Introduction to Operations Research: Models and Methods

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: 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?

(Please provide the content for this question so I can complete this section.)

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.