3.1 Formulation of Problems

• What is Linear Programming used for?

  Mathematical optimization
• Linear Programming problems are characterized by linear relationships between variables and constraints
• What does the Objective Function represent in an LP problem?

  Goal to be optimized
• Constraints in Linear Programming are linear inequalities that restrict the values of decision variables.
• In the example given, what is the Objective Function?

  $Maximize: Z =$$30x +$$20y$
• Non-negativity constraints ensure that decision variables are greater than or equal to zero
• Match the Linear Programming components with their descriptions:

  Objective Function ↔️ Mathematical expression to be optimized

  Decision Variables ↔️ Variables adjusted for optimal results

  Constraints ↔️ Linear inequalities restricting variables
• The Objective Function must always be maximized in Linear Programming problems.

  False
• What are the two main goals of Linear Programming?

  Maximize or minimize
• In the desk production example, the Objective Function is to maximize profit
• Order the steps in formulating a Linear Programming problem:

  1️⃣ Define the problem and its goal

  2️⃣ Identify decision variables

  3️⃣ Formulate the Objective Function

  4️⃣ Define constraints

  5️⃣ Write non-negativity constraints
• What does the term "feasible region" refer to in Linear Programming?

  Region defined by constraints
• Decision Variables represent the quantities of resources or products to be determined in Linear Programming.
• The Objective Function represents the goal of the Linear Programming problem, which can be to maximize or minimize
• What are decision variables in a Linear Programming problem?

  Quantities to be determined
• Decision variables are adjusted to achieve the optimal solution
• What is the objective function in a Linear Programming problem?

  Mathematical expression to optimize
• Steps to formulate an objective function

  1️⃣ Identify variables

  2️⃣ Determine coefficients

  3️⃣ Combine variables and coefficients

  4️⃣ Specify optimization type
• If a company sells chairs at \$30 each and tables at \$20 each, the objective function to maximize profit is $Maximize: Z =$$30x +$$20y$
• What are constraints in a Linear Programming problem?

  Linear inequalities limiting variables
• A resource constraint limits the use of resources such as labor
• Match the constraint type with its description:

  Resource ↔️ Limits the use of resources

  Demand ↔️ Ensures minimum production

  Non-negativity ↔️ Variables must be non-negative
• What is Linear Programming (LP)?

  Mathematical optimization technique
• The objective function in Linear Programming is optimized by maximizing or minimizing a mathematical expression
• In the example provided, the objective function to maximize profit is $Maximize: Z =$$30x +$$20y$
• Steps to define the components of an LP problem

  1️⃣ Identify the objective function

  2️⃣ Define decision variables

  3️⃣ Formulate constraints
• What do decision variables represent in Linear Programming?

  Quantities to be determined
• The objective function in Linear Programming must always be maximized

  False
• Match the constraint type with its example:

  Resource ↔️ $2x +$$y \leq 40$

  Demand ↔️ $x +$$y \geq 20$

  Non-negativity ↔️ $x \geq 0, y \geq 0$
• What type of constraint ensures variables must be non-negative?

  Non-negativity constraint
• Constraints in Linear Programming are linear inequalities
• A demand constraint ensures minimum production levels are met.
• What is an example of a resource constraint in Linear Programming?

  $2x +$$y \leq 40$
• Constraints ensure that the solution adheres to limitations or requirements
• Steps to write linear constraints in Linear Programming

  1️⃣ Consider available resources

  2️⃣ Consider requirements

  3️⃣ Formulate linear inequalities
• A furniture company has 40 labor hours and 30 material units. What is the labor hours constraint if chairs (x) require 2 hours and tables (y) require 1 hour?

  $2x +$$y \leq 40$
• Non-negativity constraints ensure decision variables are zero or positive.
• Linear Programming is a mathematical optimization technique used to find the best possible solution
• What is the term for the mathematical expression to be maximized or minimized in Linear Programming?

  Objective function
• Steps to formulate an objective function in Linear Programming

  1️⃣ Identify decision variables

  2️⃣ Determine coefficients

  3️⃣ Form a linear expression

  4️⃣ Specify optimization type