European Journal of Operational Research 255 (2016) 845–855 

Contents lists available at ScienceDirect 

European Journal of Operational Research 

journal homepage: www.elsevier.com/locate/ejor 

Decision Support 

An extended goal programming methodology for analysis of a 

network encompassing multiple objectives and stakeholders 

Dylan Jones a , ∗, Helenice Florentino 

b , Daniela Cantane 

b , Rogerio Oliveira 

b 

a Centre for Operational Research and Logistics (CORL), Department of Mathematics, University of Portsmouth, UK 
b Department of Bio-Statistics, UNESP, Botucatu, São Paulo, Brazil 

a r t i c l e i n f o 

Article history: 

Received 11 August 2015 

Accepted 17 May 2016 

Available online 27 May 2016 

Keywords: 

Extended goal programming 

Multiple objective programming 

Renewable energy 

a b s t r a c t 

This paper proposes a goal programming methodology to ensure that a mix of balance and optimisation 

is achieved across a hierarchical decision network. The extended goal programming principle is used for 

this purpose. A model is constructed that provides consideration of balance and efficiency of multiple 

objectives and stakeholders at each network node level. A goal programming formulation to provide the 

decision that best meets the goals of the network is given. The proposed model is controlled by three key 

parameters that represent the level of non-compensation between objectives, level of non-compensation 

between stakeholders, and level of centralisation in the network. The methodology is demonstrated on an 

example pertaining to regional renewable energy generation and the results are discussed. Conclusions 

are drawn as to the effect of different attitudes towards compensatory behaviour between objectives and 

stakeholders in the network. 

© 2016 Elsevier B.V. All rights reserved. 

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. Introduction 

The initial goal programming model is proposed by Charnes and

ooper (1961 ) as a means of modelling the satisficing philosophy

f Simon (1957 ) in a mathematical programming framework. Since

hen, the technique of goal programming has been developed to

ncompass many variants and fields of application ( Jones & Tamiz,

010 ). The fundamental variants are lexicographic, which combines

rdering and satisficing philosophies; weighted, which combines

ptimising ( Pareto, 1896 ) and satisficing philosophies and Cheby-

hev, which combines satisficing and balancing ( Rawls, 1973 )

hilosophies. More recently advanced variants have been proposed

hat provide effective frameworks for combining philosophies and

odelling modern complex decision problems involving multiple,

onflicting goals. These include meta-goal programming ( Rodríguez

ría, Caballero, Ruiz, & Romero, 2002 ), extended goal program-

ing ( Romero, 20 01, 20 04 ) and multi-choice goal programming

 Chang, 2008 ). The meta-goal programming model proposes the

oncept of a meta-goal, a high level goal that goes beyond a single

oal and gives an overall measure of satisfaction for the decision

aker. In this context the extended goal programming model

an be seen as comprising two meta-objectives: the minimisation

f the weighted, normalised sum of unwanted deviations from
∗ Corresponding author. Tel.: + 44 2392846362. 

E-mail address: Dylan.Jones@port.ac.uk (D. Jones). 

T  

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ttp://dx.doi.org/10.1016/j.ejor.2016.05.032 

377-2217/© 2016 Elsevier B.V. All rights reserved. 
he set of goals (using the L 1 distance function and representing

fficiency and optimising principles) and the minimisation of the

aximal weighted, unwanted, normalised deviation from amongst

he set of goals (using the L ∞ 

distance function and representing

alancing and social justice principles). A parametric analysis can

e undertaken to determine the trade-off between balance and

ptimisation in decision or objective space. On the other hand,

he multi-choice goal programming model allows the decision

aker to specify multiple target values for each goal. Definitions

f the L p distance functions and their relationship within the goal

rogramming model are given in Romero, Tamiz, and Jones (1998 ).

n combination, the above goal programming variants provide a

omprehensive methodology for modelling diverse decision maker

references and underlying philosophies. However, they also have

he commonality that they focus on the expressed goals of a single

ntity, either a single decision maker or a group of decision mak-

rs with unified goals. This is a different decision making situation

o a network of stakeholders, all of whom have some influence

n the decision(s) to be made but may have different preferences

nd views on importance and compensation amongst the set of

bjectives under consideration. The methodology presented in this

aper is concerned with examining the mix between balancing

nd optimisation philosophies over a network of stakeholders.

he effects of compensatory ( L 1 ) and non-compensatory ( L ∞ 

)

ehaviour with respect to both the multiple stakeholders in dif-

erent parts of the network and multiple objectives is examined.

xtended goal programming is chosen as the base technique for

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846 D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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the methodology due to its synergies and similarities with the

required analysis. In fact, the methodology developed in this

paper can also be seen as an extension of and contribution to the

literature on extended goal programming. 

The remainder of the paper is divided into 4 sections.

Section 2 presents a more detailed discussion of extended goal

programming as well as detailing the literature on goal pro-

gramming for networks of decisions and multiple stakeholders.

Section 3 develops the methodology for, and algebraic form of, the

network extended goal programming model. Section 4 presents a

hypothetical example from the field of renewable energy planning

in order to demonstrate the methodology and discusses the results.

Finally, Section 5 draws conclusions. 

2. Relevant goal programming topics 

This Section reviews the current state-of-the-art of goal pro-

gramming in the topics of relevance to this paper. These are di-

vided into three sub-sections. The extended goal programming

model variant; the use of goal programming to model networks

of decisions and multiple stakeholders; and the inclusion of the

Rawlsian philosophies of social justice, balance, and fairness. 

