Weighted-additive fuzzy multi-choice goal programming (WA-FMCGP) for supporting renewable energy site selection decisions

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Weighted-additive Fuzzy Multi-choice Goal Programming (WA-FMCGP) for Supporting Renewable Energy Site Selection Decisions Amin Hocine , Zheng-Yun Zhuang , Noureddine Kouaissah , Der-Chiang Li PII: DOI: Reference:

S0377-2217(20)30127-2 https://doi.org/10.1016/j.ejor.2020.02.009 EOR 16329

To appear in:

European Journal of Operational Research

Received date: Accepted date:

30 March 2019 8 February 2020

Please cite this article as: Amin Hocine , Zheng-Yun Zhuang , Noureddine Kouaissah , Der-Chiang Li , Weighted-additive Fuzzy Multi-choice Goal Programming (WA-FMCGP) for Supporting Renewable Energy Site Selection Decisions, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.02.009

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Research Highlights 

Proposes the WA-FMCGP model, hybridising FGP, MCGP and WGP by taking a novel FGP+MCGP approach.



Validates the model using an exemplar in F-MODM literature having decision contexts with imprecise goals.



Compares the results with those using FP+MCGP modelling approach; identifies positive consequences of using FGP+MCGP.



Supports empirical decisions of selecting the best location for wind-farm expansion using meta WA-FMCGP.



Provides F-MODM methods for renewable-energy site-selection (RESS), compared to the MADM models used thus far.

Weighted-additive Fuzzy Multi-choice Goal Programming (WA-FMCGP) for Supporting Renewable Energy Site Selection Decisions

Amin Hocinea, Zheng-Yun Zhuangb,, Noureddine Kouaissaha and Der-Chiang Lic a

BEAR-lab, Rabat Business School, Université Internationale de Rabat, Parc Technopolis, Rabat-Shore, Morocco b,* Department of Civil Engineering, National Kaohsiung University of Science and Technology, 415 Jiangong Road, Kaohsiung, Taiwan; Tel: +886-7-3814526 ext. 15240 c Department of Industrial and Information Management, National Cheng Kung University, Tainan, Taiwan

Abstract This paper proposes a novel weighted-additive fuzzy multi-choice goal programming (WA-FMCGP) model for the imprecise decision context wherein several conflicting goals are present but each goal has multiple-choice aspiration levels (MCALs) and, around them, the fuzzinesses are expressed in terms of membership functions (MFs). The main contribution of this model is its use of an objective function that minimises the weighted-additive summation of the normalised deviations; thus, the model can adopt any minimisation process from any goal programming (GP) variant. The advantages of this FGP-MCGP (fuzzy GP – multi-choice GP) model are shown by using it to solve a numerical example from F-MODM (fuzzy MODM) literature and comparing the results with those of a recent FP-MCGP (fuzzy programming – multi-choice GP) study. The application of the model is also verified using real data (i.e., it can model and support renewable energy site selection (RESS) where the decision context is imprecise). As WA-FMCGP is largely a MODM model, through its application, this study also provides a supplementary method in contrast to the multi-attribute decision-making (MADM) model applications used thus far for RESS. Keywords: fuzzy goal programming; multi-choice goal programming; weighted goal programming; imprecise goals; renewable energy planning; civil engineering.

1. Introduction In the practice of multi-objective decision-making (MODM), the presence of uncertain goals (i.e., the decision goal is uncertain) and/or imprecise goals (i.e., the target value of a goal, or „goal target‟, is imprecise) are common. The goal target is usually on the right-hand side of a criterion constraint, and it is commonly referred to as an „aspiration level‟ (AL) in goal programming (GP), which is regarded  Corresponding author: Z.-Y. Zhuang List of e-mail addresses (by author): [email protected] (A. Hocine); [email protected] and [email protected] (Z-Y. Zhuang); [email protected] (N. Kouaissah); [email protected] (D-C. Li).

as a major branch of MODM. Because there is practical value in fuzzy set theory (Zadeh, 1965; Bellman and Zadeh, 1970) being able to offer „fuzzy ability‟ (i.e., the ability to express goal fuzziness) around the goal target, a variety of fuzzy MODM (F-MODM) models have been proposed in the literature. Generally speaking, these F-MODM models can be categorised into one of two types: 1) fuzzy goal programming (FGP) models and 2) fuzzy (linear/non-linear) programming (F(L/NL)P) models. The most salient feature of FGP is that it intends to minimise the value of the objective function where there is a summation, in some form, of the deviational variables that are introduced to solve a GP model. These „forms‟ include formulating by reference to the weighted-additive modelling (WAM) concept. The concept of WAM can be applied to various MODM models. When the concept is applied to GP, it produces a weighted goal programming (WGP) model. When this concept is then applied to FGP, the resulting model can be referred to as a „W-FGP‟ model. In contrast, F(L/NL)P maximises the value of the objective function where there is a summation, in some form, of the utility interpretations of the membership functions (MFs) that are employed to solve a fuzzy programming (FP) model (i.e., it derives the utility value from the functional membership value for each goal and then maximises the aggregated utility). Another series of studies have been published beginning in 2007, specifically, studies of multichoice goal programming (MCGP) modelling. The fundamental concept of MCGP is the multi-choice aspiration level (MCAL), which refers to the fact that a decision-maker (DM) can sometimes state several discrete values as the possibly desired ALs for each goal (i.e., the MCALs); this is again due to the uncertain decision context. In MCGP modelling, these possible and acceptable alternative target values offer a „multi-choice‟ (that is, a one-among-many choice) to find the (more or most) satisfying solution set during the solution determination process. So, in a broader sense, MCGP can be regarded as another F-MODM approach, except that its decision application context and adoption logic are heterogeneous. MCGP models are thus quite different from the other two series of F-MODM models, which utilise fuzzy set theory or the concept of MFs directly. Given such uniqueness, scholars have devoted themselves to integrating the MCAL concept into the series of F(L/NL)P models for FMODM. However, due to the recency of MCGP, relevant models have only been presented since 2012, and these models have not utilised the WAM concept when the utilities for the goals are interpreted and aggregated. Such models are called „FP+MCGP‟ models in this study. Tabrizi et al. (2012) were the first authors to present this idea, and the model that they proposed was named the „FMCGP‟ model. Their contribution was in the initial integration of MCGP and F(L/NL)P into a unified model that started to allow two or more „targets‟ (the „MCALs‟, alternatively) for each goal and to enable fuzziness around all possible targets (MCALs) of the goal. However, this initial FP+MCGP model only formulated one type of MF, the triangular MF (TMF), to express the fuzziness of the DM‟s goal.

In addition, although the model supported the setting of different triangular bottom lengths for the MFs of the different possible targets (MCALs) of the same goal, the only permitted MF shape was the symmetric TMF. From these facts, it may be reasonably supposed that the work through which this initial FP+MCGP model was established was strongly affected by the existing MCGP series of FMODM studies at that time, in spite of the base model having originated from the „F(L/NL)P‟ series of F-MODM studies. The possible flaws in Tabrizi et al.‟s model were criticised but also corrected to some extent in a later study by Mouslim et al. (2014). Specifically, Mouslim et al. improved upon Tabrizi et al.‟s model by referencing a summary produced in 2008 of the „four typical types of piecewise linear MFs‟ (Yaghoobi et al., 2008). In addition to the two types of MFs (i.e., increasing and decreasing) that had commonly been supported by the models of the F(L/NL)P series, Mouslim et al.‟s model supported the other two types of MFs: the TMF type as inherited from Tabrizi et al.‟s model and the trapezoidal type. Moreover, for these latter two types, the model allowed for the manifestation and formulation of „maximal admissible violations‟ (MAVs) for each MF. These MAVs determined different hypotenuses and, thus, the different MF shapes that are set for the MCAL of each goal. This model was called the fuzzy goal programming with multiple target levels (FGP-MTL) model by its authors. However, both the initial FMCGP model and the more recent FGP-MTL model mentioned above, if strictly scrutinised, are not FGP models but, rather, „F(L/NL)P models‟ because the objective function of both models maximises total utility, rather than minimizing total deviation as a normal GP model does. Therefore, unlike the works in the F(L/NL)P series that integrated MCGP (i.e., those involving FP+MCGP), this study aims to start from FGP in order to propose a weighted-additive fuzzy multichoice goal programming (WA-FMCGP) model. This proposed model is distinct from previously presented models for the following reasons: i. The proposed WA-FMCGP model aligns (back) with the intrinsic core aim of GP, the objective function of which is to minimise the „sum of deviations‟. Given that the fuzzinesses around the MCALs are also modelled, it constitutes a new model member for the „FGP‟ series in F-MODM. ii. The WA-FMCGP model integrates WGP, FGP, and MCGP faithfully, rather than taking the FP+MCGP approach. In other words, it is a „true GP model‟. As a consequence, the model can adopt any minimisation procedure (objective function) of any „GP variants‟, such as GP with satisfaction functions (Martel and Aouni, 1990) and meta GP (Rodriguez-Uria et al., 2002) models. This feature makes it more adept at incorporating DM preferences. For example, in the meta GP context, the model can manipulate the provided preferences toward the meta goals that are set over the multi-choice, fuzzy goals, thereby improving on the drawbacks of missing goal priorities in existing FP+MCGP models while incorporating MCGP with F-MODM, in addition to considering the WAM concept by the integration with WGP. After establishing the WA-FMCGP model, this paper will use it to solve a numerical example

