Math Models of Operations Research, MATP 4700.
Second Exam, Friday, November 8, 2019.
运筹学数学模型代写 You may use any result from your notes or a homework that is clearly stated. You may use one sheet of handwritten notes, but no other sources.
You may use any result from your notes or a homework that is clearly stated. You may use one sheet of handwritten notes, but no other sources. The exam consists of five questions, and lasts one hundred and ten minutes.
| Q1 | / 20 |
| Q2 | / 30 |
| Q3 | / 15 |
| Q4 | / 20 |
| Q5 | / 15 |
| Total | / 100 |
| Grade |
1.(20points)
(a)(10 points) Explain why, if a lower bound constraint on nonbasic variable xpis appended to optimal tableau T ∗, a single pivot on am+1,p restores optimal form if the appended row m + 1 is the minimum-ratio row in the xp column.
(b)(10 points) Explain why, if the appended row m + 1 is not the minimum-ratio row inthe xp column, one or more dual simplex pivots are also required.
2.(30 points) Consider the linearprogram 运筹学数学模型代写
minx∈R4 5x1 + 2x2 + 3x3 + 2x4
subject to −2x1 + x2 + 2x3 + 3x4 ≥ b1 (P )
x1 + 4x2 + 6x3 + 3x4 = b2
x1, x2, x3, x4 ≥ 0
where b1 > 0 and b2 > 0 are fixed parameters.
(a)(10 points) Find a dual problem (D) to(P ).
(b)(10points) Assume x¯ = (0, b2 , 0, 0) is feasible in (P ), with −2x¯1 +x¯2 +2x¯3 +3x¯4 =b1 + ∆, with ∆ > 0. Use complementary slackness to show that x¯ is optimal.
(c)(10 points) Use complementary slackness to find another optimal solution to (P).
3.(15points) Consider the transportation problem with costs as given: 运筹学数学模型代写
demand
| supply | 40 | 10 | 30 | 20 |
| 70 | 4 | 3 | 2 | 4 |
| 30 | 5 | 7 | 6 | 3 |
(a)Findan initial feasible solution using the NW corner rule.
(b)Solve the problem (hint: youshould only need one iteration).
4.(20points) 运筹学数学模型代写
(a)(10 points) A feasible solution to the following uncapacitated network flow prob- lem isindicated:
Use duality to prove that this solution is optimal.
(b)(10 points) Assume supply increases by ∆ > 0 at node 4 and demandincreases by ∆ at node 3. Use duality to determine the change in the optimal value for small values of ∆. What is the largest possible value of ∆ for which your estimate holds. What is the modified shipping schedule? 运筹学数学模型代写
5.(15points) We wish to find a minimum cost network flow in the following uncapacitated network, where all nodes are transshipment nodes and where the edge costs are given:
The primal problem requires nonnegative flows that satisfy flow conservation at each node. Formulate the dual problem and verify explicitly that it is infeasible.
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