2.1. Extended goal programming 

As described in Section 1 , the extended goal programming

model is introduced by Romero (20 01, 20 04 ) to allow a parametric

analysis of the trade-off between efficiency and balance between

the levels of achievement of the goal target values. Lexicographic

and non-lexicographic forms of the model are presented for the

cases of the presence and absence of a lexicographic ordering of

goals respectively. As this paper is concerned primarily with in-

vestigations of efficiency-balance trade-offs between stakeholders

and objectives over a decision network rather than prioritising of

objectives the non-lexicographic form of the extended goal pro-

gramming model is used. Assuming a linear form of the achieve-

ment function, percentage normalisation, and positive target values

( b i > 0 , i = 1 , . . . ., q ) ( Jones & Tamiz, 2010 ) gives the following al-

gebraic model: 

Min a = αλ + ( 1 − α) 

{ 

q ∑ 

i =1 

(
u i n i 

b i 
+ 

v i p i 
b i 

)} 

(1)

Subject to: 

u i n i 

b i 
+ 

v i p i 
b i 

≤ λ i = 1 , . . . , q (2)

f i ( x ) + n i − p i = b i i = 1 , . . . , q (3)

x ∈ F (4)

n i , p i ≥ 0 i = 1 , . . . , q (5)

Model ( 1 )–( 5 ) is defined as having q objectives and a set x of

decision variables. f i ( x ) is the achieved value of the i th objec-

tive which has an associated target value of b i . Deviational vari-

ables n i and p i denote the negative and positive deviations from

the i th target value respectively. The maximal weighted deviation

from amongst the set of unwanted deviations is denoted by λ. The

weights u i and v i are associated with the relative level of impor-

tance associated with the minimisation of the negative and pos-

itive deviational variables from the i th target value respectively.

Unwanted deviations are given a positive weight and deviations

which are not desired to be minimised are given a zero weight. α
is a parameter which controls the relative importance of efficiency
nd equity in the model. Note that whilst this model has assumed

ercentage normalisation and positive target values, other forms of

ormalisation could also be considered, which in turn could allow

or the inclusion of non-positive target values. 

The extended goal programming formulation allows for the

nclusion and combination of the optimisation, balancing, and

atisficing underlying philosophies of a single decision making

ntity ( Jones & Tamiz, 2010 ). The satisfying philosophy is evident

n the set of goals. The optimising philosophy is achieved via the

inimisation of the weighted sum of deviations (second term

n the achievement function ( 1 )) with the use of sufficiently

ptimistic target goal values. The balancing philosophy is achieved

hrough the inclusion of the maximal deviation (first) term in

he achievement function ( 1 ). Furthermore, the balance between

ptimisation (efficiency) and balance (equity) can be controlled at

ach priority level through the parameter α which can be varied

etween complete emphasis on optimisation ( α = 0 ) and complete

mphasis on balance ( α = 1 ). The EGP framework is therefore a

omprehensive tool for the inclusion of three types of underlying

hilosophies amongst a set of objectives on a single decision level

nto the goal programming framework. This framework has been

urther enhanced by since its inception. ( Jones & Jimenez, 2013 )

ropose two further meta-objectives to add to the original two of

ptimisation and balance. These are the number of goals achieved

representing a target achieving philosophy) and consistency with

airwise comparison matrix judgements in the case that they are

sed to provide the set of weights. It has hitherto been developed

n single decision layer form rather than in hierarchical network

orm, as proposed in this paper. 

.2. Networks of decisions and multiple stakeholders 

Many multiple criteria problems involve multiple stakeholders,

ho are defined as entities (organisations, individuals, or societal

roups) who are affected by the decision to be made. An indica-

ive, but not exhaustive, review of the use of goal programming for

roblems with either multiple stakeholders or a network structure

s given by the remainder of this paragraph. The initial example of

he use of goal programming to make decisions over a hierarchical

etwork is given by Charnes, Haynes, Hazleton, and Ryan (1975 )

ho formulate a model a three level network for environmental

and use planning. The model considers economic and environ-

ental objectives, as well as stakeholders including demographical

nd industrial groups, and has an assumed governmental decision

aker. Recent examples of goal programming models that consider

ultiple stakeholders include Nixon, Dey, Davies, Sagi, and Berry

2014 ) who consider the optimal location of biomass plants on a

egional level in India considering the needs of farmers, investors,

nd downstream consumers of electricity. Gebrezgabher, Meuwis-

en, and Oude Lansink (2014 ) construct an economically, socially

nd environmentally sustainable manure management system,

sing a combination of compromise programming and goal pro-

ramming based AHP that considers the needs of both farmers and

he wider society. Giménez, Bertomeu, Diaz-Balteiro, and Romero

2013 ) apply extended goal programming for Eucalyptus plantation

anagement considering economic and sustainability criteria

nd give suggestions for extensions to a multiple stakeholder

ituation. Li, Beullens, Jones, and Tamiz (2008 ) develop a two-level

ecision model of a hospital considering bed allocation at both

 departmental and hospital level to meet economic and per-

ormance goals. The above examples demonstrate that goal

rogramming is indeed a pragmatic tool for considering differ-

nt stakeholder groups and complex decisions over a network

hat may represent geographical regions, technological types,

ultiple organisations or subdivisions of an organisation, socio-

conomic groups, or communities. However, there is not yet



D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 847 

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1

 

 

 

 

 clear methodological basis as to how the interests of these

roups are represented and combined across a decision network,

specially when the mix of compensatory and non-compensatory

ehaviour amongst both stakeholders and objectives is present.

his paper makes a contribution towards the development of a

elevant methodology in this area. 