from the literature with goal targets that are quite imprecise. Specifically, it adopts a bicycle manufacturer‟s product-mix decision problem involving fuzzy goals and multiple targets, which was previously solved in a study by Mouslim et al. (2014) using the FP+MCGP approach. In solving the same problem using the rudimentary version of WA-FMCGP without WAM (i.e., for a fair comparison basis), the quality of the solution is improved in terms of several measures, even when using the „total utility‟ measure which is addressed by the FP+MCGP models but is now a sub-product of the WA-FMCGP model. Thus, confidence is gained for further empirical verification of the proposed model, such as in solving a renewable energy site selection (RESS) problem with supplied real data. The remaining paper is organised as follows. Section 2 reviews existing literature regarding the different approaches used for modelling GP problems in uncertain and imprecise decision contexts. The WA-FMCGP method is developed in Section 3. Section 4 provides a comparison. A real case study of RESS is made in Section 5. Finally, conclusions are summarised, and areas for future research are proposed in Section 6.

2. Literature Review A key element of a GP model is that its objective function is an achievement function that is, in fact, a mathematical expression containing the unwanted deviation variables (Romero, 2004; Aouni et al., 2012), rather than the values of criterion functions themselves. Therefore, each type of achievement function leads to a different solution, as the model can become a different „GP variant‟ (Romero, 1991; Tamiz et al., 1998; Chang and Zhuang, 2014). This attribute is important since the results derived (e.g., the solution to the decision vector) from a GP model are generally very sensitive to the type of the achievement function chosen. A well-known variant of GP is the WGP model (see Jones and Tamiz 2010), which integrates the WAM concept mentioned above using a standard GP model (Charnes and Cooper, 1961; Lee, 1972; Ignizio, 1985) as its basis, as in the following formulae: (WGP) K

min

 ( n   p ) i 1

i i

i

i

s.t. ( AX )i  ni  pi  bi , i  1,..., K ni , pi  0, i  1,..., K

X  Cs .

where ni and pi are, respectively, the negative and positive deviations (in terms of moderating slack and surplus variables which are also to be solved in GP) between the i-th goal‟s criteria

function, (AX)i, and the AL, bi, as set by the DM; X is the decision vector including all of the variables that are to be decided numerically; CS is the feasible solution set; if ni ≠ 0, αi = wi / ki (or αi ni = 0 otherwise), and if pi ≠ 0, βi = wi / ki (or βi pi = 0 otherwise), wherein wi and ki are, respectively, the preferential weight and the normalisation constant that are assigned while achieving (associated with) the i-th goal, as specified by the DM. In practice, the improper selection of GP variants leads to the problem of an ill-defined objective function; that is, one that does not reflect the real preference structure of the DM and does not provide a solution that is truly acceptable to the DM. Therefore, it is supposed that a model which is derived from GP will „inherit‟ such a feature (i.e., the suitable formulation of the objective function is important). However, can this concept be widened to work in F-MODM, where a model that is entirely based on GP is constructed? Moreover, will utilising this concept to construct the FGP+MCGP model improve its objective function? These questions will be examined in subsequent sections. Furthermore, when there are multiple goals present, it is difficult to obtain clear information regarding the real objectives for each of a DM‟s goals through interviews; this is one of the major drawbacks when using standard GP (Lai and Hwang, 1994). In real-life applications, there are many instances in which a DM does not have sufficient information for some parameters, in particular, the ALs to be set for the goal criteria. Thus, in recent decades, several modelling techniques in the field of GP have been proposed to address imprecise goal targets. Among those techniques, FGP is a widelyused approach that has been applied for several decades, while MCGP is a relatively new approach that was first proposed roughly a decade ago (Chang, 2007). The application of MCGP has also been applied to solve decision problems in the RE field, but not RESS (Chang, 2015).

2.1 Fuzzy Goal Programming (FGP) Approach For F-MODM, the FGP approach focuses on the deviational variables in the GP model, with the aim of minimising the summation of deviations from the criteria achievements toward the imprecise ALs of fuzzy goals. For example, some major works using this technique include, but are not limited to, studies by Hannan (1981), Kim and Whang (1998), Kim et al. (2002), and Aouni et al. (2009). The initial model presented by Hannan in 1981 is regarded as the typical (base) FGP model. Since Hannan‟s model supported only one type of MF (which limited its ability to express DM preferences), it was improved in 1998 (Kim and Whang, 1998). Two other types of MFs (i.e., leftsided and right-sided) were added to the FGP model. In the objective function, a more flexible weighted-additive style was also offered when aggregating the sum of deviational variables. Later, Kim et al. (2002) addressed the problem that limited the possible value of the deviational variables. In 2008, Yaghoobi et al. (2008) further increased the supporting range of the FGP model by adding another type of common piecewise linear MF, trapezoidal MF.

The study by Yaghoobi et al. (2008) is important for this study not because of the fourth MF type it considered but because of the MAV concept it introduced to FGP, as well as the (WGP) model it followed in formulating the WAM concept (Romero, 2001; Yaghoobi and Tamiz, 2006). Therefore, their model is used to demonstrate such a „W-FGP‟ modelling approach here. Consider the fuzzy goals of a decision, identified by i, each of which can be expressed by a piecewise linear MF. Suppose that there are a number of i0 right-sided MFs, a number of (j0–i0) left-sided MFs, a number of (k0–j0) TMFs and a number of (K–k0) trapezoidal MFs. Such a W-FGP model can be represented by the following generalised algebraic formulations: (W-FGP-YJT) i0

min

w i 1

s.t.

i

j0 K n pi ni p   w  wi  iL  Ri    i R L Δi i i0 1 Δi i  jo 1  Δi Δi 

( AX )i  pi  bi i  1,..., io ( AX )i  ni  bi i  i0  1,..., jo ( AX )i  ni  pi  bi i  j0  1,..., ko ( AX )i  pi  biu i  k0  1,..., K

( AX )i  ni  bil i  k0  1,..., K n i  iL  1 i  i0  1,..., jo Δi

i 

pi  1 i  1,..., io ΔiR

i 

ni pi   1 i  j0  1,..., K ΔiL ΔiR

i , ni , pi  0 i  1,..., K X  Cs

where wi denotes the weight of the i-th fuzzy goal; μi is a model (moderate or intermediate) variable which determines the degree of MFs for the i-th fuzzy goal; ni and pi are the negative and positive (moderating) deviational variables; X is the decision vector including all of the variables that are to be decided numerically; ΔiL and ΔiR are the MAVs for the left (L, increasing) and right (R, decreasing) MFs, respectively; bil and biu define the lower and upper bounds of the interval of total satisfaction for the trapezoidal MF; and Cs is an optional set of hard constraints, as found in the traditional linear programming (LP) approach.

