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**Standard Minimization Problems with the Dual**

Appendix simplex method

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**STANDARD MINIMIZATION PROBLEM**

A standard minimization problem is a linear programming problem with an objective function that is to be minimized. The objective function is of the form : Z= aX1 + bX2 + cX3….. where a, b, c, are real numbers and X1, X2, X3, are decision variables. Constraints are of the form: AX1 + BX2 + CX3+ …… ≥ M where A, B, C,... are real numbers and M is nonnegative

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**STANDARD MINIMIZATION PROBLEM (cont.)**

Example: Determine if the linear programming problem is a standard minimization problem Minimize Z = 4X1+ 8X2 Subject to 3X1 + 4X2 ≤ -9 X2 ≥ 5 X1,x2 ≥0

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**STANDARD MINIMIZATION PROBLEM (cont.)**

Solution Minimize Z = 4X1+ 8X2 Subject to 3X1 + 4X2 ≤ -9 X2 ≥ 5 X1,x2 ≥0 Multiply First constraint by -1

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**STANDARD MINIMIZATION PROBLEM (cont.)**

We got: Minimize Z = 4X1+ 8X2 Subject to 3X1 + 4X2 ≥9 X2 ≥ 5 X1,x2 ≥0

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The Dual For a standard minimization problem whose objective function has nonnegative coefficients, it may construct a standard maximization problem called the dual problem

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**Using Duals to Solve Standard Minimization Problems**

Example: Minimize Z= 2X1 + 3X2 Subject to X1+ X2 ≥ 12 3X1 + 2X2 ≥ 4 X1, X2≥ 0

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**Using Duals to Solve Standard Minimization Problems (cont.)**

The solution 1- construct a matrix for the problem as: 1 12 3 2 4 X1+ X2 ≥ 12 3X1 + 2X2 ≥ 4 2X1 + 3X2= Z

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**Using Duals to Solve Standard Minimization Problems (cont.)**

2- The transpose of the matrix is created by switching the rows and columns The dual problem is: Maximize Z= 12X1+ 4X2 ST X1+ 3X2 ≤ 2 X1 + 2X2 ≤ 3 X1,X2 ≥0 1 3 2 12 4 X1+ 3X2 ≤ 2 X1+ 2X2 ≤ 3 12X1+ 4X2 = Z

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**Using Duals to Solve Standard Minimization Problems (cont.)**

Then Adding in the slack variables and rewriting the objective function yield the system of equations: X1+ 3X2 + S1= 2 X1 + 2X2 + S2= 3 X1,X2 ≥0

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**Using Duals to Solve Standard Minimization Problems (cont.)**

The initial simplex tableau: Basis X1 X2 S1 S2 RHS 1 3 2 Z -12 -4

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Basis X1 X2 S1 S2 RHS 1 3 2 -1 Z 32 12 24

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Basis X1 X2 S1 S2 RHS 1 3 2 -1 Z 32 12 24 X1 value X2 value

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**Minimization in other case**

If objective function is minimization and all constraints are “<“ , the solution can be found by multiply objective function by -1 , then objective function will convert to Max and solve the problem as simplex method. Example: Min z= 3x1 – 2x2 ST X1+x2<= 12 X2<= 24 X1,X2>=0

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**Minimization in other case (cont.)**

Solution Min z= 3x1 – 2x2 ST X1+x2<= 12 X2<= 24 X1,X2>=0 Multiply by -1

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**Minimization in other case (cont.)**

Max z= -3x1 + 2x2 ST X1+x2<= 12 X2<= 24 X1,X2>=0

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**Minimization in other case (cont.)**

Basis X1 X2 S1 S2 RHS 1 12 24 Z 3 -2

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Duality Theory Every LP problem (called the ‘Primal’) has associated with another problem called the ‘Dual’. The ‘Dual’ problem is an LP defined directly.

Duality Theory Every LP problem (called the ‘Primal’) has associated with another problem called the ‘Dual’. The ‘Dual’ problem is an LP defined directly.

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