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运筹学代写 Math Models代写 MATP 4700代写

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Math Models of Operations Research, MATP 4700.

Second Exam, Tuesday, November 6, 2018.

 

运筹学代写 You may use any result from your notes or a homework that is clearly stated. You mayuse 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 mayuse 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 / 20
Q3 / 25
Q4 / 15 运筹学代写
Q5 / 20
Total / 100
Grade  

1.(20points) For any m × n matrix A, consider the following statement: either the system Ax > 0 has a solution,  运筹学代写

or the system AT y = 0, y 0, y = 0 has a solution,

but not both systems have solutions.

(a)(10points) Write down a linear program that will determine whether or not the second system has a solution.

(b)(10points) What is the dual of the linear program in part (1a)?

2.(20points) The canonical form tableau for the following linear optimization problem is in optimal form:  运筹学代写

minxR3   12x1        + 30x3

subject to   3x1 +           x2 +          6x3 = 12

                                                x1, x2, x3  0

The optimal solution is x2  = 12, x1  = x3  = 0.  It has been determined that we need to add the additional constraint x2 6, giving a modified primal problem.

(a)(10points) Add the constraint to the problem and solve the problem using dual simplex. (You should only need one ) 运筹学代写

(b)(5 points) What is the dual problem to the modified primalproblem?

(c)(5points) What are the initial and final dual iterates?

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运筹学代写

3.(25 points) Consider the linear optimizationproblem  运筹学代写

minxR2 8x1 + 3x2

subject to  x +  x2 6                (P )

3x1  2x2  12

                     x1, x2 0

(a)(5points) With the signs of the inequalities omitted, the dual problem to (P) can be written

maxyR2 6y1  12y2

subject to y1 + 3y2 ??   8        (D)

                   y1 2y2 ??   3

                        y1, y2 ??  0

Give the dual problem including the signs of the inequalities.

(b)(10points) The point x = (0, 6) is optimal for (P). Use complementary slackness to show that the dual problem has multiple optimal  Give two different dual optimal solutions.

(c)(10points) Graph the dual problem and show the set of dual optimal solutions.

4.(15points) A feasible solution x is given to the following minimum cost network flow problem with capacities:  运筹学代写

Node 1: Supply 8

Nodes 2, 3: Transshipment

Node 4: Demand 8

edge labels:

(cost cij, capacity uij, flow xij)  运筹学代写

The given solution is a basic feasible solution, with basic variables x12 = 2, x13 = 6,

x34 = 7, and nonbasic variables both at their upper bounds: x23 = 1, x24 = 1.

Solve the problem using the network flow simplex algorithm. (You should be able to solve the problem in one iteration. Be sure to show that your solution is optimal.)

5.(20points)

(a)(10points) An optimal solution to the following uncapacitated network flow prob- lem is indicated:

Node 1: Supply 20

Nodes 2, 3, 5: Transshipment

Node 4: Supply 10

Node 6: Demand 30

edge labels:

(cost cij, flow xij运筹学代写

The edges in the optimal basis are (1, 2), (2, 5), (3, 5), (4, 6), (5, 6). The optimal dual solution is y1 = 13, y2 = 6, y3 = 8, y4 = 6, y5 = 3, y6 = 0. We are considering adding an edge (1, 4). What must the cost c14 be for it to be attractive to use this edge?

(b)(10points) In a transportation problem with more demand than supply, a dummy supply node is  Optimal primal and dual solutions are given in the following transportation tableau, where the first supply node is the dummy one:

(Note that the cost of each edge is negative, since this is a revenue maximization problem written as an equivalent mimimization problem.)

Assume extra supply ∆ can be purchased at source 2. How much would you be willing to pay per unit of additional supply? What is the maximum amount you would purchase at this price? Where is the decrease in unmet demand? What is the modified shipping schedule?

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运筹学代写

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