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Questions and Answers

What does the variable 'z' represent in the given model?

  • Perimeter of the rectangle
  • Length of the rectangle
  • Area of the rectangle (correct)
  • Width of the rectangle

Which of the following describes a feasible solution in the OR model?

  • A solution that maximizes the objective function only
  • A solution that has the maximum possible values
  • A solution that satisfies all constraints (correct)
  • A solution that minimizes the objective function only

When is a solution considered optimal in an OR model?

  • When it is feasible and has the highest return
  • When it satisfies constraints but not the objective function
  • When it maximizes or minimizes the objective function and is feasible (correct)
  • When it meets all resource allocation requirements

What type of problem is described when resources are allocated to maximize returns?

<p>Maximization problem (D)</p> Signup and view all the answers

In linear programming, what is the transportation model used for?

<p>Minimizing transportation costs while distributing commodities (B)</p> Signup and view all the answers

Which of the following is not a characteristic of the linear programming model?

<p>Always yields an integer solution (A)</p> Signup and view all the answers

What best describes a minimization problem in the context of resource allocation?

<p>Minimizing costs while meeting specified constraints (D)</p> Signup and view all the answers

When multiple factories manufacture the same commodity in different capacities, which model is appropriate for optimal distribution?

<p>Transportation model (B)</p> Signup and view all the answers

What is the objective function representing the total cost of purchasing the tonics?

<p>minimize z = 50x1 + 30x2 (C)</p> Signup and view all the answers

Which constraint correctly represents the minimum daily requirement for Vitamin A?

<p>2x1 + 4x2 ≥ 40 (A)</p> Signup and view all the answers

In the context of this problem, what do x1 and x2 represent?

<p>The quantities of tonics purchased (C)</p> Signup and view all the answers

What is the correct constraint representing the minimum daily requirement for Vitamin D?

<p>3x1 + 2x2 ≥ 50 (C)</p> Signup and view all the answers

Under what condition must x1 and x2 exist in this problem?

<p>x1, x2 ≥ 0 (D)</p> Signup and view all the answers

What caused the growth of the size and complexity of organizations since the industrial revolution?

<p>The growth in division of labour and segmentation of management (A)</p> Signup and view all the answers

What major issue arises from the growing autonomy of components within an organization?

<p>Components working at cross purposes (A)</p> Signup and view all the answers

Where can the origins of operations research (OR) be traced back to?

<p>Military services during World War II (A)</p> Signup and view all the answers

What urgent need did the military face during World War II that contributed to the development of OR?

<p>To allocate scarce resources effectively (B)</p> Signup and view all the answers

What is one of the main challenges that arises from increased complexity and specialization in organizations?

<p>Difficulty in resource allocation for overall effectiveness (A)</p> Signup and view all the answers

What is operations research primarily concerned with solving?

<p>Strategic and tactical organizational problems (A)</p> Signup and view all the answers

Which of the following was NOT a goal of the teams of scientists called upon during World War II?

<p>Conduct human behavior research (C)</p> Signup and view all the answers

How did the increasing specialization of organizations impact the alignment of their components?

<p>It often caused components to prioritize individual goals over collective objectives (D)</p> Signup and view all the answers

What information does an arc Arc(i, j) carry?

<p>Transportation cost and amount to be shipped (A)</p> Signup and view all the answers

What is the main objective of the transportation model described?

<p>Minimize the total transportation cost (C)</p> Signup and view all the answers

Which of the following represents a typical supply constraint in the transportation problem?

<p>x21 + x22 ≤ 50 (D)</p> Signup and view all the answers

In the objective function, what does the variable Z represent?

<p>Total cost of shipping electricity (B)</p> Signup and view all the answers

If the shipping cost from Plant 2 to City 1 is $9, which of the following is true?

<p>Plant 2 incurs a cost of $9 for each million kwh shipped to City 1 (C)</p> Signup and view all the answers

What does the variable xij represent in the context of the transportation model?

<p>Amount of electricity sent from plant i to city j (D)</p> Signup and view all the answers

What element is implicitly expressed in the transportation tableau?

<p>Both shipping costs and supply/demand constraints (C)</p> Signup and view all the answers

If the total demand for electricity to City 1 is 45 million kwh, which statement is accurate given the constraints?

<p>The total shipped to City 1 must equal 45 million kwh (B)</p> Signup and view all the answers

What is the starting basic solution Z calculated from the given data?

<p>1080 (B)</p> Signup and view all the answers

Which method is described as an improved version of the Least-Cost Method?

<p>Vogel Approximation Method (D)</p> Signup and view all the answers

How is the penalty calculated in the Vogel Approximation Method?

<p>The difference between the two smallest shipping costs in the row or column (B)</p> Signup and view all the answers

What action is taken when a row and a column are satisfied simultaneously in VAM?

<p>Only one is crossed out, and the other is assigned zero supply or demand (D)</p> Signup and view all the answers

What is the total supply from Plant 1 according to the information provided?

