Public Management
When you perform the primal simplex method, you are supposed to use Dantzig’s rule to choose entering columns. Instead of floating point arithmetic, it is recommended to do arithmetic with fractions. Solve the problems using the methods you learned in class (Optional ways you google will not earn any point).
1. Dual Simplex Method
1-1. (5 points) Solve the following LP by Dual-Simplex:
min 5 x1 + 2 x2 + 3 x3 + 8 x4 = z,
st 3 x1 + 2 x2 ≥ 6,
2 x1 + 4 x2 + x3 + 5 x4≥ 8,
x1, x2, x3, x4 are all non-negative.
1-1-1. Coefficient Matrix
x1
x2
x3
x4
s1
s2
min
5
2
3
8
0
0
z
3
2
0
0
– 1
0
6
2
4
1
5
0
– 1
8
Ratio
1-1-2. Find the simplex tableau with respect to the initial basis B0 = {s1, s2}. Is it feasible? (Circle Yes/No) If no, perform the dual simplex method. Circle the leaving basic variable will leave out of the basis. Perform the minimum ratio test to determine the entering column. Circle the entering pivot.
x1
x2
x3
x4
s1
s2
min
z
Ratio
1-1-3. Find the simplex tableau with respect to the next basis B1. Is it feasible? (Circle Yes/No) If no, perform the dual simplex method. Circle the leaving basic variable will leave out of the basis. Perform the minimum ratio test to determine the entering column. Circle the entering pivot.
x1
x2
x3
x4
s1
s2
min
z
Ratio
1-1-4. Find the simplex tableau with respect to the next basis B2. Is it feasible? (Circle Yes/No) If no, perform the dual simplex method. Circle the leaving basic variable will leave out of the basis. Perform the minimum ratio test to determine the entering column. Circle the entering pivot.
x1
x2
x3
x4
s1
s2
min
z
Ratio
1-2. (5 points) Add the following constraint to the simplex tableau with respect to the optimal feasible basis of Problem 1-1 and go on Dual-Simplex to get an optimal solution to the LP augmented by the additional constraint:
2 x1 + 2 x2 + 4 x3 + 4 x4 ≥ 10.
1-2-1. An initial basis of the augmented problem consists of the last optimal basis and the new surplus variable s3. What is the simplex tableau with respect to the initial basis?
x1
x2
x3
x4
s1
s2
s3
min
z –
Ratio
1-2-2. Find the simplex tableau with respect to the initial basis B0 = the optimal feasible basis of Problem 1-1 augmented by s3. Is it feasible? (Circle Yes/No) If no, perform the dual simplex method. Circle the leaving basic variable will leave out of the basis. Perform the minimum ratio test to determine the entering column. Circle the entering pivot.
x1
x2
x3
x4
s1
s2
s3
min
z –
Ratio
1-2-3. Find the simplex tableau with respect to the next basis B1. Is it feasible? (Circle Yes/No) If no, perform the dual simplex method. Circle the leaving basic variable will leave out of the basis. Perform the minimum ratio test to determine the entering column. Circle the entering pivot.
x1
x2
x3
x4
s1
s2
s3
min
z –
Ratio
2. (5 points) Perform the branch-and-bound to solve the following integer programming problem:
Max z = 3 x1 + 2 x2,
s.t. 2 x1 + x2 ≤ 10,
4 x1 + 3 x2 ≤ 25,
x1 and x2 are non-negative integers.
Follow the instructions and the rules in the yellow and the orange boxes in B&B Tree Worksheet of B&Bsimple.xlsx file which is solving the homework problem (not this exam problem).
3. (10 points) Perform the branch-and-bound to solve the following integer programming problem:
Max z = 3 x1 + 2 x2,
s.t. 2 x1 + x2 ≤ 10,
4 x1 + 3 x2 ≤ 25,
x1 and x2 are non-negative integers.
Follow the instructions and the rules in the yellow and the orange boxes in B&B Tree Worksheet of B&Bsample.xlsx file which is solving the homework problem. Submit your excel file into the submission folder. Solve Subproblem 1 using the primal simplex method. Solve the other subproblems using the dual simplex method.
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