Extended lexicographic goal programming: a unifying approach

Omega 29 (2001) 63±71

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Extended lexicographic goal programming: a unifying approach Carlos Romero* ETS Ingenieros de Montes, Departamento de EconomõÂa y GestioÂn, Universidad PoliteÂcnica de Madrid, Avenida Complutense s/n, 28040, Madrid, Spain Received 15 March 1999; accepted 28 April 2000

Abstract This article starts reviewing the satis®cing philosophy of Goal Programming (GP) and interpreting their solutions from the point of view of the utility theory. This interpretation leads to a very general optimisation structure called Extended Lexicographic Goal Programming (ELGP). It is then demonstrated that there are a signi®cant number of Multiple Criteria Decision Making (MCDM) approaches that, from a logical point of view, can be reduced to the ELGP structure. The paper ends assessing the theoretical and practical advantages of the proposed uni®ed approach. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Goal Programming (GP); Multiple Criteria Decision Making (MCDM)

1. Introduction The rapid development of Multiple Criteria Decision Making (MCDM) has led to an enormous diversity of models and methods. Moreover, it is a common practice to present these models and methods in a disconnected way, giving the wrong impression that each approach is autonomous and completely independent from other MCDM approaches. This paper attempts to demonstrate that such a view of MCDM is misleading, because there are signi®cant links and connections between the di€erent methods. It also illustrates how this kind of isolated philosophy can be counterproductive and negative for the present and for the future of MCDM. It will be argued in this paper that the opposite atti-

* Tel.: +34-91-3366393; fax: +34-91-5439557. E-mail address: [email protected] (C. Romero).

tude of connecting the di€erent approaches is very worthwhile. Indeed, there are a signi®cant number of mutual bene®ts if a bridge between the di€erent MCDM approaches is established. Good examples are the relationships between Goal Programming (GP) and the Analytic Hierarchy Process (AHP). Thus, Gass [1] clearly explains how weights derived from pairwise comparisons through AHP can be fruitfully incorporated into a GP model. Similarly, several authors have also shown how GP can be a useful tool in deriving the weights from Saaty's matrix (e.g., [2]). This paper attempts to take a step further by establishing a uni®ed framework that can lead to a better understanding and presentation of the di€erent MCDM approaches. The forerunners of our research are basically studies by Ignizio [3], Romero ([4], chap. 7) and Romero et al. [5]. In the ®rst, Ignizio develops under the name of MULTIPLEX a uni®ed structure encompassing the following optimisation methods: conventional linear programming, lexicographic vector-

0305-0483/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 4 8 3 ( 0 0 ) 0 0 0 2 6 - 8

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C. Romero / Omega 29 (2001) 63±71

min, Archimedean GP, Lexicographic GP and MINMAX (Chebyshev) GP. In the other two papers, the authors establish analytical links between GP, Compromise Programming and the Reference Point Method. The purpose of this paper is to provide a unifying basis for GP and multiple objective programming models and approaches that embody concepts like the following: 1. Conventional single objective mathematical programming. 2. Non-linear and linear GP with Archimedean weights (i.e., weighted GP). 3. Lexicographic Linear GP. 4. MINMAX (Chebyshev) GP. 5. Extended GP. 6. Reference Point Method (RPM). 7. Compromise Programming (CP). 8. Interactive Weighted Tchebyche€ Procedure (IWT). The purpose of the research is conceptual. Thus, it does not attempt to provide uni®ed algorithmic bases for treating models in multiple objective programming as is done for the interactive philosophy in [6]. The organisation of the paper is the following. First, the satis®cing interpretation of GP is reviewed. After that, the GP satis®cing solutions are interpreted from the point of view of utility theory. This interpretation leads to a rather general GP structure called Extended Goal Programming (EGP), which encompasses Archimedean and MINMAX (Chebyshev) GP variants as particular cases. Finally, it is demonstrated how the EGP formulation can be transformed into a very general optimisation structure called Extended Lexicographic Goal Programming (ELGP). In this way, if ELGP is considered the primary model, it is easy to demonstrate that the above noted MCDM approaches are just secondary models of the ELGP model (i.e., they can be logically reduced to the ELGP structure). The ®nal section analyses and critically assesses the results previously obtained.