Note that this paper makes a distinction between multiple

takeholders and multiple decision makers. The latter case assumes

he responsibility for a decision lies with a group of decision mak-

rs. The methodology in this case belongs to the field of group de-

ision making. A recent example of the use of goal programming

o aid group decision making in the context of the Analytical Hier-

rchy Process is given by Wang and Li (2015 ). In contrast, this pa-

er is concerned with the case where the actual decision belongs

o a single decision maker with responsibility for considering the

eeds and opinions of multiple stakeholders who will be affected

y the decision. It is often the case that a given stakeholder will

ave more interest in, and hence place more emphasis on, a sub-

et of the objectives and/or decisions to be made. Different stake-

olders may hence have different views on the level of importance

o be assigned to individual objectives and the level of compensa-

ion between objectives and between stakeholders that should be

mployed. 

.3. Concepts of social justice, fairness and balance in goal 

rogramming 

The concept of fairness or balance is originally introduced into

he goal programming framework by Flavell (1976 ) who proposes

he Chebyshev goal programming model. This variant is based

round the L ∞ 

distance function which links to the Rawlsian

heory of social justice ( Rawls, 1973 ). Minimising the maximum

eighted, normalised deviation from a goal ensures that a balance

etween the levels of satisfaction of the goals is achieved. Cheby-

hev goal programming is hence associated with the concepts of

airness, equality, and social justice. However, it is important to

ote that the Chebyshev variant only explicitly treats the fairness

nd balance between objectives rather than between stakeholders

r different subset of objectives that may have importance in

he context of the model. The Chebyshev variant is integrated

n the variant encompassing extended goal programming model

s detailed in Section 3 . It has been practically used to control

ibration in vehicle suspension ( Li, Liang, Wang, & Dong, 2012 );

o allocate maintenance technicians to toolsets in a semiconductor

anufacturing plant ( Ignizio, 2004 ); and to select portfolios of

utual funds ( Tamiz, Azmi, & Jones, 2013 ). 

. Formulation of extended goal programming model with a 

etwork of multiple stakeholders and objectives 

This section proposes a model for parametric consideration of

fficiency and balance over hierarchical decision network consist-

ng of L layers with multiple objectives and multiple stakeholders.

t is assumed that each stakeholder is associated with one particu-

ar node in the network which could, for instance, represent a par-

icular geographical region or sub-division of an organisation. The

resented model does not however preclude extension to cover

takeholders with interests that cover multiple nodes or layers of

he network. A stakeholder will have their own preferential data

ith respect to the set of objectives regarded as important, and in

ome cases place minimal or no importance on a particular objec-

ive. Each stakeholder may also have different views on the level

f compensation between objectives. These preferential consider-

tions are incorporated into the model presented in formulation

 6 )–( 11 ). Each network layer l consists of J l nodes. The following

lgebraic model has the capacity to consider balance and efficiency
oth amongst objectives at a given node and amongst stakeholders

t a given level: 

in a = w 1 

[ 

α j 1 λ j 1 + 

(
1 − α j 1 

) K ∑ 

k =1 

( 

u 

j 1 

k 
n 

j 1 

k 

b j 
1 

k 

+ 

v j 
1 

k 
p j 

1 

k 

b j 
1 

k 

) ] 

+ 

L ∑ 

l=2 

w l 

[ 

βl D l + ( 1 − βl ) 

×
J l ∑ 

j l =1 

{ 

α j l λ j l + 

(
1 − α j l 

) K ∑ 

k =1 

( 

u 

j l 

k 
n 

j l 

k 

b j 
l 

k 

+ 

v j 
l 

k 
p j 

l 

k 

b j 
l 

k 

) } ] 

(6) 

Subject to, 

f j 
l 

k 
( x ) + n 

j l 

k 
− p j 

l 

k 
= b j 

l 

k 
k = 1 , . . . , K; j l = 1 , . . . , J l ; l = 1 , . . . , L 

(7) 

u 

j l 

k 
n 

j l 

k 

b j 
l 

k 

+ 

v j 
l 

k 
p j 

l 

k 

b j 
l 

k 

≤ λ j l k = 1 , . . . , K; j l = 1 , . . . , J l ; l = 1 , . . . , L 

(8) { 

α j l λ j l + 

(
1 − α j l 

) K ∑ 

k =1 

( 

u 

j l 

k 
n 

j l 

k 

b j 
l 

k 

+ 

v j 
l 

k 
p j 

l 

k 

b j 
l 

k 

) } 

≤ D l 

j l = 1 , . . . , J l ; l = 1 , . . . , L (9) 

x ∈ F 

n 

j l 

k 
, p j 

l 

k 
≥ 0 k = 1 , . . . , K; j l = 1 , . . . , J l ; l = 1 , . . . , L 

(10) 

λ j l ≥ 0 j l = 1 , . . . , J l ; l = 1 , . . . , L ; D l ≥ 0 l = 1 , . . . , L (11)

here f 
j l 

k 
( x ) is a function of decision variable set x that gives the

chieved value of objective k at node j l at network level l. n 
j l 

k 

nd p 
j l 

k 
are the negative and positive deviations from goal target

evel b 
j l 

k 
of objective k at node j l at network level l respectively.

 

j l 

k 
and v j 

l 

k 
are the weights associated with penalisation of nega-

ive and positive deviations from the goal target level of objective

 at node j l at network level l respectively. If a deviation is not

o be penalised then its associated weight should be set to zero.

f a particular objective is not relevant at a node then both asso-

iated weights should be set to zero. λ j l represents the maximal

eighted, normalised deviation from amongst the set of objectives

t node j l at network level l. D l represents the maximum com-

ined measure of stakeholder dissatisfaction amongst the set of

odes at network level l. These are the two key measures of bal-

nce in the model. It is also important to note that the first level

as been modelled as a separate term in the achievement func-

ion ( 6 ). This is due to the fact that it represents the centralisation

ortion of the network as opposed to the other levels, which repre-

ent the devolved decision making in the network. Hence there are

mportant philosophical and modelling reasons that justify its sep-

rate consideration. A diagrammatical illustration of the extended

etwork goal programming algebraic model ( 7 )–( 11 ) is given in Fig.

