# Introduction to Operations Research: Models and Methods

Classified in Mathematics

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## 1) What is an Inverse Matrix and How Do You Calculate It?

A matrix A^{-1} is called the inverse matrix of a matrix A (nxn) if AxA^{-1}= A^{-1}xA=E (where E is the identity unit matrix).

We calculate it by performing row operations on the augmented matrix (A | I) to transform it into (I | B). If this reduction is possible, then B=A^{-1}, which is the inverse matrix of A.

## 2) Define the Model of a Game

Games can be modeled in various forms:

**Tree Form Model (Game Tree):**Represents the game as a sequence of decisions (moves) made by players.**Normal Form Model:**Represents the game using:- List of players
- List of strategy spaces for each player
- List of payoff functions (decision matrix) defining outcomes for each combination of strategies.

**Characteristic Function Form:**Defines payoffs for all possible coalitions of players.

## 3) Define an Optimization Model

An **optimization model** is a type of mathematical model that attempts to optimize (maximize or minimize) an objective function without violating resource constraints. It is also known as **mathematical programming**. Optimization models include Linear Programming.

## 5) Describe the Main Goal of Transportation Models and Define the Transportation Model (Its Construction and Components)

**Goal:** Transportation models aim to plan the optimal distribution of goods and services from multiple supply locations to multiple demand locations.

**Model Construction:** The transportation model assumes that the quantity of goods at each location is limited. It uses the following constraints:

**Suppliers' Constraints:**Σ_{j}x_{ij}≤ a_{i}, i=1,…,m (The total amount shipped from a supplier cannot exceed its capacity)**Demanders' Constraints:**Σ_{i}x_{ij}≥ b_{j}, j=1,…,n (The total amount received by a demander must meet its demand)**Non-Negativity:**x_{ij}≥ 0 (Shipments cannot be negative)

**Criterion:** The objective is typically to minimize the total transportation cost: Σ_{i }Σ_{j} c_{ij}.x_{ij} → MIN

**Components:**

**Suppliers:**Origins of the goods**Demanders:**Destinations for the goods**Routes:**Connections between suppliers and demanders**Transport Cost:**Cost per unit of goods shipped on each route**Units Shipped:**Decision variables representing the quantity shipped on each route

## 6) Describe the Main Goal of Linear Optimization Models and Define These Models (Their Components)

**Goal:** Linear programming (LP), also called linear optimization, aims to determine the best possible outcome or solution (e.g., maximum profit or lowest cost) given a set of constraints represented as linear relationships.

**Components:**

**Decision Variables:**Quantities to be determined**Objective Function:**Mathematical expression to be maximized or minimized**Constraints:**Linear equations or inequalities representing limitations**Data:**Coefficients and constants in the objective function and constraints

## 7) For Which Types of Models is the Simple Additive Weighting (SAW) Method Used, and Describe the Steps?

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## 8) Define Game Models and Decision Models

**Game Model:**

- Represents situations of conflict or competition between intelligent, rational players (or sometimes against non-intelligent, irrational entities like nature).
- Aims to find the optimal strategy for a player in the game.
- Uses game tree, normal form, or characteristic function form for representation.

**Decision Model:**

- Represents a decision-making problem with various alternatives and uncertain outcomes.
- Helps decision-makers analyze potential consequences and choose the best course of action.
**Elements:**- Decision alternatives
- States of nature (uncertain events)
- Decision matrix (payoffs for each alternative-event combination)
- Decision criterion (e.g., maximizing expected value, minimizing risk)
- Decision environment (certainty, risk, or uncertainty)

- Can be computer-based systems that predict outcomes based on chosen actions.