<p>35 million kwh (B)</p> Signup and view all the answers

What should be done if exactly one row or column with zero supply or demand remains uncrossed-out in VAM?

<p>Stop the process (A)</p> Signup and view all the answers

What is the maximum supply from Plant 3 based on the data given?

<p>30 million kwh (C)</p> Signup and view all the answers

What do you do after identifying the row or column with the largest penalty in VAM?

<p>Identify the basic variable with the smallest shipping cost (C)</p> Signup and view all the answers

What is the primary focus of the Graphical Method in LP problems?

<p>To visualize feasible regions for solutions (B)</p> Signup and view all the answers

Which method is used to find a Basic Feasible Solution for a Transportation Problem?

<p>Vogel Approximation Method (VAM) (A), Northwest Corner Method (NWC) (C)</p> Signup and view all the answers

What distinguishes the Assignment Model from the Transportation Problem?

<p>Assignment Model has equal supply and demand (A)</p> Signup and view all the answers

In Network Optimization Models, what is the purpose of the Minimum Spanning Tree Problem?

<p>To minimize the total cost connecting all nodes (D)</p> Signup and view all the answers

What is the key component of an Inventory Model's ABC Classification?

<p>Categorization of items based on value (A)</p> Signup and view all the answers

How does the Simplex Algorithm determine optimal solutions?

<p>By using corner points of the feasible region (A)</p> Signup and view all the answers

Which of the following best describes the purpose of PERT in project management?

<p>To define task dependencies and timing (D)</p> Signup and view all the answers

What role do Symbols and Notations play in Queuing Models?

<p>To standardize communication about queue parameters (D)</p> Signup and view all the answers

In the context of inventory systems, which method is used to balance order costs and carrying costs?

<p>Basic EOQ Model (A)</p> Signup and view all the answers

Which of the following best describes the focus of the Maximum Flow Problem?

<p>To maximize the flow through a network from a source to a sink (A)</p> Signup and view all the answers

What is a characteristic feature of the Least-Cost Method in transportation problems?

<p>It prioritizes lower shipping costs while meeting demand. (C)</p> Signup and view all the answers

Which term accurately describes the sequence of events and tasks in PERT/CPM models?

<p>Critical path calculations (C)</p> Signup and view all the answers

Which of the following best defines Basic Feasible Solution in the context of the Transportation Problem?

<p>A potential allocation of routes that satisfies demand (A)</p> Signup and view all the answers

Flashcards

Linear Programming (LP)

A mathematical technique used to find the optimal solution to a problem with linear relationships.

Graphical Method (LP)

A visual method to solve LP problems by plotting constraints, identifying feasible region, and finding the optimal solution.

Simplex Algorithm

An iterative method used to solve LP problems, often by creating a table.

Transportation Problem

A special type of LP problem dealing with shipping goods from origins to destinations, minimizing cost.

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Northwest Corner Rule

A method for finding a starting solution in the transportation problem by allocating to the top left (northwest) cell.

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Least-Cost Method

A method for finding a starting solution in the transportation problem by allocating to the cell with the lowest cost.

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Vogel Approximation Method (VAM)

A method for finding a starting solution in the transportation problem; identifies the largest cost difference among rows and columns.

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Assignment Problem

An optimization technique that assigns tasks to resources to minimize cost or maximize efficiency.

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Inventory Models

Mathematical models to manage inventory levels in a business.

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ABC Classification of Inventories

Category systems for prioritizing inventory management based on value.

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EOQ Model

A common inventory model that calculates the optimal quantity of goods to order.

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Network Optimization Models

Mathematical techniques for optimizing network-related processes.

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Shortest Path Problem

A network optimization problem to find the path with the shortest distance or time between two points.

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PERT/CPM Models

Project management models that analyze project completion time, critical paths, and resource constraints.

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Operations Research (OR)

A scientific approach to managing organizations by optimally allocating resources to achieve overall goals.

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Industrial Revolution

A period of major industrial change and growth, leading to bigger organizations.

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Division of Labour

Breaking down work into smaller, specialized tasks.

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Resource Allocation

Distributing limited resources among various activities effectively.

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Autonomous Empires

Independent sections of an organization that focus on their own goals, possibly at the expense of the overall organization

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World War II

Global war that influenced the development of Operations Research.

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Military Operations Research Teams

Early teams using scientific methods to solve military resource allocation problems.

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Scientific Approach

Method of problem-solving using data, analysis and experimentation rather than intuition or random guesswork

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Objective Function

The function that needs to be maximized or minimized in an optimization problem.

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Constraints

Restrictions that limit the solutions to an optimization problem.

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Feasible Solution

A solution that satisfies all the constraints of an optimization problem.

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Optimal Solution

The feasible solution that gives the best value of the objective function.

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Linear Programming

A technique used to solve optimization problems where the objective function and constraints are linear.

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Maximization Problem

An optimization problem where the objective is to find the highest possible value.