 . There are three principal parameter sets in the model: 

(1) w l is the relative level of importance given to network level

l. The set of network level weights w = w 1 , w 2 , . . . , w L gives

the centralisation versus decentralisation strategy of the de-

cision maker. It is suggested that the network level weights

w are normalised via the following equation: 

L ∑ 

w l = 1 
l=1 



848 D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 

Fig. 1. Diagrammatic illustration of the extended network goal programming model. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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(2) α j l gives the level of consideration of balance versus opti-

misation amongst objectives at node j l at network level l.

It is subject to the bounds 0 ≤ α j l ≤ 1 , where α j l = 0 indi-

cates the stakeholder(s) associated with that node is solely

interested in the (weighted sum) efficiency of the objectives

and α j l = 1 indicates the stakeholder(s) associated with that

node is solely interested in the (minmax) balance of the ob-

jectives. Thus α j l can be seen as a measure of consideration

of the balance and efficiency mix between objectives. 

(3) βl gives the level of consideration of balance versus op-

timisation amongst stakeholders scores at network level l.

It is subject to the bounds 0 ≤ βl ≤ 1 , where βl = 0 indi-

cates that importance at network level l is solely given to

the average stakeholder dissatisfaction and βl = 1 indicates

that importance is solely given to the maximal stakeholder

dissatisfaction at that level. Thus βl can be seen as a mea-

sure of consideration of the balance and efficiency mix be-

tween stakeholders at network level l. 

As previous stated, the decision network is assumed to be con-

trolled by a single decision maker whose role is to consider all

stakeholders in the decisions to be made. A parametric analysis
round the three key parameter sets is proposed in order that the

ecision maker gain understanding about the nature of the trade-

ffs between balance and efficiency and the effect of compensatory

ehaviour in the model. An example of a parametric analysis is

iven in the model presented in Section 4 . 

. An illustrative example 

This section develops an example decision network related to

egional development of renewable energy sources in order to test

nd illustrate the model developed in Section 3 . The model has

wo levels (regional and global) and four sets of goals relating

o energy generation, cost, environmental impact, and number of

rojects developed. Parametric analysis is then performed on a

pecific four region instance. The data used is hypothetical as the

urpose is illustration of method. The objectives and problem for-

ulation are however, inspired by the authors work on various Eu-

opean Union and São Paulo state, Brazil funded projects on the

evelopment of mathematical models for renewable energy. 



D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 849 

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∑

∑

∑

.1. Notation 

Following mathematical programming convention, the model

escription is divided into required input data; parameters to con-

rol the experimentation (the weights are considered as parame-

ers in this instance as, although the decision maker may have an

nitial estimate, some form of informal or formal sensitivity anal-

sis may be necessary; Jones, 2011 ); decision variables; and an al-

ebraic model. 

.1.1. Data 

n number of potential projects; 

J number of regions; 

K number of electricity generation types; 

Q j set of projects belonging to region j; 

Q k set of projects of electricity generation type k ; 

Q jk ( = Q j ∩ Q k ) set of projects of electricity generation type k be-

longing to region j; 

E i energy generation (average) from potential 

project i ; 

EG global energy generation target ( aim is to achieve

no less than this target ); 

E R j energy generation target for region j ( aim is to

achieve no less than these targets ); 

C i estimate annual cost for potential project i ; 

CG global cost target ( aim is to achieve no more than

this target ); 

C R j energy cost target for region j ( aim is to achieve

no more than these targets ); 

H i estimated environmental impact from potential 

project i ; 

HG global environmental impact target ( aim is to

achieve no more than this target ); 

H R j environmental impact target for region j ( aim is

to achieve no more than these targets ); 

T G k global target for number of projects of electricity

generation type k ( aim is to achieve no less than

these targets ); 

T R jk target for number of projects of electricity gener-

ation type k in region j ( aim is to achieve no less

than these targets ); 

s min 
j 

minimum number of projects to be selected in

region j. 

.1.2. Parameters 

w controls global-regional weighting; 

αG controls mix of optimisation and balance at a global

level; 

αR 
j 

controls mix of optimisation and balance in individual re-

gion j; 

β controls mix of optimisation and balance when consider-

ing set of regions; 

u G 
E 

weight associated with penalising negative deviation

from global energy target; 

v G 
C 

weight associated with penalising positive deviation from

global cost target; 

v G 
H 

weight associated with penalising positive deviation from

global environmental target; 

u G 
T k 

weight associated with penalising negative deviation

from global target for electricity generation type k ; 

u R 
E j 

weight associated with penalising negative deviation

from energy target of region j; 

v R 
C j 

weight associated with penalising positive deviation from

cost target of region j; 

v R 
H j 

weight associated with penalising positive deviation from

environmental target of region j; 
v R 
T jk 

weight associated with penalising negative deviation

from target for electricity generation type k in region j. 

