Operations Research: A Comprehensive Guide for Engineers and Decision-Makers

Posted on Sep 9, 2024 in Other subjects

Operations Research, also known as Operations Research, is a branch of mathematics that involves using mathematical models, statistics, and algorithms to aid in decision-making processes. It often focuses on analyzing complex real-world systems to improve (or optimize) their operation.

Operations research allows for the analysis of decision-making while considering resource scarcity, aiming to determine how to optimize a defined objective, such as maximizing profit or minimizing cost.

History of Operations Research

The first activities in Operations Research took place during World War II in Britain. The Military Administration assembled a group of scientists from various fields, led by A.P. Rowe, to study strategic and tactical problems related to the country’s defense. They employed a technique known as radiolocation, which involved using radio waves to locate objects, and developed civilian scientists.

The name “Operations Research” was apparently coined because the team was conducting research on military operations.

Motivated by the encouraging results achieved by the British teams, U.S. military administrators began conducting similar investigations. They brought together a select group of specialists who achieved success in their studies, tackling complex logistical problems, planning minefields at sea, and effectively utilizing electronic equipment.

After the war, industrial managers, impressed by the positive results obtained by military strategists, began applying operations research tools to solve problems arising from the increasing size and complexity of industries.

While Britain is credited with initiating operations research as a new discipline, the United States quickly took the lead in this rapidly growing field.

A second factor contributing to the impressive progress of operations research was the development of the digital computer. Its remarkable computational speed, storage capacity, and information retrieval capabilities enabled decision-makers to make quick and accurate decisions. Without the digital computer, the major problems tackled by Operations Research would not have grown to their current level of complexity.

Characteristics of Operations Research

• Employs the scientific method to investigate the problem at hand. The process begins with careful observation and problem formulation, including the collection of relevant data.
• Adopts an organizational standpoint. It aims to resolve conflicts of interest among members of the organization so that the outcome benefits the entire organization.
• Strives to find the best possible solution (called the optimal solution) to the problem under consideration. Instead of simply improving the current state of affairs, the goal is to identify the most effective course of action.

• Requires a team approach. This team should include individuals with strong backgrounds in mathematics, statistics and probability theory, economics, business administration, computer science, engineering, etc.
• Has developed a series of highly useful techniques and models for systems engineering. These include Nonlinear Programming, Queuing Theory, Integer Programming, Dynamic Programming, among others.
• Tends to represent the problem in a way that allows for quantitative analysis and evaluation, a common approach.

Importance of Operations Research

Operations Research directly addresses problems related to the conduct and coordination of operations within an organization. Its field of work is vast, encompassing issues in telecommunications, transportation, construction, manufacturing, financial planning and management, healthcare, utilities, and many other areas. It focuses on solving all problems related to management, planning, and design.

Methodology of Operations Research

Most practical problems initially presented to an Operations Research team are vaguely described. Therefore, the first step is to study the relevant system and develop a well-defined abstract problem for analysis. This involves identifying appropriate objectives, restrictions on what can be done, the interrelationships of the area under study to other areas of the organization, the different possible courses of action, time limits for making a decision, and so on. This process of defining the problem is crucial because it significantly affects the relevance of the study’s findings.

The main steps in implementing Operations Research in practice include:

1. Problem Definition: This involves defining the scope of the problem under investigation. The end result is to identify three key elements of the decision problem:
   • Description of the decision alternatives.
   • Determination of the study’s purpose.
   • Specification of the constraints under which the modeled system operates.
2. Model Construction: This involves translating the problem definition into mathematical relationships. If the mathematical relationships are too complex for an analytical solution, you can choose to simplify the model and use a heuristic approach, or employ a simulation if an approximation is acceptable.
3. Model Solution: This is often the easiest step in Operations Research, as it involves using well-defined algorithms for optimization. An important aspect of the model solution phase is sensitivity analysis.
4. Model Validation: This involves checking if the proposed model performs as intended. The model is valid if, under similar data conditions, it reproduces past performance. However, there is generally no guarantee that future operations will continue to follow past data patterns.
5. Implementation: Implementing the solution of a validated model involves translating the results into operating instructions, presented in a format understandable to those who will operate the recommended system. The Operations Research team primarily bears the responsibility for this task.

