I want to use the simplex-algorithm in matlab to solve my optimization problem. After reading the threads i underestand that the simplex-algorithm is not used for linprog, instead the dual-simplex is the default solution which essentially performs a simplex algorithm on the dual problem. To code simplex in matlab and solve almost 100 big instances maybe i should say giant internet is full of sources on primal phase 1 algorithm whereas, a two phase support method for solving linear programs numerical experiments mohand bentobache 1 2 and mohand ouamer bibi 2 1 department of. Matlab simplex tutorial sa305 spring 2012 instructor, start phase two of a 2 phase simplex algorithm mupad, download two phase simplex method c source codes two, euler s method matlab program code with c, simplex method matlab implementation github, simplex method matlab code download free open source, www researchgate net, suite of files for. Simplex Method Matlab Implementation. This is matlab implementation of the two-phase simplex method for better understanding of the algorithm. There are three modes for choosing pivots - to avoid degeneracy.
- Phase 2 Lynchburg Va
- Phase 2 Reopening New York
- Matlab Code For Phase 2 Simplex Method Pdf
- Matlab Code For Phase 2 Simplex Method Calculator
- Phase 2 Singapore
- Matlab Code For Phase 2 Simplex Methods
Two Phase Method: Minimization Example 1
Minimize z = -3x1 + x2 - 2x3
subject to
Phase 2 Lynchburg Va
x1 + 3x2 + x3 ≤ 5
2x1 – x2 + x3 ≥ 2
4x1 + 3x2 - 2x3 = 5
x1, x2, x3 ≥ 0
Solution.
If the objective function is in minimization form, then convert it into maximization form.
Changing the sense of the optimization
Any linear minimization problem can be viewed as an equivalent linear maximization problem, and vice versa. Specifically:
Minimizecjxj = Maximize(- cj)xj
If z is the optimal value of the left-hand expression, then -z is the optimal value of the right-hand expression.
Maximize z = 3x1 – x2 + 2x3
subject to
x1 + 3x2 + x3 ≤ 5
2x1 – x2 + x3 ≥ 2
4x1 + 3x2 - 2x3 = 5
x1, x2, x3 ≥ 0
Converting inequalities to equalities
x1 + 3x2 + x3 + x4 = 5
2x1 – x2 + x3 – x5 = 2
4x1 + 3x2 - 2x3 = 5
x1, x2, x3, x4, x5 ≥ 0
Where:
x4 is a slack variable
x5 is a surplus variable
The surplus variable x5 represents the extra units.
Now, if we let x1, x2 and x3 equal to zero in the initial solution, we will have x4 = 5 and x5 = -2, which is not possible because a surplus variable cannot be negative. Therefore, we need artificial variables.
Phase 2 Reopening New York
x1 + 3x2 + x3 + x4 = 5
2x1 – x2 + x3 – x5 + A1 = 2
4x1 + 3x2 - 2x3 + A2 = 5
x1, x2, x3, x4, x5, A1, A2 ≥ 0
Where A1 and A2 are artificial variables.
Phase 1 of Two Phase Method
In this phase, we remove the artificial variables and find an initial feasible solution of the original problem. Now the objective function can be expressed as
Maximize 0x1 + 0x2 + 0x3 + 0x4 + 0x5 + (–A1) + (–A2)
subject to
x1 + 3x2 + x3 + x4 = 5
2x1 – x2 + x3 – x5 + A1 = 2
4x1 + 3x2 - 2x3 + A2 = 5
x1, x2, x3, x4, x5, A1, A2 ≥ 0
Initial basic feasible solution
The intial basic feasible solution is obtained by setting
x1 = x2 = x3 = x5 = 0
Then we shall have A1 = 2 , A2 = 5, x4 = 5
Two Phase Method: Table 1
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| cj | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ||
|---|---|---|---|---|---|---|---|---|---|
| cB | Basic variables B | x1 | x2 | x3 | x4 | x5 | A1 | A2 | Solution values b (= XB) |
| 0 | x4 | 1 | 3 | 1 | 1 | 0 | 0 | 0 | 5 |
| -1 | A1 | 2 | -1 | 1 | 0 | -1 | 1 | 0 | 2 |
| -1 | A2 | 4 | 3 | -2 | 0 | 0 | 0 | 1 | 5 |
| zj–cj | -6 | -2 | 1 | 0 | 1 | 0 | 0 |
Key column = x1 column
Minimum (5/1, 2/2, 5/4) = 1
Key row = A1 row
Pivot element = 2
A1 departs and x1 enters.
Table 2
A2 departs and x2 enters.
Here, Phase 1 terminates because both the artificial variables have been removed from the basis.
Phase 2 of Two Phase Method
Matlab Code For Phase 2 Simplex Method Pdf
The basic feasible solution at the end of Phase 1 computation is used as the initial basic feasible solution of the problem. The original objective function is introduced in Phase 2 computation and the usual simplex procedure is used to solve the problem.
Table 3
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Matlab Code For Phase 2 Simplex Method Calculator
| cj | 3 | -1 | 2 | 0 | 0 | ||
|---|---|---|---|---|---|---|---|
| cB | Basic variables B | x1 | x2 | x3 | x4 | x5 | Solution values b (= XB) |
| 0 | x4 | 0 | 0 | 33/10 | 1 | -9/10 | 33/10 |
| 3 | x1 | 1 | 0 | 1/10 | 0 | -3/10 | 11/10 |
| -1 | x2 | 0 | 1 | -4/5 | 0 | 2/5 | 1/5 |
| zj-cj | 0 | 0 | -9/10 | 0 | -13/10 |
Phase 2 Singapore
Table 4
| cj | 3 | -1 | 2 | 0 | 0 | ||
|---|---|---|---|---|---|---|---|
| cB | Basic variables B | x1 | x2 | x3 | x4 | x5 | Solution values b (= XB) |
| 0 | x4 | 0 | 9/4 | 3/2 | 1 | 0 | 15/4 |
| 3 | x1 | 1 | 3/4 | -1/2 | 0 | 0 | 5/4 |
| 0 | x5 | 0 | 5/2 | -2 | 0 | 1 | 1/2 |
| zj-cj | 0 | 13/4 | -7/2 | 0 | 0 |
Matlab Code For Phase 2 Simplex Methods
Don't be impatient. The next table is the last table.