.1.3. Decision variables 

The following sets of decision and deviation variables are

pecified 

 i = 

{
1 I f project i selected 
0 otherwise 

i = 1 , . . . , n 

λG maximal deviation from set of global normalised,

weighted goals; 

λR 
j 

maximal deviation from set of global normalised,

weighted goals in region j; 

D maximal measure from amongst the set of regions (the

worst performing region); 

n G 
E 

cegative deviation from global energy target; 

p G 
E 

positive deviation from global energy target; 

n G 
C 

negative deviation from global cost target; 

p G 
C 

positive deviation from global cost target; 

n G 
H 

negative deviation from global environmental target; 

p G 
H 

positive deviation from global environmental target; 

n G 
T k 

negative deviation from global electricity generation type

k target; 

p G 
T k 

positive deviation from global electricity generation type

k target; 

n R 
E j 

negative deviation from energy target for region j; 

p R 
E j 

positive deviation from energy target for region j; 

n R 
C j 

negative deviation from cost target for region j; 

p R 
C j 

positive deviation from cost target for region j; 

n R 
H j 

negative deviation from environmental target for region

j; 

p R 
H j 

positive deviation from environmental target for region

j; 

n R 
T jk 

negative deviation from electricity generation type k tar-

get for region j; 

p R 
T jk 

positive deviation from electricity generation type k tar-

get for region j. 

.1.4. Algebraic model 

The algebraic form of the two-layer network extended goal pro-

ramming is given by Eqs. (12) –( 26 ). 

in a = w 

[
αG λG + 

(
1 − αG 

)(u 

G 
E n 

G 
E 

EG 

+ 

v G C p 
G 
C 

CG 

+ 

v G D p 
G 
D 

HG 

+ 

K ∑ 

k =1 

u 

G 
T k 

n 

G 
T k 

T G k 

) ] 

+ ( 1 − w ) 

[ 

βD + ( 1 − β) 

J ∑ 

j=1 

αR 
j λ

R 
j + 

(
1 − αR 

j 

)(u 

R 
E j 

n 

R 
E j 

E R j 

+ 

v R 
C j 

p R 
C j 

C R j 

+ 

v R 
H j 

p R 
H j 

H R j 

+ 

K ∑ 

k =1 

v R 
T jk 

n 

R 
T jk 

T R jk 

) ] 

(12)

Subject to, 

n 
 

i =1 

E i s i + n 

G 
E − p G E = EG (13) 

n 
 

i =1 

C i s i + n 

G 
C − p G C = CG (14) 

n 
 

i =1 

H i s i + n 

G 
D − p G D = HG (15) 



850 D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 

Fig. 2. Diagrammatic illustration of the renewable energy planning example. 

 

 

 

 

 

 

 

α

 

∑
 

s

 

n

 

 

w  

e  

d  

t  

t  

a  

u  

f  

g  
∑ 

i ∈ Q k 
s i + n 

G 
T k − p G T k = T G k k = 1 , . . . , K (16)

∑ 

i ∈ Q j 
E i s i + n 

R 
E j − p R E j = E R j j = 1 , . . . , J (17)

∑ 

i ∈ Q j 
C i s i + n 

R 
C j − p R C j = C R j j = 1 , . . . , J (18)

∑ 

i ∈ Q j 
H i s i + n 

R 
H j − p R H j = H R j j = 1 , . . . , J (19)

∑ 

i ∈ Q jk 
s i + n 

R 
T jk − p R T jk = T R jk k = 1 , . . . , K; j = 1 , . . . , J (20)

u 

G 
E n 

G 
E 

EG 

≤ λG , 
v G C p 

G 
C 

CG 

≤ λG , 
v G H p 

G 
H 

HG 

≤ λG , 

K ∑ 

k =1 

u 

G 
T k 

n 

G 
T k 

T G k 

≤ λG (21)

u 

R 
E j 

n 

R 
E j 

E R j 

≤ λR 
j , 

v R 
C j 

p R 
C j 

C R j 

≤ λR 
j , 

v R 
H j 

p R 
H j 

H R j 

≤ λR 
j , 

K ∑ 

k =1 

v R 
T jk 

n 

R 
T jk 

T R jk 

≤ λR 
j 

j = 1 , . . . , J (22)
R 
j λ

R 
j + 

(
1 − αR 

j 

)( 

u 

R 
E j 

n 

R 
E j 

E R j 

+ 

v R 
C j 

p R 
C j 

C R j 

+ 

v R 
H j 

p R 
H j 

H R j 

+ 

K ∑ 

k =1 

v R 
T jk 

n 

R 
T jk 

T R jk 

) 

≤ D 

j = 1 , . . . , J (23)

 

i ∈ Q j 
s i ≥ s min 

j j = 1 , . . . , J (24)

 i = 0 or 1 i = 1 , . . . , n ; n 

G 
E , p 

G 
E , n 

G 
C , p 

G 
C , n 

G 
H , p 

G 
H , λ

G ≥ 0 ; n 

G 
T k , 

p G T k ≥ 0 k = 1 , . . . , K (25)

 

R 
E j , p 

R 
E j , n 

R 
C j , p 

R 
C j , n 

R 
H j , p R H j , λ

R 
j ≥ 0 j = 1 , . . . , J; n 

R 
T jk , p R T jk ≥ 0 

j = 1 , . . . J k = 1 . . . K (26)

where Eq. (12) gives the achievement function to be minimised

hich is a specific form of achievement function ( 6 ). The param-

ters and decision variables in the second term have one less in-

ex that those of ( 6 ) as the network has two levels rather than

he generic L level case and hence the second term pertains solely

o a single (second) network layer. The unwanted deviational vari-

bles in Eqs. (13 )–( 20 ) are underlined, these are hence the set of

nwanted deviational variables to be minimised by achievement

unction ( 12 ). Eqs. (13) –( 15 ) give the global level goals for energy

eneration, annual cost, and environmental impact respectively.



D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 851 

Table 1 

Project data for specific instance. 

