# 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-1xA=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: Σj xij ≤ ai , i=1,…,m (The total amount shipped from a supplier cannot exceed its capacity)
• Demanders' Constraints: Σi xij ≥ bj , j=1,…,n (The total amount received by a demander must meet its demand)
• Non-Negativity: xij ≥ 0 (Shipments cannot be negative)

Criterion: The objective is typically to minimize the total transportation cost: Σi Σj cij.xij → 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.