Math 484.2   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 < 2, x  >  3

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


27. For every  t > -1, the linear equation    log(1 + t2)  = ty   has a solution for y.

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


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

x1  x2   x3  x4  x5   1

1   -1    0  -5   7   0  = u

2   -2   -4   0  -6   4  = v

3   -3   -1   6   5   0  -> min.

Here are all choices for the pivot entry that are consistent with the simplex method:  

(A)  -1, (B) -4, (C) 1,  (D) -1, -4, (E) none of the above.



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

x1  x2   x3  x4  x5   1

1    2    0  -5   7  -1  = u

8   -2   -4   0  -6   4  = v

3   -3   -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) -2,  (D) 1, 2, 7,  (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  -5   7   0  = u

2   -2    0   0  -6   0  = v

0    0   -1   0   5   0  -> min.

Here are all choices for the pivot entry that are consistent with the simplex method:  

(A)  -1, (B) -4, (C) 1,  (D) -1, -4, (E) none of the above.



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 = -2 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

1

-3

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   x3+ty 3-> min, x2  1,   y2≤ 9

as a function of the parameter  t  has the following slopes: (A)  -1, (B) 1,(C) 1,-1, (D) 3, 0.

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

The minimum is  -t + 3t  when t ≤ 0 and -1 - 3t  when t ≥ 0. So the slopes are 3 and -3 

(two bonus points to be included into b2 at Angel for this).


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

as a function of the parameter  t  has the following slopes: (A) -3, (B) 3, (C) -3, 3. 


35. The optimal value of the  job assignment problem

3

1

3

1

2

2

3

1

3

2

1

2

3

3

5

1

4

2

3

2

2

2

2

2

1

1

3

1

3

3

1

2

2

1

1

4

is (A)  5, (B) 5.5,  (C) 9, (D) none of the above.


36. The optimal value  for the transportation problem

2

1

3

0

1

2

1

1

1

7

1

2

2

1

1

2

1

3

1

6

2

2

1

3

1

1

3

1

2

7

0

1

1

1

1

1

2

3

0

14

6

1

6

1

6

1

6

1

6

demand\supply

 is (A) 21, (B) 23, (C) 24, (D) none of the above.

37, There is a linear program with the feasible region given by  |x| + |y| = 2 and exactly one optimal solution x= 1, y =1.  

(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 ≥  2.  (A) True. (B) False.


40. x > 2 provided that   x ≥  3.  (A) True. (B) False.