2.2 Multi-choice Goal Programming (MCGP) Approach The MCGP modelling approach was proposed by Chang (2007) to deal with the imprecise goals in the field of GP. Compared with the previous two series of F-MODM models, both the application context and the logic behind this approach are quite distinct. MCGP allows a DM to set MCALs for each goal, and in its original form, a goal can be associated with several possible discretely spanned ALs to avoid under- or overestimating the real level that a left-hand side criterion function can achieve. The righthand side of each goal criterion function is a summation of several multiplicative terms, each of which

is associated with a possible choice of AL, and this involves the introduction of some binary variables to control „which choice of AL ought to be (properly) selected (and appear) when the optimal solution is obtained‟. So, in a broader sense, it is also an F-MODM approach that fits the unique decision context. The general (and original) model of MCGP with the WAM concept integrated can be mathematically expressed as follows: (MCGP) K

min

w  p  n  i

i

i

i 1

n

s.t. ( AX )i  ni  pi  bij Sij ( B) i  1,..., K j 1

Sij ( B)  Ri ( x) i  1,..., K ni , pi  0 i  1,..., K X  Cs

where Sij(B) represents a function of binary serial number; bij is the j-th possible AL (targets) w.r.t. the i-th goal; Ri(x) is the function of resource limitations; ni and pi are, respectively, the negative and positive (moderating) deviational variables for the i-th goal; wi is a model parameter, connoting the priority for the i-th goal, in terms of goal weight; and X is the decision vector including all of the variables that are to be decided numerically. The rapid development of MCGP has produced many extensions in both modelling and formulation methods. In earlier developments of MCGP, the subject matter to be fuzzified was on the right-hand side of the goal criteria constraints, which is the AL. This description is true for the initial model above (i.e., the MCALs that will possibly be chosen are discrete), the revised MCGP model (Chang, 2008), and the improved model which „connects the dots‟ (i.e., several MCALs of a goal) as a continuous span like a utility function (Chang, 2011). Later modelling works have mainly extended the MCAL concept of MCGP to solve or formulate other types of problems. Some of these models are multi-segment GP (MSGP), multi-coefficient GP, and percentage GP (%GP) models (Liao, 2009; Chang et al., 2012a; Chang et al., 2012b), among others (Jadidi et al., 2015).

2.3 Current Works Integrating MCGP with F-MODM As of this paper‟s writing, there have been other models proposed in the literature which merged the concept of MCGP into F(L/NL)P, with the latter being one of the two main model series of F-MODM. However, few relevant works have attempted to integrate MCGP with FGP (let alone with WGP), which is the other main model series of F-MODM (see 2.1). To the best of the authors‟ knowledge, the model proposed by Tabrizi et al. (2012) was the first to merge the MCAL concept into F-MODM. This study will call their model „(FP-MCGP-Tabrizi-et-al)‟.

In their model, the authors adopted the max-min approach proposed by Zimmermann (1978) to tackle the „multi-choice yet fuzzy goals‟ (FMCGs), and their study is renowned for its pioneering contribution to merging these models. However, their model supported only one type of MF, the TMF. Before the model was proposed, many studies had voiced doubts about this approach, including those by Martel and Aouni (1998), Chen and Tsai (2001), Pal and Moitra (2003), and Yaghoobi and Tamiz (2007b). Although this model‟s limited supported types of MFs might be due to its following the style of the F(L/NL)P series of studies (e.g., the earlier initial study of fuzzy LP, or FLP, by (Tiwari et al., 1987)), such a drawback deterred its use in practical applications because the various fuzzy preferences of the DM could not be expressed comprehensively. More critically, because the (FPMCGP-Tabrizi-et-al) model originated from and followed the F(L/NL)P model series of F-MODM, it aimed to optimise the utility values of goals interpreted from the MF values, so the objective function of the model maximised the total aggregated utility. Thus, (FP-MCGP-Tabrizi-et-al) should, methodologically speaking, be seen as an „FP+MCGP‟ model, in contrast to its given description as an „FMCGP‟ model, because a „true FMCGP‟ model should be a „true GP‟ model whose objective is to minimise the sum of deviations incurred due to the distance between the value of a goal criteria function and the AL (target value) of that goal. The drawback of having a lack of supported MF types in the (FP-MCGP-Tabrizi-et-al) model was promptly addressed in another model that also took the FP+MCGP approach (Mouslim et al., 2014). In the present study, that model is called „(FP-MTL-Mouslim-et-al)‟ or simply „(FP-MTL)‟, because it integrated the concept of „multiple target levels‟ (MTL), which should be equivalent to MCAL. Due to space limitations, only the (FP-MTL) model is discussed in this study, omitting the (FP-MCGPTabrizi-et-al) model. For a clear illustration, in the constraints section, only the overlapped function of both models is presented in the constraints section, which are the constraints designed to handle goals that have the TMF type. This also helps in understanding the (FP-MCGP-Tabrizi-et-al) model. (FP-MTL-Mouslim-et-al) 

max

U

i

i   1

s.t. ( AX )i  pi  ni 

r

  b , i  (  1),..., * * il il

l 1

r

 =1, i  (  1),..., * il

l 1

 r p  n U i  1   il*  i   i    , i  (   1),...,  il    l 1  il U i  0, i  (   1),...,  << Constraint for fuzzy multi-target goals in terms of the other three types of MF >> X  Cs is a feasible set



where λil (l=1,…,r) is a binary variable indicating that the model has decided to choose which MCAL (e.g., biL for the L-th MCAL) from a pool of r MCALs (bil (l=1,…,r)) for the i-th goal; til is the l-th MTL (MCAL) for goal i (i.e., at the peak of the TMF); Ui is the utility of goal i formed for the DM as determined by some decision vector X; il (  ilR in the subsequent text) and il- (

 ilL ) are, respectively, the MAVs (in F(L/NL)P, also called „maximum allowable deviations‟ rather than „maximum admissible violations‟) that the DM has set for the l-th MTL (MCAL) of the i-th goal, each of which represents the fuzziness on a side of the TMF; and pi and ni are, respectively, the positive and negative distances between the goal criteria function (AX)i on the left hand side and the possibly chosen MTL (biL) (these are similar to the deviational variables in GP; however, they are not used in the objective function for the minimisation process but used for the determination of the utility value Ui). In the above (FP-MTL) model, when there are several MTLs (MCALs) for a goal, the shape of the TMFs that are formulated near these MTLs is similar to that shown in Figure 1, which depicts one of the four types of „common piecewise linear MFs‟ as summarised in Yaghoobi et al. (2008) (see also subsection 3.1 further). Also, in the above model, (β+1) and δ are, respectively, the lower and upper bounds of the goal index, which defines which subset of the entire set of goals in the decision problem has the TMF MF type. Furthermore, when „<< constraints for fuzzy multi-target goals in terms of the other three types of MF >>‟ are removed from the model, let β=0 and δ=K (total number of goals) and let „ il =il , i, l ‟, the (FP-MTL) model will degenerate to a special case, which is exactly the initial (FP-MCGP-Tabrizi-et-al) model of the FP+MCGP model series. However, these observations are not as important as the following points.

1

…

Fig. 1. Triangular multi-target fuzzy membership function.

First, the (FP-MTL) model is still an FP+MCGP model. This classification is justified based on the form of its objective function, wherein the aggregated total utility is maximised. When modelling in this way, the model‟s given name „FGP-MTL‟ leads to ambiguity, as it is similar to that previously discussed with regard to (FP-MCGP-Tabrizi-et-al). One such true GP model should be developed from taking an „FGP+MTL‟ approach, instead of the „FP+MTL‟ approach. Since, as discussed, the concept of MTL should be equivalent to that of MCAL, the required model should be „FGP+MCAL‟ or „FGP+MCGP‟. This is, once again, reflective of the main aim of this study, which is to formulate a true FMCGP model from „FGP+MCGP‟ rather than „FP+MCGP‟. Second, the objective function adopted in (FP-MTL) makes the improvements of this model very

difficult, at least in terms of hybridising it with other existing approaches. In fact, this is one of the main reasons why this field of research has not progressed and gained popularity among researchers. This point has inspired the authors to take the WAM concept for the modelling work of WA-FMCGP into account based on the FGP+MCGP approach and to show how it can easily adopt another GP variant, which is (WGP). In conclusion, it is notable from past literature that, in order to integrate the concept of MCGP, most of the existing works in F-MODM began with the F(L/NL)P model series. Therefore, methodologically speaking, the resulting models were FP+MCGP models. As relevant model integration studies that begin with the FGP model series (i.e., FGP+MCGP) are rare, the proposed WA-FMCGP model should fill that gap. Furthermore, if the proposed model offers better solution quality (than the FP+MCGP models) for the same F-MODM problem in the encountered highly imprecise decision context, it would be a supplement to F-MODM. The following two sections serve these purposes.