Models of Operations Research

A decision model should be viewed as a tool for summarizing a decision problem in a way that allows for the identification and systematic evaluation of all available decision alternatives. After this evaluation, a decision is reached by selecting the alternative deemed best among all options.

• Transportation Model: This is a specialized type of linear programming related to transporting an item from its sources (plants) to its destinations (warehouses). The objective is to determine the transportation program that minimizes the total transportation cost while meeting supply and demand constraints.
• Mathematical Model: This is used when the objective function and constraints of the model can be expressed in quantitative or mathematical form as functions of the decision variables.
• Simulation Model: Simulation models differ from mathematical models in that the relationships between input and output are not explicitly shown. Instead, a simulation model divides the system into basic or elementary modules, which are then linked together through well-defined logical relationships. Calculations then move from one module to another until an output result is obtained.
• Allocation Model: The phrase “the best person for the job” aptly describes the allocation model. This is illustrated by allocating workers with different levels of training to various positions. A position that matches a worker’s skills costs less than one where the employee is not working optimally. The model’s aim is to determine the optimal allocation (minimum cost) of workers to jobs.

• Network Models: A wide range of situations can be modeled and solved using networks (nodes connected by branches) in Operations Research. Some possible applications of networks include:
  • Designing a marine pipeline. The model’s aim is to minimize the cost of constructing the pipeline.
  • Determining the shortest route between two cities on a road network.
  • Determining the maximum capacity (in tons) of a network of coal slurry pipelines.
  • Determining the minimum flow program from oil fields to refineries through a pipeline network.
  • Determining the schedule (start and end dates) of activities in a construction project.
• Queueing Models: These models study waiting lines (queues). While not strictly optimization techniques, they determine efficiency measures for waiting lines, such as average waiting time in the queue, average service time, and utilization of service facilities. Some examples of queueing models include:
  • Single-server queue.
  • Multi-server queue.
  • Queue with finite capacity.
• Inventory Models: An important factor in formulating and solving an inventory model is whether the demand for a commodity (per unit time) is deterministic (known with certainty) or probabilistic (can be described by a probability distribution).
• Forecast Models: When making decisions, plans are prepared for the future. Therefore, the data describing the decision’s status should represent what will happen in the future. For these cases, forecast models are necessary. For example, in inventory control, decisions are based on the nature of the controlled item’s demand over a specific planning horizon. Similarly, financial planning requires predicting the pattern of cash flow over time.

Decision Under Uncertainty

The difference between making decisions under low risk and making decisions under uncertainty is that in the case of uncertainty, the probability distribution for state s_j, j = 1, 2, …, n, is unknown or cannot be determined. This lack of information has led to the development of the following criteria for analyzing decision problems:

1. Maximin (Minimax)
2. Laplace
3. Savage
4. Hurwicz

These criteria differ in the degree of conservatism that the decision-maker exhibits when dealing with uncertainty.

• The Laplace Criterion: This criterion is based on the principle of insufficient reason. Since no probability distributions for states of nature (s_j) are known, there is no reason to believe that they are distinct. Therefore, alternatives are evaluated with the optimistic assumption that each state is equally likely to occur, meaning P(s₁) = P(s₂) = … = P(s_n) = 1/n. If the remuneration, represented as v(a_i, s_j), represents a gain, the best alternative is the one that provides:

  

  If v(a_i, s_j) represents a loss, then minimization replaces maximization.
• The Maximin (Minimax) Criterion: This criterion is based on a conservative attitude, focusing on the best possible conditions among the worst. If v(a_i, s_j) represents a loss, select the action corresponding to the minimax criterion:

  max min (v(a_i, s_j))

