Multi-criteria optimization. How to solve it?

[Ted111]

http://tinypic.com/r/1570ojk/9
Please see the image. I don't understand how to solve such a problem. It has three criteria. Any hints or guide would be appreciated or even a solution. This is actually just an example from the book, but the book doesn't even solve the question, so I don't even have an example that I can follow.

http://tinypic.com/r/1570ojk/9
http://tinypic.com/r/1570ojk/9

 

[fresh_42]

Seems like the simplex algorithm could help.

 

[Ted111]

Seems like the simplex algorithm could help.

Are you saying that I should make a matrix and solve that?

 

[fresh_42]

I have not enough data to answer this. What exactly is your objective function, what are your constraints? I assumed they were linear and thus the simplex algorithm would be an appropriate method to solve it, as the value which you are looking for is in one of its vertices.

This is not really a question of general engineering so I'll move it to the homework section. But you are supposed to show some effort.
To describe exactly the task and what you know so far about optimization problems would be a good start.

 

[Ted111]

I have not enough data to answer this. What exactly is your objective function, what are your constraints? I assumed they were linear and thus the simplex algorithm would be an appropriate method to solve it, as the value which you are looking for is in one of its vertices.

This is not really a question of general engineering so I'll move it to the homework section. But you are supposed to show some effort.
To describe exactly the task and what you know so far about optimization problems would be a good start.

Go to this link (Page 12-13):
http://www.math.hcmus.edu.vn/~nvthuy/om/Multicriteria%20Optimization.pdf

 

[Ted111]

Go to this link (Page 12-13):
http://www.math.hcmus.edu.vn/~nvthuy/om/Multicriteria%20Optimization.pdf

All I can come up with is that the decision variables are price, petrol consumption, and power.

 

[Ray Vickson]

All I can come up with is that the decision variables are price, petrol consumption, and power.

No, that is not true; they are not decision variables, they are given input data items.

This problem is NOT one of linear programming, unless your decision variables are $x_V$ = fraction of a Volkswagen to purchase, etc. Thus, the simplex method would be of no use.

What you have is a "tradeoff" problem, which is to choose the car having the most desirable bundle of attribute values. Here, the attributes are price, consumption and power. For example, the attribute bundle for a Volkswagen is (16.2, 7.2, 66.0), which specifies the "score" that Volkswagen has on each of the three attributes. Presumably, the scoring system has been devised so that a higher score is better than a lower one, but we are not told that for sure. (Some hard work might have been done already to develop the scores, but we can just use them now as input for further analysis.)

Typical approaches to such problems are (i) determination of the "efficient" set (Pareto-optimality); (ii) Lexicographic ordering; and (iii) linear tradeoffs.

In (i) we try to cut down the decision space by eliminating choices that are dominated by others, so if $s_{ij}$ is the score of choice $i$ on attribute $j$ we eliminate choice $b$ if there is another choice $a$ that is uniformly better, that is, if $s_{aj} \geq s_{bj}$ for all $j$ and at least one of these inequalities is strict.

In (ii), we assign priorities to the three attributes, and then order the choices so that choice $a$ is better than choice $b$ if $s_{a1} > s_{b1}$ or $s_{a1}= s_{b1}$ and $s_{a2} > s_{b2}$, etc.

In method (iii) we assign "weights" $w_j > 0$ to attributes $j = 1,2,3$ and then evaluate $S_i = w_1 s_{i1} + w_2 s_{i2} + w_3 s_{i3}$ for each choice $i$. We then take the one having the highest $S_i$, using some other criterion/criteria for tie-breaking if two or more have the same $S$-value. Since the choice of weighs is highly personal---how important to you is price vs. power, etc.---there is no convincing "right" answer. Mr. Smith may make one choice that is optimal for him while Ms. Jones may make another, and neither can argue convincingly that the other is wrong. Unfortunately, that's life when multiple-criterion choices are encountered, unless you are working for a corporation that essentially tells you what weights to use when making a decision.

 

[Ted111]

No, that is not true; they are not decision variables, they are given input data items.

This problem is NOT one of linear programming, unless your decision variables are $x_V$ = fraction of a Volkswagen to purchase, etc. Thus, the simplex method would be of no use.

What you have is a "tradeoff" problem, which is to choose the car having the most desirable bundle of attribute values. Here, the attributes are price, consumption and power. For example, the attribute bundle for a Volkswagen is (16.2, 7.2, 66.0), which specifies the "score" that Volkswagen has on each of the three attributes. Presumably, the scoring system has been devised so that a higher score is better than a lower one, but we are not told that for sure. (Some hard work might have been done already to develop the scores, but we can just use them now as input for further analysis.)

Typical approaches to such problems are (i) determination of the "efficient" set (Pareto-optimality); (ii) Lexicographic ordering; and (ii) linear tradeoffs.

In (i) we try to cut down the decision space by eliminating choices that are dominated by others, so if $s_{ij}$ is the score of choice $i$ on attribute $j$ we eliminate choice $b$ if there is another choice $a$ that is uniformly better, that is, if $s_{aj} \geq s_{bj}$ for all $j$ and at least one of these inequalities is strict.

In (ii), we assign priorities to the three attributes, and then order the choices so that choice $a$ is better than choice $b$ if $s_{a1} > s_{b1}$ or $s_{a1}= s_{b1}$ and $s_{a2} > s_{b2}$, etc.

In method (iii) we assign "weights" $w_j > 0$ to attributes $j = 1,2,3$ and then evaluate $S_i = w_1 s_{i1} + w_2 s_{i2} + w_3 s_{i3}$ for each choice $i$. We then take the one having the highest $S_i$, using some other criterion/criteria for tie-breaking if two or more have the same $S$-value. Since the choice of weighs is highly personal---how important to you is price vs. power, etc.---there is no convincing "right" answer. Mr. Smith may make one choice that is optimal for him while Ms. Jones may make another, and neither can argue convincingly that the other is wrong. Unfortunately, that's life when multiple-criterion choices are encountered, unless you are working for a corporation that essentially tells you what weights to use when making a decision.

Thanks! Ray. That information helps.