3. Formulating the Proposed Model — WA-FMCGP 3.1 Types of MFs Adopted from FGP As discussed, in standard GP, the AL for each goal should be predetermined and provided. However, in real-life decision-making, determining precise ALs for goals is a difficult task because there are many situations wherein a DM is unable to specify (or state) those ALs precisely. When the FGP approach was initially developed, Narasimhan (1980) formulated FGP by taking the concept of MF, in which the value of the function is defined on the [0,1] interval, so that it reached „1‟ when the goal was fully achieved and DMs were totally satisfied; otherwise, it received a value between [0,1). In FGP, the values of these MFs depend on the decision vector (i.e., the values of the decision variables, in the universe of discourse) and are not simply crisps (i.e., either 0 or 1). As a representative work of F(L/NL)P+MCGP, the study by Mouslim et al. (2014) explored ways to introduce those MTLs to fuzzy MFs. These allowed the fuzziness around the multiple targets (i.e., MCALs) for each goal. In that work, the algebraic structures of the „four most common linear fuzzy MF types‟ in Yaghoobi et al. (2008) were adopted and used to formulate the FMCGs. For example, the triangular multi-target fuzzy MF in Figure 1 can be formulated further, as: 0  1     ( AX ) i  1     0

( AX )i  bim  ΔimL



bim  ( AX )i Sim ( B) ΔimL

bim  ΔimL  ( AX )i  bim i  j0  1,..., k0



( AX )i  bim Sim ( B) ΔimR

bim  ( AX )i  bim  ΔimR

r m1 r m1

( AX )i  bim  ΔimR

The other three types of multi-target fuzzy MF are outlined in Figures 2–4, with relevant formulations given. In the equations formulated above for Figure 1 or within Figures 2–4, ΔimR and ΔimL are the MAVs allowed around those MCALs. They are either subjectively chosen by the DM or determined by the tolerances in a technical process (Yaghoobi and Tamiz, 2007a). S im (B ) represents a function of the binary serial number, B, that is set for the m-th possible target (i.e., the m-th MCAL) of the i-th FMCG, as in the study by Chang (2007); other variables are as defined in FGP.

…

1

1    ( AX ) i  1   0 

( AX ) i  bim ( AX ) i  bim Sim ( B ) m 1 ΔimR



r

bim  ( AX ) i  bim  Rim i  1,..., i0 ( AX ) i  bim  Rim

Fig. 2. Right-sided multi-target fuzzy membership function.

…

1

1    ( AX ) i  1   0 

( AX )i  bim bim  ( AX ) i Sim ( B ) m 1 ΔimL



r

bim  Lim  ( AX ) i  bim

i  i0  1,..., j0

( AX ) i  bim  Lim

Fig. 3. Left-sided multi-target fuzzy membership function.

…

1

0   1    ( AX ) i 1  1    0

l ( AX )i  bim  ΔimL l bim  ( AX )i Sim ( B) m1 ΔimL



r

l l bim  ΔimL  ( AX )i  bim l u bim  ( AX )i  bim

u ( AX )i  bim Sim ( B) R m1 Δim



r

i  k0  1,..., K

u u bim  ( AX )i  bim  ΔimR u ( AX )i  bim  ΔimR

Fig. 4. Trapezoidal multi-target fuzzy membership function.

The proposed WA-FMCGP model should also consider at least these four types of MF during modelling. An identical functional aspect in F-MODM should be guaranteed because it will be compared later to the other FP+MCGP model (i.e., FP-MTL).

3.2 The Semantic Problem Definition for WA-FMCGP Following the symbolic conventions, the core concepts of the proposed WA-FMCGP can be sorted by reference to WGP (see section 2), FGP (see section 2.1), and MCGP (see section 2.2). Doing so entails writing down the following semantic model that optimises the K multi-choice yet fuzzy goals (i.e., „fuzzy multi-choice goals‟ alternatively, or simply, the „FMCGs‟) simultaneously. This serves as a general basis of the problem to be modelled and solved: (WA-FMCGP-Semantic) OPTIMISE in terms of wi, i{1,…,K}

 AX i  bi1 or bi 2  AX i ~ bi1 or bi 2  AX i  bi1 or bi 2  AX i  bil1 , biu1 

or,...,or bir

i  1,..., i0

or,..., or bir

i  i0  1,..., j0

or,...,or bir

i  j0  1,..., k0

or bil2 , biu2  or,...,or  birl , biru 

i  k0  1,..., K

X  Cs where OPTIMISE means finding the optimal decision vector X such that all fuzzy goals are satisfied (Hannan, 1981; Yaghoobi and Tamiz, 2007a); (AX )i is the matrix multiplication format (Chang et al., 2012b) of the criterion function for the i-th objective; for i, i  1,..., k0 , bim is the multiple imprecise (possible) ALs for the i-th fuzzy goal, where m=1,…,r; for i, i  k0  1,..., K , blim and buim denote the imprecise lower and upper bounds for the i-th fuzzy goal, respectively, where m=1,…,r; CS is an optional set of hard constraints, as found in traditional LP; the operator symbols indicate the type of a fuzzy goal, while „  ‟, „  ‟, and „  ‟, respectively, denoting that

the i-th goal‟s LHS criterion functional value „is approximately less than‟, „is approximately greater than‟, and „is approximately equal to‟ the MCALs (bir) of this goal on the RHS; likewise, operator „  ‟ denotes that the i-th goal‟s LHS value „is approximately between‟ some intervals given for the MCALs of this goal, while each of the closed intervals is delimited by an upper bound and a lower bound, i.e., [ birl , biru ].

3.3 WA-FMCGP: The General Model The general (WA-FMCGP) model is formulated based on (WA-FMCGP-Semantic). It is formulated with reference to the features of FGP and MCGP, while the method for dealing with the goal weights is determined by reference to the work of Yaghoobi, Jones and Tamiz (Yaghoobi et al., 2008), which strictly followed Romero‟s (WGP) model in integrating WGP and FGP. In brief, the model aims to minimise the weighted-additive summation of deviations induced by the distance from the criterion function to the possible MCALs of each goal, while there also exists some fuzziness around each AL target. Therefore, the proposed WA-FMCGP model is formulated as follows: (WA-FMCGP) i0

min

w p t i 1

s.t.

i

R i i



j0

 wnt

i i0 1

L i i i

 w n t K



i

i  jo 1

L i i

 pi tiR 

( AX ) i  pi   i1bi1 or  i 2 bi 2 or,..., or  ir bir

i  1,..., io

( AX ) i  ni   i1bi1 or  i 2 bi 2 or,..., or  ir bir

i  i0  1,..., jo

( AX ) i  ni  pi   i1bi1 or  i 2 bi 2 or,..., or  ir bir

i  j0  1,..., k o

( AX ) i  pi   i1bi1 or  i 2 bi 2 or,..., or  ir bir

i  k 0  1,..., K

( AX ) i  ni   i1bi1 or  i 2 bi 2 or,..., or  ir bir

i  k 0  1,..., K

u

u

l

tiR tiL tiR tiL

  λ  λ  λ

 λi1 i1 i1

       

ΔiR1 ΔiL1 ΔiR1 ΔiL1

u

l



l





 λi 2 ΔiR2  ,...,  λir ΔirR  λi 2 ΔiL2  ,...,  λir ΔirL  λi 2 ΔiR2  ,...,  λir ΔirR  λi 2 ΔiL2  ,...,  λir ΔirL

i1 R i  piti  1 L i  niti  1 i  ni tiL  pi tiR

  

