In the paper [Standard goal programming with fuzzy hierarchies: a sequential approach, Soft Computing, First online: 22 March 2015], it has been assumed that the normalized deviations should lie between zero and one. In some cases, this assumption may not be valid. Therefore, additional constraints must be incorporated into the model to ensure that the normalized deviations should not exceed one. This modification is illustrated by the given numerical example.

Fuzzy Goal Programming Imprecise Hierarchy Normalized Deviations
1. Introduction

The problem of fuzzy goal programming when the importance relation between the fuzzy goals is vague has initially been investigated by Aköz and Petrovic  and followed by Li and Hu  and Cheng  . A suggested sequential approach in fuzzy goal programming, when the importance hierarchy of the goals is imprecise, has been presented by Arenas-Parra et al.  . In their article, the model of goal programming with fuzzy hierarchy (GPFH) is given as

Maximize   λ ∑ i = 1 k ( 1 − n i m i − f i _ ) + ( 1 − λ ) ∑ ( i , j ) = 1     i ≠ j k ∑ r = 1 3 b R ˜ r ( i , j ) μ R ˜ r ( i , j )

subject to:

f i ( x ) + n i − p i = m i ,     i = 1 , ⋯ , k ,

1 − ( n i m i − f i _ − n j m j − f j _ ) ≥ μ R ˜ 1 ( i , j ) ,       if   b R ˜ 1 ( i , j ) = 1 ,

1 − ( n i m i − f i _ − n j m j − f j _ ) 2 ≥ μ R ˜ 2 ( i , j ) ,       if   b R ˜ 2 ( i , j ) = 1 , (1)

n j m j − f j _ − n i m i − f i _ ≥ μ R ˜ 3 ( i , j ) ,       if   b R ˜ 3 ( i , j ) = 1 ,

0 ≤ μ R ˜ r ( i ,   j ) ≤ 1 ,     r = 1 , 2 , 3 ,

n i , p i ≥ 0 ,     n i × p i = 0 ,     i = 1 , ⋯ , k ,

x ∈ X ,

where 0 ≤ λ ≤ 1, and fi (x) is an ith linear function of an x vector of decision variables, i = 1 , ⋯ , k . Also, ni and pi are the negative and positive deviations, respectively, where mi is the aspiration level and f i _ is the anti-ideal value for the ith fuzzy goal constraint. Moreover, b R ˜ r ( i ,   j ) (r = 1, 2, 3) is a binary variable associated with the membership function of the rth importance relation (slightly, moderately, significantly) of the ith goal more than the jth goal; while μ R ˜ r ( i ,   j ) is the membership function of the rth imprecise relation between the ith and the jth fuzzy goals. Finally, X is a set of system constraints which define the feasible set of the problem.

This model is implemented for each class of Phase I. Hence, it is assumed that the normalized deviation for the ith fuzzy goal constraint must lie between zero and one i.e.,

0 ≤ n i / ( m i − f i _ ) ≤ 1. (2)

This assumption may be violated, especially when the anti-ideal value is close to the aspiration level. In this case, n i / ( m i − f i _ ) may exceed one, due to a small denominator value, which means that the value of the achieved goal is worse than the anti-ideal value of that goal. Accordingly, for each class, the following constraints should be incorporated in the GPFH model:

n i ≤ m i − f i _ , (3)

if the negative deviation is required to be minimized for the ith fuzzy goal constraint, i.e., if fi (x) ≥ mi; or

p i ≤ f i _ − m i , (4)

if the positive deviation is required to be minimized for the ith fuzzy goal constraint, i.e., if fi (x) ≤ mi.

Notably, constraints (3) and (4) correspond to the non-negativity of the membership functions of the fuzzy goal constraints given by Aköz and Petrovic  .

Proposition: The normalized deviations constraints might limit the feasible set of the problem. This may worsen the value of the achievement function of each class, and therefore affect the results of the suggested sequential approach.

In the next section, this note is verified by the given illustrative example.

