For the following assignments, respond to the following questions in which first three are True-False questions and the reamining are multiple choice with the choices under the questions labeled choices:
1. Sensitivity ranges can be computed for all objective parameters and all constraints' right-hand sides. (T/F)
2. If the optimal value of a decision variable of a linear programming maximization problem is positive, then the reduced cost corresponding to that variable will be nonzero. (T/F)
3. The shadow price of a resource changes whenever the right-hand side of that constraint is changed to a value outside the sensitivity range for that constraint quantity. (T/F)
4. The linear programming problem below describes the daily production process to determine the number of units of product 1 (x1), product 2 (x2), and product 3 (x3) that should be manufactured in order to maximize the total daily profit. The company operates 12 hours (720 minutes) per day (constraint 1) and uses two types of materials in the production: wood and steel (constraints 2 and 3). Furthermore, constraint 4 describes the market requirement for products 1 and 3. Suppose that the profit for each unit of product 1 has increased to $32.50. Then the optimal profit
Choices:
- Will not change
- Will increase by $240
- Will increase by $300
- Will increase by $600
- None
5. The linear programming problem below describes the daily production process to determine the number of units of product 1 (x1), product 2 (x2), and product 3 (x3) that should be manufactured in order to maximize the total daily profit. The company operates 12 hours (720 minutes) per day (constraint 1) and uses two types of materials in the production: wood and steel (constraints 2 and 3). Furthermore, constraint 4 describes the market requirement for products 1 and 3. Assume that one of the decision variables has an optimal value of zero. At least how much should the per unit profit for this product increase for the optimal value to change?
Maximize Profit Z = 30x1 + 20x2 + 36x3
Subject to:
Production time (minutes): 2x1 + x2 + 3x3 ≤ 720
Wood (lbs.): 1.5x1 + x2 + 2x3 ≤ 600
Steel (lbs.): 8x1 + 3x2 + 10x3 ≤ 3000
Demand: x1 + x3 ≥ 200
x1, x2, x3 ≥ 0
Choices:
- it should increase by at least $4
- it should increase by at least $5
- it should increase by at least $10
- it should increase by at least $12
- none
6. The linear programming problem below describes the daily production process to determine the number of units of product 1 (x1), product 2 (x2), and product 3 (x3) that should be manufactured in order to maximize the total daily profit. The company operates 12 hours (720 minutes) per day (constraint 1) and uses two types of materials in the production: wood and steel (constraints 2 and 3). Furthermore, constraint 4 describes the market requirement for products 1 and 3. Suppose that the profit for each unit of product 2 has increased to $25. Then the optimal profit _____________________.
Maximize Profit Z = 30x1 + 20x2 +36x3
Subject to:
Production time (minutes): 2x1 + x2 + 3x3 ≤ 720
Wood (lbs.): 1.5x1 + x2 + 2x3 ≤ 600
Steel (lbs.): 8x1 + 3x2 + 10x3 ≤ 3000
Demand: x1 + x3 ≥ 200
x1, x2, x3 ≥ 0
Choices:
- will not change,
- will increase by $300
- will increase by $600
- will increase by $1200
- none
7. Consider the following linear programming minimization problem. What is the sensitivity range for the right-hand side of the second constraint?
Minimize Cost Z = 2x1 + 8x2
Subject to:
Ingredient 1: 8x1 + 6x2 ≥ 64
Ingredient 2: 2x1 + 4x2 ≥ 32
Demand: x2 ≥ 2
x1, x2 ≥ 0
Choices:
- 33 or larger
- 32 or smaller
- 21 or larger
- 21 or smaller
- none
8. Consider the following linear programming minimization problem. What is the sensitivity range for the objective parameter of x2?
Minimize Cost Z = 2x1 + 8x2
Subject to:
Ingredient 1: 8x1 + 6x2 ≥ 64
Ingredient 2: 2x1 + 4x2 ≥ 32
Demand: x2 ≥ 2
x1, x2 ≥ 0
Choices:
- 4 or larger
- 8
- 4 to 8
- 8 or larger
- none
9. Consider the following linear programming problem. If the available labor hours are decreased to 950 hours, how much will the optimal profit decrease?
Maximize Profit Z = 6x1 + 4x2
Subject to:
Material 1: 2x1 + 3x2 ≤ 700 lbs.
Material 2: 2x1 + x2 ≤ 500 lbs.
Labor Hrs: 3x1 + 4x2 ≤ 1000 Hrs.
x1, x2 ≥ 0
Choices:
10. Consider the following linear programming problem. Management has decided to increase the available amount of one of the resources. Which of the three resources (the amount of material 1, the amount of material 2, or the number of labor hours) should be increased?
Maximize Profit Z = 6x1 + 4x2
Subject to:
Material 1: 2x1 + 3x2 ≤ 700 lbs.
Material 2: 2x1 + x2 ≤ 500 lbs.
Labor Hrs: 3x1 + 4x2 ≤ 1000 Hrs.
x1, x2 ≥ 0
Choices:
- Material 1
- Material 2
- labor hrs
- none