that of the process and that of the problem.

1. Perspective of the process. It focuses on the stages of decision making. Within this perspective, the objective is to select the best decision alternative. It approaches the decision process as a sequence of three simple stages:

1. Definition of the problem What are the possible alternative solutions to the problem What is the best alternative solution (choice)

His emphasis is on the search for alternative media. It is an approach criticized for being concerned with the procedure and not with the content of the decision.

2. Perspective of the Problem.- It is oriented towards solving problems.

From the perspective of the problem, the decision maker applies quantitative methods to transform the decision process as rationally as possible, concentrating on the definition and elaboration of the equation of the problem to be solved.

MATHEMATICAL MODELS IN ADMINISTRATION

Mathematical theory seeks to build mathematical models capable of simulating real situations in the company.

The model is the representation of something or the standard of something to be done.

In mathematical theory, the model was used as a simulation of future situations and evaluations of the probability of their happening.

A.- STRUCTURED PROBLEMS

A structured problem is one that can be perfectly defined since its main variables are known

The structured problem can be subdivided into three categories:

1. Decisions with certainty.- The variables and their consequences are deterministic.Decisions under risk.- The variables are known and the relationship between the consequence and the action is known in probabilistic terms.Decisions under uncertainty.- The variables are known, but the probabilities to evaluate the consequence of an action are unknown or are not determined with any degree of certainty.

B.- NON-STRUCTURED PROBLEMS

The unstructured problem cannot be clearly defined as one or more of its variables is unknown or cannot be determined with any degree of confidence. The mathematical model can treat structured and unstructured problems with advantages, because:

1. It allows to discover a better situation It discovers relationships of the problem It allows to treat the problem as a whole and to consider all the main variables simultaneously It is susceptible of expansion by stages and includes factors left in the verbal descriptions It uses techniques of objective and logical mathematics It leads to a solution safe and qualitative It allows immediate responses and on a gigantic scale through computers and electronic equipment.

C.- TYPES OF DECISION

Depending on structured and unstructured problems, decision-making techniques (programmed and unscheduled) work as follows:

OPERATIONS RESEARCH

The operations research branch (IO) comes from scientific management which added mathematical methods such as computer technology and a broader orientation.

The IO adopts the scientific method as a structure for solving problems with a strong emphasis on object judgment.

Definitions of OR vary from specific mathematical techniques to the scientific method itself. In general, these definitions include three basic aspects common to the IO approach to managerial decision making.

1. Systematic vision of the problems to be solved. Use of the scientific method in problem solving. Use of specific techniques of statistics, probability and mathematical models to help the decision maker to solve problems.

The IO approaches the analysis of operations of a system and not only as a particular problem, the IO uses:

1. The probability in the IO approach for decisions under conditions of risk and uncertainty. Statistics in data systematization and analysis to obtain solutions. Mathematics in the formulation of quantitative models.

IO is "the application of scientific methods, techniques and instruments to problems that involve the operations of a system, in order to provide those who control the system with optimal solutions for the problem in question."

Mathematics aims to transform the decision process in organizations into scientific, rational and logical.

The IO methodology uses six phases:

1. Formulate the problem.- With the analysis of the system and its objectives and action alternatives. Construct a mathematical model. to represent the system- The model expresses the system the system as a set of variables, of which one by one at least, is subject to control.Deduce a solution of the model.- The optimal solution of a model by means of the prosees analytical or numerical process Test the model and the model solution - Build the model that represents reality and that must be able to accurately predict the effect of changes in the system and the overall efficiency of the system Establish control over the solution.- the solution of a model will be adequate as long as the uncontrolled variables retain their values ​​and the relationships between the variables remain constant.Put the solution into operation (deployment). The solution needs to be tested and transformed into a series of operational processes.

The main IO techniques are:

• Game theory Graph theory Linear programming Dynamic programming Statistical analysis and probability calculus.

1. Game theory

Game theory proposed by mathematicians Johann Von Neumann (1903-1957) and Oscar Morgenstern 1902-1962) proposes a mathematical formulation for the strategy and analysis of conflicts.

The conflict situation occurs when one player wins and another loses, as the targets in the crosshairs are invisible, antagonistic and incompatible with each other.

The number of available Strategies is finite and therefore innumerable. Each strategy describes what will be done in any situation.

Game theory applies when:

1. The number of participants is finite Each participant has a finite number of possible courses of action. Each participant knows the courses of action. Each participant knows the courses of action available to the adversary, although they do not know what the course of action will be chosen by him. The two parties intervene each time and the game is "zero sum", that is to say, purely competitive, the benefits of one player are the losses of the other, and vice versa.

When the participants choose their respective courses of action, the outcome of the game will show the finite losses or gains, which are dependent on the chosen courses of action.

Game theory has its own terminology.