Project ( s i ) Region Type Energy ( E i ) Cost ( C i ) Environmental Impact ( H i ) 

1 1 1 34 12 5 

2 1 2 23 15 6 

3 1 1 24 18 7 

4 2 3 25 19 9 

5 2 1 56 45 3 

6 2 2 12 12 4 

7 3 2 14 34 6 

8 3 3 19 31 9 

9 3 2 72 64 2 

10 3 3 54 43 4 

11 4 3 12 14 7 

12 4 3 96 85 8 

13 4 1 54 45 9 

E  

o  

s  

t  

s  

p  

e  

d  

m  

r  

g  

g  

d  

c  

i  

c  

e  

t  

w  

l  

E  

v  

n  

l  

t  

o

4

 

b  

l  

s  

t  

f  

g

T

 

e  

o  

b  

n  

a  

Fig. 3. Effect of variance of parameter w on set of measures. 

Fig. 4. Effect of variance of parameter α on set of measures. 

o  

c  

d  

c  

c  

0  

a  

P  

o  

c  

w  

s  

a  

o  

e  

T  

w  

s

4

 

a  

m  

p

 

quation set ( 16 ) gives the goals for the global number of projects

f the K different electricity generation types selected. Equation

ets ( 17 )–( 19 ) give the J regional level goals for energy genera-

ion, annual cost, and environmental impact respectively. Equation

et ( 20 ) gives the J regional level goals for the global number of

rojects of the K different electricity generation types selected. In-

quality set ( 21 ) ensures that the weighted, normalised, unwanted

eviation from each global goal target is less than or equal to the

aximal global value ( λG ). Inequality set ( 22 ) ensures that for each

egion j the weighted, normalised, unwanted deviation from each

oal target is less than or equal to the maximal value for that re-

ion ( λR 
j 
). Note that at both a global and at a regional level the

eviations from the target numbers of projects to be funded are

onsidered as a set rather than separately by technology type. This

s to avoid over-emphasis being placed on this set of goals when

alculating the maximal deviation. Inequality set ( 23 ) ensures that

ach region’s composite score (i.e. the parametric combination of

he worst case and average deviations) is less than or equal to the

orst case regional score ( D ). Inequality set ( 24 ) ensures that at

east the minimal number of projects is selected in each region.

qs. ( 25 ) and ( 26 ) give the set of sign restrictions for deviation

ariables in the model. It is noted that this model is a mixed bi-

ary problem that can be solved by an integer programming so-

ution algorithm. The number of binary variables is equivalent to

he number of potential projects ( n ). A diagrammatic illustration

f algebraic model ( 12 )–( 26 ) is given in Fig. 2. 

.2. Experimentation on specific instance 

In order to demonstrate the effects of the level of compensatory

ehaviour between objectives and between stakeholders and the

evel of centralisation in decision making, a specific four region in-

tance of model ( 12 )–( 26 ) is constructed. The data giving region,

ype, energy output, cost and environmental impact of each project

or this instance is given in Table 1 . Furthermore, the global and re-

ional targets and minimal projects per region are set as follows: 

EG = 350 ; CG = 60 ; HG = 15 ; T G k = 4 , k = 1 , . . . , 3 ;
E R j = 80 , j = 1 , . . . , 4 ;
C R j = 25 , j = 1 , . . . , 4 ; H R j = 5 , j = 1 , . . . , 4 ; s min 

j = 1 , 

j = 1 , . . . , 4 ;
 R jk = 1 , j = 1 , . . . , 4 , k = 1 , . . . , 3 . 

An experimental analysis with respect to the three key param-

ters w, α, and β is conducted. In order to control the number

f executions, the assumption that the level of compensatory

ehaviour between objectives remains constant throughout the

etwork is made. That is, α = αG = αR 
j 

∀ j. This is a reasonable

ssumption in the context of energy planning, where no regional
r central stakeholder would wish to be seen as more or less

ompensatory in its approach than others, leading to a settling

own around a common level of tolerance of compensation. These

onsiderations give a three-parameter model, each of which is dis-

retised into six points on its zero to one range (0.01, 0.2, 0.4, 0.6,

.8, 0.99). Note that the values of 0.01 and 0.99 have been used

s the end points rather than 0 and 1 in order to avoid potential

areto inefficiency at the meta-objective level. This leads to 216

ptimisations, for which the average and maximal deviations at

entral and regional level are measured. This is in order to judge

hether the parameters giving emphasis on centralisation, optimi-

ation, and balance are working effectively and to draw conclusions

bout their effects. As the min purpose is to investigate the effect

f varying w, α, and β , an equal weight solution is used in order to

nsure that all stakeholders and objectives are equally considered.

he models are solved via the LINGO software on a PC machine

ith 3.10 GHz processor speed and 4GB RAM, with all models

olving in less than the minimal recording time of one second. 

.2.1. Results 

Figs.3 –5 show the effects of varying the single parameters w, α,

nd β respectively, with each observation compromising of the

ean of the relevant measure over the 36 values of the other two

arameters. The four measures used are: 

• AGND: Average global normalised deviation: 

( 
n G 

E 
EG + 

p G 
C 

CG + 

p G 
H 

HG + 

3 ∑ 

k =1 

n G 
Tk 

T G k 
) / 4 . 

• MGND: Maximum global weighted normalised deviation:

Max ( 
n G 

E 
EG , 

p G 
C 

CG , 
p G 

H 
HG , 

3 ∑ 

k =1 

n G 
Tk 

T G k 
) . 



852 D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 

Fig. 5. Effect of variance of parameter β on set of measures. 