  If v(a_i, s_j) represents profit, using the maximin criterion, it is defined by:

  min max (v(a_i, s_j))

• The Hurwicz Criterion: This criterion is designed to reflect decision-making attitudes ranging from the most optimistic to the most pessimistic. Define 0 ≤ α ≤ 1 and assume that v(a_i, s_j) represents a gain. Then, the action you select must be associated with:

  max α max (v(a_i, s_j) + (1 – α) min v(a_i, s_j))

  The parameter α is the index of optimism. If α = 0, the criterion is conservative because it applies the standard minimax. If α = 1, the test produces optimistic results because it seeks the best of the best conditions. You can adjust the degree of optimism (or pessimism) by selecting an appropriate value of α in the interval (0,1). In the absence of a strong preference for optimism or pessimism, an appropriate choice would be α = 0.5.

  If v(a_i, s_j) represents a loss, then the criterion changes to:

  α min min (v(a_i, s_j) + (1 – α) max v(a_i, s_j))

Standards for Success in Operations Research (OR)

• The successful use of OR is a problem-solving approach rather than a collection of associated quantitative methods.

• It is relatively expensive, meaning it should not be used for all problems, but only for those where the potential profit outweighs the costs.

• To achieve appropriate use of OR, you must first understand the methodology for solving problems and the techniques involved in this form of solution. This knowledge is essential for knowing when to use or not use OR in different circumstances.

Formulation of the Model

Once we ensure that the problem definition has been built specifically and correctly, we proceed with the model formulation. The model, usually mathematical, must be formulated in a way that expresses the essence of the problem.

The mathematical model is based on equations and inequalities stated in terms of variables, which express the essence of the problem to be solved. These variables are defined in relation to the model of the problem.

After identifying the variables depending on the problem, we proceed to mathematically determine the two parts that constitute the model:

• The effectiveness measure, which allows us to know the level of achievement of the objectives, is generally a function called the objective function.
• The constraints of the problem, called constraints, are a set of equalities and inequalities that constitute barriers and obstacles to achieving the objective.

A mathematical model is an abstract idealization of a problem, which often leads to approximations and assumptions. We must ensure that the model is always a valid representation of the problem.

Model Validation

The validity of a model requires a high correlation between model predictions and reality. To achieve this, it is important to conduct a significant number of tests on the model and, if necessary, make relevant amendments. While model validation is included at the end of this document, the majority of model validation is done during the model construction stage.

Optimal Solution

Once the model has been tested and appropriate corrections have been applied, it is ready to start generating valid solutions. However, the real goal and purpose of operations research is to find the best solution to a problem. In the case of an economic problem, the objective function is to maximize performance at the lowest cost.

Limitations of Operations Research

• It is often necessary to make simplifications to the original problem to handle it and obtain a solution. Most models only consider a single objective, while organizations often have multiple objectives.

• There is a tendency to disregard all restrictions on a practical problem. This is because teaching and training methods often focus on applying this science to small problems for practical reasons. This can lead students to develop a simplistic and naive view of applying these techniques to real-world problems.

• Cost-benefit analysis of implementing solutions defined by Operations Research is rarely performed. Sometimes, the potential benefits are outweighed by the costs incurred in developing and implementing a model.

Importance of Operations Research (OR) to Systems Engineers

In an Operations Research team, adequate scientific and artistic research abilities are crucial. If one aspect is lacking, it is likely to hinder the effective use of operations research in practice.

• Operations Research in Systems Engineering is primarily used in the areas of coordinating operations and activities of the organization or system being analyzed. This is achieved through the use of models that describe the interactions between the components of this system and its environment.
• The “Research” in Operations Research refers to the use of a similar approach to the way research is conducted in established scientific fields. The “Operations” part is because it solves problems related to the conduct of operations within an organization.