1. player.- Each participant involved.Match (or dispute). When each player chooses a course of action. Strategy.- Decision rule by which the player determines her course of action. The player does not always know the opponent's strategy. Mixed strategy.- When the player uses all her available courses of action in a fixed proportion. Pure strategy.- When the player uses only one course of action. Matrix.- It is the table The numbers in the matrix represent the values ​​won by the player. Negative values ​​translate losses.

2.- TAIL THEORY

The queue theory is the theory that takes care of choke points and waiting times, that is, of the delays observed in some service point.

In queuing theory the points of interest are: customer waiting time; the number of customers in the queue; and the ratio between the waiting time and the service provision time.

In a queue situation, there are the following components:

1. Customers or operations A passage or service point through which customers or operations must pass An entry process (imputa) A discipline on the queue A service organization

3.- GRAPHICS THEORY

Graph Theory is based on networks and arrow diagrams for various purposes. It offers planning and programming techniques through networks (APM, PERT, etc.) used in SS construction and industrial assembly activities. Both PERT (Rebién Technique Evaluation Program) and APM (Critical Path Method) are arrow diagrams that identify the critical path by establishing a direct relationship between time and cost factors, indicating the “economic optimum” of a project.

The Neopert is a simplified variation of the Pert, making it possible to save time in its preparation.

The networks or arrow diagrams are applied in projects that involve several operations and stages, several resources, different organs involved, deadlines and minimal costs.

The networks or arrow diagrams have the following advantages:

1. Execution of the project in the shortest time and at the lowest cost Allow the interrelation of the stages and operations of the project Optimal distribution of available resources and facilitate their redistribution in case of modifications Provide alternatives for project execution and facilitate decision making They identify “critical” tasks or operations that do not offer slack in time for their execution, and thus fully concentrate on them. The “critical” tasks or operations affect the term for the completion of the overall project. They define responsibility of the organs or people involved in the project.

4.- LINEAR PROGRAMMING

Linear programming (PL) is a mathematical technique that allows you to analyze production resources to maximize profits and minimize cost. It is a problem-solving technique that requires the definition of the values ​​of the variables involved in the decision to optimize an objective to be achieved within a set of limitations or restrictions, which constitute the rules of the game. Such problems involve resource allocation, linear relationships between the decision variables, objective to be achieved, and constraints.

The allocation problem involves situations such as scheduling production to maximize profits, mixing ingredients of a product to minimize costs, selecting an excellent portfolio of investments, assigning sales personnel in a territory, or defining an intermodal transport network with the lowest cost and faster.

The PL has characteristics such as:

1. Find the optimal position in relation to a target. The purpose is to minimize costs and maximize benefits based on the pre-established objective. It involves the choice between alternatives or a combination of those alternatives. It considers limits or restrictions that surround the decision. The variables must be quantifiable and have linear relationships with each other.

5.- DYNAMIC PROGRAMMING

Dynamic programming is applied in problems that have several interrelated stages, where a suitable decision for each of the stages must be adopted, without losing sight of the final objective. Only when the effect of each decision is evaluated is the final choice made.

6.- PROBABILITY AND STATISTICAL ANALYSIS

Statistical analysis is the mathematical method used to obtain the same information with the least amount of data. One of its best known applications is statistical quality control (CEQ) in the production area. Statistical methods allow the maximum amount of information to be produced from the available data.

The application of statistics to quality problems began with Malter A. Shewhart in the course of World War II.

a.- Statistical quality control

The initial idea was to apply statistical methodology in the quality inspection and reaching the assured quality in order to obtain conformity with the specifications and provide a high degree of reliability, durability and performance in the products.

Statistical quality control is based on techniques for determining the moment when tolerated errors in production begin to exceed tolerance limits, this is when corrective action becomes necessary.

Statistical quality control aims to locate deviations, errors, defects or failures in the production process, comparing performance with the established standard. This comparison can be made in three ways:

1. 100% quality control. Corresponds to total quality inspection. Total quality control (QC) is part of the production process and all products are inspected. Quality control by sampling. It is the one that is made by batches of samples collected for inspection. Sample control replaces total control as it does not interfere with the production process. If the sample is approved, the entire lot is approved. The sample is rejected, the whole lot must be inspected Random quality control. It is the probabilistic QC and consists of inspecting only a certain percentage of products or work in a random way.

b.- Total quality

JM Juran (born in 1904). He extended the quality concepts to the entire company with his full quality control.

While statistical quality control is applied only at the operational level, and preferably in the production and manufacturing area, total quality extends the concept of quality to the entire organization, from the operational level to the institutional level, covering the entire staff from the office and factory base as a whole.

The advantages of TQC are:

1. Reduction of waste Reduction of time cycles and results times Improvement of the quality of results (products or services).

Both constitute incremental approaches for excellence in the quality of products and processes, in addition to providing a formidable cost reduction.