2. A satis®cing interpretation of a GP model GP, as any other decision making approach, can be based on di€erent philosophies. Ignizio [7] was the ®rst to interpret and to formulate GP models within a Simonian ``satis®cing'' philosophy. From an analytical point of view, a satis®cing philosophy within a GP context implies that decision-makers (DM) are interested only in minimising the non-achievement of several goals. This fundamental idea leads to the following analytical framework. Let us consider a general setting where q goals are considered. The structure of the generic ith goal is the following:

… gi † fi …x† ‡ ni ÿ pi ˆ ti where fi (x) is the mathematical expression for the ith attribute (i.e., a function of the vector x of decision variables); ti is the target value for the ith attribute, i.e., the achievement level for the ith attribute considered satisfactory by the DM; ni is the negative deviation variable, i.e., quanti®cation of the underachievement of the ith goal; and pi is the positive deviation variable, i.e., quanti®cation of the over-achievement of the ith goal. Once the q goals have been formulated, the next step consists of detecting the unwanted deviation variables. These variables are unwanted in the sense that they are the ones a DM wants to minimise. To illustrate this idea let us consider the following cases: 1. The goal derives from a ``more is better'' attribute (i.e., satisfy fi (x) r ti). In this case, DM does not want under-achievements with respect to target ti. Consequently, the unwanted deviation variable would be the negative one (ni) and would have to be minimised. 2. The goal derives from a ``less is better'' attribute (i.e., satisfy fi (x) R ti). In this case, DM does not want over-achievements with respect to target ti. Consequently, the unwanted deviation variable would be the positive one ( pi) and would have to be minimised. 3. The goal derives from an attribute which wants to be achieved exactly (i.e., satisfy fi(x)=ti). In this case, the DM neither wants over-achievements nor under-achievements with respect to target ti. Hence, both the negative variable ni and the positive one pi are equally unwanted, making it necessary to minimise the sum ni+pi. Let us assume that the unwanted deviation variables for a given problem are as follows: n1 , p2 , . . . , …ni ‡ pi †, . . . , nq That is, goal g1 is of the ®rst type ( f1(x) r t1), goal g2 of the second type … f2 …x†Rt2 †, . . ., goal gi of the third type … fi …x† ˆ ti †, etc. The formulation of a GP model implies the minimisation of a function of the former unwanted deviation variables: min g…n1 , p2 , . . . , …ni ‡ pi †, . . . , nq † The minimisation process can be accomplished with di€erent methods, each one leads to a di€erent GP variant. Basically, there are only three GP variants reported in the literature. The most widely used is Lexicographic Goal Programming (LGP), which attaches pre-emptive priorities to the di€erent goals in order to minimise the unwanted deviation variables in a lexicographic order. The second one is Weighted Goal Pro-

C. Romero / Omega 29 (2001) 63±71

gramming (WGP), which attempts to minimise a composite objective function formed by a weighted sum of unwanted deviation variables. The third is MINMAX (Chebyshev ) Goal Programming, which attempts to minimise the maximum deviation from the stated goals. An extensive survey [8] shows that around 75% of the applications reported use the LGP option, 20% the WGP one and only 5% the MINMAX (Chebyshev) option (see also [9]). In most of the cases the GP variant is chosen without explicitly justifying the reason for the election. It then seems as if the choice of the corresponding GP variant were a mere question related to the analyst's taste or to the availability of the corresponding computer codes. However, the selection of the right GP variant or mix of variants is a crucial matter if we want the GP model to capture the essential features of the reality modelled. In fact, there is a di€erent DM's philosophy on preferences for each GP variant. Hence, the variant or mix of variants which best re¯ect the DM's actual preferences should be chosen [10]. This matter is clari®ed in the next section by providing a utility interpretation for the di€erent GP variants.

65

In order to connect the above utility structure with GP, we resort to the widely used change of variables suggested by Charnes and Cooper [11]: ni ˆ 12 ‰j ti ÿ fi …x† j ‡…ti ÿ fi …x††Š

…2†

pi ˆ 12 ‰j ti ÿ fi …x† j ÿ…ti ÿ fi …x††Š

…3†

By adding (2) and (3) and then by subtracting (3) from (2) we get: ni ‡ pi ˆj ti ÿ fi …x† j

