The problem of fuzzy goal programming when the importance relation between the fuzzy goals is vague has been investigated by Aköz and Petrovic [1] and followed by Li and Hu [2] and Cheng [3]. 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. [4]. 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 f_i (x) is the i^th linear function of the x vector of decision variables, i = 1 , ⋯ , k . Also, n_i and p_i are the negative and positive deviations, respectively, while m_i is the aspiration level and f i _ is the anti-ideal value for the i^th fuzzy goal constraint. Moreover, b R ˜ r ( i ,   j ) (r = 1, 2, 3) is a binary variable associated with the membership function of the r^th importance relation (slightly, moderately, significantly) of the i^th goal more than the j^th goal; while μ R ˜ r ( i ,   j ) is the membership function of the r^th imprecise relation between the i^th and the j^th 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 i^th 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 i^th fuzzy goal constraint, i.e., if f_i (x) ≥ m_i; or

p i ≤ f i _ − m i , (4)

if the positive deviation is required to be minimized for the i^th fuzzy goal constraint, i.e., if f_i (x) ≤ m_i.

Notably, constraints (3) and (4) correspond to the non-negativity of the membership functions of the fuzzy goal constraints given by Aköz and Petrovic [1].

Proposition: The constraints of the normalized deviations 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.

The GPFH model (Phase I) is solved using the following example that is given by Arenas-Parra et al. [4]:

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 these 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 λ_I 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 constraints of the normalized deviations leads to a worse value of A F I * , which verifies the proposition.

The constraints of the normalized deviations 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.