Download CH 18 Simplex Based Sensitivity Analysis and Duality and more Exams Mathematics in PDF only on Docsity!
True / False
- The range of optimality is useful only for basic variables. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Remember
- The range of optimality is calculated by considering changes in the cj − zj value of the variable in question. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- As long as the objective function coefficient remains within the range of optimality, the variable values will not change although the value of the objective function could. a. True b. False ANSWER: True POINTS: 1
DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- If the simplex tableau is from a maximization converted from a minimization, the signs and directions of the inequalities that give the objective function ranges will need to be adjusted to apply to the original coefficients. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- The ranges for which the right-hand-side values are valid are the same as the ranges over which the dual prices are valid. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- A dual price is associated with each decision variable.
TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- The range of optimality for a basic variable defines the objective function coefficient values for which that variable will remain part of the current optimal basic feasible solution. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Remember
- The dual price is the improvement in value of the optimal solution per unit increase in a constraint's right-hand-side value. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.2 Duality KEYWORDS: Bloom's: Remember
- As long as the actual value of the objective function coefficient is within the range of optimality, the current basic feasible solution will remain optimal. a. True b. False
ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- We can often avoid the process of formulating and solving a modified linear programming problem by using the range of optimality to determine whether a change in an objective function coefficient is large enough to cause a change in the optimal solution. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- Within the concept of duality is the original formulation of a linear programming problem known as the primal problem. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18.
- A one-sided range of optimality a. always occurs for nonbasic variables. b. always occurs for basic variables. c. indicates changes in more than one coefficient. d. indicates changes in a slack variable's coefficient. ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Understand
- A linear programming problem with the objective function 3x 1 + 8x 2 has the optimal solution x 1 = 5, x 2 = 6. If c 2 decreases by 2 and the range of optimality shows 5 ≤ c 2 ≤ 12, the value of Z a. will decrease by 12. b. will decrease by 2. c. will not change. d. cannot be determined from this information. ANSWER: a POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Apply
- The improvement in the value of the optimal solution per unit increase in a constraint's right-hand side is a. the slack value.
b. the dual price. c. never negative. d. the 100% rule. ANSWER: b POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Remember
- The dual variable represents the a. marginal value of the constraint. b. right-hand-side value of the constraint. c. artificial variable. d. technical coefficient of the constraint. ANSWER: a POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.2 Duality KEYWORDS: Bloom's: Understand
- The range of feasibility indicates right-hand-side values for which a. the value of the objective function will not change. b. the values of the decision variables will not change. c. those variables that are in the basis will not change. d. more simplex iterations must be performed. ANSWER: c POINTS: 1
POINTS: 1
DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.2 Duality KEYWORDS: Bloom's: Apply
- Given the simplex tableau for the optimal primal solution, a. the values of the dual variables can be found from the cj − zj values of the slack/surplus variable columns. b. the values of the dual surplus variables can be found from the cj − zj values of the primal decision variable columns. c. the value of the dual objective function will be the same as the objective function value for the primal problem. d. All of these are correct. ANSWER: d POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.2 Duality KEYWORDS: Bloom's: Understand
- An LP maximization problem with all less-than-or-equal-to constraints and nonnegativity requirements for the decision variables is known as a. a canonical form for a minimization problem. b. a canonical form for a maximization problem. c. always unbounded. d. None of these are correct. ANSWER: b POINTS: 1 DIFFICULTY: Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18.
NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 18.2 Duality KEYWORDS: Bloom's: Remember Subjective Short Answer
- For the following linear programming problem Max Z 0.5x 1 + 6x 2 + 5x 3 s.t. 4x 1 + 6x 2 + 3x 3 ≤ 24 1x 1 + 1.5x 2 + 3x 3 ≤ 12 3x 1 + x 2 ≤ 12 the final tableau is x 1 x 2 x 3 s 1 s 2 s 3 Basis cB 0.5 6 5 0 0 0 x 2 6 1 1 0 0.22 −0.22 0 2. x 3 5 0 0 1 0.11 −0.44 0 2. s 3 0 2.33 0 0 −0.22 0.22 1 9. zj 4 6 5 0.77 0.88 0 29. cj − zj 0.5 0 0 −0.77 −0.88 0 a. Find the range of optimality for c1, c2, c3, c4, c5, and c 6. b. Find the range of feasibility for b 1 , b 2 , and b 3. ANSWER: a. c 1 ≤ 4 2.5 < c 2 ≤ 10 3 < c 3 ≤ 12 c 4 ≤ 0. c 5 ≤ 0. −1.5 ≤ c 6 ≤ 3.