  



i  1,..., io



i  i0  1,..., jo

  

i  1,...,i 0 i  i0  1,...,j o i  j0  1,...,K i  j0  1,...,K

1 i  j0  1,..., K i1  i 2 ,..., ir  1 i  1,..., K ir  0,1 i  1,..., K i , ni , pi  0 i  1,..., K X  Cs where wi denotes the weight of the i-th fuzzy goal and μi is a moderate variable which represents the utility level that is derived from the degree of MF for the i-th fuzzy goal; (AX)i is the matrix multiplication format of the criterion function for the i-th objective; X is the decision vector including all the variables that are to be decided numerically; pi and ni are the positive and negative (moderating) deviational variables, respectively; λi1, λi2, …, λir are the binary variables which ensure that one AL must be chosen for each goal i; ΔiR and ΔiL are, respectively, the

MAVs that are allowed on the right and left sides of the MF binding to the r-th MCAL of the i-th goal (i.e., bir); tiR and tiL are continuous variables which represents a coefficient that can be regarded as the „equivalent concentration‟ or „normality‟ in e.g., chemistry, that is to be multiplied with the determined wi prior to the solution time and/or the absolute distance from the criterion function value that is determined for the i-th goal ((AX*)i) to the default (original) AL value (target value) that is selected by using λi1, λi2, …, λir during solution time (i.e., pi and/or ni); and the other variables are as defined in (W-FGP-YJT). When examining the above model concerning the objective measure, the most salient feature is that the model utilises the WAM concept from WGP to aggregate its objective measure. More importantly, the objective measure aggregates several forms of deviations, positive (pi), negative (ni), or both (ni + pi), according to the DM preferences, and then minimises the summation of those deviations, which are both weighted and normalised. This feature distinguishes the (WA-FMCGP) model from previous FP+MCGP models, which maximise total utility that is totalled without using the WAM concept.

3.4 Extended WA-FMCGP Formulation: The Meta WA-FMCGP Model As discussed previously, the (WA-FMCGP) model is a „true GP model‟. Consequently, the model can adopt (or to be integrated with) any minimisation procedure (or objective function) which belongs to other GP variants, such as lexicographic GP and meta GP (Jones and Jimenez, 2013). The latter GP variant applies to situations in which the DM sometimes provides unusual preferences about the deviational variables in order to gain more control about the achievement statuses of the goals. Such a concept could be easily converted into the proposed (WA-FMCGP) model (i.e., a meta WA-FMCGP model). However, before giving the „(Meta WA-FMCGP)‟ model, the core concepts of meta GP will be briefly reviewed. Meta GP, with its original name as “[GP]2”, allows a DM to set some meta goals and impose those implicit meta goals on existing explicit goals by introducing additional constraints to a GP model‟s original deviational variables. From the methodological aspect of GP, a meta goal can be achieved by means of a lexicographic or a weighted structure, as deemed appropriate by the DM. In terms of the mathematical formulation of GP, there are three types of meta-goals (Rodriguez-Uria et al., 2002): Type 1 meta-goal: The sum of percentages of the unwanted deviations (against the AL) should not be greater than a certain bound, Q(1):

w

i

isk( i )

di  QK(1) . bi

Type 2 meta-goal: The maximum percentage of deviation of some goals of concern (i.e., the watched goals) should not be greater than a certain bound, Q(2): wi

di  Di  0, bi

i  sl(2)

Di  Q (2) .

Type 3 meta-goal: Among all or some watched goals, the percentage of the unachieved goals (against the total number of goals) should not be greater than a certain bound, Q(3) : d i  Ri yi  0, i  S r(3)  s   i 1 yi .  Qr(3) ,  (3) card ( S ) r   y  0,1, i  S r(3)  i where wi is a preferential weight for each imposed explicit goal considered in sk(1) ; di is every

undesired deviation from each imposed explicit goal considered in sl( 2 ) ; bi is the AL (target) of each imposed explicit goal considered in sl( 2 ) and can be viewed as a normalisation constant; D is an extra continuous variable that measures the maximum deviation; yi is a binary variable for each imposed explicit goal i considered in sr(3) ; and Ri is a sufficiently large arbitrary number. As seen above, introducing meta GP can improve on the drawbacks of missing goal priority for any GP variant in F-MODM, so too for the proposed (WA-FMCGP) model (since it is also a GP variant; see 3.3). In addition to the WAM concept that has been used and formulated to realise the goal priority, the resulting (Meta WA-FMCGP) model can offer more flexibility in manipulating DMs‟ preferences over FMCGs. Thus, this hybrid model can be formulated as follows: (Meta WA-FMCGP) (2) (2) (3) (3) min 1(1) ,..., r(1) 1 ,1 ,..., r 2 ,1 ,..., r 3 

s.t. i0

w p t i 1

i

R i i

i0



j0

 wnt

i i0 1 j0

w p t

R i i



Dl  

(2) l



i 1

i

L i i i

 wnt

i i0 1 (2) l

L i i i



 w n t

 pi tiR    s(1)   s(1)  Qs(1) , s  1,..., r1

 w n t

 pi tiR   Dl  0, i  sl(2) , l  1,..., r 2

K

i  jo 1



i

L i i

K

i  jo 1

i

L i i

 Q , l  1,..., r 2 (2) l

di  Ri yi  0, i  Sr(3) r  1,..., r 3  s   i 1 yi   k(3)  k(3)  Qr(3) r  1,..., r 3 .  (3) card ( S ) r   y  0,1 , i  S (3) r  1,..., r 3 r  i

<< All the system constraints of WA-FMCGP >>. where σs, σl, σr, φs, φl and φr are the meta positive and negative deviation variables for the meta goals; all other variables are as defined in previous discussions.

4. Modelling the Numerical Example and Comparisons

This section solves a numerical example in order to show that the proposed model behaves appropriately (i.e., that it is valid) and demonstrates the advantages of (WA-FMCGP) through comparisons. Section 5 will show further practical applications of the model.

4.1 The Numerical Example A decision problem presented in the literature is adopted as the numerical exemplar of this study in order to show the effectiveness of (WA-FMCGP) by comparing its results with the previous solution set obtained by the FP+MCGP model. The problem is a bicycle manufacturer‟s product-mix decision case (Mouslim et al., 2014) in which the DM knew the set of goals but not the precise AL for each goal and could give several possible MCALs for each goal and state the MAVs around these MCALs. Despite the two models compared here being different because (WA-FMCGP) is an FGP+MCGP model, the problem is identical in terms of highly imprecise goal targets (i.e., the FMCGs with MAVs) in F-MODM. Consider the following F-MODM problem with several FMCGs: (P-Exemplar) OPTIMIZE

FMCG1: FMCG2: FMCG3: FMCG4: s.t.

x1  4 x2  4 x3  100 or 150 4 x1  5 x2  1x3  240 or 280 3x1  1x2  4 x3  250 or 300

x1  x2  6x3 200,250 or 280,340

8 x1  3x2  4 x3  750 3x1  5x2  2 x3  350 4 x1  7 x2  5 x3  500 x1 , x2 , x3  0

(C1) (C2) (C3) (C4)

In addition, in this decision case, the four FMCGs have an equal priority, and there are MAVs associated with every possible MCAL on the right-hand side of the four goal statements. These goal statements have different „fuzzifier types‟ as set by the DM, which also determine the types of MF in the above (P-Exemplar). Except for the fuzzifier types, to define the fuzziness around the ALs of each goal completely, the associated MAVs are given as follows:  MAVs for FMCG1: 30 and 20, respectively;  MAVs for FMCG2: 12 and 20, respectively;  MAVs for FMCG3: (10, 5) and (15, 20);  Fourth fuzzy goal: (4, 5) and (3, 2).

4.2 Solution Obtained Using FP-MCGP Model Applying the FP+MCGP model (i.e., the (FP-MTL-Mouslim-et-al) model reviewed in subsection 2.3) to solve (P-Exemplar), the optimal solutions obtained are as follows: (Solution Set I)

   

Decision variable values: ( x1 , x2 , x3 )  (71, 0, 22 ) ; Degree of MFs: ( 1 , 2 , 3 , 4 )  (0.55, 0.95,1,1) ; Deviational variable values: ( p1 , n2 , p2 , n3 , p3 , n4 )  (9,0,1,0,0,0) ; Binary variables: ( λ11 , λ12 , λ21 , λ22 , λ31 , λ32 , b31 , b32 , λ41 , λ42 )  (0,1, 0,1,1, 0,1, 0,1, 0) .

