review article fair optimization and networks: a surveyof fairness was early recognized with respect...

Review ArticleFair Optimization and Networks A Survey

Wlodzimierz Ogryczak1 Hanan Luss2 MichaB Pioacutero34

Dritan Nace5 and Artur Tomaszewski4

1 Institute of Control and Computation Engineering Warsaw University of Technology 00-665 Warsaw Poland2Department of Industrial Engineering and Operations Research Columbia University New York NY 10027 USA3Department of Electrical and Information Technology Lund University 22100 Lund Sweden4 Institute of Telecommunications Warsaw University of Technology 00-665 Warsaw Poland5 Laboratoire Heudiasyc Universite de Technologie de Compiegne 60203 Compiegne France

Correspondence should be addressed to Wlodzimierz Ogryczak wogryczakelkapwedupl

Received 7 March 2014 Revised 3 August 2014 Accepted 6 August 2014 Published 3 September 2014

Academic Editor Yuri N Sotskov

Copyright copy 2014 Wlodzimierz Ogryczak et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Optimization models related to designing and operating complex systems are mainly focused on some efficiency metrics such asresponse time queue length throughput and cost However in systemswhich servemany entities there is also a need for respectingfairness each system entity ought to be provided with an adequate share of the systemrsquos services Still due to system operations-dependant constraints fair treatment of the entities does not directly imply that each of them is assigned equal amount of theservicesThat leads to concepts of fair optimization expressed by the equitable models that represent inequality averse optimizationrather than strict inequality minimization a particular widely applied example of that concept is the so-called lexicographicmaximin optimization (max-min fairness) The fair optimization methodology delivers a variety of techniques to generate fairand efficient solutions This paper reviews fair optimization models and methods applied to systems that are based on some kindof network of connections and dependencies especially fair optimization methods for the location problems and for the resourceallocation problems in communication networks

1 Introduction

System design and optimization often lead to diverse alloca-tion problems where limitedmeansmust be assigned to com-peting agents or activities so as to achieve the best overallsystem performance Depending on the context the alloca-tion decisions may pertain to costs tasks goods or otherresources that can be assigned to one or several agents (actu-ally most allocation problems can be interpreted as resourceallocation problems) Such problems arise in numerousapplications of considerable complexity with system compo-nents being users stakeholders and their coalition systemseconomic and governmental institutions policy systemsenvironmental systems [1 2] and so forth Very often com-plex systems that involve resource allocation can essentiallybe treated as systems of systems [3 4]

The generic resource allocation problemmay be stated asfollows Each activity is measured by an individual perfor-mance function that depends on the resource levels assigned

to that activity A larger function value is considered betterlike in the case when the performance ismeasured in terms ofassigned system capacity quality of service level serviceamount available and so forth In practical applications onecan distinguish different variants of the general allocationproblem depending on whether the resource is divisible ornot In particular one-to-one allocation of indivisible resou-rces lead to the well-known assignment problem whilemany-to-many allocation problems arise in task schedulingwhere a task can be assigned in parallel to several agents witheach agent being potentially in charge of several tasks

Most approaches to allocation problems are focused onefficiency-based objectives However the maximization ofeither total or average results across all relevant agents mayrequire compromising individual agents for the good ofothers as long as everyonersquos good is taken impartially intoaccount Thus with the increasing awareness of systeminequity resulting from solely pursuing efficiency a num-ber of fairness or equity oriented approaches have been

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 612018 25 pageshttpdxdoiorg1011552014612018

2 Journal of Applied Mathematics

developed a particular example is models of resource allo-cation that try to achieve some form of fairness in resourceallocation patterns [5] In general the models relate to theoptimization of systems which serve many users and thequality of service provided to every individual user definesthe optimization criteria That pattern applies among othersto telecommunication and Internet networks in those net-works it is important to allocate network resources such asavailable bandwidth so as to provide equitable performanceto all services and all origin-destination pairs of nodes [6 7]Still there aremany other pressing examples of systemswherefair distribution of resources is required Problems of efficientand fair resource allocation arise in complex systems of sys-tems when the system combines a number of component sys-tems such as resource supply systems utilization systems atdemand sites or users stakeholders and their coalitionsystems economic and governmental institutions policysystems and environmental systems Actually addressingfairness in particular types of systems of systems has become agreat challenge of the 21 century [8] as fairly dividing limitednatural resources (such as the fossil fuels the clean water andthe environments capacity to absorb greenhouse gases) isperceived as being of utmost importance

Essentially fairness is an abstract sociopolitical conceptthat implies impartiality justice and equity In order to ensurefairness in a given system all system entities have to beequally well provided with the systemrsquos services For examplethe issue of equity iswidely recognized in the analysis of locat-ing public services where the clients of a system are entitledto fair treatment according to community regulations In thatcontext the decisions often concern the placement of servicecenters or other facilities at such positions that all users aretreated in an equitable way with respect to certain criteria[9] In particular location of the facilities pertaining to publicservices such as police and fire departments and emergencymedical facilities should provide fair response time to alldemand locationswithin ametropolitan area Similarly waterresources should be allocated fairly [10]

As far as technical systems are concerned the importanceof fairness was early recognized with respect to problems ofallocation of bandwidth in telecommunication networks [1112] (resulting in many models and methods of fair optimiza-tion [7]) flight scheduling [13] and allocation of takeoff andlanding ldquoslotsrdquo at airports [14] In such areas as allocation ofresources in high-tech manufacturing and optimal allocationof water and energy resources the context of fair resourceallocation was additionally enriched by considering possiblesubstitutions among the resources models with such substi-tutions are presented in [5 Ch 4] and [15ndash17]

In general complex systems require mathematical pro-gramming models in order to describe the dependencies andto enable system optimization Many such models are basedon some kind of network of connections and dependenciesIn particular wide range of systemmodels are related to somekind of network flows that express realizations of competingactivities [18] This applies to telecommunication systemspower distribution systems transportation systems logisticssystems and so forthThe discrete location problems can alsobe viewed in terms of such network system [19 20]

The general purpose of this paper is to review fairoptimizationmodels and algorithms supporting efficient andfair resource allocation in problems related to such networkmodels The particular focus is on location-allocation prob-lems and allocation problems related to communication net-works since in those areas the fair optimization concepts havebeen extensively developed and widely applied

The paper is organized as follows In the next sectionwe present methodological foundations of fair optimizationmodels In Section 3 the most important models and meth-ods of fair optimization in communication networks arereviewed Section 4 aims at reviewing applications of fair-ness optimization in location and allocation problems Thecomputational complexity issues are addressed in Section 5The paper is concluded by addressing the most importantdirections of the development of fair optimization method-ology for network systems

2 Fairness Equity and Fair Optimization

21 Efficiency and Equity The generic allocation problemdeals with a system comprising a set 119868 of119898 services (activitiesagents) and a given set 119876 of allocation patterns (allocationdecisions) For each service 119894 isin 119868 a function 119891

119894(x) of the allo-

cation pattern x isin 119876 is defined This function measures theoutcome (effect) 119910

119894= 119891119894(x) of allocation pattern x for service

119894 In applications we consider this measure that usuallyexpresses the service quality In general outcomes can bemeasured (modeled) as service time service costs andservice delays as well as in a more subjective way In typicalformulations a larger value of the outcome means a bettereffect (higher service quality or client satisfaction) Other-wise the outcomes can be replacedwith their complements tosome large number Therefore without loss of generality wecan assume that each individual outcome 119910

119894is to be maxi-

mized which allows us to view the generic resource allocationproblem as a vector maximization model Consider

max f (x) x isin 119876 (1)

where f(x) is a vector-function that maps the decision space119883 = 119877

119899 into the criterion space 119884 = 119877119898 and 119876 sub 119883 denotes

the feasible set We consider complex systems represented bymathematical programming models and specifically modelsbased on some network of connections and dependencies

An outcome vector y is attainable if it expresses outcomesof a feasible solution x isin 119876 (ie y = f(x)) The set of all theattainable outcome vectors is denoted by 119860 Note that ingeneral convexity of the feasible set 119876 and concavity of theoutcome function f do not guarantee convexity of the corre-sponding attainable set 119860 Nevertheless the multiple criteriamaximization model (1) can be rewritten in the equivalentform

max y 119910119894le 119891119894 (x) forall119894 x isin 119876 (2)

where the attainable set 119860 is convex whenever 119876 is convexand functions 119891

119894are concave

Model (1) only specifies that we are interested in maxi-mization of all objective functions 119891

119894for 119894 isin 119868 = 1 2 119898

Journal of Applied Mathematics 3

In order to make it operational one needs to assume somesolution concept specifyingwhat itmeans tomaximizemulti-ple objective functionsThe solution conceptsmay be definedby properties of the corresponding preference model [21]The commonly used concept of the Pareto-optimal solutionsas feasible solutions for which one cannot improve anycriterion without worsening another depends on the rationaldominance which may be expressed in terms of the vectorinequality

Simple solution concepts for multiple criteria problemsare defined by aggregation (or utility) functions119892 119884 rarr 119877 tobe maximized Thus the multiple criteria problem (1) isreplaced with the maximization problem Consider

max 119892 (f (x)) x isin 119876 (3)

In order to guarantee the consistency of the aggregatedproblem (3) with the maximization of all individual objectivefunctions in the originalmultiple criteria problem (or Pareto-optimality of the solution) the aggregation function must bestrictly increasing with respect to every coordinate

The simplest aggregation functions commonly used forthe multiple criteria problem (1) are defined as the totaloutcome 119879(y) = sum

119898

119894=1119910119894 equivalently as the mean (average)

outcome 120583(y) = 119879(y)119898 = (1119898)sum119898

119894=1119910119894or alternatively

as the worst outcome 119872(y) = min119894=1119898

119910119894 The mean

(total) outcome maximization is primarily concerned withthe overall system efficiency As based on averaging it oftenprovides a solution where some services are discriminated interms of performance On the other hand the worst outcomemaximization that is the so-called max-min solution con-cept

max min119894=1119898

119891119894 (x) x isin 119876 (4)

is regarded as maintaining equity Indeed in the case of asimplified resource allocation problem with knapsack con-straints the max-min solution

max min119894=1119898

119910119894

119898

sum

119894=1

119886119894119910119894le 119887 (5)

takes the form 119910119894= 119887sum

119898

119894=1119886119894for all 119894 isin 119868 thus meeting the

perfect equity requirement1199101= 1199102= sdot sdot sdot = 119910

119898 In the general

case with possible more complex feasible set structure thisproperty is not fulfilled [22 23] Nevertheless if there exists aPareto-optimal vector y isin f(119876) satisfying the perfect equityrequirement 119910

1= 1199102= sdot sdot sdot = 119910

119898 then y is the unique optimal

solution of the max-min problem (4) [24]Actually the distribution of outcomesmaymake themax-

min criterion partially passive when one specific outcome isrelatively very small for all the solutions For instance whileallocating clients to service facilities such a situation maybe caused by existence of an isolated client located at aconsiderable distance from all the facilities Maximization ofthe worst service performances is then reduced to maximiza-tion of the service performances for that single isolated clientleaving other allocation decisions unoptimized For instancehaving four outcome vectors (1 1 1) (8 1 1) (1 8 1) and (8

