The revised simplex method is known to be mathematically an equivalent approach towards standard simplex method and it simply differs in implementation. There is no need to maintain a tableau which can easily explicit and represent different constraints that can be adjusted to basic variables which helps to maintain any kind of representation of basis. The matrix may represent any constraints. While getting into depth of matrix oriented approach it is possible to look for greater computation efficiency and finally help in understanding sparse matrix operations.
There are varied methods available which comes with entire table and does not need calculations for iteration. The information that appears to be necessary while developing a new table is Ci -Zi value. The key column as well as current basic variables comes with solution value constraints. It is the information which can be obtained directly while making use of certain properties related to matrix. You can adopt different methods which are ultimately developed and make computation easier and simplified:
- The revised simplex method
- The decomposition method
- The bounded variables method
Problem formulation
It is said that linear programming problem can easily be converted into standard form:
where A ∈ Rm×n in such a case without any loss of generality it is said that constraint matrix A comes with full row rank and also problems are feasible which means that at least one x ≥ 0 and it refers as Ax = b. In such situation A is rank deficient and can be redundant constraints and the problem is infeasible.
Some duality problems
Min Z = 2x1 + 2 x2
Subject to 2 x1 + 4 x1 ≥ 1
x1 + 2 x1 ≥ 1
2 x1 + x2 ≥ 1
Solve it with the help of duality
Sol: Maximize Z = a + b + c
Subject to 2a + b + c ≤ 2
4a + 2b + c ≤ 2
The problem has now enjoying maximization which will also introduce slack variables.
Maximize Z = a + b + c + 0S1 + 0S2 + 0S3
Subject to 2a + b + c + S1 + 0S2 = 2
4a + b + c + 0S1 + S2 = 2
The Simplex table can be written as follows
| Cj → | 1 | 1 | 1 | 0 | 0 | Min Ratio | ||
| ↓ | Basic Variables | Solution Values | a | b | c | S1 | S2 | |
| 0 | S1 | 2 | 2 | 1 | 2 | 1 | 0 | |
| 0 | S2 | 2 | 4 | 2 | 1 | 0 | 1 | |
| Zj | 0 | 0 | 0 | 0 | 0 | 0 | ||
| (Cj-Zj) | 1 | 1 | 1 | 0 | 0 | |||
Since it is regarded that values of (Cj – Zj) are equal which says 1,1,1 and can enter variables which taking it into key column.
Links of Previous Main Topic:-
- Concept of capital expenditure
- Learning objectives and chapter outline
- Limitations of operations research
- Linear programming learning objectives and outline of chapter
- Introduction learning objectives
- Duality in linear programming lpp
- Dual problems when primal in standard form
- Interpreting primal dual optimal solutions
- Dual simplex method
- Revised simplex method
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