4.3 Solution Obtained Using WA-FMCGP The same (P-Exemplar) decision problem using the proposed (WA-FMCGP) model with an identical symbolic system can be formulated as follows: (P-Exemplar-WA-FMCGP)

Min a  p1t1R  n2t2L  n3t3L  p3t3R  n4t4L  p4t4R s.t.

x1  4 x2  4 x3  p1  100 λ11  150 λ12   μ1  p1t1R  1   for FMCG1 R t1   λ11 30    λ12 20    λ11  λ12  1  4 x1  5 x2  1x3  n1  240 λ21  280 λ22   μ2  n2t2L  1   for FMCG2 L t2   λ21 12    λ22 20    λ21  λ22  1 

3x1  1x2  4 x3  n3  p3  250 λ31  300 λ32   μ3  n3t3L  p3t3R  1   L t3   λ31 10    λ32 15  for FMCG3  R t3   λ31 5   λ32 20    λ31  λ32  1 

x1  x2  6 x3  p4  250 λ41  340 λ42  x1  x2  6 x3  n4  200 λ41  280 λ42   μ4  n4t4L  p4t4R  1   for FMCG4 L t4   λ41 4    λ42 5  R  t4   λ41 3   λ42 2    λ41  λ42  1 Constraint equations (C1) – (C4). Solving the above (P-Exemplar-WA-FMCGP) using the written model in LINGO (Schrage, 2009) yields another optimal solution set, as follows: (Solution Set II)  Decision vector (decision variable values): ( x1 , x2 , x3 )  (71, 0, 21) ;  Degree of MFs: ( 1 , 2 , 3 , 4 )  (0.64,1,1,1) ;

 Deviational variable values: ( p1 , n2 , p2 , n3 , p3 , n4 )  (7,0,0,0,0,0) ;  Binary variables: ( λ11 , λ12 , λ21 , λ22 , λ31 , λ32 , λ41 , λ42 )  (0,1,1, 0, 0,1,1, 0) . At first glance, it can be seen that the decision vector in this solution set suggests another productmix portfolio (Zhuang and Chang, 2017), which differs from that indicated in Solution Set (I). However, the next subsection provides more insightful comparisons.

4.4 Comparisons Based on Solution Sets (I) and (II), Table 1 summarises and compares the relevant information for when (FP-MTL-Mouslim-et-al) and (WA-FMCGP) were individually applied to solve the same (PExemplar), an F-MODM problem with the imprecise goal targets. Table 1. Comparison of the results from solving the same exemplar problem with the two models. Solutions

Item

Decision Variable Values

x1

Utility in Terms of Degree of MF

WA-FMCGP Model Solution Set (II)

71

71

x2

0

0

x3

22

21

1

0.55

0.64

2

0.95

1

3

1

1

4

1

1

3.5

3.64

p1

9

7

n2

0

0

p3

1

0

n3

0

0

p3

0

0

n4

0

0

10

7

Total

Deviational Variable Values (Surpluses and Slacks)

FP-MTL Model Solution Set (I)

Total

First, the (WA-FMCGP) model performed better because Solution Set (II) yields a greater number of satisfied goals. In comparison with Solution Set (I), which is produced using FP+MCGP where there were two remaining unmet goals (FMCG1 and FMCG2), FMCG2, FMCG3, and FMCG4 were completely achieved with the use of WA-FMCGP, in terms of the degree of MF. In Solution Set (II), only FMCG1 was not completely met, while the rest of the goals (FMCG2, FMCG3 and FMCG4) were all satisfied and completely achieved. Therefore, the overall results in Solution Set (II) are preferable to those in Solution Set (I) for the DM. Second, the advantage of the WA-FMCGP model is also clear if deviations are observed. In Solution Set (II), the total deviation (  ( ni  pi ) ) is 7, while it is 10 in Solution Set (I). As a 1 4

deviation usually connotes the absolute goal achievement in a TLTB (the less, the better) fashion, this salient gap demonstrates the superiority of the proposed model in terms of solution quality. Third, and most importantly, when the total utility that a model can bring to the DM is measured, it is clear that the proposed WA-FMCGP model outperformed FP+MCGP. Solution Set (I) provided a total utility of 3.5 (out of 4), while Solution Set (II) yielded a total utility of 3.64, an improvement of 4%. This outcome is surprising because the FP+MCGP model, (FP-MTL-Mouslim-et-al), pursues optimality according to the „total utility maximisation‟ logic in the objective function stated at its origin. In contrast, the (WA-FMCGP) model pursues optimality according to a logic of another nature, a „total deviation minimisation‟ logic, such that in the solution set, the interpreted utility values for the goals are merely the model sub-products. However, as utility is the main aim of FP+MCGP (while being only a sub-product of FGP+MCGP), and given that for the same F-MODM problem the latter yields a greater total utility, the effectiveness of the proposed (WA-FMCGP) model (and of the FGP+MCGP „pure GP‟ approach) should be axiomatic. The utility levels assessed for the goals in the optimal solution sets can also be scrutinised individually to provide further evidence. In Solution Set (II), the pro-rata-based utility of FMCG2 (μ2), 1, is at the completely satisfactory level. In contrast, in Solution Set (I), it (μ2) has a value of 0.95, meaning it does not provide perfect utility. Moreover, although both models failed to completely satisfy FMCG1 (i.e., μ1≠1 in both solution sets), the proposed model yielded a utility level (0.64) that was more satisfactory than that yielded in Solution Set (I) (0.55). Therefore, while in both solution sets FMCG3 and FMCG4 were satisfied with full utility, in general, the (WA-FMCGP) model enhanced the goal achievements for individual FMCGs, and thus reached a higher total utility level. In other words, in terms of these utility measures, the proposed model yielded improved solution quality, bringing greater satisfaction to the DM by producing a solution that was closer to the ideal. One noteworthy matter is that in the original definition of (P-Exemplar) (see subsection 4.1), the FMCGs were assumed to have equal importance. For a fair comparison, the (P-Exemplar-WAFMCGP) model as formulated in subsection 4.3 takes the EWI (equally weighted importance) assumption (González-Pachón et al., 2019). However, because the (WA-FMCGP) model also integrates the WAM concept of (WGP) in its objective (see subsection 3.3), it is possible to further explore the effect of assigning different weights (relative importance) to the deviational variables w.r.t. the FMCGs, based on some observable DM preference. In this case, the model may lead to a different solution, and there is a fair chance of obtaining a more satisfying solution set for the DM; that is, one that is even superior to Solution Set (II). In any case, the discussion presented in this section verifies the validity of using the proposed (WA-FMCGP) model. Comparisons demonstrated its advantages over the existing F(L/NL)P series model that also integrated MCGP by solving an identical F-MODM problem facing up to the uncertain

goals with highly imprecise targets. Overall, in terms of several measures used in this section, the proposed (WA-FMCGP) model, a „true GP‟ model originated from the FGP series of F-MODM, was preferable to the FP+MCGP models, which originated from the FP series and aimed to maximise the total utility. The positive results obtained from solving an identical F-MODM problem establish a solid foundation for progressing further (i.e., to the application of WA-FMCGP to a RESS problem with empirical data).

5. Application to a Practical Decision This section verifies the (WA-FMCGP) model‟s applicability and demonstrates its practical effectiveness by tackling a RESS problem and answering the question: “subject to the annual capacity expansion budget, w.r.t. the imprecise goal (target) expressions, where should the wind farm capacity in Algeria be expanded?” According to the collected data (the map of Algeria‟s annual wind speed), the south-western region has the greatest potential for green electricity generation. This information is shown in Figure 5 (Chellali et al., 2011; Zhuang and Hocine, 2018). The five places (i.e., the alternative cities that have wind farms in-production or pilot running) under consideration by DMs for the construction of wind farm capacity expansion are as follows: Tindouf, Adrar, Insala, Hassi R‟Mel, and Bordj Badji Mokhtar. These cities are shown and marked in Figure 6. A total of four criteria were considered to facilitate the decision of selecting the most suitable location for installing turbines to expand the wind farm capacity. The criteria were as follows: „electricity generation‟, „unit cost energy‟, „topography and infrastructure‟, and „social acceptability‟. Table 2 provides the criteria weights and the performance (vector) for each location.

