Business Decision-Making and Optimization Concepts

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Questions and Answers

What does the objective function $P = 115x_{1} + 90x_{2}$ represent in the context of the company's decision making?

  • The cost of materials for desk production
  • The total profit made from producing desks (correct)
  • The total amount of raw materials used
  • The number of desks produced

Which of the following is a correctly formulated constraint based on the available raw materials?

  • $10x_{1} + 20x_{2} ext{ } ewer than ext{ } 200$ (correct)
  • $15x_{1} + 20x_{2} ext{ } ewer than ext{ } 128$
  • $16x_{1} + 4x_{2} ext{ } ewer than ext{ } 128$
  • $20x_{1} + 15x_{2} ext{ } ewer than ext{ } 220$

In the context provided, what do the variables $x_{1}$ and $x_{2}$ specifically represent?

  • The quantities of wood and metal used
  • The total material availability
  • The profit margins of red and blue desks
  • The number of desks produced of each type (correct)

Which option accurately describes the 'Feasible Region'?

<p>The graphical representation of all possible solutions satisfying the constraints (C)</p> Signup and view all the answers

How is the maximum profit achieved in the given scenario?

<p>By finding the intersection of all constraints (D)</p> Signup and view all the answers

What characterizes routine decisions?

<p>They utilize standard decision procedures for common scenarios. (D)</p> Signup and view all the answers

What does bounded rationality imply in decision-making?

<p>Choices are made based on limited knowledge and choosing a 'good enough' solution. (D)</p> Signup and view all the answers

Under which circumstances should decisions typically be made?

<p>When they are considered important and cannot be reasonably delegated. (A)</p> Signup and view all the answers

What distinguishes non-routine decisions from routine decisions?

<p>Non-routine decisions are characterized by high uncertainty and non-recurring situations. (A)</p> Signup and view all the answers

In the context of objective rationality, which of the following assumptions is made?

<p>There is a perfect anticipation of the value of consequences. (A)</p> Signup and view all the answers

What do decision trees begin with in their structure?

<p>A single decision node (D)</p> Signup and view all the answers

Which method would you use to choose an alternative with the highest possible outcome?

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

What is the expected value of Alternative Y based on the provided probabilities?

<p>$4000 (D)</p> Signup and view all the answers

How does queuing theory assist businesses?

<p>By minimizing costs related to staffing levels (A)</p> Signup and view all the answers

What is indicated by the variance in expected outcomes during risk analysis?

<p>The level of risk associated with an alternative (A)</p> Signup and view all the answers

In decision making under uncertainty, what does Hurwicz's criterion involve?

<p>Assuming an optimism factor then maximizing a weighted outcome of best and worst results (B)</p> Signup and view all the answers

What is a characteristic of decision-making strategies like Maximin and Insufficient Reason?

<p>They aim to minimize potential losses (D)</p> Signup and view all the answers

What does the expected value of both alternatives X and Y indicate about their cash flow potential?

<p>They generate equal expected returns (D)</p> Signup and view all the answers

What is the formula used to calculate the Expected Value (Eᵢ)?

<p>Eᵢ = Σ (Pᵢ * Oᵢⱼ) where i runs from 1 to m. (B)</p> Signup and view all the answers

In the context of decision-making under risk, which of the following statements is true?

<p>The best alternative is determined by the highest expected value. (C)</p> Signup and view all the answers

What does the concept of Queuing Theory primarily aim to achieve?

<p>Minimize costs by determining the optimal number of servers. (D)</p> Signup and view all the answers

In the given example of product production decisions, which alternative offers the highest expected value?

<p>Producing 1000 tons. (D)</p> Signup and view all the answers

What is a common feature of a decision tree?

<p>It starts at a single decision node and radiates out with alternatives. (A)</p> Signup and view all the answers

Which alternative reflects the situation where no one likes the product?

<p>P = 0.35; Income = 0. (A)</p> Signup and view all the answers

What would be a potential outcome of constructing a mathematical model in simulation?

<p>It helps identify the best alternative based on real-world data. (B)</p> Signup and view all the answers

Which statement about the states of nature and their probabilities is true?

<p>Probabilities assigned to states of nature can lead to unexpected outcomes. (A)</p> Signup and view all the answers

In Example 2, which of the following represents the constraints of the maximization problem?

<p>x₁ + 4x₂ ≥ 8 (B), 2x₁ + x₂ ≥ 7 (D)</p> Signup and view all the answers

What is the outcome when an 'Empty Feasible Region' is identified?

<p>No feasible solutions satisfy all constraints. (C)</p> Signup and view all the answers

Which method can be used to find the optimal solution in linear programming?

<p>Evaluating the objective function at extreme points (A)</p> Signup and view all the answers

What is a characteristic of unbounded solutions in linear programming?

<p>The objective function can increase indefinitely. (A)</p> Signup and view all the answers

When using graphical methods for linear programming, what is indicated by the feasible region?