 

 

 

 

 

Table 2 

Key solutions in meta-objective and decision space. 

w Alpha Beta AGND MGND ARND MRND Projects funded 

0.2 0 .2 0 .2 0 .794 1 .400 0 .528 1 .200 1, 2, 4, 8, 11 

0.8 0 .2 0 .2 0 .566 0 .780 0 .447 0 .850 1, 6, 8, 11 

0.2 0 .2 0 .8 0 .794 1 .400 0 .528 1 .200 1, 2, 4, 8, 11 

0.8 0 .2 0 .8 0 .565 0 .780 0 .447 0 .850 1, 6, 8, 11 

0.4 0 .4 0 .4 0 .794 1 .400 0 .528 1 .200 1, 2, 4, 8, 11 

0.6 0 .6 0 .6 0 .794 1 .400 0 .528 1.200 1, 2, 4, 8, 11 

0.2 0 .8 0 .2 0 .882 1 .667 0 .562 1 .600 1, 2, 4, 6, 8, 11 

0.8 0 .8 0 .2 0 .633 0 .733 0 .440 1 .200 1, 2, 6, 10, 11 

0.2 0 .8 0 .8 0 .882 1 .667 0 .562 1 .600 1, 2, 4, 6, 8, 11 

0.8 0 .8 0 .8 0 .633 0 .733 0 .440 1 .200 1, 2, 6, 10, 11 

 

p  

m  

w  

w  

o  

i  

f  

t  

s  

m  

s

• ARND: Average regional normalised deviation:
4 ∑ 

j=1 

( 
n R 

E j 

E G j 
+ 

p R 
C j 

C G j 
+ 

p R 
H j 

H G j 
+ 

3 ∑ 

k =1 

n R 
T jk 

T G j k 
) / 16 . 

• MRND: Maximum regional normalised deviation:

Max ( 
n R 

E j 

E G j 
, 

p R 
C j 

C G j 
, 

p R 
H j 

H G j 
, 

3 ∑ 

k =1 

n R 
T jk 

T G j k 
, j = 1 , . . . , 4 ) . 

Figs. 6–8 give the effects of varying two of the three parameters

within their defined range values. The same four measures as in

Figs. 3–5 (AGND, MGND, ARND, MRND) are used. Each observation

is the mean of the six values of the third parameter. 
Fig. 6. Effect of variance of parame
Figs. 3–8 have focussed on the visualisation of the solution in

arameter space, considering a set of measures that are essentially

eta-objectives, that is they are functions of a number of un-

anted deviation variables ( Jones & Jimenez, 2013 ). This is in line

ith the focus of this paper which is that of parametric analysis

f the meta-objective space. In order to understand the solution

n the decision space ( Jones, 2011 ), Table 2 presents the solutions

ound at the eight corner points of the parameter space and

wo central points. Columns 1–3 give the solution in parameter

pace, columns 4–7 give the solution in the four dimensional

eta-objective space, and column 8 gives the solution in decision

pace in terms of the set of funded projects. 
ters w, α on set of measures. 



D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 853 

Fig. 7. Effect of variance of parameters w, β on set of measures. 

4

 

(  

i  

t  

n  

r  

a  

l  

i  

e  

n  

w  

v  

m  

i  

c  

n  

e  

i  

c  

a  

t  

o  

b  

t  

c  

w  

i  

F  

t  

o  

c  

h  

t

 

(  

f  

T  

d  

e  

o  

c

 

c  

b  

i  

F  

s  

e  

p  

i  

w  

f  

s  

a  

i  

a  
.2.2. Discussion of results 

Considering the first order effects, an increase in parameter w

 Fig. 3 ) implies a growing level of centralisation of decision mak-

ng, as the central decision maker retains a greater proportion of

he total weight and passes down a lesser proportion to the lower

etwork level. In the example, this is shown to have the effect of

educing (i.e. improving) all of the four measures, thus showing

n advantage not only on the global level but also on the regional

evel. This shows that a level of benevolent coordination is of value

n decision networks such as the one proposed in this paper. How-

ver, it is also noted that the level of improvement is more pro-

ounced on the global than on the regional level. This shows that

hilst centralisation of decision making may have symbiotic ad-

antage for the whole network, the maximum level of improve-

ent in the example is found at the central level. An increase

n parameter α ( Fig. 4 ) implies an increase in the level of non-

ompensation amongst objectives for all stakeholders across the

etwork. That is, stakeholders are less willing to accept a wors-

ning in the achieved value of one objective in order to gain an

mproved value in another objective. Fig. 4 demonstrates that in-

reasing the level of compensation amongst objectives in the ex-

mple leads to worse values for all four measures. This indicates

hat if stakeholders were to allow for more flexibility in terms

f trade-offs between objectives, then a better overall solution on

oth a global and regional scale could be achieved. An increase in

he parameter β ( Fig. 5 ) implies an increase in the level of non-

ompensation between stakeholders at different nodes in the net-

ork. That is, stakeholders are less willing to accept a worsening

n their position in order that the overall position is improved.
ig. 5 shows that increasing the level of non-compensation be-

ween stakeholders in the example leads to an improvement in all

f the four measures. This shows that in order to build an effective

onsensus in the example, it is beneficial to ensure that all stake-

olders are effectively represented and not overly disadvantaged to

he gain of others. 

A common feature of all of the three single parameter effects

 Figs. 3–5 ) is that the sensitivity to parameter change is higher

or the two maximal measures than for the two average measures.

his indicates that determining the correct parametric mix or con-

ucting sufficient parametric sensitivity analysis is important in

nsuring that the “worst off” stakeholders in any decision are not

verly disadvantaged. This will help to promote the building of a

onsensus amongst stakeholders. 

