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Final Exam Aug 16, 2001 MAP 5326 S Hudson
Name
Show all your work. Use the space provided, or leave a note. Don’t use a calculator or your own extra paper.
- Consider this problem. Maximize Z = x 1 − x 2 + 2x 3 , subject to
2 x 1 − 2 x 2 + 3x 3 ≤ 5 x 1 + x 2 − x 3 ≤ 3 x 1 − x 2 + x 3 ≤ 2 and xj ≥ 0 for j = 1, 2 , 3.
a) Use the fundamental insight (use B−^1 etc) to complete the final simplex tableau below. Show all work.
eqn Z x 1 x 2 x 3 x 4 x 5 x 6 rhs
Z (0) 1 2 1 1 0
x 2 (1) 0 1 3 0 14 x 6 (2) 0 2 0 0 1 1 x 3 (3) 0 1 2 0
b) What are the defining equations for the optimal solution in the final tableau?
- Consider the following IP problem. Maximize Z = −x 1 + 3x 2 subject to
− 2 x 1 + 5x 2 ≤ −1 and
0 ≤ xj ≤ 4 (with xj an integer) for j = 1, 2.
a) Solve this graphically.
b) Use the MIP branch and bound method to solve this problem by hand. For each subproblem, solve the LP relaxation graphically.
- Consider the network below (left), where the numbers on each arc are costs, and where arc BA has capacity uBA = 10 and arc CE has capacity uCE = 80. Suppose that the network simplex method leads to the spanning tree below (right). Do one more iteration, to find the next spanning tree (and label it).
- A marketing campaign has $4 million to split up and spend (in integer multiples of $ million), to increase three variables m, f 2 and f 3 according to the chart below. At least $1 million must be spent on m. Use dynamic programming to maximize m · f 2 · f 3. [this is the short version of HW problem 11.3-8(a); see me if you want to read the long version].
Millions spent Effect
m f 2 f 3
- The following table is an initial tableau for a transportation problem (perhaps from Russell’s method).
a) Fill in the 3 missing numbers in column 2.
b) Find the entering and leaving basic variables for the first iteration.
