Operations Research HWK#1 Due: Oct 3, 2007
1. The problem relates to the definition of a LP and its standard form. For the following systems, please check
(i) Which one is a LP? Which one is not? (bring your answer with a complete reason)
(ii) For those which can be classified as a LP, check if they are in standard forms.
If not, convert them into the standard forms.
(iii) Write down A, b, and C for each LP in its standard form.
(Note: A is the constraint matrix, b is the right hand side vector, and C is the cost coefficient.)
Systems to be checked:
(a)
min log(5x1+ 4x2− x3) s.t. x1− 10x2 = 5
2x2− x3 = 7 x1, x2, x3 ≥ 0.
(b)
min (log 5)x1− (cos 4)x2
s.t. e4x1− 7x2 ≤ 1
x1 + 22x2 ≥ − log 0.01 x1 ≥ 0.
(Note:e is defined to be ln e = 1.) (c)
max 7
s.t. x1− 8 ≥ 0.
(d)
max √
7 − x1+ x2
√7 s.t. x1− |x2| + 3|x3| ≤ 9
2x1+ 5x2− |x3| = 1 x2 ≥ 0.
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(e)
max 5x1+ 5x2+ 3x3 s.t. x1 + 3x2+ x3 ≤ 3
−x1+ 3x3 ≤ 2 2x1− x2+ 2x3 ≤ 4 2x1+ 3x2− x3 ≤ 2 x1, x2, x3 ≥ 0.
2. Find the complete range for the number s and t to make the LP problem max x1+ x2
s.t. sx1+ tx2 ≤ 1 x1, x2 ≥ 0 (a) have an optimal solution.
(b) be infeasible.
(c) be unbounded.
(d) have multiple optimal solutions.
3. (a) Give an example to show there exists a bounded LP with an unbounded feasible domain.
(b) Give an example to show there exists a bounded LP with infinitely many optimal solutions.
(c) Give an example to show there exists a bounded LP with an unbounded opti- mal solution set.
(d) Show that it is impossible to have a LP having a bounded feasible domain but with an unbounded optimal solution set.
4. Use the graphic method to find an optimal solution of the following systems:
(a)
min x + 2y s.t. x + y = 1
x, y ≥ 0.
(b)
min x + 2y
s.t. |x − 1| + y ≤ 1 x, y ≥ 0.
(c)
min y − 1 4x
s.t. ||x − 1| − 2| + |y − 3| ≤ 3 x, y ≥ 0.
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Please draw your picture as clearly as you can. Different colors are highly recom- mended. Be sure all the feasible domains, objective functions, optimal solutions must be found on your drawings.
5. For the following three minimization problems (f1, s1), (f2, s2), (f3, s3) (f1, s1) : min −2x1+ x2
s.t. x1+ x2 ≤ 1 x1, x2 ≥ 0.
(f2, s2) : min −2x1 + x2 s.t. x1+ x2 + x3 = 1
x1, x2, x3 ≥ 0.
(f3, s3) : min 3x2+ 2x3 − 2 s.t. x2+ x3 ≤ 1
x2, x3 ≥ 0.
(a) Draw a three-dimensional graph for the above three systems.
(b) Show that the three systems are equivalent.
(ps. Two optimization problems (P1) and (P2) are equivalent if the optimal solution of one problem can be transformed to that of the other so that solving one the problems also gives the optimal solution of the other.)
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