ORGANIZATIONAL STRATEGY

Although mathematical theory has not been characterized by forays into organizational strategy, it was concerned with the typical competition of games, where the basic elements of strategic competition are as follows.

1. Ability to understand competitive behavior with a system in which competitors, customers, money, people and resources continuously interact Ability to use that understanding to predict how a given strategic move will alter the competitive balance Resources that can be permanently invested in new uses even if the consequent benefits only appear in the long term. Ability to foresee risks and profits with sufficient accuracy and certainty to justify the corresponding investment. Willingness to act.

THE NEED FOR PERFORMANCE INDICATORS

One of the greatest contributions of mathematical authors was the contribution of financial and non-financial indicators to measure or evaluate organizational performance or part of it, such as departmental, financial or accounting, business indicators, evaluation of human performance, and so on.

1.- Why measure?

Performance indicators are the vital signs of an organization as they allow to show what it does and what the results of its actions are.

The measurement system is a model of reality and can take various forms, such as periodic reports, graphs, or online information system, etc.

Setting up a performance measurement system generally follows a schedule.

The main advantages of a measurement system are:

1. Evaluate performance and indicate necessary corrective actions Support performance improvement Maintain convergence of purposes and coherence of efforts in the organization through the integration of strategies, actions and measurements.

2.- What to measure?

Organizations use measurement, evaluation and control in three main areas:

1. Results. That is, the concrete and final results that are intended to be achieved within a certain period, such as day, week, month or year. That is, the conduct or the instrumental means that it is intended to place in practice. Critical success factors. That is, the fundamental aspects for the organization to be very successful in its results or in its performance.

3.- Six-Sigma

Sigma is a measure of statistical variation. When applied to an organizational process, it refers to the frequency with which a certain operation or transaction uses more than the minimum resources to satisfy the customer.

The 6-sigma program uses several techniques in a step-by-step process method to achieve well-defined goals. The main difference is that with the 6-sigma since quality is not sought for quality, but it is intended to improve all the processes of an organization. In practice, 6-sigma differs from total quality in four basic areas:

1. Greater breadth of application. Most of TQM is applied within the product and manufacturing area and not in the project, finance, and so on. The 6-sigma is for the entire organization. Motorota fixes bulletins of cycle time, defect data and improvement goals in the dining rooms and bathrooms. Simpler implementation structure. The black belts are fully dedicated to change and are out of the everyday. Management rewards or punishes for business improvement. Deeper tools. In addition to the TQM tools, the 6-sigma goes deep to describe the current situation and foresee the future. There is a strong dose of applied statistics and a better understanding of how the processes behave, an auxiliary software and a map for the application of the tools.Application of tools allows to clarify problems and improve. Strong link with the health (financial) of business. The 6-sigma addresses the objectives of the company and certifies that all key areas for the future health of the company contain measurable measures with better measurements and detailed plans and application.

The 6-sigma seeks organizational effectiveness in three dimensions that must work together:

1. Waste reduction. Through the concept of exact entrepreneurship, without surpluses, only the essentials, or future time effort, or reduction of the time cycle or even elimination of what has no value for the customer, printing speed to the company. defects. It is the 6-sigma itself, Involvement of people. Through the so-called "human architecture".

4.- THE BALANCE SCORE CARD (BSC)

Measures and indicators significantly affect the behavior of people in organizations.

What an organization defines as an indicator is what will be obtained as results. The central point of the systems and measures traditionally used in organizations focuses purely on financial or quantitative aspects, and tries to control behaviors.

The BSC is a management method focused on organizational balance and is based on four basic perspectives, which are as follows:

1. Finance. Analyze the business from a financial point of view. This point involves financial and accounting indicators and measures that allow evaluating the organization's behavior against points such as profit, return on investments, value added to equity and other indicators that the organization adopts as relevant to its business. Customers. Analyze the business from the customer's point of view. It includes indicators and measures such as satisfaction, market share, trends, customer retention and acquisition of potential customers, as well as value added to products / services, market position, level of services added to the community by which customers indirectly contribute, etc. Internal processes. Analyze the business from the internal point of view of the organization.It includes indicators that guarantee the intrinsic quality of products and processes, innovation, creativity, reproduction capacity and optimization with demands, logistics and optimization of flows, as well as the quality of information, internal communication and interfaces. Learning / organizational growth. Analyze the business from the point of view of what is essential to reach the future successfully.Analyze the business from the point of view of what is essential to achieve the future successfully.Analyze the business from the point of view of what is essential to reach the future successfully.

These perspectives can be as many as the organization needs to choose depending on the nature of its business, purposes, style of action, and so on. The BSC seeks balanced strategies and actions in all areas that affect the business of the organization as a whole, allowing efforts to be directed towards the areas of greater competence and detecting and indicating the areas for eliminating incompetence.