A Note on Standard Goal Programming with Fuzzy Hierarchies: A Sequential Approach ()

Maged George Iskander^{}

Faculty of Business Administration, Economics and Political Science, The British University in Egypt, El-Sherouk City, Egypt.

**DOI: **10.4236/ajor.2016.61009
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Faculty of Business Administration, Economics and Political Science, The British University in Egypt, El-Sherouk City, Egypt.

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.

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Iskander, M. (2016) A Note on Standard Goal Programming with Fuzzy Hierarchies: A Sequential Approach. *American Journal of Operations Research*, **6**, 71-74. doi: 10.4236/ajor.2016.61009.

1. Introduction

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

$\text{Maximize}\text{\hspace{0.17em}}\lambda {\displaystyle {\sum}_{i=1}^{k}\left(1-\frac{{n}_{i}}{{m}_{i}-\underset{\_}{{f}_{i}}}\right)}+\left(1-\lambda \right){\displaystyle {\sum}_{\begin{array}{l}\left(i,j\right)=1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}i\ne j\end{array}}^{k}{\displaystyle {\sum}_{r=1}^{3}{b}_{{\stackrel{\u02dc}{R}}_{r}\left(i,j\right)}{\mu}_{{\stackrel{\u02dc}{R}}_{r}\left(i,j\right)}}}$

subject to:

${f}_{i}\left(x\right)+{n}_{i}-{p}_{i}={m}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,k,$

$1-\left(\frac{{n}_{i}}{{m}_{i}-\underset{\_}{{f}_{i}}}-\frac{{n}_{j}}{{m}_{j}-\underset{\_}{{f}_{j}}}\right)\ge {\mu}_{{\stackrel{\u02dc}{R}}_{1}\left(i,j\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{b}_{{\stackrel{\u02dc}{R}}_{1}\left(i,j\right)}=1,$

$\frac{1-\left(\frac{{n}_{i}}{{m}_{i}-\underset{\_}{{f}_{i}}}-\frac{{n}_{j}}{{m}_{j}-\underset{\_}{{f}_{j}}}\right)}{2}\ge {\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(i,j\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{b}_{{\stackrel{\u02dc}{R}}_{2}\left(i,j\right)}=1,$ (1)

$\frac{{n}_{j}}{{m}_{j}-\underset{\_}{{f}_{j}}}-\frac{{n}_{i}}{{m}_{i}-\underset{\_}{{f}_{i}}}\ge {\mu}_{{\stackrel{\u02dc}{R}}_{3}\left(i,j\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{b}_{{\stackrel{\u02dc}{R}}_{3}\left(i,j\right)}=1,$

$0\le {\mu}_{{\stackrel{\u02dc}{R}}_{r}\left(i,\text{\hspace{0.17em}}j\right)}\le 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}r=1,2,3,$

${n}_{i},{p}_{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{n}_{i}\times {p}_{i}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,k,$

$x\in X,$

where 0 ≤ *λ *≤ 1, and *f _{i }*(

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

$0\le {n}_{i}/\left({m}_{i}-\underset{\_}{{f}_{i}}\right)\le 1.$ (2)

This assumption may be violated, especially when the anti-ideal value is close to the aspiration level. In this case, ${n}_{i}/\left({m}_{i}-\underset{\_}{{f}_{i}}\right)$ 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:

${n}_{i}\le {m}_{i}-\underset{\_}{{f}_{i}},$ (3)

if the negative deviation is required to be minimized for the *i** ^{th}* fuzzy goal constraint,

${p}_{i}\le \underset{\_}{{f}_{i}}-{m}_{i},$ (4)

if the positive deviation is required to be minimized for the *i** ^{th}* fuzzy goal constraint,

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.

2. Illustrative Example

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

Goal 1: $4{x}_{1}+2{x}_{2}+8{x}_{3}+{x}_{4}\le 35$

Goal 2: $4{x}_{1}+7{x}_{2}+6{x}_{3}+2{x}_{4}\ge 100$

Goal 3: ${x}_{1}-6{x}_{2}+5{x}_{3}+10{x}_{4}\ge 120$

Goal 4: $5{x}_{1}+3{x}_{2}+2{x}_{4}\ge 70$

Goal 5: $4{x}_{1}+4{x}_{2}+4{x}_{3}\ge 40$

subject to:

$\begin{array}{l}7{x}_{1}+5{x}_{2}+3{x}_{3}+2{x}_{4}\le 98,\\ 7{x}_{1}+{x}_{2}+2{x}_{3}+6{x}_{4}\le 117,\\ {x}_{1}+{x}_{2}+2{x}_{3}+6{x}_{4}\le 130,\\ 9{x}_{1}+{x}_{2}+6{x}_{4}\le 105,\\ {x}_{i}\ge 0,\text{\hspace{0.17em}}i=1,\cdots ,4,\end{array}\}X$

where *Class I *contains goals (1, 2, and 4). Accordingly, the assumed anti-ideal values for these goals are
$\underset{\_}{{f}_{1}}=261.33$ ,
$\underset{\_}{{f}_{2}}=0$ ,
$\underset{\_}{{f}_{4}}=0$ . 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:

