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Questions and Answers
Is the solution x1 = 2, x2 = 5 feasible for the equation 0.4x1 + 0.6x2 = 0?
Is the solution x1 = 2, x2 = 5 feasible for the equation 0.4x1 + 0.6x2 = 0?
- Yes, as it satisfies the production requirements.
- No, because the coefficients must be equal.
- No, because the equation leads to a contradiction. (correct)
- Yes, because both variables are positive.
What does the equation 0.4x1 + 0.6x2 = 0 imply about the values of x1 and x2?
What does the equation 0.4x1 + 0.6x2 = 0 imply about the values of x1 and x2?
- At least one of the variables must be greater than one.
- Both variables must equal zero. (correct)
- At least one of the variables must be negative.
- Both x1 and x2 must be positive.
If 0.4x1 + 0.6x2 = 0 and x1 = 0, what must x2 be for the equation to hold true?
If 0.4x1 + 0.6x2 = 0 and x1 = 0, what must x2 be for the equation to hold true?
- Any positive number.
- Any negative number.
- It must also be zero. (correct)
- It must be less than zero.
What is the significance of the coefficients 0.4 and 0.6 in the equation 0.4x1 + 0.6x2 = 0?
What is the significance of the coefficients 0.4 and 0.6 in the equation 0.4x1 + 0.6x2 = 0?
In the context of the equation 0.4x1 + 0.6x2 = 0, what can be inferred about the production levels when x1 and x2 are both positive?
In the context of the equation 0.4x1 + 0.6x2 = 0, what can be inferred about the production levels when x1 and x2 are both positive?
For the production equation $0.4x_1 + 0.6x_2 = 0$, what must be true if $x_1$ is a positive number?
For the production equation $0.4x_1 + 0.6x_2 = 0$, what must be true if $x_1$ is a positive number?
If a production problem is defined by $0.4x_1 + 0.6x_2 = 0$, what does this imply about the relationship between $x_1$ and $x_2$?
If a production problem is defined by $0.4x_1 + 0.6x_2 = 0$, what does this imply about the relationship between $x_1$ and $x_2$?
For the equation $0.4x_1 + 0.6x_2 = 0$, which of the following pairs $(x_1, x_2)$ is a feasible solution?
For the equation $0.4x_1 + 0.6x_2 = 0$, which of the following pairs $(x_1, x_2)$ is a feasible solution?
If the production equation is $0.4x_1 + 0.6x_2 = 0$ and we know that $x_2 = -4$, what is the value of $x_1$?
If the production equation is $0.4x_1 + 0.6x_2 = 0$ and we know that $x_2 = -4$, what is the value of $x_1$?
In a production model represented by $0.4x_1 + 0.6x_2 = 0$, if both $x_1$ and $x_2$ are constrained to be non-negative, which of these is true?
In a production model represented by $0.4x_1 + 0.6x_2 = 0$, if both $x_1$ and $x_2$ are constrained to be non-negative, which of these is true?
Flashcards
Feasibility in Linear Programming
Feasibility in Linear Programming
A solution is feasible if all constraints in the linear programming problem are satisfied. In this case, we are given that the constraint is 0.4x1 + 0.6x2 = 0. We need to determine if the values x1 = 2 and x2 = 5 satisfy this constraint. Substituting these values into the constraint, we get: 0.4(2) + 0.6(5) = 0.8 + 3 = 3.8. As the result is not 0, the solution is not feasible.
Constraint in Linear Programming
Constraint in Linear Programming
A constraint in linear programming represents a limitation or restriction on the variables in the problem. These constraints define the feasible region, which represents all the possible solutions that satisfy these limitations.
Variables in Linear Programming
Variables in Linear Programming
Variables in linear programming represent the quantities or amounts that we are trying to determine in the problem. They are typically represented by letters like x1, x2, ..., xn.
Solution in Linear Programming
Solution in Linear Programming
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What is a feasible solution in linear programming?
What is a feasible solution in linear programming?
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What is a constraint in linear programming?
What is a constraint in linear programming?
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What are the variables in linear programming?
What are the variables in linear programming?
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How do you check if a solution is feasible?
How do you check if a solution is feasible?
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Is the solution x1 = 2, x2 = 5 feasible for the constraint 0.4x1 + 0.6x2 = 0?
Is the solution x1 = 2, x2 = 5 feasible for the constraint 0.4x1 + 0.6x2 = 0?
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Study Notes
Linear Programming, Transportation, and Production Planning
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Constraint 1: A production problem uses the constraint 0.4x₁ + 0.6x₂ ≤ 300. Optimal solution uses 300 labor hours. The constraint is binding because the left-hand side equals the right-hand side.
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Constraint 2: A resource constraint is 5x₁ + 2x₂ ≤ 50. The solution only uses 45 units. The constraint is not binding because the left-hand side does not reach 50.
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Constraint 3 (New): A company with two suppliers and three warehouses has a constraint for Warehouse 3 as x₁₃ + x₂₃ = 150. This means the total shipments to Warehouse 3 must be 150.
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Shadow Price: A shadow price of $10 indicates that adding one unit of the resource increases profit by $10.
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Shadow Price of $0: A shadow price of $0 means the constraint is non-binding, and additional resource availability will not affect the objective function.
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Transportation Problem Objective Function: A company's transportation problem with costs (C₁₁ = 5, C₁₂ = 6, C₂₁ = 4) has an objective function to minimize cost (Z = 5x₁₁ + 6x₁₂ + 4x₂₁).
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Inventory Holding Cost Objective Function: A factory producing two products with costs of $12 and $18 per unit and inventory holding costs of $3 and $4 per unit, respectively, has an objective function to minimize total cost (Z = 12xA + 18xB + 3sA + 4sB).
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Minimization Problem Objective Function: The goal of an objective function in a minimization problem is to minimize resource use and minimize total cost.
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Feasibility (Constraint 1): 4x₁ + 3x₂ ≤ 24. Examples of feasible solutions: x₁ = 2, x₂ = 4; x₁ = 3, x₂ = 2
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Feasibility (Constraint 2): 5x₁ + 2x₂ ≥ 20. x₁ = 2, x₂ = 5 is not feasible.
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Feasibility (Constraint 3): Additional examples of x1 and x2 values that satisfy 4x₁ + 3x₂ <= 24 are provided
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