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Minimization Problem

An optimization problem where the objective is to find the lowest possible value.

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Decision Variables

The unknown quantities in a linear programming problem that represent the choices to be made. They are typically represented by letters like x1, x2, etc.

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Feasible Region

The set of all possible solutions that satisfy all the constraints of a linear programming problem. It is the area within the boundaries defined by the constraints.

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Transportation Tableau

A table that summarizes the key information in a transportation problem: supply at each source, demand at each destination, and the cost of transporting one unit between each source and destination.

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Decision Variable (xij)

The amount of goods shipped from source i to destination j. It's what we need to determine to find the optimal solution in a transportation problem.

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Objective Function (Minimize Z)

The mathematical expression that represents the total cost of transportation (to be minimized) in a transportation problem. It's a sum of costs for all shipping routes.

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Supply Constraint

A limitation on the amount of goods that can be shipped from a source. Each source has a maximum capacity.

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Demand Constraint

A restriction on the amount of goods that can be received at a destination. Each destination has a specific requirement.

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Transportation Cost (cij)

The cost of shipping one unit of goods from source i to destination j. This cost depends on the distance and other factors.

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Penalty

In VAM, the penalty is the difference between the two smallest shipping costs in a row or column. It represents the potential cost incurred by not allocating units to the cheapest cell.

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Basic Variable

A variable in a transportation problem that represents a specific amount of goods shipped from a particular source (plant) to a particular destination (city).

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Highest Possible Value

In VAM, when you identify a basic variable with the smallest shipping cost, you assign the maximum possible units to that cell, subject to supply and demand constraints.

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Cross-Out

After allocating the highest possible value to a basic variable, you cross out the corresponding row or column. This indicates that either the plant's supply or the city's demand is satisfied.

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Zero Supply or Demand

When a row or column is satisfied, it is crossed out. But if a row (or column) has zero supply (or demand) remaining, it can be crossed out without allocating further units.

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Uncrossed-Out

In VAM, if a row or column remains uncrossed-out, it indicates that either the plant still has supply or the city still has demand. You need to allocate units to these remaining locations.

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Study Notes

Operations Research Lecture Notes

  • Notes compiled by Jane Aduda, November 6, 2013

Contents

  • List of Figures (Page v)
  • List of Tables (Page viii)
  • Course Outline

Introduction

  • History of Operations Research (Page 3)
  • Nature of Operations Research (Page 4)
  • Operations Research Models (Page 6)
  • Solving the OR Model (Page 8)

Linear Programming

  • Basic Assumptions (Page 12)
  • Mathematical Formulation of a LP model (Page 13)
    • General Linear Programming Model (Page 14)
  • Resource Allocation Models (Page 15)
    • Maximization Problems (Page 15)
    • Minimization Problems (Page 18)

Solving LP Problems

  • Graphical Method (Page 22)
  • Simplex Computations (Page 27)
    • Algebraic Determination of Corner Points (Page 29)
    • Simplex Algorithm (Page 30)
  • Transportation Problem (Page 39)
    • Finding Basic Feasible Solution for Transportation Problem (Page 43)
    • Methods for balanced TP (Page 43)
    • Northwest Corner Method (NWC) (Page 44)
    • Least-Cost Method (Page 45)
    • Vogel Approximation Method (VAM) (Page 46)
    • Iterative Computations of the Transportation Algorithm (Page 47)
    • Maximization using Transportation Algorithm (Page 47)
  • Assignment Model (Page 55)
    • Unbalanced assignment model (Page 63)
    • Maximization using Assignment algorithm (Page 66)

Inventory Models

  • Types of Inventory (Page 68)
  • ABC Classification of Inventories (Page 70)
    • A-Class Items (Page 71)
    • B-Class Items (Page 72)
    • C-Class Items (Page 72)
  • Lot/Order Size Model with no Shortages or Basic EOQ Model (Page 77)
  • Derivation of Basic EOQ Model (Page 77)

Network Optimization Models

  • Terminologies used in Networks(Page 81)
  • The Shortest Path Problem (Page 85)
  • The Minimum Spanning Tree Problem (Page 86)
  • The Maximum Flow Problem (Page 87)
  • The Minimum Cost Flow Problem (Page 89)

PERT/CPM Models for Project Management

  • Basic difference between PERT and CPM (Page 91)
  • PERT (Program Evaluation Review Technique) (Page 91)
  • CPM (Critical Path Method) (Page 91)
    • CPM Network Components & Precedence Relationship (Page 92)
  • Critical Path Calculations (Page 95)
  • Determination of the Critical Path (Page 96)
  • Project Management PERT(Page 98)

Waiting Line Theory or Queuing Model

  • Queuing System or Process (Page 101)
  • Input Process (Page 102)
  • Service Mechanism or Service Facility (Page 103)
  • Queuing Problems (Page 105)
  • Symbols used in Queuing Models (Page 106)
    • Notations (Page 107)

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