_{1}

^{*}

Multi-Objective Optimization (MOO) techniques often achieve the combination of both maximization and minimization objectives. The study suggests scalarizing the multi-objective functions simpler using duality. An example of four objective functions has been solved using duality with satisfactory results.

Multi-Objective Optimization helps in making decisions in presence of usually conflicting objectives. Scalarizing techniques have been popularly used for solving multi-objective optimization problems. Several new scalarizing techniques [

The mathematical form of Sen’s MOO technique [

Optimize Z = [ Max . Z 1 , Max . Z 2 , ⋯ , Max . Z r , Min . Z r + 1 , ⋯ , Min . Z s ]

Subject to:

A X = b and X ≥ 0

The individual optima are obtained by optimizing each objective separately as:

Z optima = [ θ 1 , θ 2 , ⋯ , θ s ]

The Primal Multi-Objective Function is formulated as:

Maximize Z = ∑ j = 1 r Z j | θ j | − ∑ j = r + 1 s Z j | θ r + 1 |

Subject to:

A X = b and X ≥ 0

θ j ≠ 0 for j = 1 , 2 , ⋯ , s .

where, θ j is the optimal value of jth objective function.

All the objective functions are converted into either maximizing or minimizing form as described below:

Maximize Z_{j} or Minimize Z_{j}

Subject to:

A X = b and X ≥ 0

The minimization objective function can be converted into maximization objective function by multiplying −1. Similarly the maximization objective can be converted into minimization objective function by multiplying −1. The Multi-Objective Function is formulated as:

Maximize Z = ∑ j = 1 s Z j | θ j |

or

Minimize Z = ∑ j = 1 s Z j | θ j |

Subject to:

A X = b and X ≥ 0

θ j ≠ 0 for j = 1 , 2 , ⋯ , s .

where, θ j is the optimal value of jth objective function.

Step I: Convert all the objective functions either maximization of minimization mode.

Step II: Formulate multi-objective function as explained in 2.2

Step III: Optimize the multi-objective function under the same constraints.

The following example has been solved with duality technique.

Example

Max . Z 1 = 12500 X 1 + 25100 X 2 + 16700 X 3 + 23300 X 4 + 20200 X 5

Max . Z 2 = 21 X 1 + 15 X 2 + 13 X 3 + 17 X 4 + 11 X 5

Min . Z 3 = 370 X 1 + 280 X 2 + 350 X 3 + 270 X 4 + 240 X 5

Min . Z 4 = 1930 X 1 + 1790 X 2 + 1520 X 3 + 1690 X 4 + 1720 X 5

Subject to:

X 1 + X 2 + X 3 + X 4 + X 5 = 4.5

2 X 1 ≥ 1.0

3 X 4 ≥ 1.5

The above problem can be converted with all the four objective functions either maximization of minimization mode as detailed below:

Max . Z 1 = 12500 X 1 + 25100 X 2 + 16700 X 3 + 23300 X 4 + 20200 X 5

Max . Z 2 = 21 X 1 + 15 X 2 + 13 X 3 + 17 X 4 + 11 X 5

Max . Z 3 = − 370 X 1 − 280 X 2 − 350 X 3 − 270 X 4 − 240 X 5

Max . Z 4 = − 1930 X 1 − 1790 X 2 − 1520 X 3 − 1690 X 4 − 1720 X 5

or

Min . Z 1 = − 12500 X 1 − 25100 X 2 − 16700 X 3 − 23300 X 4 − 20200 X 5

Min . Z 2 = − 21 X 1 − 15 X 2 − 13 X 3 − 17 X 4 − 11 X 5

Min . Z 3 = 370 X 1 + 280 X 2 + 350 X 3 + 270 X 4 + 240 X 5

Min . Z 4 = 1930 X 1 + 1790 X 2 + 1520 X 3 + 1690 X 4 + 1720 X 5

The problem was solved with multi-objective function of both maximization and minimization mode. It is very clear from

This necessitates the need of multi-objective optimization. Both the solutions of multi-objective optimization are exactly the same and achieving all the four objectives simultaneously. Hence the multi-objective optimization problems can be solved by formulating multi-objective function after converting all the objective functions in either maximizing or minimizing mode.

Objective Function | Individual Optimization | Multi-Objective Optimization | ||||
---|---|---|---|---|---|---|

Max.Z_{1} | Max.Z_{2} | Min.Z_{3} | Min.Z_{4} | Maximization Mode | Minimization Mode | |

X_{i} | 0.5, 3.5, 0, 0.5, 0 | 4, 0, 0, 0.5, 0 | 0.5, 0, 0, 0.5, 3.5 | 0.5, 0, 3.5, 0.5, 0 | 0.5, 0, 0, 4, 0 | 0.5, 0, 0, 4, 0 |

Z_{1} | 105750 | 61650 | 88600 | 76350 | 99450 | 99450 |

Z_{2} | 71.5 | 92.5 | 57.5 | 64.5 | 78.5 | 78.5 |

Z_{3} | 1300 | 1615 | 1160 | 1545 | 1265 | 1265 |

Z_{4} | 8075 | 8565 | 7830 | 7130 | 7725 | 7725 |

One of the important advantages of the duality theory is presented in the paper for solving MOO problems. It is established that duality makes easier the formulation of multi-objective function. However, it is needed only when optimization is done for a set of both maximization and minimization objective functions.

The author declares no conflicts of interest regarding the publication of this paper.

Sen, C. (2019) Duality in Solving Multi-Objective Optimization (MOO) Problems. American Journal of Operations Research, 9, 109-113. https://doi.org/10.4236/ajor.2019.93006