2. Illustrative Example

The GPFH model (Phase I) is solved using the following example that is given by Arenas-Parra et al.  :

Goal 1: 4 x 1 + 2 x 2 + 8 x 3 + x 4 ≤ 35

Goal 2: 4 x 1 + 7 x 2 + 6 x 3 + 2 x 4 ≥ 100

Goal 3: x 1 − 6 x 2 + 5 x 3 + 10 x 4 ≥ 120

Goal 4: 5 x 1 + 3 x 2 + 2 x 4 ≥ 70

Goal 5: 4 x 1 + 4 x 2 + 4 x 3 ≥ 40

subject to:

7 x 1 + 5 x 2 + 3 x 3 + 2 x 4 ≤ 98 , 7 x 1 + x 2 + 2 x 3 + 6 x 4 ≤ 117 , x 1 + x 2 + 2 x 3 + 6 x 4 ≤ 130 , 9 x 1 + x 2 + 6 x 4 ≤ 105 , x i ≥ 0 ,   i = 1 , ⋯ , 4 , } X

where Class I contains goals (1, 2, and 4). Accordingly, the assumed anti-ideal values for those goals are f 1 _ = 261.33 , f 2 _ = 0 , f 4 _ = 0 . Also, the GPFH model for Class I assumes that Goal 1 is moderately more important than Goal 2; and Goal 2 is moderately more important than Goal 4. Finally, the parameter λI is set equal to 0.8.

Thus, the model for Class I is as follows:

Maximize   A F I = λ I ( 1 − P 1 226.33 + 1 − n 2 100 + 1 − n 4 70 ) + ( 1 − λ I ) [ μ R ˜ 2 ( 1 , 2 ) + μ R ˜ 2 ( 2 , 4 ) ]

subject to:

4 x 1 + 2 x 2 + 8 x 3 + x 4 + n 1 − p 1 = 35 ,

4 x 1 + 7 x 2 + 6 x 3 + 2 x 4 + n 2 − p 2 = 100 ,

5 x 1 + 3 x 2 + 2 x 4 + n 4 − p 4 = 70 ,

1 − ( p 1 226.33 − n 2 100 ) 2 ≥ μ R ˜ 2 ( 1 , 2 ) ,

1 − ( n 2 100 − n 4 70 ) 2 ≥ μ R ˜ 2 ( 2 , 4 ) ,

0 ≤ μ R ˜ 2 ( 1 , 2 ) ≤ 1 ,       0 ≤ μ R ˜ 2 ( 2 , 4 ) ≤ 1 ,

n k , p k ≥ 0 ,       n k × p k = 0 ,     k = 1 , 2 , 4 ,

x ∈ X .

The given note is verified by just resolving the GPFH model for Class I in Phase I. Assume that the anti-ideal values of the first and the fourth fuzzy goal constraints f 1 _ and f 4 _ are 40 and 63 instead of 261.33 and 0, respectively. In this case, the normalized p1 is p1/5, while the normalized n4 becomes n4/7.

Then, the solution obtained is: μ R ˜ 2 ( 1 ,   2 ) = 0.463 , μ R ˜ 2 ( 2 ,   4 ) = 1 , p1 = 0.375, n2 = 0, n4 = 9, G1 = 35.375, G2 = 100, G4 = 61, A F I * = 1.604 . Hence, n4/7 = 1.286, which is greater than 1.

Accordingly, by incorporating the following three constraints:

p 1 ≤ 5 ,

n 2 ≤ 100 ,

n 4 ≤ 7 ,

and by solving the model, the solution becomes: μ R ˜ 2 ( 1 ,   2 ) = 0.325 , μ R ˜ 2 ( 2 ,   4 ) = 1 , p1 = 1.750, n2 = 0, n4 = 7, G1 = 36.750, G2 = 105, G4 = 63, A F I * = 1.585 .

It is realized that incorporating the normalized deviations constraints leads to a worse value of A F I * , which verifies the proposition.

3. Conclusion

The normalized deviations constraints must be included in the GPFH model in all classes of Phase I as well as in Phase II to ensure that the achieved value of each goal should never become worse than the anti-ideal value of that goal.

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