# Objective function of a linear program with multiple variables

## Main Question or Discussion Point

Hello,
Please I need help to find the objective function of a linear program (attachement : example).
I tried to figure it out from the formula provided in (attachement : formula) but I couldn't understand it, it's written (MIN(lambda)wj) I think it's the key to my question ! ( Full file is attached too )
Any help as little as it could be, would be most welcomed!

V. Vamitha et al., International Journal of Fuzzy Mathematics and Systems 2(3), 217-230 (2012)

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I am not familiar with the notation around the min_lambda w_j ... . But given the statement starts with "for each j" and contains only one w_j the only sensible way to read this for me is "for each j, find a joint solution for w_j and the lambda_jk that minimizes w_j and obeys the additional constraints mentioned."

Thank you Very much Timo for your help, I appreciate it.
But I'm still confused about the objective function formula ...

Ibix
I'm also struggling with the notation, but it looks to me like you haven't shown us the objective function. They equations in the images all look like constraints to me - they're inequalities. The objective function looks to me to be ##w_j##, which is presumably defined somewhere. Although defining constraints in terms of the objective function seems odd to me.

You don't have a link to the paper? A copy on arxiv.org, maybe?

Thank you very much Ibix for your time, actually this set of equations is from the multivariate markov's chain model, here is the link of the full text : https://www.ripublication.com/ijfms/ijfmsv2n3_02.pdf

If the objective is to minimize wj then suitable objective functions would be "f(wj) = wj", "f(wj) = 1.0 * wj" or "f(wj, {lambda_jk}) = 1.0 * wj + 0.0 * lambda_j1 + 0.0 * lambda_j2 + ...".

Thanks Timo for your response, well I already tried that objective function and tried also "0.0 * wj ..+ 1.0 * lambda_j1 + 1.0 * lambda_j2 + + 1.0 * lambda_j3 + 1.0 * lambda_j4 ". but couldn't get the same solution of the paper (((lambda's estimators for the inequalities that I posted are the 4th row of the solution attached)))
so it seems that it's another objective function.

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Possibly. It could also be that there are multiple values for the lambda that lead to the same wj. Are your wj correct? Another option is that you made an error somewhere. I recommend to
1) Manually check that your solution satisfies the constraints. lambda>0 and sum(lambda)=1 can be seen directly from just looking at the numbers (-> the solution of the paper satisfies them); the other two relations are at least simple to verify.
2) Verify that you have the same Xj and B as the paper. On first glance, they are not explicitly given by the paper. So double-check your calculations.

And of course the paper could be wrong.

I really appreciate you help Timo, thanks a lot. for the points that you mentionned :

1) I didn't get lambda's that sum up to 1, I got some odd values, so I removed them all.
2) Yes that's the problem the paper tries to aggregate the results so they removed important steps for the solution, I was able to rewrite the other coefficients in the inequalities following the steps mentionned by the paper, but unfortunately I got different results.

As you said nothing proves that their results are right !