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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

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

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

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

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

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

## Introduction to Mathematical Optimization for Business Problems

Mathematical optimization is a powerful tool used in business to make better decisions by maximizing or minimizing certain objectives, subject to constraints. It involves finding the best possible solution from all feasible solutions. In the context of business, optimization problems often arise in various areas such as finance, operations management, marketing, and supply chain management.

Here’s an introduction to the key concepts and techniques used in mathematical optimization for business problems:

1. Objective Function: This function represents what you want to maximize or minimize. In business, it could be maximizing profit, minimizing costs, optimizing resource allocation, or maximizing efficiency.
2. Decision Variables: These are the variables that you can control or adjust to achieve your objective. For example, in production planning, decision variables could be the quantity of each product to produce.
3. Constraints: These are the limitations or restrictions that must be considered in the decision-making process. Constraints can include resource limitations, capacity constraints, demand constraints, etc.
4. Feasible Region: The set of all possible combinations of decision variables that satisfy all constraints. Solutions outside this region are not feasible.
5. Optimal Solution: The solution that maximizes or minimizes the objective function while satisfying all constraints. It’s the best possible outcome given the available options and constraints.
6. Linear Programming (LP): LP is a common optimization technique used in business to solve problems where both the objective function and constraints are linear. It’s widely applied in areas like production planning, resource allocation, and transportation logistics.
7. Integer Programming (IP): In some cases, decision variables are required to take integer values (i.e., whole numbers) rather than continuous values. IP extends linear programming to handle such discrete decision variables.
8. Non-linear Programming (NLP): When the objective function or constraints are non-linear, more advanced techniques like NLP are used to find optimal solutions. NLP is often applied in problems involving non-linear cost functions, demand curves, or production functions.
9. Heuristic and Metaheuristic Algorithms: For complex optimization problems where traditional methods struggle to find optimal solutions in a reasonable amount of time, heuristic and metaheuristic algorithms like genetic algorithms, simulated annealing, and particle swarm optimization can be employed.
10. Software Tools: Various software tools and libraries are available to solve optimization problems efficiently. Some popular ones include MATLAB, Python’s SciPy library, IBM CPLEX, and Gurobi.

In summary, mathematical optimization provides a systematic approach to decision-making in business by finding the best possible solutions to complex problems. By formulating problems mathematically and applying optimization techniques, businesses can improve efficiency, reduce costs, and make more informed decisions.