…4†

ni ÿ pi ˆ ti ÿ fi …x†

…5†

By using (4) for the terms in the objective function and (5) for the terms in the constraints, then the utility model (1) becomes equivalent to the following Archimedean GP model: min

q X W i p …ni ‡ pi † p iˆ1

Subject to: 3. A utility interpretation of a GP model The satis®cing interpretation of GP presented in the previous section is not the only possible logical framework for this approach. Thus in this section, GP will be analysed from the point of view of the utility theory. This interpretation provides insights that clarify matters for the right election of a GP variant or mix of variants. To undertake this task, let us start with the following optimisation problem: max ÿ

q X W i p j ti ÿ fi …x† j p iˆ1

Subject to: x2F

…1†

Except for the following, all other variables and symbols of the above structure have been de®ned: Wi is the parameter attached to the discrepancy between the achievement of the ith goal and its aspiration level. This weight re¯ects preferential as well as normalising purposes; p is the real number belonging to the interval [1, 1) or 1; and F is the feasible or opportunity set. If the objective function de®ned in (1) is interpreted as a piecewise utility function, then it has a separable and additive structure in the q objectives. Moreover, for p = 1 a linear structure in the q attributes is obtained, a quadratic structure for p = 2, and so on.

fi …x† ‡ ni ÿ pi ˆ ti

i 2 f1, . . . , qg

ni r0 pi r0 x2F

…6†

It should be noted that the rigorous equivalence between models (1) and (6) also requires nipi=0 for every i. But this condition underlies Eqs. (2) and (3). In fact, if we multiply (2) and (3) term by term, then condition nipi=0 is straightforwardly obtained (see [11]). This condition will be used in Section 5. Therefore, an Archimedean GP model has a clear utility interpretation: ``it implies the maximisation of a separable and additive utility function in the q attributes considered''. The utility function can be linear or not depending upon the value of the parameter p. Dyer [12] proposes a similar utility interpretation of an Archimedean GP model. Therefore, it would seem reasonable to test the separability between attributes before the decision problem is modelled with the help of this GP variant. This task can be undertaken by resorting to a relatively simple test where the DM faces trade-o€s instead of random lotteries [13]. By setting p=1 in (1) the model turns into the following optimisation problem:   min max Wi j ti ÿ fi …x† j iˆ1, ..., q

66

C. Romero / Omega 29 (2001) 63±71

Subject to: x2F

…7†

The utility function in model (7) has a MINMAX structure. In other words, it corresponds to a utility function where the maximum deviation is minimised. By introducing in (7) the change of variables (4) and (5) the following equivalent model is obtained (e.g., [14]): min D Subject to: Wi …ni ‡ pi †RD fi …x† ‡ ni ÿ pi ˆ ti

i 2 f1, . . . , qg

ni r0 pi r0 x2F

…8†

where D is the maximum deviation. As a result, a MINMAX (Chebyshev) GP model also has a clear utility interpretation: ``it implies the minimisation of the maximum disutility, what leads to the optimisation of a MINMAX utility function where the maximum deviation is minimised''.

4. The concept of extended goal programming The utility formulation of the Archimedean and MINMAX (Chebyshev) GP models undertaken in the preceding section suggests the following generalisation: Achievement function:

iˆ1

Subject to …1 ÿ Z †…ai ni ‡ bi pi †RD i 2 f1, . . . , qg

ni r0 pi r0 x2F

5. Extended lexicographic goal programming: primary and secondary models The EGP formulation suggests possible theoretical extensions of its analytical structure. With this purpose in mind, a more general framework that EGP called Extended Lexicographic Goal Programming (ELGP) is proposed in what follows. First, the analytical structure of ELGP is expounded and after that it is demonstrated how an important number of di€erent MCDM approaches can be considered, from a logical point of view, just as a special class of models of this general ELGP formulation. To be more precise, it will be demonstrated how a signi®cant number of MCDM approaches (secondary models ) can be reduced to ELGP ( primary model ). The following formulation for ELGP is proposed:

q X min…1 ÿ Z †D ‡ Z …ai ni ‡ bi pi † p

fi …x† ‡ ni -pi ˆ ti

where parameter ai and bi are the weights re¯ecting preferential and normalising purposes attached to the negative and positive deviation variables of ith goal, respectively. Parameter Z weights the importance attached to the minimisation of the weighted sum of unwanted deviation variables. For Z = 0, we have a MINMAX (Chebyshev) GP model, for Z = 1 an Archimedean GP model and for other values of parameter Z belonging to the interval (0, 1) intermediate solutions between the solutions provided by the two GP options considered. It should be noted that if p = 1 we have linear programming structures from an algorithmic point of view. Structure (9) can be denominated as Extended Goal Programming (EGP) and presents some attractive properties. First, it encompasses the Archimedean and the Chebyshev GP variants in a uni®ed format. Moreover, the utility meaning associated to an EGP model is appealing, at least for the following reasons. It can be said that from a utility perspective the Archimedean and Chebyshev solutions represent two opposite poles. Thus, the Archimedean solution provides the maximum aggregate achievement (maximum eciency ), while the Chebyshev option provides the most balanced solution between achievements of di€erent goals (maximum equity ). The ®rst solution (Archimedean Pole) can be extremely biased towards some of the goals, whereas the second (Chebyshev Pole) can provide poor aggregated performance between the di€erent goals. Precisely for these reasons, there are many cases in which the EGP formulation provides a good compromise between the two opposite views of optimising eciency and equity with regards to the achievements of the goals under consideration.

…9†

Achievement function:

C. Romero / Omega 29 (2001) 63±71

" X Lex min a ˆ l1 D1 ‡ m1 …ai ni ‡ bi pi † p , . . . ,

approaches cited in the introductory section of this paper. The mathematical structure of these approaches appears in the Appendix placed at the end of the paper. Table 1 shows the parameters speci®cation for the di€erent approaches. In fact, by substituting in model (10) its parameter values by the corresponding speci®cations shown in Table 1 the approaches shown in the Appendix are straightforwardly obtained. The following clari®cations should be made with respect to Conventional Mathematical Programming, Reference Point Method, Compromise Programming and Interactive Weighted Tchebyche€ Procedure. Regarding Conventional Mathematical Programming the speci®cations introduced in Table 1 lead initially to the following GP model:

i2h1

lj Dj ‡ mj

X …ai ni ‡ bi pi † p , . . . , i2hj

lQ DQ ‡ mQ

X i2hQ

…ai ni ‡ bi pi † p

#

Subject to lj …ai ni ‡ bi pi †RDj

i 2 hj

fi …x† ‡ ni ÿ pi ˆ ti

i 2 f1, . . . , qg

67

j 2 f1, . . . , Qg

min…a1 n1 ‡ b1 p1 †

ni r0 pi r0

Subject to:

x2F

…10†

Although model (10) looks like the lexicographic version of model (9) its meaning and purpose is very di€erent. Thus, model (10) has been built with the intention of being a primary model for an important number of MCDM approaches. In this sense it is important to remark that parameters mj and Z play a di€erent role. In fact, parameter Z was introduced to implement trade-o€s between eciency and equity through the achievement function of model (9). It will be shown in the next pages how parameter mj performs not only this function but also other di€erent functions in the corresponding reduction process. In (10) hj represents the index set of goals placed in the jth priority level and lj is a control parameter that takes the value 0, when the MINMAX (Chebyshev) option is not considered, otherwise it takes a positive value. It will be shown that the ELGP primary model given by (10) encompasses at least the set of MCDM

f1 …x† ‡ n1 ÿ p1 ˆ t1 x2F

…11†

where t1 is an anchor value. In (11) we have p1=0, because t1 is an anchor value, so we have n1 ˆ t1 ÿ f1 …x†: By substituting this expression in the objective function of (11), the usual conventional single objective mathematical programming model is obtained. As for the Reference Point Method (RPM), the speci®cations introduced in Table 1 lead initially to the following model: min D ‡ e

q X Ki …ni ÿ pi † iˆ1

Subject to:

Table 1 Parameters speci®cation in the extended lexicographic goal programming model lj

mj

ti

p

ai

bi

Approach

0 0 0 1 (1ÿZ ) 1 0 1 1

1 1 1 0 Z e 1 0 Z

t1 ti ti ti ti ti ti ti  t i ˆ ti ‡ e

1 p 1 1 p 1 1 1 1

ai ai ai ai ai Ki Ki Ki Ki

bi bi bi bi bi ÿKi 0 0 0

Conventional Mathematical Programming Archimedean GP (Case Q = 1) Lexicographic GP (Case Qr2) MINMAX (Chebyshev) GP Extended GP Reference Point Method Compromise Programming (L1 bound) Compromise Programming (L1 bound) Interactive Weighted Tchebyche€ Procedure

