Mathematical Optimization for Business Problems Cognitive Class Exam Answers

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Introduction to Mathematical Optimization for Business Problems

Mathematical optimization, or mathematical programming, involves selecting the best solution from a set of feasible alternatives. It uses mathematical models to describe the problem and finds the optimal solution by systematically evaluating possible solutions.

Types of Optimization Problems:

1. Linear Programming (LP):
   • Involves optimizing a linear objective function subject to linear equality and inequality constraints.
   • Example: Resource allocation, production planning.
2. Integer Programming (IP):
   • Extends linear programming by adding constraints that require some or all variables to be integers (0, 1, 2,…).
   • Example: Capital budgeting, job scheduling.
3. Mixed-Integer Programming (MIP):
   • Combines discrete (integer) and continuous variables in optimization problems.
   • Example: Facility location, portfolio optimization.
4. Nonlinear Programming (NLP):
   • Deals with optimizing objectives and constraints that are nonlinear functions.
   • Example: Supply chain optimization with nonlinear costs or demands.
5. Convex Optimization:
   • Focuses on optimizing convex functions over convex sets.
   • Example: Portfolio selection, risk management.

Steps in Mathematical Optimization:

1. Formulating the Problem:
   • Define the decision variables, objective function (what to maximize or minimize), and constraints (limitations or requirements).
2. Choosing the Optimization Method:
   • Select an appropriate optimization technique based on the problem structure (LP, IP, NLP, etc.).
3. Implementing the Model:
   • Use specialized software (e.g., MATLAB, Python libraries like scipy.optimize, commercial solvers like CPLEX or Gurobi) to set up and solve the optimization model.
4. Interpreting Results:
   • Analyze the optimal solution to make informed business decisions.

Applications in Business:

• Supply Chain Management: Optimize inventory levels, distribution routes.
• Finance: Portfolio optimization, risk management.
• Marketing: Resource allocation in advertising campaigns.
• Operations: Production planning, scheduling.
• Logistics: Vehicle routing, facility location.

Challenges:

• Complexity: Large-scale problems can be computationally intensive.
• Data Quality: Optimization models rely on accurate input data.
• Modeling Assumptions: Simplifications may affect real-world applicability.

Conclusion:

Mathematical optimization provides powerful tools for businesses to improve efficiency, reduce costs, and make data-driven decisions. By understanding the problem structure and leveraging appropriate techniques, businesses can gain competitive advantages in various domains.

Mathematical Optimization for Business Problems Cognitive Class Certification Answers

Module 1 – The Big Picture Quiz Answers

Question 1: True or false? Constraint Programming is particularly useful for solving scheduling problems and certain combinatorial optimization problems.

• False
• True

Question 2: True or false? A feasible solution can be, but is not guaranteed to be, an optimal solution.

• False
• True

Question 3: True or false? Objective functions always start with the words “maximize” or “minimize”.

• False
• True

Module 2 – Linear Programming Quiz Answers

Question 1: True or false? The following constraint is valid for a linear programming problem, where x and y are variables and z is a data item: 2x + 3y less than or equal to z²

• False
• True

Question 2: True or false? Hard constraints can be converted to soft constraints to help resolve infeasibilities.

• False
• True

Question 3: True or false? If a constraint is non-binding, its dual price will be zero.

• False
• True

Module 3 – Network Models Quiz Answers

Question 1: True or false? In a transportation problem, if all the capacities and demands are integer, then you can declare the variables to be continuous even though they are integer.

• False
• True

Question 2: True or false? The critical path is the shortest path in the network.

• False
• True

Question 3: True or false? A sequence of arcs connecting two nodes is called a path.

• False
• True

Module 4 – Beyond Simple LP Quiz Answers

Question 1: True or false? A piecewise linear function can be used to approximate convex nonlinear functions.

• False
• True

Question 2: True or false? The branch and bound method begins with LP relaxation.

• False
• True

Question 3: True or false? Mixed-integer programming is often used for investment planning.

• False
• True

Module 5 – Modelling Practice Quiz Answers

Question 1: True or false? Data sparsity can be exploited to create only the essential variables and constraints, thus reducing memory requirements.

• False
• True

Question 2: True or false? It is important to always use integer variables when a model involves the production of whole items.

• False
• True

Question 3: True or false? It is important to use penalties only when absolutely necessary.

• False
• True

Mathematical Optimization for Business Problems Final Exam Answers

Question 1: What are the two types of objects used to describe network structures?

• Nodes and arcs
• Arcs and chains
• Nodes and chains

Question 2: What are the possible reasons for an infeasible model?

• Real-world conflict
• Incorrect data
• Incorrect formulation
• All of the above

Question 3: True or false? Mathematical programming and constraint programming are the techniques you can apply using CPLEX.

• False
• True

Question 4: True or false? Binary variables are also known as Boolean variables.

• False
• True

Question 5: What is the first step of a typical optimization model development cycle?

• The identification of objectives, variables, and constraints
• The scope definition
• The creation of a prototype

Question 6: True or false? Basic variables take zero values in an iteration or final solution of the Simplex method.

• False
• True

Question 7: True or false? An unbounded variable always influences the solvability of a model.

• False
• True

Question 8: True or false? Flow conservation constraints are typically used in network models.

• False
• True

Question 9: True or false? Nonlinear terms and absolute values are not permitted in linear programming.

• False
• True

Question 10: True or false? Piecewise linear programming is used when dealing with functions consisting of several nonlinear segments.

• False
• True

Question 11: True or false? Very large linear programming models are often non-sparse.

• False
• True

Question 12: What does an optimization-based solution involve?

• An optimization engine
• Data
• An optimization model
• All of the above

Question 13: True or false? When all arcs in a chain are directed in such a way that it is possible to traverse the chain following the directions of arcs, it is called a path.

• False
• True

Question 14: A region is convex if …

• A straight line connecting two points inside the region passes outside it to get from one point to the other
• Any straight line between two points inside the region remains entirely in the region
• All of the above

Question 15: True or false? The scale of numbers used in an LP problem can affect computational time.

• False
• True