8 1) available they are all optimal in the corresponding max-min optimization as the third outcome cannot be better than1 Maximization of the first and the second outcome is thennot supported the max-min solution concept allowing oneto select (1 1 1) as the optimal solution This is a clear case ofinefficient solution where one may still improve other out-comes while maintaining fairness by leaving at its bestpossible value the worst outcomeThemax-min solutionmaybe then regularized according to the Rawlsian principle ofjustice Rawls [25 26] considers the problem of rankingdifferent ldquosocial statesrdquo which are different ways in which asociety might be organized taking into account the welfare ofeach individual in each society measured on a single numer-ical scale Applying the Rawlsian approach any two statesshould be ranked according to the accessibility levels of theleast well-off individuals in those states if the comparisonyields a tie the accessibility levels of the next-least well-offindividuals should be considered and so on Formalizationof this concept leads us to the lexicographic maximin opti-mization model or the so-called max-min fairness where thelargest feasible performance function value for activities withthe smallest (ie worst) performance function value (this isthe maximin solution) are followed by the largest feasibleperformance function value for activities with the secondsmallest (ie second worst) performance function valuewithout decreasing the smallest value and so forth The lexi-cographic maximin solution is known in the game theory asthe nucleolus of amatrix game It originates froman idea pre-sented by Dresher [27] to select from the optimal (max-min)strategy set of a player a subset of optimal strategies whichexploit mistakes of the opponent optimally It has been laterrefined to the formal nucleolus definition [28] and gener-alized to an arbitrary number of objective functions [29]The concept was early considered in the Tschebyscheffapproximation [30] as a refinement taking into account thesecond largest deviation the third one and further to be hier-archically minimized Actually the so-called strict approx-imation problem on compact ordered sets is resolved byintroducing sequential optimization of the norms on sub-spaces Luss and Smith [31] published the first paper on lex-icographic maximin approach for resource allocation prob-lems with continuous variables and multiple resource con-straintsWithin the communications or network applicationsthe lexicographic maximin approach has appeared alreadyin [11 12] and now under the name max-min fair (MMF)is treated as one of the standard fairness concepts [7] Thelexicographic maximin has been used for general linearprogramming multiple criteria problems [32ndash34] as well asfor specialized problems related to multiperiod resourceallocation with and without substitutions [5 Ch 5] and [35ndash39]

In discrete optimization it has been considered for variousproblems [40 41] including the location-allocation ones [42]Luss [43] presented an expository paper on equitable resourceallocations using a lexicographic minimax (or lexicographicmaximin) approach while [44] provides wide discussion ofvarious models and solution algorithms in connection withcommunication networksThe recent book by Luss [5] bringstogether much of the equitable resource allocation research


from the past thirty years and provides current state of art inmodels and algorithm within wide gamut of applications

Actually the original introduction of the MMF in net-working characterized the MMF optimal solution by the lackof a possibility to increase of any outcome without decreasingof some smaller outcome [12] In the case of convex attainableset (as considered in [12]) such a characterization representsalso lexicographic maximin solution In nonconvex case aspointed out in [45] such strictly defined MMF solution maynot exist while the lexicographic maximin always exists andit covers the former if it exists (see [46] for wider discus-sion) Therefore the MMF is commonly identified with thelexicographic maximin while the classical MMF definition isconsidered rather as an algorithmic approach which isapplicable only for convex models We follow this in theremainder of the paper Indeed while for convex problems itis relatively easy to form sequential algorithms to execute lexi-cographic maximin by recursive max-min optimization withfixed smallest outcomes (see [5 31ndash33 43 44 46 47]) fornonconvex problems the sequential algorithms must be builtwith the use of some artificial criteria (see [24 40 42 4448] and [5 Ch 7]) Some more discussion is provided inSection 24

22 FromEquity to FairOptimization Theconcept of fairnesshas been studied in various areas beginning from politicaleconomics problems of fair allocation of consumption bun-dles [25 49ndash52] to abstract mathematical formulation [5354] Fairness is essentially an abstract sociopolitical conceptof distributive justice that implies impartiality and equity indistribution of goods In order to ensure fairness in a systemall system entities have to be equally well provided with thesystemrsquos services Therefore in systems analysis and oper-ational research fairness was usually quantified with theso-called inequality measures to be minimized [55ndash60] orfairness indices [61 62] Typical inequalitymeasures are somedeviation type dispersion characteristics They are inequalityrelevant which means that they are equal to 0 in the caseof perfectly equal outcomes while taking positive values forunequal ones The simplest inequality measures are basedon the absolute measurement of the spread of outcomes ordeviations from the mean like the mean absolute differencemaximum absolute difference standard deviation (variance)mean absolute deviation and so forth Relative inequalitymeasures are frequently used For instance measures arenormalizezd by mean outcome like the Gini coefficientwhich is the relative mean difference

Complex systems require usuallymathematical program-ming models in order to describe the dependencies and tomake possible system optimization Many such models arebased on some network of connections and dependencies Awide range of systemsmodels is related to some flowswithin anetwork expressing realizations of competing activities [18]This applies to communication systems power distributionsystems transportation systems logistics systems and soforth Among others the discrete location problems can beviewed in terms of such network system [19 20] Typicallyfairness is considered in relation to division of a given amount

(the cake division problem) imposing a consistency require-ment the reference points must sum to the total amountavailable to the agents A methodology capable to modeland solve fair allocation problems in the context of systemoptimization must take into account possible increase ofthe amount Unfortunately direct minimization of typicalinequality measures contradicts the maximization of indi-vidual outcomes and it may lead to inferior decisions Themax-min fairness represented by lexicographic maximinoptimization meets such needs This specific concept may begeneralized to concepts of fairness expressed by the equitableoptimization [9 24 43 63ndash65] representing inequality averseoptimization rather than inequality minimization Since theterm equitable optimization or equitable resource allocationis frequently used as limited to the lexicographic maximinoptimization (see [5]) we use the term fair optimization toexpress wider class of equitable approaches

The concept of fair optimization is a specific refinementof the Pareto-optimality taking into account the inequalityminimization according to the Pigou-Dalton approach Firstof all the fairness requires impartiality of evaluation thusfocusing on the distribution of outcome valueswhile ignoringtheir orderingThat means that in the multiple criteria prob-lem (1) we are interested in a set of outcome values withouttaking into account which outcome is taking a specific valueHence we assume that the preference model is impartial(anonymous symmetric) In terms of the preference relationit may be written as the following axiom

(119910120587(1)

119910120587(2)

119910120587(119898)

) cong (1199101 1199102 119910

119898)

for any permutation 120587 of 119868

(6)

whichmeans that any permuted outcome vector is indifferentin terms of the preference relation Further fairness requiresequitability of outcomes which causes that the preferencemodel should satisfy the (Pigou-Dalton) principle of trans-fers The principle of transfers states that a transfer of anysmall amount from an outcome to any other relatively worse-off outcome results in a more preferred outcome vector As aproperty of the preference relation the principle of transferstakes the form of the following axiom

1199101198941015840 gt 11991011989410158401015840 997904rArr y minus 120576e

1198941015840 + 120576e11989410158401015840 ≻ y

for 0 lt 120576 lt 1199101198941015840 minus 11991011989410158401015840

(7)

The rational preference relations satisfying additionallyaxioms (6) and (7) are called hereafter fair (equitable) rationalpreference relations We say that outcome vector y1015840 fairly(equitably) dominates y10158401015840 if and only if y1015840 is preferred to y10158401015840 forall fair rational preference relations In other words y1015840 fairlydominates y10158401015840 if there exists a finite sequence of vectorsy119895 (119895 = 1 2 119904) such that y1 = y10158401015840 y119904 = y1015840 and y119895 isconstructed from y119895minus1 by application of either permutation ofcoordinates equitable transfer or increase of a coordinate Anallocation pattern x isin 119876 is called fairly (equitably) efficient orsimply fair if y = f(x) is fairly nondominated Note that eachfairly efficient solution is also Pareto-optimal but not viceverse


In order to guarantee fairness of the solution con-cept (3) additional requirements on aggregation (utility)functions need to be introduced The aggregation functionmust be symmetric that is for any permutation 120587 of 119868119892(119910120587(1)

119910120587(2)

119910120587(119898)

) = 119892(1199101 1199102 119910

119898) as well as being

equitable (to satisfy the principle of transfers) 119892(1199101 119910

1198941015840 minus

120576 11991011989410158401015840 + 120576 119910

119898) gt 119892(119910

1 1199102 119910

119898) for any 0 lt

120576 lt 1199101198941015840 minus 11991011989410158401015840 Such functions were referred to as (strictly)

Schur-concave [66] In the case of a strictly increasing andstrictly Schur-concave function every optimal solution tothe aggregated optimization problem (3) defines some fairlyefficient solution of allocation problem (1) [64]

Both simplest aggregation functions the mean and theminimum are symmetric although they do not satisfy strictlythe equitability requirement For any strictly concave andstrictly increasing utility function 119906 119877 rarr 119877 theaggregation function119892(y) = sum

119898

119894=1119906(119910119894) is a strictlymonotonic

and equitable thus defining a family of the fair aggregations[64] Consider

max

119898

sum

119894=1

119906 (119891119894 (x)) x isin 119876 (8)

Various concave utility functions 119906 can be used to definethe fair aggregations (8) and the resulting fair solutionconcepts In the case of the outcomes restricted to positivevalues one may use logarithmic function thus resultingin the proportional fairness (PF) solution concept [67 68]Actually it corresponds to the so-called Nash criterion [69]whichmaximizes the product of additional utilities comparedto the status quo Again in the case of a simplified resourceallocation problem with knapsack constraints the PF solu-tion

max

119898

sum

119894=1

log (119910119894)

119898

sum

119894=1

119886119894119910119894le 119887 (9)

takes the form 119910119894

= 119887119886119894for all 119894 isin 119868 thus allocating the

resource inversely proportional to the consumption of par-ticular activities

For positive outcomes a parametric class of utility func-tions

119906 (119910119894 120572) =

1199101minus120572

119894

(1 minus 120572) if 120572 = 1

log (119910119894) if 120572 = 1

(10)

may be used to generate various fair solution concepts for120572 gt 0 [70] The corresponding solution concept (8) called120572-fairness represents the PF approach for 120572 = 1 while with120572 tending to the infinity it converges to the MMF For largeenough 120572 one gets generally an approximation to the MMFwhile for discrete problems large enough 120572 guarantee theexactMMF solution Such away to identify theMMF solutionwas considered in location problems [40 42] as well as tocontent distribution networking problems [71 72] Howeverevery such approach requires to build (or to guess) a utilityfunction prior to the analysis and later it gives only onepossible compromise solution For a common case of upper

W( )

y

u( ) = u( )

B( )

y2 = y1

y2

y1

y

yy

y

Figure 1 The fair dominance structures 119882(y) the set of outcomesfairly dominated by y and119861(y) the set of outcomes fairly dominatingy

bounded outcomes 119910119894

le 119906lowast one may maximize power

functions minussum119898

119894=1(119906lowastminus119910119894)119901 for 1 lt 119901 lt infinwhich is equivalent

tominimization of the corresponding119901-normdistances fromthe common upper bound 119906

lowast [64]Figure 1 shows the structure of fair dominance for two-

dimensional outcome space For any outcome vector ythe fair dominance relation distinguishes set 119882(y) of dom-inated outcomes (obviously worse for all fair rational prefer-ences) and set119861(y) of dominating outcomes (obviously betterfor all fair rational preferences) Some outcome vectorsremain neither dominated nor dominating (in white areas)and they can be differently classified by various specific fairsolution concepts The lexicographic maximin assigns theentire interior of the inner white triangle to the set of pre-ferred outcomes while classifying the interior of the externalopen triangles as worse outcomes Isolines of various utilityfunctions split the white areas in different ways For instancethere is no fair dominance between vectors (1 100) and(2 2) and the MMF considers the latter as better while theproportional fairness points out the former On the otherhand vector (2 99) fairly dominates (1 100) and all fairnessmodels (includingMMF and PF) prefer the former One maynotice that the set 119882(y) of directions leading to outcomevectors being dominated by a given y is in general not a coneand it is not convex Although when we consider the set 119861(y)of directions leading to outcome vectors dominating given ywe get a convex set