68

C. Romero / Omega 29 (2001) 63±71

Ki …ni ÿ pi †RD fi …x† ‡ ni ÿ pi ˆ ti

leads to the following initial model for the L1 bound of the compromise set: i 2 f1, . . . , qg

min D

ni r0 pi r0

Subject to:

x2F

…12†

Ki ni RD

However, from the general ith goal we have: ni ÿ pi ˆ ti ÿ fi …x†

…13†

By substituting (13) into model (12), a close relative of the classic RPM formulation introduced by Wierzbicki [15] and presented in the Appendix is obtained. In fact, in classic RPM the weight for over-achievement is set much smaller than the weight for under-achievement. Now, as the ELGP model allows for separate weights (ai and bi) the exact RPM model can be reproduced for a concrete DM problem. It should also be noted that the use of negative weights in a GP model might look strange. However, this practice is valid from a logic point of view and has already been implemented in the literature. Thus, when Ogryczak [16] makes a similar reduction of RPM to a GP structure considers that to debate if a GP model with negative weights is still GP or not is only a scholastic problem ([16], p. 43). Model (12) can present computational problems because of its inherent capacity to provide unbounded solutions. However, this problem can be avoided by imposing constraints ni pi ˆ 0 for every i or by resorting to a GP solver with the facility for basis restriction [17]. In CP, the ®rst speci®cation introduced in Table 1 leads to the following initial model for the L1 bound of the compromise set: min

q X Ki ni

ni r0 pi r0 x2F

…15†

In (15) we again have pi=0 for every i and hence ni ˆ ti ÿ fi …x†: By substituting this expression in the ®rst block of constraints of (15), the usual CP model for the L1 bound of the compromise set shown in the Appendix is obtained (see [18,19]). With respect to the Interactive Weighted Tchebyche€ Procedure (IWT) procedure it should be noted that only a structural comparison will be undertaken. Indeed, it might be regarded as problematic to reduce to an optimisation structure as ELGP interactive methodologies as IWT. However, the comparison has full meaning from a structural point of view. The speci®cations introduced in Table 1 lead to the following initial model: min D ‡ Z

q X Ki ni iˆ1

Subject to:

fi …x† ‡ ni ÿ pi ˆ t i

Subject to:

i 2 f1, . . . , qg

ni r0 pi r0

i 2 f1, . . . , qg

x2F

ni r0 pi r0 x2F

i 2 f1, . . . , qg

Ki ni RD

iˆ1

fi …x† ‡ ni ÿ pi ˆ ti

fi …x† ‡ ni ÿ pi ˆ ti

…14†

In (14) we have pi=0 for every i, because ti is an anchor value, so we have ni ˆ ti ÿ fi …x†: By substituting this expression in the objective function of (14), the usual CP model for the L1 bound of the compromise set shown in the Appendix is obtained (see [18,19]). The second CP speci®cation introduced in Table 1

…16†

In (16) once more, pi=0 for every i, because t is a i strict higher bound, so we have ni ˆ t i ÿ fi …x†: By substituting this expression in the objective function and in the ®rst block of constraints of (16), the usual analytical structure of an IWT model as is shown in the Appendix is obtained (see [20]). Finally, it should also be pointed out that if the cases presented in the rows 4, 5 and 6 of Table 1 the number of priority levels is larger than one, then we

C. Romero / Omega 29 (2001) 63±71

have lexicographic versions of MINMAX (Chebyshev) GP, Extended GP and RPM models, respectively (see [21] for another version of a lexicographic RPM model).

6. Concluding remarks The uni®ed formulation referred to as ELGP and presented in this paper is tentative for the following two reasons. First, it is sensible to conjecture that there are more comprehensive structures, which encompass ELGP as a secondary model. Second, the sample of approaches derived as secondary models of ELGP is not necessarily a complete list and can be enlarged. It should also be noted that from a preferential point of view the proposed ELGP presents some attractive features. Thus, due to its lexicographic character ELGP allows the modelling of problems where preferences among some of the attributes under consideration are non-continuous (i.e., trade-o€s among attributes are not ®nite) which is quite realistic in some problem situations (e.g., [10], p. 196). However, the lexicographic character is not exclusive or even necessary within an ELGP formulation. Thus, within each priority level, continuous preferences are modelled as a balance between additive preferences (eciency ) and MAXMIN preferences (equity ). Although all the constituents of this article are in themselves well-known MCDM methodologies, their integration into a unifying approach serves to clarify the close relationship between them. This kind of link has usually been neglected in the literature. Moreover, the following theoretical and practical points seem to derive from this research: 1. From a theoretical point of view, all the valid results for ELGP are also valid for its secondary models. For instance, the development of a computer code to solve ELGP models seems worthwhile as a uni®ed code applicable to most of the common MCDM approaches. 2. The proposed uni®ed approach stresses similarities between MCDM approaches that can lead to increased clarity and precision in future dialogues, helping to reduce separations between advocates of di€erent approaches. Thus, ELGP can become a useful teaching tool to introduce students to MCDM by avoiding the common presentation based upon a completely disconnected ``pigeon hole'' system of MCDM methods. 3. It seems helpful for practitioners to be aware that, whatever the MCDM model they are building, they are actually formulating a particular case (secondary model ) of the general ELGP structure ( primary

