Dear Milan Dordevic, Rade Tešić, Srdjan Todorović , Miloš Jokić , Dillip Kumar Das, Željko Stević, and Sabahudin Vrtagic

Reference is made to your paper

“Development of Integrated Linear Programming Fuzzy‐Rough MCDM Model for Production Optimization”

I read it and my comments are:

1- I n the abstract you say “Exactly such a problem is solved in this paper, which integrates linear programming and a Multi‐Criteria Decision‐Making (MCDM) model”

In reality, Linear Programming (LP) is part of MCDM. As a matter of fact, it was the first method of MCDM, created by Kantorovich in 1940.

In “First, linear programming was applied to optimize production and several potential solutions lying on the line segment AB were obtained”

This is correct, and some times it happens, but only when the objective function has the same slope as a criterion. This is your case, and a-b t is a Pareto frontier, with infinite optimal solutions between a and b, if the alternatives are not finite.

2. In page 2 “This model includes qualitative and quantitative indicators, which is an advantage considering that the disadvantage of various multi‐objective programming models is that they are basically mathematical and often ignore qualitative and subjective factors”

In reality, practically all MCDM methods work with quantitative and qualitative criteria, however not LP, that works only with quantitative criteria or indicators, as you call them.

3. In page 2 “fuzzy analytic hierarchy process”

You can’t apply fuzzy in the AHP method. It was expressly said in writing by Saaty, its creator, because AHP is already fuzzy.

4- In “I” Figure 1, in my opinion, there is a sequence mistake, since determination of criteria must precede LP, not as is shown in the figure. You can’t solve a LP scenario if you don’t have the criteria.

5- In “IV” Figure 1, what is the gain in comparing rankings from different methods addressing the same problem? What information can you extract from this comparison?

6- In page 2 “When applying LP for the optimization and management of production processes in this special case, several potential solutions are obtained instead of one which is usually the case”

Again, LP can give several optimal solutions ONLYwhen the mono-objective function is parallel to a criterion. LP was designed to give, if it exists, only one optimal solution, like maximize benefits, or minimize costs, but not the two at the same time. However, you can use maximize a benefit or minimize a cost, at the same time, if they are criteria. The method will try to find a solution that balances both criteria, that is, it will find a compromise solution.

By the way, if you use LP, you don’t need weights. These are determined in each iteration of the LP method. DM preferences can be applied after a mathematically correct results are reached.

7- What are ‘Rough MCDM methods”? You did not explain this concept. The same for rough numbers. Remember that not all readers in RG , probably most, are not mathematicians

8- In page 9, Figure 2 where is the objective function Z?. In reality, it coincides with criterion C2

Look at your equations. Criteria C2 or Equation 2 is:

C2 = A + 0.5 B = 3000

Look at your objective equation

Z = 2000 A + 1000 B

Z = A + 0.5 B

The only difference is that C2 has a goal (3000), i.e., it is definite, while Z is not.

The Figure is correct, but the objective function must be identified, if not, the reader will be asking where is it.

Z can be displaced to the right parallel to itself, because it is maximizing. Since it is indefinite it can take any value. Suppose that you assign it the value 3000. It means that whatever its initial position in the A-B coordinates system, at that value Z equals C2, which is what you have in your diagram.

Thus, this parallelism between Z and C2 has been forced by establishing that Z and C2 have the same slope (1 and 0.5). Therefore a-b constitutes a Pareto Front where all pairs of A and B are optimal.

But remember that this is a particular case. Most PL problems determine the optimal value when the Z line tangents one vertex of the polygon.

9- In page 9 “The optimal value of the objective function is unique because it is six million regardless of how many products A and how many products B will be produced, but there are infinitely many admissible solutions that provide this function value”

Exactly

10- In page 9 “Since we have as a solution many points that represent the optimal solution, a multicriteria model can be applied further”

I don’t understand. You already applied MCDM using LP. Why do you need to apply another method?

What is the ‘t’ value? In may opinion you should explain that it is a parameter or percentage.

11- On page 10 “3.3. Determining the Significance of Criteria Using the IMF SWARA Method”

What do you need that for if you already have the solution?

12- There is something that puzzles me, and it is why to develop so a complex procedure when the same result of A21, can be reached just putting ‘=’ instead of ‘≤’, by indicating in this inequation of yors.

1.5 x1 + 1.5 x2 = 6000

Since x1 = x2, I don’t see why you say that the best alternative is x1.

Obviously, the authors must have had reasons to follow a complex mechanism, when it can be replaced by a simple operation. I would like to hear from them about this.

In my opinion the article is very valuable, but very difficult to understand, mainly because it appears that the authors take for granted that readers don’t need any clarification.

Hope it helps

Nolberto Munier