Certainly any fair solution concept usually leads to somedeterioration of the system efficiency when comparing to thesole efficiency optimization This is referred to as the price offairness and it was quantified as the relative difference withrespect to a fully efficient solution that maximizes the sumof all performance functions (total outcome) [73] that is theprice of fairness concept 119865 on the attainable set 119860 is definedas

POF (119865 119860) =

(sum119898

119894=1119910119879

119894minus sum119898

119894=1119910119865

119894)

sum119898

119894=1119910119879

119894

(11)

where y119879 is the outcome vectormaximizing the total outcome119879(y) on 119860 while y119865 denotes the outcome vector maximizing


the fair optimization concept119865(y) on119860 Formula (11) is appli-cable only to the problems with a positive total outcomemdashthis however is a common case for attainable sets of modelsbased on some network of connections and dependenciesBertsimas et al [73] examined the price of fairness for a broadfamily of problems focusing on PF and MMF models Theyshown that for any compact and convex attainable sets119860withequal maximum achievable outcome which are greater than0 the price of proportional fairness is bounded by

POF (PF 119860) le 1 minus2radic119898

119898 (12)

and the price of max-min fairness is bounded by

POF (MMF 119860) le 1 minus4119898

(119898 + 1)2 (13)

Moreover the bound under PF is tight if radic119898 is integer andthe bound under MMF is tight for all 119898 Similar analysisfor the 120572-fairness [74] shows that the price of 120572-fairness isbounded by

POF (120572119865 119860) le 1 minus min120578isin[1119898]

1205781+1120572

+ 119898 minus 120578

1205781+1120572 + (119898 minus 120578) 120578

cong 1 minus 119874 (119898minus120572(120572+1)

)

(14)

The price of fairness strongly depends on the attainable setstructure One can easily construct problems where any fairsolution is alsomaximal with respect to the total outcome (noprice of fairness occurs) In [75] the 120572-fairness concept fornetwork flow problems was analyzed and a class of networkswas generated with the property that a fairer allocation isalways more efficient In particular it implies that max-minfairness may achieve higher total throughput than propor-tional fairness

23 Multicriteria Models The relation of fair dominance canbe expressed as a vector inequality on the cumulative orderedoutcomes [63] The latter can be formalized as follows Firstwe introduce the ordering map Θ 119877

119898rarr 119877

119898 such thatΘ(y) = (120579

1(y) 1205792(y) 120579

119898(y)) where 120579

1(y) le 120579

2(y) le sdot sdot sdot le

120579119898(y) and there exists a permutation 120587 of set 119868 such that

120579119894(y) = 119910

120587(119894)for 119894 = 1 119898 Next we apply cumulation to

the ordered outcome vectors to get the following quantities

120579119894(y) =

119894

sum

119895=1

120579119895(y) for 119894 = 1 119898 (15)

expressing respectively the worst outcome the total ofthe two worst outcomes and the total of the three worstoutcomes Pointwise comparison of the cumulative orderedoutcomes Θ(y) for vectors with equal means was extensivelyanalyzed within the theory of equity [76] or themathematicaltheory of majorization [66] where it is called the relation ofLorenz dominance or weak majorization respectively Itincludes the classical results allowing to express an improve-ment in terms of the Lorenz dominance as a finite sequence

of Pigou-Dalton equitable transfers It can be generalized tovectors with various means which allows one to justify thefollowing statement [63 77] Outcome vector y1015840 isin 119884 fairlydominates y10158401015840 isin 119884 if and only if 120579

119894(y1015840) ge 120579

119894(y10158401015840) for all 119894 isin 119868

where at least one strict inequality holdsFair solutions to problem (1) can be expressed as Pareto-

optimal solutions for the multiple criteria problem withobjectives Θ(f(x)) Consider

max (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (16)

Hence the multiple criteria problem (16) may serve as asource of fair solution concepts Note that the aggregationmaximizing themean outcome corresponds tomaximizationof the last objective 120579

119898(f(x)) in problem (16) Similarly the

max-min corresponds to maximization of the first objective1205791(f(x)) As limited to a single criterion they do not guarantee

the fairness of the optimal solution On the other hand whenapplying the lexicographic optimization to problem (16)

lex max (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876

(17)

one gets the lexicographic maximin solution concept that isthe classical equitable optimization model [5] representingthe MMF

For modeling various fair preferences one may use somecombinations of the criteria in problem (16) In particular forthe weighted sum aggregation on getssum119898

119894=1119904119894120579119894(y) which can

be expressed with weights 120596119894= sum119898

119895=119894119904119895(119894 = 1 119898) allo-

cated to coordinates of the ordered outcome vector that isas the so-called ordered weighted average (OWA) [78 79]

max

119898

sum

119894=1

120596119894120579119894 (f (x)) x isin 119876 (18)

If weights 120596119894are strictly decreasing and positive that is 120596

1gt

1205962

gt sdot sdot sdot gt 120596119898minus1

gt 120596119898

gt 0 then each optimal solutionof the OWA problem (18) is a fairly efficient solution of(1) Such OWA aggregations are sometimes called orderedordered weighted averages (OOWA) [80] When looking atthe structure of fair dominance (Figure 1) the piece-wiselinear isolines of the OOWA split the white areas of outcomevectors remaining neither dominated nor dominating (cfFigure 2)

When differences between weights tend to infinity theOWA model becomes the lexicographic maximin [81] Onthe other hand with the differences between subsequentmonotonic weights approaching 0 the OWA model tends tothe mean outcome maximization while still preserving fairoptimizations properties (cf Figure 3)

To the best of our knowledge the price of fairness relatedto the fair OWA models has not been studied till now TheOWA aggregation may model various preferences from themax to the min Yager [78] introduced a well appealingconcept of the andness measure to characterize the OWA


W( )

B( )

OOWA

y2 = y1

y2

y1

y

y

y

Figure 2 The fair dominance structure and the ordered OWAoptimization

y2 = y1

y2

y1

1205962 = 1205961

1205961 ≫ 1205962

1205961 gt 1205962

1205961 gt 1205962

Figure 3 Variety of fair OWA aggregations

operators The degree of andness associated with the OWAoperator is defined as

andness (120596) =sum119898

119894=1((119898 minus 119894) (119898 minus 1)) 120596119894

sum119898

119894=1120596119894

(19)

For themin aggregation representing theOWAoperator withweights 120596 = (1 0 0) one gets andness(120596) = 1 whilefor the max aggregation representing the OWA operator withweights120596 = (0 0 1) one has andness(120596) = 0 For the total(mean) outcome one gets andness((1119898 1119898 1119898)) =

12 OWA aggregations with andness greater than 12 areconsidered fair and fairer when andness gets closer to 1 Agiven andness level does not define a unique set of weights 120596Various monotonic sets of weights with a given andnessmeasure may be generated (cf [82 83] and referencestherein)

Thedefinition of quantities 120579119896(y) is complicated as requir-

ing ordering Nevertheless the quantities themselves canbe modeled with simple auxiliary variables and linear con-straints Althoughmaximization of the 119896th smallest outcomeis a hard (combinatorial) problem The maximization of thesum of 119896 smallest outcomes is a linear programming (LP)problem as 120579

119896(y) = max

119905(119896119905minussum

119898

119894=1max119905minus119910

119894 0)where 119905 is an

unrestricted variable [84 85] This allows one to implementthe OWA optimization quite effectively as an extension of theoriginal constraints and criteria with simple linear inequali-ties [86] (without binary variables used in the classical OWAoptimization models [87]) as well as to define sequentialmethods for lexicographic maximin optimization of discreteand nonconvex models [48] Various fairly efficient solutionsof (1) may be generated as Pareto-optimal solutions tomulticriteria problem

max (1205781 1205782 120578

119898) (20a)

st x isin 119876 (20b)

120578119896= 119896119905119896minus

119898

sum

119894=1

119889119894119896

119896 = 1 119898

(20c)

119905119896minus 119889119894119896

le 119891119894 (x) 119889

119894119896ge 0

119894 119896 = 1 119898

(20d)

Recently the duality relation between the generalizedLorenz function and the second order cumulative distribu-tion function has been shown [88] The latter can also bepresented as mean shortfalls (mean below-target deviations)to outcome targets 120591

120575120591(y) =

1

119898

119898

sum

119894=1

(120591 minus 119910119894)+ (21)

It follows from the duality theory [88] that one may com-pletely characterize the fair dominance by the pointwisecomparison of the mean shortfalls for all possible targetsOutcome vector y1015840 fairly dominates y10158401015840 if and only if 120575

120591(y1015840) le

120575120591(y10158401015840) for all 120591 isin 119877 where at least one strict inequality

holds In other words the fair dominance is equivalent to theincreasing concave order more commonly known as theSecond Stochastic Dominance (SSD) relation [89]

For 119898-dimensional outcome vectors we consider all theshortfall values are completely defined by the shortfalls forat most119898 different targets representing values of several out-comes 119910

119894while the remaining shortfall values follow from the

linear interpolation Nevertheless these target values aredependent on specific outcome vectors and one cannot defineany universal grid of targets allowing to compare all possibleoutcome vectors In order to take advantages of the multiplecriteria methodology one needs to focus on a finite set oftarget values Let 120591

1lt 1205912lt sdot sdot sdot lt 120591

119903denote the all attainable

outcomes Fair solutions to problem (1) can be expressed asPareto-optimal solutions for the multiple criteria problemwith objectives 120575

120591119895

(f(x)) Consider

min (1205751205911

(f (x)) 1205751205912

(f (x)) 120575120591119903

(f (x))) x isin 119876

(22)

Hence the multiple criteria problem (22) may serve asa source of fair solution concepts When applying the


lexicographic minimization to problem (22) one gets thelexicographic maximin solution concept that is the classicalequitable optimization model [5] representing the MMFHowever for the lexicographicmaximin solution concept onesimply performs lexicographic minimization of functionscounting outcomes not exceeding several targets [42 48]

Certainly in many practical resource allocation problemsone cannot consider target values covering all attainableoutcomes Reducing the number of criteria we restrict oppor-tunities to generate all possible fair allocations Neverthelessone may still generate reasonable compromise solutions [24]In order to get a computational procedure one needs either toaggregate mean shortages for infinite number of targets or tofocus analysis on arbitrarily preselected finite grid of targetsThe former turns out to lead us to the mean utility optimiza-tion models (8) Indeed classical results of majorization the-ory [66] relate themean utility comparison to the comparisonof the weighted mean shortages Actually the maximizationof a concave and increasing utility function 119906 is equivalentto minimization of the weighted aggregation with positiveweights 119908(120585) = minus119906