$\text{Maximize}\text{\hspace{0.17em}}A{F}_{I}={\lambda}_{I}\left(1-\frac{{p}_{1}}{226.33}+1-\frac{{n}_{2}}{100}+1-\frac{{n}_{4}}{70}\right)+\left(1-{\lambda}_{I}\right)\left[{\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(1,2\right)}+{\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(2,4\right)}\right]$

subject to:

$4{x}_{1}+2{x}_{2}+8{x}_{3}+{x}_{4}+{n}_{1}-{p}_{1}=35,$

$4{x}_{1}+7{x}_{2}+6{x}_{3}+2{x}_{4}+{n}_{2}-{p}_{2}=100,$

$5{x}_{1}+3{x}_{2}+2{x}_{4}+{n}_{4}-{p}_{4}=70,$

$\frac{1-\left(\frac{{p}_{1}}{226.33}-\frac{{n}_{2}}{100}\right)}{2}\ge {\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(1,2\right)}$ ,

$\frac{1-\left(\frac{{n}_{2}}{100}-\frac{{n}_{4}}{70}\right)}{2}\ge {\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(2,4\right)}$ ,

$0\le {\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(1,2\right)}\le 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le {\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(2,4\right)}\le 1,$

${n}_{k},{p}_{k}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{n}_{k}\times {p}_{k}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,2,4,$

$x\in X.$

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
$\underset{\_}{{f}_{1}}$ and
$\underset{\_}{{f}_{4}}$ are 40 and 63 instead of 261.33 and 0, respectively. In this case, the normalized *p*_{1} is *p*_{1}/5, while the normalized *n*_{4} becomes *n*_{4}/7.

Then, the solution obtained is:
${\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(1,\text{\hspace{0.17em}}2\right)}=0.463$ ,
${\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(2,\text{\hspace{0.17em}}4\right)}=1$ , *p*_{1} = 0.375, *n*_{2} = 0,* n*_{4} = 9, *G*_{1} = 35.375, *G*_{2} = 100, *G*_{4} = 61,
$A{F}_{I}^{*}=1.604$ . Hence, *n*_{4}/7 = 1.286, which is greater than 1.

Accordingly, by incorporating the following three constraints:

${p}_{1}\le 5,$

${n}_{2}\le 100,$

${n}_{\text{4}}\le \text{7},$

and by solving the model, the solution becomes:
${\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(1,\text{\hspace{0.17em}}2\right)}=0.325$ ,
${\mu}_{{\stackrel{\u02dc}{R}}_{2}\left(2,\text{\hspace{0.17em}}4\right)}=1$ , *p*_{1} = 1.750, *n*_{2} = 0,* n*_{4} = 7, *G*_{1} = 36.750, *G*_{2} = 105, *G*_{4} = 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.

3. Conclusion

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.

References

- 1. Aköz, O. and Petrovic, D. (2007) A Fuzzy Goal Programming Method with Imprecise Goal Hierarchy. European Journal of Operational Research, 181, 1427-1433. http://dx.doi.org/10.1016/j.ejor.2005.11.049
- 2. Li, S. and Hu, C. (2009) Satisfying Optimization Method Based on Goal Programming for Fuzzy Multiple Objective Optimization Problem. European Journal of Operational Research, 197, 675-684.http://dx.doi.org/10.1016/j.ejor.2008.07.007
- 3. Cheng, H.-W. (2013) A Satisficing Method for Fuzzy Goal Programming Problems with Different Importance and Priorities. Quality and Quantity, 47, 485-498. http://dx.doi.org/10.1007/s11135-011-9531-0
- 4. Arenas-Parra, M., Bilbao-Terol, A. and Jiménez, M. (2015) Standard Goal Programming with Fuzzy Hierarchies: A Sequential Approach. Soft Computing. http://dx.doi.org/10.1007/s00500-015-1644-2

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
Aköz, O. and Petrovic, D. (2007) A Fuzzy Goal Programming Method with Imprecise Goal Hierarchy. European Journal of Operational Research, 181, 1427-1433. http://dx.doi.org/10.1016/j.ejor.2005.11.049 |

[2] |
Li, S. and Hu, C. (2009) Satisfying Optimization Method Based on Goal Programming for Fuzzy Multiple Objective Optimization Problem. European Journal of Operational Research, 197, 675-684. http://dx.doi.org/10.1016/j.ejor.2008.07.007 |

[3] |
Cheng, H.-W. (2013) A Satisficing Method for Fuzzy Goal Programming Problems with Different Importance and Priorities. Quality and Quantity, 47, 485-498. http://dx.doi.org/10.1007/s11135-011-9531-0 |

[4] |
Arenas-Parra, M., Bilbao-Terol, A. and Jiménez, M. (2015) Standard Goal Programming with Fuzzy Hierarchies: A Sequential Approach. Soft Computing. http://dx.doi.org/10.1007/s00500-015-1644-2 |

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