Multi - goal and multi-objective optimizations are similar techniques to achieve multiple conflicting goals/objectives simultaneously. There are several techniques for solving multi-goal and multi-objective optimization problems. The present study proposed the possibility of convertibility in solving multi-goal and multi-objective optimization problems.
The Multi-Goal Optimization (MGO) methods have been proposed [
Multi-Objective Optimization (MOO) is also alternatively used for achieving multiple conflicting objectives. An appropriate method of MOO was proposed by Chandra Sen in 1982 [
The improved MGO method is described as:
Minimize Z = ∑ i = 1 m ( d i + + d i − ) / g i
Subject to:
Goal Constraints:
∑ j = 1 n a i j X j − d i + + d i − = g i for i = 1 , ⋯ , m
System constraints:
∑ j = 1 n a i j X j = b i for i = m + 1 , ⋯ , p
where: gi are n goals to be achieved, i = 1 , 2 , ⋯ , m .
There are “m” Goals, “p” System constraints and “n” decision variables:
Z = Objective function/Summation of all deviations;
aij = the coefficient associated with jth variable in ith Goal/constraint;
Xj = the jth decision variable;
gi = the right hand side value of ith goal;
bi = the right hand side value of ith constraint;
d i − = negative deviational variation from ith goal (under achievement);
d i + = positive deviational variation fromith goal (over achievement).
Z i = [ g i , g 2 , ⋯ , g m ]
Max . Z = ∑ i = 1 r Z i g i − ∑ j = r + 1 m Z j g j
Subject to:
A X ≥ / ≤ / = b
X ≥ 0
where;
Zj are the goals to be achieved.
The combined objective function has been formulated with the scalarized objective functions by their respective goals.
Max . Z i = [ Z 1 , Z 2 , ⋯ , Z r ] and Min . Z j = [ Z r + 1 , ⋯ , Z m ]
Max . Z = ∑ i = 1 r Z i θ i − ∑ j = r + 1 m Z j θ j
Subject to:
A X ≥ / ≤ / = b
X ≥ 0
where:
Zi and Zj are the objective functions to be maximized and minimized respectively;
θ i and θ j are the individual optima of ith & jth objective function.
Minimize Z = ∑ i = 1 m ( d i + + d i − ) / θ i
Subject to:
Goal Constraints:
∑ j = 1 n a i j X j − d i + + d i − = θ i for i = 1 , ⋯ , m
System constraints:
∑ j = 1 n a i j X j = b i for i = m + 1 , ⋯ , p
where: gi are n goals to be achieved, i = 1 , 2 , ⋯ , m .
There are “m” Goals, “p” System constraints and “n” decision variables:
Z = Objective function/Summation of all deviations;
aij = the coefficient associated with jth variable in ith Goal/constraint;
Xj = the jth decision variable;
θ i = the right hand side value of ith goal;
bi = the right hand side value of ith constraint;
d i − = negative deviational variation from ith goal (under achievement);
d i + = positive deviational variation from ith goal (over achievement).
Example 1:
Goal Z 1 = 6 X 1 + 9 X 2 + 3 X 3 + 8 X 4 + 4 X 5 ≥ 42
Goal Z 2 = 400 X 1 + 300 X 2 + 700 X 3 + 600 X 4 + 500 X 5 ≥ 3200
Goal Z 3 = 14 X 1 + 16 X 2 + 15 X 3 + 12 X 4 + 11 X 5 ≤ 50
Goal Z 4 = 70 X 1 + 100 X 2 + 95 X 3 + 80 X 4 + 90 X 5 ≤ 300
Subject to:
X 1 + X 2 + X 3 + X 4 + X 5 = 5
X 1 ≥ 1
2 X 4 ≥ 1
X 1 , X 2 , X 3 , X 4 , X 5 ≥ 0
Solution:
Example 1 was solved by scalarizing the objective functions by respective goals as suggested in the methodology. The example has also been solved using Sen’s MOO method for comparative analysis. The results are presented in
The results of individual optimization of all the four goals are different indicating the conflicts amongst goals. All the four goals have been achieved as 41.5, 4150, 58.50, and 355 which are very close to their goals of 42, 3200, 50, and 300 respectively. However, these solutions are all different. The results of the MGO method in MOO mode are the same. The achievements of the goals were 38, 2800, 62, and 390 with respect to their targets of 40, 3000, 60, and 300. All four goals have been achieved to their acceptable levels.
Example 2:
Max . Z 1 = 370 X 1 + 550 X 2 + 450 X 3 + 500 X 4
Max . Z 2 = 90 X 1 + 60 X 2 + 70 X 3 + 80 X 4
Max . Z 3 = 25 X 1 + 20 X 2 + 35 X 3 + 30 X 4
Subject to:
X 1 + X 2 + X 3 + X 4 = 7.5
X 2 ≥ 0.5
X 1 , X 2 , X 3 , X 4 ≥ 0
Solution:
The MOO problem was solved using Sen’s MOO method and MGO mode. All the objectives have been optimized individually also. The results are presented in
It is clear from
| Item | Individual Goal Optimization | MGO Method | MOO Mode | |||
|---|---|---|---|---|---|---|
| Z1 | Z2 | Z3 | Z4 | |||
| Goals | 42 | 3200 | 50 | 300 | ||
| Xi | 1, 3.5, 0, 0.5, 0 | 1, 0, 3.5, 0.5, 0 | 1, 0, 0, 0.5, 3.5 | 4.5, 0, 0, 0.5, 0 | 1, 0, 0, 4, 0 | 1, 0, 0, 4, 0 |
| Z1 | 41.5 | 20.50 | 24 | 31 | 38 | 38 |
| Z2 | 1750 | 3150 | 2450 | 2100 | 2800 | 2800 |
| Z3 | 76 | 72.50 | 58.50 | 69 | 62 | 62 |
| Z4 | 460 | 442.50 | 425 | 355 | 390 | 390 |
| Item | Individual Optimization | MOO Method | MGO Mode | ||
|---|---|---|---|---|---|
| Max. Z1 | Max. Z2 | Min. Z3 | |||
| Xi | 0, 7.5, 0, 0 | 7, 0.5, 0, 0 | 0, 0.5, 7, 0 | 0, 0.5, 0, 7 | 0, 0.5, 0, 7 |
| Z1 | 4125 | 2865 | 3425 | 3775 | 3775 |
| Z2 | 450 | 660 | 520 | 590 | 590 |
| Z3 | 150 | 185 | 255 | 220 | 220 |
solved in MGO mode are very clear that all three objectives have been achieved simultaneously. The results indicate that both the methods for solving MGO or MOO problems are equally efficient.
The present analysis has been carried out to explore the possibilities of solving MGO problems using the MOO method and vice versa. Two suitable examples have been solved using modified methodologies. The solutions of proposed modified optimizations are very encouraging and shall be highly useful in various applications.
The authors declare no conflicts of interest regarding the publication of this paper.
Gangwar, S. and Sen, C. (2022) Convertibility in Multi-Goal and Multi-Objective Optimization. American Journal of Operations Research, 12, 11-17. https://doi.org/10.4236/ajor.2022.121002