10158401015840(120585) (due to concavity of 119906 the second

derivative is negative)

24 Methodologies for Solving Lexicographic Maximin Prob-lems Consider the following resource allocation problem

lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942

) le sdot sdot sdot le 119891119894119898

(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)

119897119894le 119909119894le 119906119894 forall119894 isin 119868 (23d)

where the performance functions are strictly increasing andcontinuous and 119886

119894119895ge 0 for all 119894 and 119895 The lexicographic

maximization objective function jointly with the orderingconstraints defines the lexicographic maximin objectivefunction (this is equivalent to defining the objective func-tion using the ordering mapping Θ) Consider Figure 4which presents a network that serves point-to-point demandsbetween nodes 1 and 2 nodes 3 and 4 and nodes 3 and 5The numbers on the links are the link capacities for example4 Gbs on links (1 3) Suppose demand between a node-pair can be routed only on a single path where this path isgiven as part of the input for example the path selectedbetween nodes 1 and 2 uses links (1 3) and (3 2) Theproblem of finding the lexicographic maximin solution ofdemand throughputs between various node-pairs subject tolink capacity constraints (which serve as the resource con-straints) can be formulated by (23a)ndash(23d)

It turns out that for various performance functionssuch as linear functions and exponential functions thelexicographic maximin solution of (23a)ndash(23d) is obtainedby simple algebraicmanipulations of closed-form expressionsand the computational effort is polynomial This facilitatessolving very large problems in negligible computing time For

3

2

4

1

5

2

34

11

2

Figure 4 A single path for each demand

3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1

Figure 5 Multiple path for demand between nodes 1 and 2

other functions where the solution cannot be derived usingclosed-form expressions somewhat more computations arerequired in particular function evaluations complementedby a one-dimensional numerical search are employed (see [5Ch 3] and [31 90 91]) Algorithms for problem (23a)ndash(23d)serve as building blocks for more complex problems suchas for problems with substitutable resources for multiperiodproblems and for content distribution problems (see [5 Chs4ndash6])

Now consider the cases of performance functions thatare nonseparable where each of the functions 119891

119894(119909119894) in (23a)

and (23b) is replaced by 119891119894(x) thus depending on multiple

decision variables Consider Figure 5 which shows threepossible paths for the demand between nodes 1 and 2 Thethroughput between this node-pair is simply the sum of flowsalong these three paths

Even for linear performance functions (eg throughputsin communication networks) the computational effort issignificantly larger as the algorithm for finding the lexico-graphic maximin solution requires solving repeatedly linearprogramming problems (see [5 Chs 34 and 62] [7 Ch 8]and [32 33 44 92])

Next consider the case of a nonconvex feasible regionfor example with discrete decision variables For exampleconsider a communication network (as in Figure 5) wherethe demand between any node-pair can flow along multiplepaths but only one of these paths may be selected (here theselected path for each demand is a decision variable) Theresulting formulation includes 0-1 decision variables [7]Again the objective is to find the lexicographic maximin


e1 e21205921 1205922 1205923

Figure 6 A network example illustrating fairness issues

solution of the throughputs where each demand uses onlyone path All the solution methods above do not apply If thenumber of possible distinct outcomes 120591

1lt 1205912

lt sdot sdot sdot lt 120591119903

is small one can construct counting functions where the 119896thcounting function value is the number of times the 119896th dis-tinct worst outcome appears in the solution That means thatone introduces functions ℎ

119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)

expressing the number of values 120591119896in the outcome vector y

The lexicographic maximin optimization problem is thenreplaced by lexicographicminimization of the counting func-tions ℎ

119896(y) which is solved by repeatedly solving minimiza-

tion problems with discrete variables

lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)

subject to x isin 119876 (24b)

120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)

where 119872 is a sufficiently large constant (see [5 Ch 72] and[44 48 93]) Moreover in general binary variables may beeliminated if large numbers of auxiliary continuous variablesand constrains are added leading to the formulation based on(22) (see [5 Ch 72] and [44 48 93 94])

When the number of distinct outcomes is large wecan solve the lexicographic maximin problem by solvinglexicographic maximization problems in the format of prob-lems (20a)ndash(20d) (see [5 Ch 73] and [44 48 64 94ndash96])Again the solution method adds many auxiliary variablesand constraints to the formulation

3 Fairness in Communication Networks

31 Fairness and Traffic Efficiency Fairness issues in commu-nication networks becomemost profound when dealing withtraffic handling Roughly speaking whenever the capacity ofnetwork resources such as links and nodes is not sufficientto carry the entire offered traffic a part of the traffic must berejectedThen a natural question arises how the total carriedtraffic traffic should be shared between the network usersin a fair way at the same time assuring acceptable overalltraffic carrying efficiency This kind of problems arise forexample in the Internet for elastic traffic sources which frommathematical point of view can be treated as generatinginfinite traffic Thus the total traffic that can eventually becarried by the network should be fairly split into the trafficflows assigned to individual demandsThis issue is illustratedby the following example [7]

Example 1 Consider a simple network composed of twolinks in series depicted in Figure 6 There are three nodes

(V1 V2 V3) two links (119890

1 1198902) and three demand pairs (119889

1=

V1 V2 1198892

= V2 V3 1198893

= V1 V3) The demands

generate elastic traffic that is each of them can consumeany bandwidth assigned to its path Suppose that the capacityof the links is the same and equal to 15 (119888

1= 1198882

=

15) Let 119883 = (1198831 1198832 1198833) be the path-flows (bandwidth)

assigned to demands 1198891 1198892 1198893 respectively Clearly such a

flow assignment is feasible if and only if 1198831 1198832 1198833ge 0 and

1198831+1198833le 1198881 1198832+1198833le 1198882 For the three basic traffic objectives

the solutions are as follows

(i) max-min fairness (lex max Θ(1198831 1198832 1198833)) 119883

1=

1198832= 1198833= 075 (119879(119883) = 225)

(ii) proportional fairness (max log1198831+ log119883

2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and

(iii) throughput maximization (max1198831+1198832+1198833) 1198831=

1198832= 15 119883

3= 0 (119879(119883) = 3)

Above 119879(119883) denotes the throughput that is 119879(119883) =

1198831+ 1198832+ 1198833 Clearly the MMF solution is perfectly fair

from the demand viewpoint but at the same the worst interms of throughput This is because the ldquolongrdquo demand 119889

3

consuming bandwidth on both links gets the same flowas the ldquoshortrdquo demands 119889

1 1198892 each consuming bandwidth

on its direct link The PF solution increase the flow ofshort demands at the expense of the long demand This isacceptably fair for the demands and increases the throughputFinally the 119879(119883) maximization solution is unfair (the longdemand gets nothing) but by assumption maximizes thethroughput

Note that in this example the price of max-min fairnesscalculated according to formula (11) is 14 which is equal tothe upper bound (13) Similarly the price of proportionalfairness 16 is close to its upper bound (12) However the priceof fairness strongly depends on the network topology In [75]the authors demonstrate a class of networks such that an 120572-fair allocation with higher 120572 is always more efficient in termsof total throughput In particular this implies that max-minfairness may achieve higher throughput than proportionalfairness

In the networking literature related to fairness the aboveMMFandPF objectives are themost popularThe throughputmaximization objective is rarely used as totally unfairInstead a reasonable modification consisting in lexicograph-ical maximization of the two ordered criteria (min(119883) 119879(119883))

is used where min(119883) denotes the minimal element of thedemand vector 119883

Considering MMF besides optimization objectivesdirectly related to traffic handling objectives related to linkloads are commonly considered in communication networkoptimization In this case the traffic volumes of demands tobe realized are fixed We shall come back to this issue lateron

32 Generic Optimization Models The considered networkis modeled with a graph G(VE) undirected or directedcomposed of the set of nodesV and the set of links E Thus


each link 119890 isin E represents an unordered pair V 119908 (undi-rected graphs) or an ordered pair (V 119908) (directed graphs) ofnodes V 119908 isin V and is assigned the nonnegative unit capacitycost 120585

119890which is a parameter and the maximum capacity 119888(119890)

which is a given constant (possibly equal to +infin) Whenlink capacities are subject to optimization they becomeoptimization variables denoted by 119910

119890 119890 isin E The cost of the

network is given by the quantity 119862 = sum119890isinE 120585119890119910119890 The traffic

demands are represented by the setD Each demand 119889 isin D ischaracterized by a directed pair (119900(119889) 119905(119889)) composed of theoriginating node 119900(119889) and the terminating node 119905(119889) and aminimum value ℎ(119889) (a parameter possibly equal to 0) of thetraffic volume that has to be carried from 119900(119889) to 119905(119889)Demand volumes and link capacities are expressed in thesame units

Each demand 119889 has a specified set of admissible pathsP(119889) (called the path-list) composed of selected elementarypaths from 119900(119889) to 119905(119889) in graphG (Recall that an elementarypath does not traverse any node more than once) Paths inP(119889) used to realize the demand (traffic) volumes areassigned flows 119909

119901 119901 isin P(119889) which are optimization vari-

ables Each value 119909119901

specifies the reference capacity(expressed in the same units as link capacity and demandvolume) reserved on path 119901 isin P(119889) The set of all admissiblepaths is denoted by P = ⋃

119889isinD P(119889) The maximumpath-lists that is path-lists P(119889) containing all elementarypaths from 119900(119889) to 119905(119889) will be denoted by P(119889) 119889 isin Dwith P = ⋃

119889isinD P(119889) The set of all paths in P traversina simple network composed of two links in series depictedin Figure 6 There are three nodes (V

1 V2 V3) two links

(1198901 1198902) and three demand pairs (119889

1= V1 V2 1198892

= V2 V3

1198893

= V1 V3) The demands generate elastic traffic that

is each of them can consume any bandwidth assigned toits path Suppose that the capacity of the links is the sameand equal to 15 (119888

1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be

the path-flows (bandwidth) assigned to demands 1198891 1198892 1198893

respectively 119892 a given link 119890 isin E will be denoted by Q(119890)Note that in an undirected graph the links can be traversedby paths in both directions while in a directed graphmdashonlyin the direction of the link

Let 119883119889

= sum119901isinP(119889) 119909119901 denote the total flow assigned to

demand 119889 isin D that is traffic of demand 119889 carried in thenetwork and let 119883 = (119883

119889 119889 isin D) Besides let 119884

119890=

sum119901isinQ(119890) 119909119901 be the link load induced by the path-flows Then

the generic feasibility set (optimization space) of a trafficallocation problem (TAP) can be specified as follows

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)

The set X specifies the domain of a path-flow variable andis problem-dependent Two typical cases are X = R

+and

X = Z+ Note that in the undirected graph the path-flows

through a link sum up to the link load no matter in whichdirection they traverse the link

The three cases of TAP considered in Example 1 above canbe now formulated as follows

(i) TAPMMF lex max Θ(119883) subject to (25a)ndash(25e)(ii) TAPPF max 119871(119883) = sum

119889isinD log119883119889subject to (25a)ndash

(25e) and(iii) TAPTM lex max (119872(119883) = min

119889isinD119883119889 119879(119883) =

sum119889isinD 119883

119889) subject to (25a)ndash(25e)

Observe that the third case above is actually different fromthe third case considered in Example 1 as now throughputmaximization is the secondary objective in lexicographicalmaximization