69

model ). In this way, an improved understanding of the linkages between various approaches is provided. 4. It has been argued elsewhere that reliance upon a single GP variant is not generally justi®ed [10,14]. The ELGP structure gives theoretical support to this kind of assertion. In fact, in most real-life cases the correct modelling of a DM's preferences requires a mix of GP variants which take into account the non-continuity of preferences between some goals as well as additive and balanced compromises between others.

Acknowledgements The author wishes to thank Francisco R. FernaÂndez, Kiyotada Hayashi and the reviewers for helpful comments. The English language was checked by Christine Mendez. Preliminary versions of this paper were presented at the First Meeting of the Spanish MCDM Group (Madrid, October 1997) and at The Third Multiobjective Programming and Goal Programming Conference (MOPGP98) (QueÂbec, June 1998). This research was supported by ``ComisioÂn Interministerial de Ciencia y TecnologõÂ a'' and by ``ConsejerõÂ a de EducacioÂn y Cultura, Comunidad de Madrid''.

Appendix A. Analytical structure of some MCDM models (secondary models) 1. Non-linear and linear weights (weighted GP) min

GP

with

Archimedean

q X …ai ni ‡ bi pi † p iˆ1

Subject to: fi …x† ‡ ni ÿ pi ˆ ti ni r0

i 2 ‰f1, . . . , qg

pi r0

x2F 2. Lexicographic Linear GP "X X Lex min a ˆ …ai ni ‡ bi pi †, . . . , …ai ni ‡ bi pi †, i2h1

...,

X i2hQ

i2hj

# …ai ni ‡ bi pi †

70

C. Romero / Omega 29 (2001) 63±71

Subject to: fi …x† ‡ ni ÿ pi ˆ ti

i 2 f1, . . . , qg

ni r0 pi r0

malising factor. 6. Compromise Programming 6.1. L1 bound min

q X Ki ‰ti ÿ fi …x†Š iˆ1

x2F

Subject to:

3. MINMAX (Chebyshev) GP

x2F where Ki is a normalising factor. The above objective function is equivalent to q max Siˆ1 Ki fi …x†, hence if weights Ki are interpreted as parameters we have a vectorial optimisation problem (weighting method ). 6.2. L1 bound

min D Subject to: …ai ni ‡ bi pi †RD fi …x† ‡ ni ÿ pi ˆ ti

i 2 f1, . . . , qg

min D Subject to:

ni r0 pi r0

Ki ‰ti ÿ fi …x†ŠRD

x2F

x2F

4. Extended GP min…1 ÿ Z †D ‡ Z

q X …ai ni ‡ bi pi † p

7. Interactive Weighted Tchebyche€ Procedure

iˆ1

q X Ki fi …x†

Subject to:

min D ÿ Z

…1 ÿ Z †…ai ni ‡ bi pi †RD

Subject to:

fi …x† ‡ ni ÿ pi ˆ ti

Ki ‰t i ÿ fi …x†ŠRD

iˆ1

i 2 f1, . . . , qg

i 2 f1, . . . , qg

x2F

ni r0 pi r0

 where t i ˆ ti ‡ e, e again being a small perturbation term and Ki a normalising factor.

x2F If p = 1 the model has a linear programming structure. If Z = 0 the EGP model turns into a MINMAX (Chebyshev) GP model and if Z = 1 the EGP model turns into a weighted GP model. 5. Reference Point Method min D ÿ e

i 2 f1, . . . , qg

q X

Ki fi …x†

iˆ1

Subject to: Ki ‰ti ÿ fi …x†ŠRD

i 2 f1, . . . , qg

x2F where e is a small perturbation term and Ki a nor-

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