Math 484.1  Nov  7, 2013.     Midterm 2.
15 problems, 5 pts each.   Write your name here  Vaserstein  and return the page even if you wrote nothing else on it.

26. The linear program   x+y -> min,  x ≥ 0, y ≤ 3

(A) is infeasible, (B) is unbounded, (C) has an optimal solution, (D) is not a linear program.

27. For every  t, the linear equation      tz = sin(z)  has a solution for z.

(A) True.  (B)  False.  (C) This is not a linear equation for z.

28. The linear program  is given by the standard row tableau

x1 x2  x3  x4 x5  1

1  -1   0   0  7  0  = u

2  -2   0   0 -6  4  = v

3  -3  -5   6  5  0  -> min.

Here are all choices for the pivot entry that are consistent with the simplex method: (A)  -1, (B) -6, (C) 3,  (D) -1, -6, (E) none of the above.

29.  The linear program  is given by the standard row tableau

x1 x2   x3  x4  x5   1

1  -1    0   0   7   0  = u

2  -2    0   0  -6  -4  = v

3   1   -1   6   5   0  -> min.

Here are all choices for the pivot entry that are consistent with the simplex method:  (A)  1, (B) 2, (C) 3,  (D) 1, 2,  (E) none of the above.

30.   The linear program  is given by the standard row tableau

x1 x2  x3  x4  x5    1

1  -1   0   0   7    0 = u

2  -2   0   0  -6    0 = v

3   1   1   6   5   -5 -> min.

Here are all choices for the pivot entry that are consistent with the simplex method: (A)  1 in x1 column, (B) 2, (C) -1,   (D) 7, (E) none of the above.

The tableau is optimal.

31. For the linear program given by the standard row tableau

 x y z 1 1 -1 0 -2 = u 1 -2 1 0 -> min

x=0,  y = 0, z = 0, u = -1 is (A) the basic solution, (B) an optimal solution, (C)  none of the above.

32. The linear program given by the standard row tableau

 x y z 1 1 -1 0 -2 = u 2 -2 1 0 -> min

(A) is infeasible, (B) has an optimal solution, (C) is unbounded, (D)  none of the above.

33. The optimal value of the   program   (sin x)2+t(cos y)2-> min, x2  1,   y2 ≤4

as a function of the parameter  t  has the following slopes: (A) 2, (B) -2,(C) 2, -2, (D) -1, 0.

None of the above answers is correct (5 pts for this).

The minimum is  t   when t ≤ 0 and 0 when t ≥ 0. So the slopes are 1 and 0 (two bonus points to be included into b2 at Angel for this).

34. The optimal value of the   program   x+y -> max, |x| ≤ 3t+1,   |y| ≤ 2t

as a function of the parameter  t  has the following slopes: (A) -3, (B) 5, (C) -3, 3. (D) none of the above

35. The optimal value of the  job assignment problem

 3 0 3 1 2 1 3 1 3 1 2 1 3 3 5 2 4 1 3 2 4 2 1 2 1 1 3 1 3 1 2 2 2 1 1 1

is (A)  5, (B) 5.5,  (C) 6, (D) none of the above.

36. The optimal value  for the transportation problem

 2 3 3 2 1 2 3 4 1 6 1 2 2 1 1 2 3 3 1 9 2 2 1 3 1 3 3 4 0 7 3 3 1 1 2 3 2 3 2 12 6 1 6 1 6 1 6 1 6 demand\supply

is (A) 30, (B) 35, (C) 37, (D) none of the above.

37, There is a linear program with the feasible region given by  x2 ≤ 1, y2 ≤ 4 and exactly one optimal solution

x= 1, y = 2.  (A) True. (B) False.

38. For any feasible tableau with 2 columns, the simplex method terminates in at most one pivot step.

(A) True. (B) False

39. x > 2 only if x ≥  0.

(A) True. (B) False.

40.  x > 2  provided that x ≥  0.

(A) True. (B) False.