When X = R+ all the three problems are convex and

as such can be approached effectively by means of the algo-rithms described in [7 44 46] For the TAPPF version see[67] In fact TAPTM is a two level linear program possiblycombined to a single LP [23] and TAPMMF can be solvedas a series of linear programs [32 33 44 97] Optimizationapproaches to TAPPF are presented in [67]

Certainly the feasible set (25a)ndash(25e) can be furtherconstrained to consider more restricted routing strategiesThe most common restriction is imposed by the single-pathrequirement that each 119883

119889is carried entirely on one selected

path Then the feasibility set must be augmented by thefollowing constraints

sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)

In (26a)ndash(26c) 119906119901 119901 isin P are additional binary routing

variables and 119872 is a ldquobig 119872rdquo constant In this setting theabove defined TAP problems become essentially mixed-integer programming problems (FTPPF after a piece-wiseapproximation of the logarithmic function) and in the caseofMMFmust be treated by the general approach described inSection 23 as problem (20a)ndash(20d) (see also [44 48 64 94ndash96] and [5 Ch 73])

We note that when the routing paths are fixed that iswhen |P(119889)| = 1 119889 isin D then TAPMMF becomesthe classical fair allocation (equitable resource allocation)problem considered in Section 24 (see [12 Sec 652] and[5 Ch 61]) This version of the problem can be efficientlysolved in polynomial time by the so called water-fillingalgorithmbased on the bottleneck link characterization of theproblem (see [45] and Section 37) In fact the bottleneckcharacterization of this TAPMMF problem can be directlyformulated as an integer programming problem (with binaryvariables) as demonstrated in [92]Themodular flow versionof the problem is considered in [98]


An interesting version of the single-path TAPMMFproblem is considered in [99] that uses the bottleneck formu-lation of [92] In that problem the routes are optimized so toachieve the maximum traffic throughput while maintainingthe MMF demand traffic assignment

The above specified problems use the noncompact link-path formulationwhere the optimization variables are relatedto the routing paths Hence whenwewish to consider all pos-sible elementary paths then the number of variables 119909

119901 119901 isin

P becomes exponential with the size of the network In thiscase path generation algorithm should be applied (this is easyin the case of linear programs) or the problems should bereformulated in the node-link notation using link-flow vari-ables instead of the path-flow variables used in (25a)ndash(25e)

33 Selected Specific Models In this section we will discussseveral specific network optimization models related to var-ious aspects of fairness An interesting case arise when thetraffic demands ℎ(119889) 119889 isin D are considered as given and thedesign objective is to balance the load of the links aimingat minimizing the average packet delay in the network Thecommonly known formulation of such load balancing is asfollows

min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)

119903 isin R 119909119901isin X 119901 isin P (27d)

Using the MMF notion it is easy to define a load balancingproblem that is stronger than problem (27a)ndash(27d) which infact find the maximum element of the MMF vector 119877 = (119903

119890

119890 isin E) expressing the relative link loads

lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)

Some variants of the problem given by (28a)ndash(28d) werestudied in [100 101]

Another version of the MMF load balancing problem(28a)ndash(28d) maximizes the unused link capacity 119884 = (119884

119890

119890 isin E) in a fair way relevant to circuit switching

lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)

Above we have considered flow allocation problemsassuming given link capacity When the link capacity is sub-ject to optimization that is whenwe simultaneously optimizepath-flows and link capacities then we deal with dimension-ing problems An example of such a problem (with a budgetconstraint) is as follows

lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)

where 119861 gt 0 is a given budget for the total link cost Notethat we have skipped constraint (25b) which has established alower bound on the demand traffic allocation in formulation(25a)ndash(25e) If no additional constraints are enforced (as(25b)) then the optimal solution of (30a)ndash(30e) is trivial Foreach demand 119889 isin D the optimal traffic 119883

119889= 119883lowast is the same

and realized on the cheapest path 119901(119889) isin P119889with respect to

the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)

When the PF objective

min sum

119889isinD

log 119883119889 (32)

instead of the MMF objective (30a) is considered then theoptimal solution is as follows (see [7 68 102])

119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)

so the total optimal flow119883lowast

119889allocated to demand119889 is inversely

proportional to the cost of its shortest path (and allocated tothis path)


More complicated optimization problems including linkdimensioning were treated in [7 Ch 13] (see also [103 104])For the MMF optimization problems related to wirelessnetworks (in particular to Wireless Mesh Networks) thereader can refer to [105]

34 Extended Fairness Objectives While the MMF and PFobjectives are the most popular in the networking literaturerelated to fairness there are also attempts to find variousfair solutions taking advantages of the multicriteria fairoptimization models presented in Section 23 In particularthe OWA aggregation (18) was applied to the networkdimensioning problem for elastic traffic [95] as well as to theflow optimization in wireless mesh networks [106]

Example 2 Consider the simple network from Example 1composed of two links in series depicted in Figure 6 Thereare three demand pairs (119889

1= V1 V2 1198892

= V2 V3 1198893

=

V1 V3) generating elastic traffic where119883 = (119883

1 1198832 1198833) are


respectively Note that the ordered OWA maximization withdecreasing weights 120596 = (04 035 025) results in bandwidthallocation 119883

1= 15 119883

2= 15 119883

3= 0 thus representing

themaximum throughputOrderedOWAmaximizationwithdecreasing weights 120596 = (06 03 01) results in bandwidthallocation 119883

1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution

It was demonstrated that allocations representing theclassical fairness concepts (MMF and PF) were easy toachieve [95] On the other hand in order to find a largervariety of new compromise solutions it was necessary toincorporate some scaling techniques originating from the ref-erence pointmethodology Actually it is a commonflawof theweighting approaches that they provide poor controllabilityof the preference modeling process and in the case of multi-criteria problems with discrete (or more general nonconvex)feasible sets they may fail to identify several compromiseefficient solutions In standard multicriteria optimizationgood controllability can be achieved with the direct use ofthe reference point methodology [107] based on reservationand aspiration levels for each of the activitiesThe reservationlevels are the required activity levels whereas the aspirationlevels are the desired levels commonly referred to as refer-ence points The reference point methodology applied to thecumulated ordered outcomes (16) was tested on the problemof network dimensioning with elastic traffic [96 108] Thetests confirmed the theoretical advantages of the methodVarious (compromise) fair solutions for both continuous andmodular problems could be easily generated

Multiple criteria model of the mean shortfalls to allpossible targets (22) when applied to network dimensioningproblem for elastic traffic results in a model with criteriathat measure actual network throughput for various levels(targets) of flows [109] Thereby the criteria can easily beintroduced into the model Experiments with the referencepoint methodology applied to the multiple target throughputmodel confirmed the theoretical advantages of the methodVarious (compromise) fair solutions were easily generated

despite the fact that the single path problem (discrete one)was analyzed

Both the multiple criteria models with the lexicographicoptimization of directly defined artificial criteria introducedwith some auxiliary variables and linear inequalities providescorresponding implementations for the MMF optimizationindependently from the problem structure The approachesguarantee the exact MMF solution for a complete set of crite-ria and their applicability is limited to rather small networksIn [94] there were developed some simplified sequentialapproaches with reduced number of criteria thus generatingeffectively approximations to the MMF solutions Compu-tational analysis on the MMF single-path network dimen-sioning problems showed the approximated models allowedto solve within a minute problems for networks with 30nodes and 50 links providing very small approximationerrors thus suggesting possible usage in many practicalapplications

35 Fairness on the Session Level One of themajor challengesof the Internet is to provide high performance of data trans-port Basically the problem is how to obtain high utilizationof network resources and to ensure required quality ofcommunications services Those two goals result in a poten-tial trade-off as when the amount of data sent through thenetwork is too high links become overloaded and the qualityof service deteriorates

The overload occurs when the amount of data loading theoutgoing link of the Internet router is higher than the one thatcan actually be carried When that happens the linkrsquos queueof packets becomes longer and potentially the queuersquos bufferfinally overflows That causes the increase of packet delayand delay variations and may also cause packet loss Bothphenomena are perceived by the pair of communicatingInternet applications as low quality of data transport

Let S be the set of Internet sessions which are packetflows between pairs of Internet applications Let function119897 S 997891rarr R

+define the average packet length of the session

expressed in bits and for each 119904 isin S let variable119909119904denote the

packet rate of session 119904 Then for each 119904 isin S 119909119904119897(119904) is an

average bit-rate of session 119904Let E be the set of network links and for each 119904 isin S let

E(119904) denote the set of links that are used by session 119904 and foreach 119890 isin E let S(119890) denote the set of sessions that use link119890 Then the load of link 119890 isin E is equal to sum

119904isinS(119890) 119909119904119897(119904) Letfunction 119888 E 997891rarr R

+denote the capacity (the bit-rate) of the

link The following constraint expresses the fact that the totalload of any link cannot be greater than the linkrsquos capacityConsider

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)

The overload of the Internetrsquos link is a very common situationThe links can become overloaded for a number of reasonswhen the amount of traffic entering the network becomessignificantly larger when links lose some capacity due to fail-ures or when they fail completely and the packet flows mustbe rerouted to some other links that do not have sufficient


capacity Thus solving the trade-off between utilization andquality of service requires effective mechanisms of handlingoverload That is the place when the concept of fairness isused

The data between a pair of applications in the Internetcan be conveyed using one of two transport protocols userdatagram protocol (UDP) and transport control protocol(TCP)While theUDP is a connectionless data transport pro-tocol where each data packet is sent individually and thereis no interaction between the sending and the receivingapplication the TCP protocol is connection-oriented whichmeans that packets are sent within a connection that must beorganized between the sending and the receiving applicationbefore the data can be sent and can be torn down only afterthe last packet has been delivered Due to the connection-oriented character of the TCP flows there is an associationbetween the two applications which allows them to controlthe packet rate

With the flow control mechanisms of the TCP protocolthe rate at which packets are sent is adapted to network con-ditions if the amount of available bandwidth is large packetrate is being increased and when the links become over-loaded the rate is decreased thus reducing the overloadThepacket rate of the TCP session increases every time the senderapplication receives an acknowledgement that a packet hasreached the destination and the rate is decreased everytime a packet is lost While the increase is linear the decreaseis geometrical which helps to ease congestion quickly In areactive scenario the packet is lost when the packet buffer issaturated In the proactive scenario to avoid uncontrolledcongestion the random early discard (RED) mechanism ofthe router can be activated that discards randomly selectedpackets However in both cases a random packet is lost anda randomly selected session is affected

Arguably the higher the packet rate of a session the higherthe probability that packets of the session will be droppedand the packet rate of the session will be reduced Thus if anumber of sessions have their packet rate reduced due tocongestion of a given link none of the sessions is supposed togenerate packets at an average rate higher than the othersessions For each 119890 isin E let variable 119910

119890denote the maximum

packet rate on link 119890 Noticeably there is some maximumrate at which a particular application can generate packets letfunction 119903 S 997891rarr R

+define the maximum achievable packet

rate of the session Thus the packet rate of the session mustpotentially satisfy the following condition

119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)

Due to (35) the bandwidth of a single link is shared in a fairway If a link is saturated every session 119904 attains the samepacket rate 119910

119890 unless that rate is higher than the maximum

achievable rate 119903(119904) of that session Thus the session cannothave packet rate higher than any other session unless theother sessionrsquos maximum achievable rate is lower than 119910