Considering the two parameter effects, Figs. 6–8 confirm the

onclusions drawn above from the single parameter effects. It can

e seen that it most cases there is a general trend of increase

n both of the parameter directions identified in the analysis of

igs. 3–5 . This trend is more pronounced in the two maximal mea-

ures, MGND and MGRD. It is noted, however that there are some

xtreme parameter values which negate the sensitivity of the other

arameter. See for instance the w = 0 . 99 value of all four measures

n the w, β parametric analysis of Fig. 7 . There is no sensitivity to-

ards the parameter β at this level. This is an indication of the

act that if an extreme degree of centralisation is used, the sen-

itivity between stakeholders is lost. Some of the two parameter

nalyses also demonstrate areas of stability in parameter space. For

nstance, the MRND graph of the w, α analysis shows there is an

rea of low α and high w that will lead to good values of this



854 D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 

Fig. 8. Effect of variance of parameters α, β on set of measures. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t  

a  

p  

g  

e  

t  

s  

f  

m  

S  

t  

P  

p  

t  

v

 

a  

p  

t  

T  

g  

b  

i  

i  

h  

i  

t  

p  

s  
measure. This indicates a region with a high level of centralisation

and low level of non-compensation between objectives where good

values of regional balance can be found. 

With respect to the analysis of the solution in decision space

given in Table 2 , it can be seen that some solutions are repeated,

again indicating a lack of sensitivity to parameter change in some,

relatively limited regions in the parameter space. The trade-offs

taking place in the meta-objective space across the entire region

are evident from the values in the four meta-objective columns.

The last column demonstrates that, for this example, these changes

are achieved by relatively small changes in decision space, with the

addition or removal of marginal projects to a set of core projects

that appear in all solutions. It is also recognised that some of the

maximal deviations are significantly high, rising to a maximum de-

viation of 1.667 beyond the goal value. This is mainly driven by

the deliberate setting of challenging goal values in order to ensure

Pareto inefficiency does not occur ( Jones & Tamiz, 2010 ), but also

results the challenge of satisfying all stake-holders across a multi-

objective network. 

5. Conclusions 

This paper has extended the methodology of extended goal

programming to consider a decision network containing mul-

tiple stakeholders and objectives. The motivation for doing so

was the occurrence of situations that require this type of co-

ordinated network goal programme in the authors work on re-

newable energy. This should be regarded as an extension and

enhancement of the goal programming paradigm to encompass
he type of decision problems with conflicting objectives across

 network of stakeholders that are now arising in modern ap-

lications. For instance develop a single-layer extended goal pro-

ramme for offshore wind farm location that would benefit from

xtension to a decision making network if greater number and

ypes of renewable energy projects were to be included. A demon-

trative example from the renewable energy sector has been

ormulated in Section 4 and its results analysed. Although the

odel pertains to renewable energy, the methodology presented in

ections 2 and 3 is generic and hence the results could be applied

o any decision network with multiple objectives and stakeholders.

otential examples could include transportation networks, com-

uter networks, and decision making in large hierarchical organisa-

ions including defence organisations, universities, and health ser-

ice providers. 

The inclusion of the numerical example in Section 4 is intended

s a demonstrative concept. The presented results show that a

arametric analysis is capable of producing a range of solutions

hat vary across decision, objective, and meta-objective space.

he model presented in this paper can thus be used as a tool to

enerate solutions that will enhance the chances of a consensus

etween multiple stakeholders occurring. In the particular example

n Section 4 , this occurred with relatively high levels of central-

sation, low levels of non-compensation between objectives, and

igh levels of non-compensation. The prime usage of the model

s in a prescriptive sense to produce implementable solutions

o complex multi stakeholder multi-objective decision network

roblems. It can, however, also be used in a more descriptive

ense to simulate the effects of different levels of centralisation,



D. Jones et al. / European Journal of Operational Research 255 (2016) 845–855 855 

n

b  

o

 

a  

i  

(  

p  

o  

c

A

 

f  

P  

a  

p  

w  

h

R

C  

C  

C  

 

F

G  

 

G  

 

I  

 

J  

J  

 

J
L  

 

L  

 

N  

P
R

R  

R  

R  

 

S

T  

 

R  

W  

 

on-compensation between objectives and non-compensation 

etween stakeholders on the solutions generated in decision,

bjective and meta-objective space. 

This paper has presented the model as a complete technique,

nd it can be used as such. However, it also retains the flexibil-

ty associated in goal programming described in Jones and Tamiz

2010 ). This includes the ability to incorporate different underlying

hilosophies and to combine with other techniques from the fields

f Operational Research and/or artificial intelligence to enhance de-

ision making capability. 

cknowledgments 

The authors wish to gratefully acknowledge financial support

rom grants: 2014/04353-8, 2014/01604-0 and 2015/07293-9, São

aulo Research Foundation (FAPESP), for their funding of the first

uthor’s research visit to UNESP, Brazil in 2014 where the research

resented in this paper was primarily developed. The authors also

ish to thank the three anonymous referees whose comments

elped shape the final version of this paper. 

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	An extended goal programming methodology for analysis of a network encompassing multiple objectives and stakeholders
	1 Introduction
	2 Relevant goal programming topics
	2.1 Extended goal programming
	2.2 Networks of decisions and multiple stakeholders
	2.3 Concepts of social justice, fairness and balance in goal programming

	3 Formulation of extended goal programming model with a network of multiple stakeholders and objectives
	4 An illustrative example
	4.1 Notation
	4.1.1 Data
	4.1.2 Parameters
	4.1.3 Decision variables
	4.1.4 Algebraic model

	4.2 Experimentation on specific instance
	4.2.1 Results
	4.2.2 Discussion of results


	5 Conclusions
	 Acknowledgments
	 References