Mathematical Model Flashcards
(21 cards)
What are the basic types of equations studied under linear algebra in this context?
Linear Equations: Equations of the first degree (e.g., ax + b = 0).
Quadratic Equations: Second-degree equations (e.g., ax² + bx + c = 0).
Simultaneous Equations: Two or more equations solved together for common unknowns.
Which methods are used to formulate and solve equations?
Elimination Method: Cancel out variables by adding or subtracting equations.
Substitution Method: Solve one equation for a variable and substitute it into another.
Matrix Methods: Use matrices and operations such as inversion or Gaussian elimination to solve linear systems.
What are the key cost and revenue concepts modeled using linear algebra?
Cost and Marginal Cost: Total cost function and the incremental cost of producing one additional unit.
Revenue and Marginal Revenue: Total revenue function and the extra revenue from selling one more unit.
Profit Analysis: Calculation of profit as revenue minus cost and analyzing changes using marginal figures.
How are production, cost, expenditure, sales, and profits analyzed as functions?
Functional Relationships: Expressed as equations relating output to inputs (time, price, etc.).
Time Analysis: Studying changes in production, costs, and profits over time.
Price Analysis: Graphical and algebraic methods to assess break-even points.
Graphical Approaches: Use of cost curves and break-even diagrams for interpretation
List four advantages of using control charts in quality control.
Provides real-time monitoring of processes.
Helps distinguish between common cause and assignable cause variations.
Aids in early detection of process shifts.
Facilitates continuous improvement in production or service processes.
What are four disadvantages or limitations of control charts?
May require significant training to interpret correctly.
Can be resource-intensive to maintain.
Interpretation can be subjective if not standardized.
May not capture complex or non-normally distributed data behavior.
Which control charts are commonly used, and for what are they applied?
Sample Mean Charts: Monitor the average of subgroups.
Range Charts: Track the variability within subgroups.
Proportion Charts (p-Charts): Monitor the proportion of defective items in a sample.
Usage: Each type addresses different aspects of process quality.
What is the difference between common cause and assignable cause variations?
Common Cause: Natural, inherent variations present in every process.
Assignable Cause: Unusual variations that can be traced to specific factors (e.g., machine malfunction).
Interpretation: Control charts help identify which type of variation is affecting a process.
What are the key elements of a linear programming problem?
Problem Model: A mathematical representation of a real-world scenario.
Objective Function: The function to maximize or minimize (e.g., profit, cost).
Constraints: Limitations or conditions (e.g., resource availability, time).
Feasible Region: The set of all possible solutions satisfying the constraints.
List four advantages and four limitations of linear programming.
(Advantages):
Optimizes resource allocation.
Provides a clear decision framework.
Handles multiple constraints simultaneously.
Offers visual insight via graphical methods.
A (Limitations):
Requires all relationships to be linear.
May oversimplify complex real-world scenarios.
Assumes certainty in data and conditions.
Can become computationally intensive for large problems.
What are the two main methods used to solve linear programming problems?
Graphical Method: Used for problems with two decision variables; visually identifies the feasible region and optimal point.
Simplex Method: An algorithmic approach for higher-dimensional problems.
Additional Concepts: Includes understanding primal and dual problems and comparing their solutions.
What are the essential elements of decision theory in this context?
A:
State of Nature: Possible scenarios that affect decisions.
Event: An occurrence or outcome within the state of nature.
Decision Alternatives: The different courses of action available.
Payoff: The benefits or costs associated with each alternative.
What types of decision making are covered under decision theory?
Decision making under certainty (known outcomes).
Decision making under uncertainty (unknown probabilities).
Decision making under risk (known probabilities).
Decision making in conflict situations (competitive or adversarial environments).
Describe the use of payoff tables and decision trees in decision making.
Payoff Tables: Arrange possible outcomes and payoffs in a matrix to compare decision alternatives.
Decision Trees: Diagrammatically represent decisions, chance events, and outcomes in a tree structure, aiding in sequential decision making.
Application: They help identify optimal strategies and visualize complex decision processes.
What rules or strategies are commonly applied in decision theory?
Bayes Decision Rule: Involves updating probabilities based on new information.
Expected Payoff: Calculation of average payoff weighted by probability.
Expected Loss & Opportunity Loss: Analysis of potential regrets or missed opportunities.
Strategies: Use of maximin, maximax, and minimax regret rules to choose among alternatives.
What is queuing theory, and why is it important?
Definition: A mathematical study of waiting lines or queues.
Purpose: Optimizes service processes and resource allocation by analyzing queue behavior.
Components: Involves arrival rates, service rates, and queue discipline.
Application: Widely used in customer service, manufacturing, and network traffic management.
What are the fundamental concepts in game theory?
Game Theory: The study of strategic interactions among rational decision makers.
Strategy: A planned set of actions a player takes.
Payoff: The reward or loss resulting from a strategic decision.
Optimum Game: A game where an optimal (or saddle point) solution exists.
What are two-person and zero-sum games?
Two-Person Games: Involve two players whose decisions affect each other.
Zero-Sum Games: One player’s gain is exactly balanced by the other’s loss.
Application: Useful for modeling competitive situations such as market rivalries.
Analysis Techniques: Include finding saddle points, dominance strategies, and equilibrium outcomes
What is network analysis in the context of mathematical modeling
Definition: A method for planning and controlling complex projects by mapping out activities and their dependencies.
Purpose: Identifies the most critical tasks (the critical path) and calculates total project duration.
Usage: Helps in project scheduling and resource allocation.
Visualization: Often uses network diagrams and Gantt charts.
What are key terms in network analysis?
Network: A diagram that represents activities and their interconnections.
Dummy Activity: A placeholder activity showing logical dependencies without consuming time or resources.
Critical Path: The longest path through the network that determines the shortest possible project duration.
Float (Slack): The amount of time an activity can be delayed without affecting the overall project time.
Cost Slope: The rate of change of project cost with time.
Activity vs. Event: Activities are tasks; events are milestones marking the end or beginning of activities.