119890

And only if a link is not saturated every session attains itsmaximum achievable packet rate However since in generalsessions use multiple network links on a given link a sessioncan in fact have a lower packet rate than other sessions that

use that particular link That results from the fact that thepacket rate of the session can be reduced even more due tocongestion on some other link Thus condition (35) mustactually be replaced with the following one

119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)

That condition can be interpreted as follows For any session119904 isin S the sessionrsquos packet rate 119909

119904attempts to approach the

maximum achievable packet rate 119903(119904) However on any link119890 isin E(119904) that is used by session 119904 the value of 119909

119904cannot

exceed the maximal packet rate 119910119890 that is attained by the

sessions that use that particular linkThus the sessionrsquos packetrate 119909

119904can only attain the minimal of those rates min

119890isinE(119904)119910119890

unless that minimal rate is still higher than 119903(119904) in that casethe packet rate of 119904 just approaches 119903(119904)

Considering conditions (34) and (36) it can now beseen that the flow control mechanism of the TCP protocolmaximizes the vector of the packet rates of individual sessions119909 equiv (119909

119904 119904 isin S) in a fair wayConsider

lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)

The max-min fairness property of the packet rates vectormeans that the packet rates of the data sessions are increasedup to their maximum values unless links become overloadedand in the case of a link overload the data sessions on the linkdecrease their rate to the common highest feasible valueThistype of behaviour appears to have far reaching consequencesfor solving the problem of packet network design that carryelastic traffic when the aim of the design is controlling thequality of services when the capacity of links changes [110]

36 Content DistributionNetworks Bandwidth allocation forcontent distribution through networks composed of multipletree topologies with directed links and a server at the root ofeach tree is another problemof fair network optimization [111112] and [5 Ch 6] Content distribution over networks hasbecome increasingly popular It may be related for instanceto a video-on-demand application where multiple programscan be broadcasted from each server Each server broadcastsalong a tree topology where these trees may share links andeach link has a limited bandwidth capacity Figure 7 presentsa network with two trees and servers at the root nodes 1 and 2The server at node 1 can broadcast programs 1 2 and 3 andthe server at node 2 can broadcast programs 4 5 and 6The numbers adjacent to the links are the link capacities andthe numbers adjacent to the nodes are the programs reque-sted for example links (1 3) have a capacity of 100Gbs andprograms 2 3 and 5 are requested at node 7

These models are fundamentally different from multi-commodity network flowmodels since they do not have flow


1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150

Figure 7 Content distribution from two servers

conservation constraints as each link carries at most onecopy of a program On the other hand the models have tree-like ordering constraints for each program as the allocatedbandwidth for a given program cannot be increased whenmoving farther away from the broadcasting server For eachrequested program at any node there is associated a perfor-mance function that represents satisfaction from the video-on-demand service and depends on the bandwidth availablefor that program on the incoming link to the node Fair opti-mization with respect to all nodes and programs requestedperformance values is needed In [111] the MMF model isintroduced and a lexicographic max-min algorithm is pre-sented As shown in [113] the algorithm can be implementedin a distributed mode where most of the computations aredone independently and in parallel at all nodes while someinformation is exchanged among the nodes More complexcontent distribution models and corresponding algorithmsare discussed in [114ndash116]

37 Fairness Issues in the IP Traffic In its beginnings theInternet suffered from severe deficiencies due to congestionThe answer came from new features added to TCP namelyemploying control admission and additive increasemulti-plicative decrease algorithms that led to congestion avoidanceand fair rate allocationThemain idea behind was putting thecontrol traffic mechanisms at the end-nodes and combiningboth packet scheduling with admission control which willlead to fair bandwidth sharing Plenty of studies have beendone on the behavior of the network when such congestionavoidance algorithms are employed They have shown thatthis leads to some kind of max-min fair sharing in verysimplified networks [117] and to proportional fairness forlarge networks ([67] etc) This difference is mainly due toend-to-end delays which can be significantly different in largescale networks At this point an important topic is how to getclose to maximal throughput while keeping a high level offairness In [118] there are investigated the performances ofnetworks handling elastic flows (in contrast to stream flowsthey adjust their rate to the available bandwidth) It is

shown that in linear networks under random traffic pat-terns ensuringmax-min fairness results in better throughputperformances comparing to proportional fairness while theconverse holds for persistent flows All these works aresituated at the session level and refer to traffic demand as theproduct of the flow arrival rate with the average flow size Atthis stage a more global solution would come by combiningsession level decisions (see Section 35) with higher leveldecisions as routing and load balancing Hence relations ofrate adaptation and congestion control in TCP networks withrouting and network design have been the subject of severalworks over the last decade Among them somework has beendevoted to the static routing case (connections and corre-sponding routing paths are given) where source rates aresubject to changes In [12] there is presented the water-fillingalgorithm for achieving a MMF distribution of resources toconnections for the fixed single path routing case (where eachconnection is associated with a particular fixed path) Themain idea behind the algorithm is to uniformly increase theindividual allocations of connections until one or more linkbecomes congested Then the connections that cannot beimproved are removed from the network together with thecapacities they occupy the process continues until all con-nections are removed In [119] the problem of MMFbandwidth-sharing among elastic traffic connections whenrouting is not fixed has been considered in an offline contextThe proposed iterative algorithm can be seen as an extensionof the water-filling algorithm given in [12] except that therouting is not fixed and at each iteration a new routingis computed while the previously saturated links and thecorresponding fair sharing remain fixed until the end of thealgorithm

Load balancing is in a way a problem dual to MMFrouting (seeTAPMMF in Section 32) as one focuses onmin-max fair load sharing instead of max-min fair bandwidthallocation to demands Achieving load-balancing in a givennetwork consists in distributing the demand traffic (load)fairly among the network links while satisfying a given set oftraffic constraints Fair load sharing means that not onlythe maximal load among links is minimized but rather that


the sorted (in the nonincreasing order) vector of link loadsis minimized lexicographically like in formulation (28a)ndash(28d) In contrast to the max-min fair routing problem likeTAPMMF the link load-balancing problem assures fairnessin the min-max senseThe problem arises in communicationnetworks when the operator needs to define routing withrespect to a given traffic demandmatrix such that the networkload is fairly distributed among the network links Theproblem can be easily modeled and solved by conventionalmethods in LP using MMF properties for linear link loadsThis approach can be applied to more general link load func-tions (especially nonlinear frequently used in telecommuni-cations) In practice the loaddelay functions considered bynetwork operators are usually nonlinear A well-knownloaddelay function called the Kleinrock function is givenby 119891119890(119862119890minus 119891119890) It can be shown that any routing achieving

min-max fairness for the relative load function (ie 119891119890119862119890)

achieves also min-max fair load for the Kleinrock functionThis idea is generalized for general link load functions as(120572 minus 1)

minus1(1 minus 119891

119890119862119890)1minus120572 and (120572 minus 1)

minus1(119891119890minus 119862119890)1minus120572 where 119891

119890

and 119862119890give respectively the flow and capacity on link 119890 and

120572 is a given constant see [120] for further detailsThe problem of fairness is more complex when dealing

with wireless networks and has been addressed in a numberof papers during the last decade A range of problems can bedistinguisheddepending on the network characteristics fromwireless mesh networks ad-hoc networks sensor networksrandom-access networks opportunistic ones and so forthAs for conventional wired networks a fundamental problemin wireless networks is to estimate their throughput capacityand then to develop protocols to utilize the network close tothis capacity without causing congestion in the network andunfairness Then the first idea that comes in mind toaddress the fairness problem in wireless networks is theclassic approach to manage congestion inherited from wirednetworksThen nodesflows will have preassigned fair sharesand applying admission control would allow ensuring fairsharing In wireless networks this cannot be applied becauseof interference which constrains the set of links that cantransmit simultaneously while in ad-hoc networks nodes androutersmobility renders the problem evenmore complicatedIn WSN (wireless sensor networks) the fairness problembecomes on one hand closely connected to fair data gather-ing that is serving the sources equitably and on the otherhand it is connected to ensuring aware energy consumingbecause of the reduced energy capacity of nodes in suchnetworks Then the main constraint that one has to deal withis the so-calledMAC (medium access control) constraint Letus recall briefly what MAC constraint is we start by itsdefinition Two basic definitions can be distinguished theprotocol and the physical one The protocol definition ofinterference assumes that two links which are less than 119896

(generally 119896 is taken 2) hops away from each other interferepotentially and cannot be scheduled in the same time slotTheindicated number of hops refers to the number of hopsbetween the sender nodes of these links On the other handthe physical definition is based on the signal-to-interferenceand noise ratio (SINR) constraint where the transmissionlinks that do not satisfy the SINR constraint cannot be

scheduled simultaneously Hence this constraint leads to newconnected problems namely synchronization and schedulingGiven the abovemost of the work related to these strategies isdedicated to scheduling The basic version of a time slotallocation problem aims to find a slot allocation for all nodesin the networkwithminimal number of slots such that 119896 hopsneighboring nodes are not allocated to the same time slotTherespective optimization problem is the chromatic graph onewhich aims to minimize the number of colors for coloringthe nodes such that two neighbor elements do not use thesame colorThe problem becomesmore difficult if one desiresto achieve fairness between connections or sources All thisyields the max-min fair scheduling In [121] the authors con-sider scheduling policies formax-min fair allocation of band-width in wireless ad-hoc networks They formalize the max-min fair objective under wireless scheduling constraints andpropose a fair scheduling which assigns dynamic weights tothe flows such that the weights depend on the congestion inthe neighborhood and schedule the flows which constitutea maximum weighted matching While in [122] the authorspropose a quite different alternativeTheir method is inspiredfrom per-flow queuing in wired networks and consists of aprobabilistic packet scheduling scheme achieving max-minfairness without changing the existing IEEE 80211 mediumaccess control protocolWhen awireless node is ready to senda packet the packet scheduler of the node is likely to select thequeue whose number of packets sent in a certain time is thesmallest and when no packet is available the transmission isdelayed by a fixed duration In [123] the authors investigatesimple queuing models for random traffic and discuss theirinterest for both wired and wireless transmissions

With respect toWSN the rate allocation problem for dataaggregation in wireless sensor networks can be posed withtwo objectives the first is maximizing the minimum (max-min) lifetime of an aggregation cluster and the second achiev-ing fairness among all data sourcesThe two objectives cannotbe maximized simultaneously and an approach would beto solve recursively first the max-min lifetime for the aggre-gation cluster and next the fairness problem In [124] theauthors use this approach and formulate the problem ofmax-imizing fairness among all data sources under a given max-min lifetime as a convex optimization problem Next theycompute the optimal rate allocations iteratively by a lexico-graphic method In a recent paper [125] the authors addressthe problem of scheduling MMF link transmissions inwireless sensor networks jointly with transmission powerassignment Given a set of concurrently transmitting linksthe considered optimization problem seeks for transmissionpower levels at the nodes so that the SINR values of activelinks satisfy themax-min fairness property By guaranteeing afair transmission medium (in terms of SINR) other networkrequirements such as the scheduling length the throughput(directly dependent on the number of concurrent links in atime slot) and the energy savings (no collisions and retrans-missions) can be directly controlled

4 Location and Allocation Problems

41 Inequality Measures The spatial distribution of publicgoods and services is influenced by facility location decisions


and the issue of equity (or fairness) is important in manylocation decisions In particular various public facilities (orpublic service delivery systems) like schools libraries andhealth-service centers require some spatial equity whilemaking location-allocation decisions [126 127]