<p>All possible solutions that do not violate the constraints. (A)</p> Signup and view all the answers

Which statement is true about the objective function in linear programming?

<p>It can be either maximized or minimized based on requirements. (C)</p> Signup and view all the answers

What does the presence of multiple constraint lines in a graphical representation suggest?

<p>A feasible region may form with several potential solutions. (B)</p> Signup and view all the answers

In which scenario does the optimal solution lie within the feasible region?

<p>At an extreme point of the feasible region. (C)</p> Signup and view all the answers

Flashcards

Decision Variables

The number of units of each product that will be produced. For example, in the desk problem, x1 represents the number of red desks and x2 represents the number of blue desks.

Objective Function

A mathematical expression that represents the quantity you want to maximize or minimize. In the desk example, the objective function is P = 115x1 + 90x2, where P is the profit.

Constraint Functions

Mathematical expressions that represent limitations or restrictions on the decision variables. In the desk problem, the constraints represent the limited resources available.

Feasible Region

The graphical representation of all possible combinations of decision variables that satisfy all constraint functions.

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Linear Programming

A linear programming technique that involves finding the optimal solution to a problem by manipulating a series of equations and inequalities. It helps determine the best possible outcome based on available resources and constraints.

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Constraints

A set of inequalities or equations that limit the values of variables in a linear programming problem.

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Optimal solution

A point within the feasible region that yields the optimal value for the objective function.

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Unbounded solution

A special case in linear programming where there is no limit to how high or low the objective function can be. The feasible region is unbounded.

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Mathematical approach

The process of finding the optimal solution to a linear programming problem by analyzing the objective function at all extreme points of the feasible region.

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Graphical approach

A visual method for solving linear programming problems by identifying the extreme point of the feasible region that yields the optimal value for the objective function.

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Extreme points

The corners or vertices of the feasible region, which represent possible solutions to the problem.

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Decision Making Under Risk

Decision making under risk involves selecting the best alternative when faced with multiple future possibilities, each with its own probability of occurrence. The best alternative is chosen based on the highest expected value calculated by considering the potential outcomes and their associated probabilities.

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Expected Value

An expected value represents the average outcome of a decision when all potential future states and their probabilities are taken into account. It is calculated by multiplying the value of each outcome by its probability and summing the results.

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Decision Tree

A decision tree visually represents the decision-making process under risk. It starts with a decision node branching out into different alternatives, each ending with a chance node representing various possible future states with associated probabilities and outcomes.

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Queuing Theory

Queuing theory examines the flow of customers or items through a system with limited resources, particularly focusing on minimizing delays and optimizing service utilization by determining the optimal number of servers.

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Simulation

Simulation analysis is a technique that uses a mathematical model to mimic a real-world situation with various inputs and parameters. It helps analyze different scenarios, predict outcomes, and determine the best alternative based on observed results.

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Resource Allocation Problem

The problem of optimizing resource allocation based on constraints and a specific objective, often involving a mathematical model with decision variables, an objective function, and constraint functions, which represent limitations on the available resources.

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Risk as Variance

The risk associated with an outcome is measured by the dispersion or spread of its possible values, represented by its variance.

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Maximax

A decision-making approach under uncertainty where the alternative with the highest potential outcome is chosen.

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Maximin

A decision-making approach under uncertainty where the alternative with the highest worst-case outcome is chosen.

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Insufficient Reason

A decision-making approach where equal probabilities are assigned to all possible states of nature and the expected value is calculated.

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Objective Rationality

A decision-making process that focuses on making the best possible choice, taking into account all possible alternatives and their consequences.

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Bounded Rationality

A decision-making approach that recognizes the limits of our knowledge and opts for the 'good enough' solution rather than the 'perfect' one.

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Decision Making under Certainty

A decision-making situation where the outcomes of each choice are known with certainty.

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Decision Making under Uncertainty

A decision-making situation where the outcomes of each choice are unknown and unpredictable.

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Study Notes

Linear Programming

  • Linear programming is used to maximize or minimize a linear function, subject to linear constraints.
  • The function to be optimized is called the objective function.
  • Constraints are limitations on the resources or other factors.
  • Decision variables represent the unknowns to be determined.
  • Constraints can be inequalities or equalities.
  • The feasible region is the set of all points that satisfy all the constraints of a linear programming problem.
  • The optimal solution is the point within the feasible region that gives the best value for the objective function.

Example

  • A company makes two types of desks: red and blue.
  • The company has limited resources (wood, metal, and plastic).
  • The company wants to maximize profit.
  • Variables: $x_1$ = number of red desks and $x_2$ = number of blue desks
  • Objective function (profit): $P = 115x_1 + 90x_2$
  • Constraints:
    • $10x_1 + 20x_2 \le 200$ (wood)
    • $4x_1 + 16x_2 \le 128$ (metal)
    • $15x_1 + 10x_2 \le 220$ (plastic)
    • $x_1, x_2 \ge 0$ (non-negative)

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