− 1 ≤ c 5 ≤ 1 c 6 ≤ 0 b. 4 ≤ b 1 b 2 ≤ 7 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Analyze
- Write the dual of the following problem: Min Z = 2x 1 − 3x 2 + 5x 3 s.t. −3x 1 + 2x 2 + 5x 3 ≥ 7 2x 1 − x 3 ≥ 5 4x 2 + 3x 3 ≥ 8 ANSWER: Max Z 2 = 7y 1 + 5y 2 + 8y 3 s.t. −3y 1 + 2y 2 ≤ 2 2y 1 + 4y 3 ≤ − 3 5y 1 − y 2 + 3y 3 ≤ 5 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.02 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.2 Duality KEYWORDS: Bloom's: Create
- For the following linear programming problem
the final tableau is
c. Find the dual prices. ANSWER:
- Max 10x 1 + 12x
- s.t. 1x 1 + 2x 2 ≥ - 5x 1 + 8x 2 ≤ - 1x 1 + 1x 2 ≤ - x 1 , x 2 ≥ - x 1 x 2 s 1 s 2 s
- Basis cB
- x 2 12 0 1 −2.5 −0.5
- x
- s 3 0 0 0 −1.5 −0.5 - zj - cj − zj 0 0 − 10 −
- a. Find the range of optimality for c 1 and c
- b. Find the range of feasibility for b 1 , b 2 , and b - −∞ < c 2 ≤ a. 7.5 ≤ c 1 < ∞ - b. 32 ≤ b 1 < - 160 ≤ b 2 ≤ - c. u 1 = 20 ≤ b 3 < ∞ - u 2 = - u 3 =
- POINTS:
- LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. DIFFICULTY: Challenging
Basis cB − 100 − 110 − 95 0 0 x 3 − 95 0 −1.5 1 1 −1.5 75 x 1 − 100 1 2 0 − 1 1 50 zj − 100 −57.5 − 95 5 42.5 −12, cj − zj 0 −52.5 0 − 5 −42. a. What would the new solution be if the right-hand-side value in the first constraint had been 325? b. What would the new solution be if the right-hand-side value for the second constraint had been 220? ANSWER: a. x 3 = 50, x 1 = 75, Z = 12, b. x 3 = 30, x 1 = 80, Z = 10, POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Evaluate
- Creative Kitchen Tools manufactures a wide line of gourmet cooking tools from stainless steel. For the coming production period, there is demand of 1200 for eight- quart stock pots and unlimited demand for three-quart mixing bowls and large slotted spoons. In the following model, the three variables measure the number of pots, bowls, and spoons to make. The objective function measures profit. Constraint 1 measures steel, constraint 2 measures manufacturing time, constraint 3 measures finishing time, and constraint 4 measures the stock pot demand. Max 5x 1 + 3x 2 + 6x 3 s.t. 3x 1 + 1x 2 + 2x 3 ≤ 15, 4x 1 + 4x 2 + 5x 3 ≤ 18, 2x 1 + 1x 2 + 2x 3 ≤ 10, x 1 ≤ 1200 x 1 , x 2 , x 3 ≥ 0
The final tableau is as follows: x 1 x 2 x 3 s 1 s 2 s 3 s 4 Basis cB 5 3 6 0 0 0 0 s 1 0 0 − 2 −1.75 1 −0.75 0 0 1500 s 4 0 0 1 1.25 0 0.25 0 1 3300 s 3 0 0 − 1 −0.5 0 −0.5 1 0 1000 x 1 5 1 1 1.25 0 0.25 0 0 4500 zj 5 5 6.25 0 1.25 0 0 22, cj − zj 0 − 2 −0.25 0 −1.25 0 0 a. Calculate the range of optimality for c 1 , c 2 , and c 3. b. Calculate the range of feasibility for b 1 , b 2 , b 3 , and b 4. c. Suppose that the inventory records were incorrect and the company really has only 14,000 units of steel. What effect will this have on your solution? d. Suppose that a cost increase will change the profit on the pots to $4.62. What effect will this have on your solution? e. Assume that the cost of time in production and finishing is relevant. Would you be willing to pay a $1.00 premium over the normal cost for 1000 more hours in the production department? What would this do to your solution? ANSWER: a. 4.80 ≤ c 1 < ∞ −∞ < c 2 ≤ 5 −∞ < c 3 ≤ 6. b. 13,500 ≤ b 1 < ∞ 480 ≤ b 2 ≤ 20, 9000 ≤ b 3 < ∞ −∞ < b 4 ≤ 4500 c. This would affect only the amount of slack,
all xi ≥ 0 The final tableau is as follows: x 1 x 2 x 3 x 4 x 5 x 6 s 1 s 2 Basis cB − 20 − 14 − 15 − 28 − 20 − 25 0 0 x 6 − 25 0 0 0 0.5 0.333 1 0 −0.033 666. s 1 0 0 0 0 −17.5 −31.67 0 1 −0.833 6666. x 1 − 20 1 0 0 1 0 0 0 0 2000 x 2 − 14 0 1 0 0 1 0 0 0 2000 x 3 − 15 0 0 1 −0.5 −0.333 0 0 0.033 1333. zj − 20 − 14 − 15 − 25 −17.33 − 25 0 0.33 68, cj − zj
a. Calculate the range of optimality for all of the objective function coefficients. b. Calculate the range of feasibility for the first two right-hand sides. c. How much less expensive would it have to be to buy frames before you would consider it? d. How much more expensive would legs have to be to make before you would change your solution? e. What would the total cost be if the cost to make a panel increased by $3.00? f. What would you be willing to pay for more production time? g. What would happen to the total cost if the amount of assembly time decreased by 2000 hours? ANSWER: a. −∞ < c 1 ≤ 23 −∞ < c 2 ≤ 16. 9 ≤ c 3 ≤ 25 25 ≤ c 4 < ∞ 17.33 ≤ c 5 < ∞
15 ≤ c 6 ≤ 31 b. 173,333.33 ≤ b 1 < ∞ 50,000 ≤ b 2 ≤ 98, 6666.67 ≤ b 3 ≤ 2380. 0 ≤ b 4 ≤ 2210. 1333.33 ≤ b 5 < ∞ c. $3.00 per frame d. $2.67 additional e. Cost would increase by 1333.33(3) = 4000. f. nothing g. 0.33(2000) = 666.67 more cost POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.18.01 - 18. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 18.1 Sensitivity Analysis with the Simplex Tableau KEYWORDS: Bloom's: Evaluate
- Write the dual to the following problem. Min 12x 1 + 15x 2 + 20x 3 + 18x 4 s.t. x 1 + x 2 + x 3 + x 4 ≥ 50 3x 1 + 4x 3 ≥ 60 2x 2 + x 3 − 2x 4 ≤ 10 x 1 , x 2 , x 3 , x 4 ≥ 0 ANSWER: Max 50u 1 + 60u 2 − 10u 3