The generic discrete location problem may be stated asfollows There is given a set of 119898 clients (service recipients)Each client is represented by a specific point There is alsogiven a set of 119899 potential locations for the facilities and thenumber (or the maximal number) 119901 of facilities to be locatedis given (119901 le 119899) This means discrete location problems ornetwork location problems with possible locations restrictedto some subsets of the network vertices [128] The maindecisions to be made can be described with the binaryvariables 119909

119895(119895 = 1 2 119899) equal to 1 if location 119895 is to

be used and equal to 0 otherwise To meet the problemrequirements the decision variables 119909

119895have to satisfy the

following constraints

119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)

where the equation is replaced with the inequality (le) if119901 specifies the maximal number of facilities to be locatedFurther the allocation decisions are represented by theadditional variables 119909

1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal

to 1 if location 119895 is used to service client 119894 and equal to0 otherwise The allocation variables have to satisfy thefollowing constraints

119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)

In the capacitated location problem the capacities of thepotential facilities are given which implies some additionalconstraints

Let 119889119894119895

ge 0 (119894 = 1 2 119898 119895 = 1 2 119899) denote thedistance between client 119894 and location 119895 (travel effort or othereffect of allocation client 119894 to location 119895) For the standarduncapacitated location problem it is assumed that all thepotential facilities provide the same type of service and eachclient is serviced by the nearest located facilityThe individualobjective functions then can be expressed in the linear form

119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)

These linear functions of the allocation variables are applica-ble for the uncapacitated as well as for the capacitated facilitylocation problems In the case of location of desirable facilitiesa smaller value of the individual objective function means abetter effect (smaller distance)This remains valid for locationof obnoxious facilities if the distance coefficients are replacedwith their complements to some large number 1198891015840

119894119895= 119889 minus 119889

119894119895

where 119889 gt 119889119894119895for all 119894 = 1 2 119898 and 119895 = 1 2 119899

Generally replacing the distances with their utility values orso-called proximity measures for example 119906

119894119895= exp (minus120573119889

119894119895)

[129] Therefore we can assume that each function 119891119894is to be

minimized as stated in the multiple criteria problem [130]Further some additional client weights 119908

119894gt 0 are

included into locationmodel to represent the service demand(or clients importance) Integer weights can be interpretedas numbers of unweighted clients located at exactly the sameplace The normalized client weights 119908

119894= 119908119894sum119898

119894=1119908119894for 119894 =

1 119898 rather than the original quantities 119908119894 In the case of

unweighted problem (all 119908119894= 1) all the normalized weights

are given as 119908119894= 1119898

Note that constraints (38) take a very simple form of thebinary knapsack problem with all the constraint coefficientsequal to 1 [131] Indeed the location problem may be viewedas a resource allocation problem on network It may beconsidered as capacities allocation to links from an artificialsource to potential locations nodes while flows are routedfrom the source to all client nodes through the the potentiallocation nodes [19 20]

Equity is usually quantified with the so-called inequalitymeasures to be minimized Inequality measures were pri-marily studied in economics [57 76] The simplest inequalitymeasures are based on the absolute measurement of thespread of outcomes Variance is the most commonly usedinequality measure of this type and it was also widelyanalyzed within various location models [132 133] Howevermany various measures have been proposed in the literatureto gauge the level of equity in facility location alternatives[58] like the mean absolute difference also called the Ginirsquosmean difference [9 59] Consider

119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)

or like the mean absolute deviation

MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)

In economics one usually considers relative inequal-ity measures normalized by mean outcome Among manyinequality measures perhaps the most commonly accepted isthe Gini index (Lorenz measure) 119863(y)120583(y) a relative mea-sure of themean absolute difference which has been also ana-lyzed in the location context [134ndash136] One can easily noticethat a direct minimization of typical inequality measures(especially relative ones) contradicts the minimization ofindividual outcomes As noticed by Erkut [134] it is rather acommonflawof all the relative inequalitymeasures that whilemoving away from the clients to be serviced one gets bettervalues of the measure as the relative distances become closerto one-another As an extreme one may consider an uncon-strained continuous (single-facility) location problem andfind that the facility located at (or near) infinity will provide(almost) perfectly equal service (in fact rather lack of service)to all the clients Unfortunately the same applies to alldispersion type inequality measures including the upper


2

100100

1205921

1205922 1205923

Figure 8 A network location problem for Example 3

semideviations This can be illustrated by a simple exampleof location problem on network

Example 3 Consider a single facility location on a (triangu-lar) network of 3 nodes two nodes V

1and V2are close one to

the other 11988912

= 2 and one remote node V3with 119889

13= 11988923

=

100 (see Figure 8) Most of the demand is equally distributedin V1and V

2 That means the normalized values of weights

take values 1199081= 1199082= (1 minus 120576)2 and 119908

3= 120576 with a very small

positive value 120576 While locating facility at node V1(or V2) one

gets distance 0 for (1 minus 120576)2 demand distance 2 for (1 minus 120576)2

demand and large distance 100 for only 120576 demand However120583(V1) = 1 + 99120576 and MAD(V

1) = (1 minus 120576)(1 + 99120576) and in

terms of MAD minimization it is beaten by remote locationV3 Indeed locating facility at node V

3one gets distance 0 for

only 120576 demand while getting large distance 100 for (1 minus 120576)

demand thusmuchworser than for V1 Nevertheless 120583(V

3) =

100(1minus120576) andMAD(V3) = 200120576(1minus120576) Hence for small values

of 120576 MAD(V3) lt MAD(V

1) Actually for sufficiently small

values 120576 (eg 0 lt 120576 lt 1200) location V3is a global MAD

minimizer on the entire network (when allowing location onedges in addition to the nodes)

For typical inequality measures a simplified bicriteriamean-equity model is computationally very attractive sinceboth the criteria are well defined directly for the weightedlocation problem without necessity of its disaggregation butit may result in solutions which are inefficient It turns outthat under the assumption of bounded trade-offs the bicri-teria mean-equity approaches for selected absolute inequalitymeasures (maximum upper deviation mean semideviationor mean absolute difference) comply with the rules ofequitable (fair) optimization [9 137] In other words severalinequality measures can be combined with the mean itselfinto the optimization criteria generalizing the concept of theworst outcome and generating equitably consistent under-achievement measures Simple sufficient conditions forinequality measures to keep this consistency property havebeen introduced in [137]

This applies in particular to themean absolute difference(41) generating a proper fair solution concept

119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)

for any 0 lt 120572 le 1 Similar result is valid for themean absolutedeviation (42) but not for the variance [24 137]

42 Lexicographic Minimax and Ordered Medians Althoughminimization of the inequality measures contradicts theminimization of individual outcomes the inequality mini-mization itself can be consistently incorporated into loca-tional models The notion of equitable multiple criteriaoptimization [63] introduces the preference structure thatcomplies with both the outcomes minimization and with theinequality minimization rules [57 76] The equitable opti-mization is well suited for the locational analysis [9 137 138]The equitably (fair) efficiencymodels presented in Section 23apply also to the minimized outcomes as commonly con-sidered in location-allocation problems The equitable min-imization can be modeled with the standard multiple criteriaoptimization applied to the cumulative ordered outcomesexpressing respectively the worst outcome the total of thetwo worst outcomes the total of the three worst outcomesand so forth However in the case of minimization the worstoutcome means the largest rather than the smallest Hencethe corresponding model takes form

min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898

119895=119898minus119894+1120579119895(y) The minimax called the center

solution concept represents only the first criterion while thetotal outcome criterion called the median solution conceptis focused on the last criterion Several cent-dian solutionconcepts combining these two criteria have been considered(see [139] and references therein) For unweighted locationproblems a compromise solution concept was introduced bySlater [140] as the so-called 119896-centrumwhere the sum of the 119896largest distances is minimized Consistently with typical dis-tribution characteristics The 119896-centrum concept is restrictedto unweighted problems Although some weights are used toscale the specific distances [141] (which may be consideredas a definition of distance dependent outcomes) the demandweights as defining the distribution of clients are not con-sidered Ogryczak and Zawadzki [142] introduced a para-metric generalization of the 119896-centrum concept applied toweighted problems by taking into account the portion ofdemand related to the largest outcomes (distances) ratherthan the specific number of worst outcomes Namely for aspecified portion 120573 of demand the entire 120573 portion (quantile)of the largest outcomes is taken into account and their averageis considered as the (worst) conditional 120573-mean outcome


According to this definition the concept of conditionalmedian is based on averaging restricted to the portion of theworst outcomes For the unweighted location problems and120573 = 119896119898 the conditional 120573-mean represents the average ofthe 119896 largest outcomes thus modeling the 119896-centrum solu-tion concept

However in order to guarantee the equitable efficiencyof a selected location pattern one needs to take into accountall the ordered outcomes (all the criteria in (44) The entiremultiple criteria ordered model is rich with various equitablyefficient solution concepts [64 142 143] For the weightedsum aggregation one gets the OWA aggregation sum

119898

119894=1120596119894120579119894(y)

(18) called the ordered median solution concept [144] Ifthe OWA weights are strictly increasing and positive that is0 lt 120596

1lt 1205962

lt sdot sdot sdot lt 120596119898minus1

lt 120596119898 then each optimal

solution of the OWA problem (18) is an equitably (fairly)efficient location pattern Although the cumulated orderedoutcomes can be expressed with linear programming models[85] these approaches requires the disaggregation of locationproblemwith the demandweightswhich usually dramaticallyincreases the problem size

When applying the lexicographic optimization to prob-lem (44)

lex min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876

(45)

one gets the lexicographic minimax solution concept calledalso lexicographic center [42] as a lexicographic refinementof the center solution concept The lexicographic minimaxlocation may be converted to a lexicographic minimizationobjective by constructing counting functions that count foreach possible distinct outcome the number of occurrencesof the specified outcome It is quite simple to construct suchcounting functions for the discrete location problem (see[42 48] or [5 Ch 72])

The lexicographic maximin approach can be applied tovarious location problemsThe sensor location problem is anextension of the equitable facility location problem [5 Ch73] and [145] Consider a set 119873 of nodes that need to bemonitored as protection against undesired intrusion and aset 119872 of nodes where sensors can be placed Let 119868

119895be the

subset of nodes in 119872 that can monitor node 119895 isin 119873 andlet 119870 be the number of available sensors that can be placedamong nodes in 119872 The protection level provided to node 119895

is represented by the number of sensors that monitor node 119895The sought after solution is the lexicographic maximin solu-tionwith respect to the number of sensors that protects nodes119895 isin 119873 Figure 9 presents a problem with four locations thatneed to be monitored (119873 = 2 3 4 5) and four locationswhere sensors can be placed (119872 = 1 3 4 5) The links thatconnect the nodes represent subsets 119868

119895 Two sensors can be

placed among the nodes in 119872 (119870 = 2)

1

3

4

5

2

3

4

5

Candidate sensor locations

Sensitive locations

M N

2

2

2

1

Figure 9 The sensor location problem

The formulation of this problem is as follows

lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952

le sdot sdot sdot le 119883119895119899

(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)

119909119894isin 0 1 forall119894 isin 119872 (46e)

