Notes, summaries, assignments, exams, and problems for Mathematics

Genetic Algorithm Implementation Functions

The following Python functions demonstrate key components of Genetic Algorithms, including crossover, selection, and population initialization.

Four-Point Crossover Method

This function performs a four-point crossover using four parents to generate offspring.

def four_point_cross(parent1, parent2, parent3, parent4):
    pos_alea = sorted(list(np.random.choice(range(1, parent1), 4, replace=False)))
    hijo1 = parent1[:pos_alea[0]] + parent2[pos_alea[0]:pos_alea[1]], parent1[:pos_alea[1:2]], parent2[pos_alea[2]:pos_alea[3]], parent1[:pos_alea[3:]]
    hijo2 = parent2[:pos_alea[0]] + parent3[pos_alea[0]:pos_alea[1]], parent2[:pos_alea[1:2]], parent3[pos_alea[2]:pos_alea[3]], parent2[:pos_alea[3:]]
    hijo3

Saw-Tooth Diagram for Inventory Management

The Saw-Tooth Diagram is a design used for input control and monitoring of stock. This graph serves to manage inventories, ordering, and the volume of flow itself.

The optimal quantity is determined according to the purchase price, storage costs, and handling volume. The optimal lot (often related to the Economic Order Quantity or EOQ, here referred to as PDID) includes the purchase price, carrying costs, PDID costs, overhead costs, and costs incurred until the item reaches the point of sale or placement.

Henry Gantt and the Gantt Chart

Henry Gantt was an American engineer belonging to the classical school of organization who planned American military transport during World War I, using bar-line graphs.... Continue reading "Project Scheduling Diagrams: Gantt, CPM, PERT, and Inventory Control" »

Inventory: Key Products Waiting for Use

1. The STOCKS: key products in a situation of waiting for later use.

2. Types of Stock

Depending on the degree of transformation of materials

Depending on the degree of transformation of materials:

• Raw materials: Goods, parts or components found in the warehouse expected to be transformed into products suitable for final use or consumption.
• Work in progress (WIP): Materials in intermediate manufacturing stages that are undergoing transformation. These are ongoing products that will progress from raw material to finished product and usually require inspection or quality control before reaching the finished product stage.
• Semi-finished products: Manufactured materials used by the business that are not yet for

Least Squares Adjustment: Fundamental Models

1. Conditional Equations Model

This section outlines the formulation and solution for conditional equations in least squares adjustment.

• B: Jacobian matrix of the conditional equations with respect to observations.
• A: Vector representing the sum or constant terms of the conditional equations.
• P: Weight matrix of observations, calculated as 1 / σ₀².
• Q: Covariance matrix of observations, Q = P&supminus;¹.
• Qe: Covariance matrix of the conditional equations, Qe = B * Q * B&supT;.
• K: Vector of Lagrange multipliers, K = -Qe&supminus;¹ * A.
• V: Vector of residuals, V = Q * B&supT; * K.
• L̂: Adjusted observations, L̂ = V + Lobserved.

Linearized Conditional Observation Equation

A common form of... Continue reading "Least Squares Adjustment Models and Error Propagation" »

Quality Management Tools and Techniques

Brainstorming

Brainstorming is based on the ideas and contributions of individuals and is used to define the causes of problems, take interim measures, and achieve innovative ideas. This mode can be:

• Structured: Each person in the group gives their idea in turn (e.g., in a circle).
• Unstructured: Free flow, where each person gives their idea when they think of it.

Cause-Effect Diagram (Ishikawa Fishbone)

This diagram is used to identify and sort all the causes of a problem to specify the source and fix it. Its use is widespread and helps to:

• Identify the true causes.
• Summarize existing relationships.
• Promote process improvement and foster creativity.
• Gain a more global perspective.
• Equalize members and help analyze

Accounting: It uses information systems to register technical summaries, classify, and analyze transactions based on operations and along the timeline of the company. Accounting Steps: Identification and economic analysis of differences, quantifying the value of the operation, and booking entries to decide (journals, ledgers, and auxiliary) for the preparation of accounts, balance sheets, and analysis of balances. Characteristics of Accounting: Order, adequacy, relevance, objectivity, rigor, and standardization. Normalization: Accounting is understandable for all persons who need the information it contains. Standard Accounting Plan Types: General Types of Accounting: Microaccounting: Economic domestic, profit entities without profit motive,... Continue reading "Understanding Accounting Principles and Practices" »

Fundamentals of 2D Analytical Geometry

This document outlines key concepts in the analytical geometry of straight lines within a two-dimensional plane.

Coordinate Systems and Vector Representation

An affine coordinate system (SdR) is defined by a triple (O, u, U), where O is a fixed point in the Euclidean plane (E2) serving as the origin, and (u, U) forms a basis for the vector space R² (V2R). A point P in E2 is uniquely determined by its coordinates (x, y) relative to the chosen basis, which represent its position vector OP.

Points and Segments

The coordinates of the midpoint M of a segment AB are obtained by averaging the coordinates of its endpoints. If A = (x_A, y_A) and B = (x_B, y_B), then the midpoint M = ((x_A + x_B) / 2, (y_A + y_B) / 2).

Equations

Stock Break

The stock break occurs when existing stocks are insufficient to cover ordinary consumption needs. There are two variants:

• Low stock fulfilled: This indicates that the current stock has proven insufficient.
• Low stocks reached: Low stocks have reached the established minimum level.

In the first case, the low stock level is outdated, and the company must study the current situation and its needs to determine a new minimum stock level. In the second case, provisioning has been carried out, and it is necessary to determine why the established low levels were not respected.

Store managers often prepare monthly stock break reports containing the following data:

• Date of the report
• Report name and product code
• Product description
• Material stock unit

Data Visualization: Types and Applications

Graphics are a visual representation of data, prioritizing simplicity, easy interpretation, and adherence to standards.

Classification of Graphics

• Structural graphics: Representing a single set of data.
• Relational graphics: Connecting two sets of data.
• Special graphics.

Types of Graphics

• Organizational Charts: Represent the structure of an organization.
• Classification Plans: Represent elements of a total set, subdivided into smaller subsets. These can be square or pyramidal.
• Schedules: Use current statistics, ordering data over time to identify peaks and trends.
• Histograms: Represent class intervals or monthly values, highlighting extreme values.
• Function Tables: Establish mutual relationships between two sets

Microsoft Excel Spreadsheet Fundamentals

Microsoft Excel is an electronic spreadsheet that serves as a tool for performing various operations and mathematical calculations. A spreadsheet is an application or program that allows for data entry and processing. These sheets are also referred to as worksheets.

Core Components

• Rows: Horizontal lines identified by numbers ranging from 1 to 65,536.
• Columns: Vertical lines identified by letters running from A through IV (totaling 256 columns).
• Cells: The intersection of a row and a column, identified by the column letter and row number (e.g., A1, AB256, B25).
• Range: A group of cells extending along one or more rows or columns.
• Active Cell: The cell where the cursor is currently located, identified by dark