Fig. 5. Map of annual wind speed in Algeria.

Hassi Adrar

Insala

Tindou

Bordj Badji Mokhtar Fig. 6. Alternative locations for wind farm site expansion construction. Table 2. Evaluating the possible wind-farm sites for expansion construction according to the considered criteria.

Criteria (Criteria Set) Potential locations (decision variables mapping)

Energy generation(a) (GW)

Unit cost energy(a)

Social acceptability

Topography and infrastructure

Weights (Criteria Weight Vector, or CWV) 0.4

0.3

0.15

0.15

Tindouf (X1)

42.82

0.1310

6.5

5

Adrar (X2)

49.01

0.1145

7

6

Insala (X3)

49.08

0.1143

6

5.5

Hassi R‟Mel (X4)

58.68

0.0956

5

8

Bordj Baji Mokhtar (X5)

57.56

0.0975

5

7

(a)

Obtained from Benmemdejahed and Mouhadjer (2016).

In Table 2, the decision variables of the problem are defined as follows:

1, if location j isselected, Xj  0, otherwise where j = 1 for Tindouf, j = 2 for Adrar, j = 3 for Ain Salah, j = 4 for Hassi R‟Mel and j = 5 for Bordj Badji Mokhtar. Many criteria are used in renewable energy planning to measure the performance of renewable energy sources. However, since several quantitative and qualitative criteria have been adopted and because DM judgments are not crisp, it is relatively difficult for DMs to provide precise numerical values for their preferences, particularly for qualitative criteria (e.g., social acceptability). Such criteria are generally expressed by linguistic variables (such as “high” or “moderate”) and then converted to scale-based numerical values (e.g., 5 and 4 for “very high” and “relatively high”). Therefore, assigning

them a single crisp value (or parameter) is unsuitable. While it is generally accepted that random variables (e.g., realisation time) should be treated with stochastic methods, consensus fails on whether there is enough data available, which is key to formulating probabilistic distribution (see Kirkwood, 1992). Unfortunately, in the design stage of new systems, the information to have precise probability distributions of these probabilistic variables is not always sufficient due to immeasurability or assumptions. The robust optimisation theory has been developed as a supporting tool to mitigate the impact of such uncertainty. However, robust optimisation methodologies have a critical drawback. According to Bertsimas et al. (2013), most robust optimal solutions are highly conservative, aiming to hedge against all possible worst-case realisations of the uncertainty. In some cases, this can lead to overly conservative solutions. For example, if a renewable energy plant is designed too conservatively, then the project may be too expensive or needlessly complex. In such cases, it is assumed that these random variables are controllable and that they are addressed through other uncertainty theories (such as fuzzy set or interval), which are more appropriate than their alternatives because they require less information. Thus, due to limited decision practices, the DM is not entirely sure about the precise values of the ALs (targets) for the goals but has stated them in an FMCG fashion. Specific to the studied RESS decision case here, the possible MCALs for each FMCG and its type of fuzziness (when expressed in terms of the MF) are described in Table 3. The MAVs around each MCAL (target value) are listed in Table 4. Table 3. Possible aspiration levels (ALs) and the MF type for/of each FMCG. The FMCGs

Type of goal

Multi-choice aspiration level (MCAL)

FMCG1: Energy generation (GW)

Left

60 or 65

FMCG2: Unit cost energy

Right

0.0945 or 0.0950

FMCG3: Social acceptability

Triangular

6.5 or 7

FMCG4: Topography and infrastructure

Trapezoidal

[4.5,7] or [5,8]

Table 4. Admissible violations (tolerances) around each possible AL for each fuzzy goal. Goals

Tolerances (MAVs) around the possible MCALs

Energy generation

15 and 10

Unit cost energy

0.02 and 0.02

Social acceptability

(15,10) and (15,20)

Topography and infrastructure

(1.1,1.1) and (1.2,1.2)

Thus, following the semantic style used to express the (P-Exemplar), the encountered RESS decision problem can be stated as follows: (P-RESS)

OPTIMISE in terms of wi, i{1,…,K} FMCG1: 0.1310 X 1  0.1145 X 2  0.1143 X 3  0.0956 X 4  0.0975 X 5  0.0945 or 0.0950 FMCG2 : 42.82 X 1  49.01X 2  49.08 X 3  58.68 X 4  57.57 X 5  60 or 65 FMCG3: 6.5 X 1  7 X 2  6 X 3  5 X 4  5 X 5  6.5 or 7 FMCG4 : 5 X 1  6 X 2  5.5 X 3  8 X 4  7 X 5  [4.5,8] or [5,8]

s.t., X 1  X 2  X 3  X 4  X 5  1 and {The MAVs around each possible AL in Table 4}. The above (P-RESS) problem was also modelled using (WA-FMCGP) as another (P-RESS-WAFMCGP) model, with constraints taking model parameters from Tables 2, 3, and 4 under the EWI assumption. The formulation details of this model are omitted here due to space limitations because the formulation process is akin to the process detailed above for (P-Exemplar-WA-FMCGP). The (PRESS-WA-FMCGP) model was then solved using LINGO, and the optimal solution set obtained is as follows: (Solution Set III)  Decision vector (variable values): X *  ( X1, X 2 , X 3 , X 4 , X 5 )  (0, 0, 0,1, 0) ;  Degrees of MFs (also the utilities interpreted): ( 1 , 2 , 3 , 4 )  (0.91,0.98,0.76,1) ;  Deviational variable values: (n1 , p2 , n3 , p3 , n4 , p4 )  (1.32,0.0006,3.5,0,0,0) . The obtained solution for the decision vector shows that Hassi R‟Mel is the best location for expanding wind farm capacity in Algeria (i.e., X 4 1 ). This desired knowledge is the most important empirical implication for both the DM and wind-farm site construction practitioners. Thus, the solution set, when subject to the constraints given, fully satisfies the DM in terms of only one goal (FMCG4). However, each of the goals can be achieved to a satisfaction level of „≥ 90%‟ in terms of the MF-degree measure, except for the third goal (FMCG3). In addition, using this optimal solution achieves energy generation (FMCG1), unit cost energy (FMCG2), topography and infrastructure (FMCG3), and social acceptability (FMCG4) with a pro-rata average of 91%. As in the (WA-FMCGP) model, the degree of MF provides a utility interpretation for each goal; so in terms of total utility, adopting the capacity expansion strategy suggested by the solution set may yield a total utility level of 3.65 (out of 4). A final observation is that when exercising a capacity expansion plan according to this RESS solution, the DM can expect a total deviation of only 4.8206. All of these results mean that DMs should be highly satisfied with the choice suggested by the (P-RESS-WAFMCGP) model because the strategic plan closely follows their ideals. However, the selection of Hassi R‟Mel to be the location for wind farm expansion is not an applicable scenario, at least in the short and medium terms. This is because this area is one of the most profitable oil fields in Algeria. Thus, replacing (covering) this area with wind farms will directly affect government revenue (Himri et al., 2009) while simultaneously increasing negative effects regarding social acceptability (Kraft and Kraft, 1978). Therefore, some sensitivity analysis for evaluating

whether a solution can further improve is required, and there are numerous possible methods. For example, examining further consequences by either changing the weights of the criteria accordingly or asking DMs to provide additional information about their preferences to control the achievement statuses of goals using the deviation variables is appropriate. This study made the latter sensitivity analysis. Suppose a DM would like to see the potential solutions when some restrictions to the deviation variables have been imposed because the DM has a proposed meta goal. Based on Solution Set (III), the DM stated that “social acceptability should be fully satisfied”. In terms of methodology, this is doubtless a meta goal. With such a meta goal in mind, and formulating the problem above using the (Meta WAFMCGP) model proposed in subsection 3.4, the following optimal solution set can be obtained: (Solution Set IV)  Decision vector (variable values): X *  ( X 1 , X 2 , X 3 , X 4 , X 5 )  (0,1,0,0,0) ;  Degrees of MFs (also the utilities interpreted): ( 1 , 2 , 3 , 4 )  (0.267,0.35,1,1) ;  Deviational variable values: (n1 , p2 , n3 , p3 , n4 , p4 )  (10.99,0.019,0,0,0,0) . The results show that achieving full social acceptability is addressed, and the (Meta WA-FMCGP) model achieves this watched goal completely. Meanwhile, the potential location for wind farm expansion has moved to Adrar, which is also a viable option if and only if achieving the social acceptability decision criterion (FMCG3) is the only concern. However, in contrast to Solution Set (III), as making a new RE construction expansion plan according to this different solution set would strongly erode the goal achievement statuses of both FMCG1 (for energy generation throughput) and FMCG2 (for the unit cost for energy production), the DM should consider the trade-offs. When dealing with MODM problems, the concept of Pareto (Paretian) efficient solutions should be considered. However, among the various MODM methods, GP was not developed to obtain a single or all non-dominated solutions but as a method for locating satisfactory solutions on the optimal frontier in the objective (goal) space for complex, real-world problems. The extensive study by Zhuang and Hocine (2018) supported just this logic, inspecting how to explain and choose the most satisfactory solution from several optimal solutions that are exactly on, or near, the efficiency frontier. Therefore, obtained GP solutions should be judged solely on how well they meet the goals of the DM and whether they produce a practical solution to the decision problem (Jones & Tamiz, 2010; Romero, 1991). Either (WA-FMCGP) or the (Meta WA-FMCGP) model proposed in this study serves these purposes because they are hybrid GP models in nature and are formulated on a commensurable basis in that all of the integrated „sub-models‟ are GP models in the intrinsic. So, based on such properties, it should be noted that some concepts of the so-called „efficiency detection and restoration techniques‟ (see Tamiz & Jones, 1996) can be introduced and explored in future works.

6. Conclusions This paper proposed a novel F-MODM model for solving a challenging and uncertain MCDM problem with several conflicting goals. In the cases modelled and solved, the decision involved imprecise goal targets whose form are quite complicated (i.e., the FMCGs with MAVs); that is, DMs could not state their preference about goal targets precisely but could specify several MCALs for each goal, the type and „direction‟ of goal fuzziness and the MAVs alongside each possible MCAL allowed for the fuzziness. The (WA-FMCGP) model was established and proposed subject to this decision context. As the (WA-FMCGP) model seamlessly integrates WGP, FGP, and MCGP, it can function as a comprehensive model that can deal with many uncertain decision cases wherein the goal targets are highly imprecise. It is a „true hybrid GP‟ model because of the form of its objective function, wherein the unwanted deviations incurred are normalised and aggregated by considering the WAM concept and because all three of the integrated models may share a commensurable basis, which is GP. It is also a „true FMCGP‟ model because, unlike the existing models which took the FP+MCGP approach during their formulation (i.e., integrating the MCAL concept with FP, while the objective function is to minimise total utility without the use of the WAM concept), this study takes the FGP+MCGP approach (i.e., integrating the MCAL concept with FGP). As F(L/NL)P and FGP are usually regarded, methodologically speaking, as the two main model series for F-MODM, these two model series are heterogeneous in their origins. As such, these two hybrid approaches (i.e., F(L/NL)P+MCGP and FGP+MCGP) differ. This difference addresses the initial novelty of this study for the fields of FGP (Mirzaee, 2018), GP (Tanino et al., 2013), F-MODM, MODM (de Oliveira et al., 2018; Afrapoli et al., 2019), and thus of multi-objective optimisation (Tsionas, 2019) in the broadest sense, in that while there have been studies of FP+MCGP, studies of FGP+MCGP are rare. The proposed (WA-FMCGP) model fills this gap. Note that some recent developments were intentionally selected and used here for contradistinction. The model validation work in Section 4 demonstrated the advantages of the FGP+MCGP approach (i.e., WA-FMCGP) over existing FP+MCGP models when they were used to solve the same numerical exemplar (a known product-mix decision problem with highly imprecise goal targets presented in previous literature) in F-MODM. Thus, the proposed model also provides methodological supplements to the field F-MODM, since the role of fuzzy set theory has been reconfirmed just in this decade due to the popularities of AI, big data analytics, and thus data-driven decision-making. Solutions to an empirical RESS decision in Section 5 further verified the practical applicability of the proposed model using real data as the model parameters. Unlike the problem of designing a suitable power generation portfolio for the initial construction work undertaken at several known places given a large budget, the problem in question required determining one suitable place for capacity expansion subject to a small annual budget (relative to the initial construction fund deposited).

However, as the use of such „relatively small but still large‟ budget is usually critical for sustainable development (a wrong decision might lead to unrecoverable consequences), the DM preferences should be treated very carefully every time when one such decision is to be made. The resulting model parameterised based on (WA-FMCGP) was shown to be effective for the encountered RESS decision by suggesting installing additional wind turbines at the most suitable site in Hassi R‟Mel (among the five places with in-production or pilot-running turbines). From the perspective of decision-making practices, such an application scenario addresses another niche of the study: a real application of the proposed model was explored (i.e., WA-FMCGP is practically effective) and the application of an approach to a new problem domain (i.e., applying MODM, rather than MADM, to RESS) was provisioned. This result may extend the knowledge base for renewable energy (construction) planning in that the literature is abundant with MADM methods (Ehrgott and Gandibleux, 2002; Figueira et al., 2005; Zopounidis and Pardalos; Hwang and Yoon, 2012; Kahraman et al., 2015; Emrouznejad and Marra, 2017; Zhuang et al., 2019) and that these methods have been applied to site selection problems in renewable energy planning (Wang et al., 2009; San Cristóbal, 2012; Reisi et al., 2018). However, the use of MODM on RESS is scant. Since the effect of randomness was unexplored in the empirical case study, this would be a topic worthy of further exploration (e.g., using and further integrating the chance constraint technique to explore whether a solution can be further improved upon or not). The probabilistic chance constraint programming (CCP) concept initiated in the 1970s (Charnes and Cooper, 1961; Prekopa, 1970) has been applied to GP later and enriched after year 2000 (De et al., 1982; Ballestero, 2001; Aouni et al., 2012). As can be imagined, integrating the chance constraints into the proposed model and replacing the FGP part of the model with stochastic GP are both viable options. This would lead to a „StochasticMCGP‟ model, which would (still) be a GP model worthy of inquiry. ACKNOWLEDGMENT

The authors sincerely thank the following funding institutions for their grants in support of this research: Ministry of Science and Technology, Taiwan (Project No.: MOST108-2410-H-992-046), Czech Science Foundation (GACR), Czech Republic (Project No.: 17-19981S), and UM Research Fund, Macau (Project No.: CPG2015-00017-FST). REFERENCES Afrapoli, A. M., Tabesh, M., & Askari-Nasab, H. (2019). A multiple objective transportation problem approach to dynamic truck dispatching in surface mines. European Journal of Operational Research, 276(1): 331–342. Aouni, B., Abdelaziz, F.B., & La Torre, D. (2012). The stochastic goal programming model: theory and applications. Journal of multi-criteria decision analysis, 19: 185–200. Aouni, B., Martel, J.M., & Hassaine, A. (2009). Fuzzy goal programming model: an overview of the current state-of-the-art. Journal of multi-criteria decision analysis, 16: 149–161. Ballestero, E. (2001). Stochastic goal programming: A mean–variance approach. European Journal of Operational Research, 131(3): 476-481. Bellman, R.E., & Zadeh, L.A. (1970). Decision making in a fuzzy environment. Management Science, 17(2): 141–164. Benmemdejahed, M, & Mouhadjer, S. (2016). Evaluation of wind energy cost and site selection for a wind-farm in the south of Algeria. Technologies and Materials for Renewable Energy, Environment and Sustainability, 1758, 030001; doi: 10.1063/1.4959397.

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