In Figure 9 a unique optimal solution has sensors at nodes 1and 5 implying that nodes 2 3 and 5 are monitored by bothsensors while node 4 is monitored by only one sensor Notethat in general there may be multiple optimal solution Thisproblem can be solved by constructing counting functions asdescribed in Section 24 However whereas for the equitablefacility location problem [42] the counting function for eachlocation 119895 is represented by a single constraint here therepresentation of counting functions adds a large number ofvariables and constraints into the problemNow suppose thatthe probability of detecting an intruder at node 119895 from asensor at node 119894 is 119901

119894119895gt 0 for 119894 isin 119868

119895 Then the protection

level provided to 119895 is the probability that an intruder willbe detected at node 119895 by at least one sensor from amongthose placed in the set 119868

119895 Although the formulation of this

case is similar to the formulation above the number ofpossible distinct outcomes can be much larger As discussedin Section 24 this would necessitate employing a differentsolution method that is not based on counting functions (see[145])

5 Complexity Issues

Essentially fair optimization models are based on concavepiecewise linear criteria possibly replacing a linear crite-rion of the total output maximization Such criteria imple-mentable with auxiliary linear inequalities in most casesdo not significantly affect the complexity of the original


optimization problems In particular problems representedby linear programming remain linear programs in their fairoptimization versions (single LP problem in the case offair OWA aggregations or a sequence of LP problems forthe MMF models) Certainly some specialized algorithmstaking into account the structure of the problem in handcan be more efficient than the general linear programmingtechniques Consider resource allocation problem (23a)ndash(23d) in Section 24 which have a lexicographic maximinobjective For certain performance functions (eg linear andexponential functions) a lexicographic maximin solution isobtained by manipulations of closed-form expressions in apolynomial time As presented in [5 Ch 33] depending onthe algorithm employed the computational effort for solving(23a)ndash(23d) is 119874(119898119899) or 119874(119898

2119899) where 119898 is the number of

activities in the set 119868 and 119899 is the number of resources in theset 119869 Moreover the same complexity is achieved for somecontent distribution problems in tree networks describedin Section 36 With respect to communication networksapplications a well-known example of such a specializedalgorithm is the already mentioned water-filling algorithm(see Section 32) Another example is a special case of the sin-gle source traffic allocation problem (also see Section 32) forwhich Megiddo [146] introduced a polynomial-time MMFalgorithm which applies to splittable (fractional) flows Aspresented in Section 24 there exist simple polynomial timetechniques for solving general convex MMF problems Thuswhen applied to networks problems the algorithms do notdepend on any specific traffic routing problem formulationand is sufficiently general to be applied to a broad class oftraffic routing and capacity allocation problems

Generally MMF optimization problems on convexattainable sets are characterized by polynomial complexity[92] Polynomial algorithms may be developed for variousspecific forms of load balancing problems For instance in[147] a polynomial algorithm to determine theMMF optimalbandwidth allocation in order to satisfy the communicationneeds between two private networks The algorithm is guar-anteed to converge in finite number of steps and for linearcosts its complexity is 119874(|V|

5)

Nonconvex attainable sets usually results in 119873119875-hardcomplexity of the corresponding fair optimization problemsIn the network environment this is the case of single-path flows (unsplittable flows) In particular a single-sourcemultiple-sink demand MMF optimization of single-pathflows in a directed network was proven 119873119875-hard in [148]Nilsson [149] generalized this result showing that generalMMFunsplittable-flow problems on undirected networks are119873119875-hardThis applies to the case when each demandmay useany path as well as to the case when each demand may useone path from a predefined list Actually it is proven therethat in both cases obtaining just the first entry of the sortedallocation vector (the standardmaximin) is119873119875-hard in itselfObserve that this shows that all corresponding fairnessoptimization models are 119873119875-hard as they must take intoaccount that criterion Single-path optimization problemsremain 119873119875-hard also when fairness is implemented as aconstraint rather than a criterion Amaldi et al [150] showedthat the Max-Throughput Single-Path Network Routing sub-ject to MMF flow allocation is 119873119875-hard even with equal

(unit) capacities for all links Nilsson [149] has also shownthan nonconvexity introduced by modular flows (granular)causes that even splittable traffic allocation problems become119873119875-hard Therefore there is an emerging need to developapproximate or heuristic algorithms for such problems Earlyresults in this area show that several communication networkproblems with PF or OWA fairness criteria can be effectivelyhandled by meta-heuristic approaches [80 106 151]

In location and allocation problems the general fair-ness (equitable) models may be viewed as the so-calledordered median solution concepts corresponding to theOWA criterion with monotonic weights Such a criterionmay be implementedwith simple auxiliary linear inequalitiesNevertheless even standard (median or center) multifacilitylocation problems on general networks are usually 119873119875-hardand the same remains valid for the orderedmedian problemsFor tree networks however polynomial time algorithmsexist Dynamic programming algorithm for the orderedmedian problem presented in [152] has time complexity of119874(119901119898

8) for the general problem and just 119874(119901119898

4) for the

node restricted problem Polynomial algorithms exist also forthe single facility location ordered median problems [153]with complexity 119874(119898

3log2119898) for trees and 119874(|E|1198983log2119898)

for general networksIn this survey we have not discussed in detail fair

optimization in connection to problems which can be relatedto abstract networks or analyzed with some networks Animportant wide group of such problems is related to job-shopscheduling Most approaches for the job-shop schedulingproblem deal with the makespan criterion that is themaximum completion time of all jobs Still there are variouscriteria that consider the due dates of jobs and aim atminimizing the tardiness of jobs or the fact that jobs are latethat is not completed before their due dates Actually simpleaggregations of a number of such uniform criteria are com-monly applied Each single criterion applies to one schedulingobject like job or affected agent and a need for aggregationsproviding fairness arises Note that any fair aggregation isstrictly increasing thus satisfying the condition of the so-called regular scheduling criterion (ie it is an increasingfunction of the completion times of the jobs ie it is alwaysoptimal to start and complete jobs as early as possible)The job-shop scheduling problems with regular criteria arewell studied For the nonpreemptive two machine job-shopscheduling problem with a fixed number of jobs any regularcriterion can be solved in polynomial time [154] Generallythe 119899-job119898-machine job-shop problembelongs to the class of119873119875-hard problems [155ndash157] though there are exceptions forspecific problems Nevertheless generic efficient approachesare available for approximate solving the job-shop schedulingproblem with regular criteria [158] Importance of fairnessissues has been recently recognized in just-in time sequencingproblems [13] in apportionment concepts [76 159] Very fewfair optimization approaches have been presented to job-shopscheduling although already in 1989 such approaches wereconsidered in [160] Specifically a lexicographic minimaxobjective was analyzed for the production smoothness ofmultiple feeder shops that produce components for custom-made products assembled at a final assembly shop Finally


we mention that the fair dominance rules were used in amultiobjective method to solve reentrant hybrid flow shopscheduling problem [161]

6 Concluding Remarks

In systemswhich servemany entities there is a need to respectsome fairness rules Extensive progress in fair optimizationmethodology made in the last three decades resulted in avariety of techniques enabling to generate fair and efficientsolutions In particular allocation problems related to com-munication networks and location-allocation problems arethe areas where the fair optimization concepts are extensivelydeveloped and widely applied Within the networking appli-cations the lexicographic maximin approach (or the relatedmax-min fairness approach) is the most widely used Therecent book by Luss [5] exhibits a variety of models witha lexicographic maximin objective and the correspondingalgorithms in the context of resource allocation Many ofthese models apply to communication network and locationproblems Since this approach may lead to significant lossesin the overall efficiency (eg throughput of the network) theproportional fairness or other utility based approaches (like120572-fairness) are also applied In location-allocation problemsthe fairness understood as equity is usually quantified withinequality measures to be minimized or treated with mini-max optimization called center solution conceptThe latter isapplied especially for emergency facilities location andrecently is considered with a lexicographically regularizedcriterion to lexicographic minimax The inequality measuresare scalar indices based on some measurement of the spreadof outcomes Direct minimization of the inequality measurescontradicts the optimization of individual outcomes butseveral inequality measures can be combined with the meanoutcome into the equitable criteria thus allowing to generatevarious fair solutions

Awide variety of fair optimizationmodels and algorithmssupporting efficient and fair allocation in complex systemshas been introduced and studied in the literature In mostcases they can be effectively used to generate various fair allo-cation schemes while taking into account the problem speci-ficities Nevertheless problems with discrete structure leadto massive computations questioning possibility to achieveany fair solution in a reasonable time Therefore there is aneed to develop approximate or heuristic algorithms for suchproblems

Frequently one may be interested in putting into allo-cation models some additional service importance weightsThe importance weights are easily incorporated into thescalar inequality measures [59 137 162] or the Jain fairnessindex [163] as well as in proportional fairness [7] There arealso possibilities to introduce importance weights into thegeneral fair preferences [164] and fair optimizationmodels Inparticular the OWA aggregations (18) may be extended tothe corresponding Weighted OWA (WOWA) aggregations[165 166] which still remain LP computable [167 168] whilemetaheuristic may be also applied [169] The performance

functions in a lexicographic minimax objective functionmayalso include demand weights cf [31 43 170] and [5 Ch 1]

Vector fair optimization approaches taking into accountmultiattribute outcomes are still underexplored In resourceallocation context this relates to problems withmultiple typesof resources where the users request different ratios of dif-ferent resources A typical example is datacenters processingjobs with heterogeneous resource requirements on CPUmemory network bandwidth and so forth Recently pro-posed (vector) fairness measure [171] called dominantresource fairness allocates resources according to max-minfairness on dominant resource shares Koppen et al [172]have extended the Jain fairness index [61] to the multiat-tribute case By means of a leximin procedure an allocationcan be selected where the smallest among the Jain fairnessindexes takes the largest value This extends the notion of anallocation where fairness is achieved only for a single alloca-tion metric A unifying framework addressing the fairness-efficiency tradeoff in the light of multiple types of resourceshas been developed in [173]

Another still underexplored area of fair network opti-mization is related to distributed optimization process andrelated models [174] In some equitable optimization prob-lems as shown in [113] the optimization algorithm canbe implemented in a distributed mode where most of thecomputations are done independently and in parallel at thenodes However in most cases the distributed approaches tofairness must be based on game theory rather than on directoptimization [175ndash177]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Research conducted by W Ogryczak M Pioro and ATomaszewski was supported by the National Science Centre(Poland) under Grant 201101BST702967 ldquoInteger pro-gramming models for joint optimization of link capacityassignment transmission scheduling and routing in fairmulticommodity flow networksrdquo

References

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[2] N Katoh and T Ibaraki ldquoResource allocation problemsrdquo inHandbook of Combinatorial Optimization D-Z Du and P MPardalos Eds pp 159ndash260 Kluwer Academic Dordrecht TheNetherlands 1998

[3] A P Sage and C D Cuppan ldquoOn the systems engineering andmanagement of systems of systems and federations of systemsrdquoInformation Knowledge and System Management vol 2 pp325ndash345 2001

[4] L Wang L Fang and K W Hipel ldquoFairness in resource allo-cation in a system of systemsrdquo in Proceedings of the IEEE


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[5] H Luss Equitable Resource Allocation Models Algorithms andApplications John Wiley amp Sons Hoboken NJ USA 2012

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[7] M Pioro and D Medhi Routing Flow and Capacity Design inCommunication and Computer Networks Morgan KaufmannSan Francisco Calif USA 2004

[8] A D Procaccia ldquoCake cutting not just childrsquos playrdquo Communi-cations of the ACM vol 56 no 7 pp 78ndash87 2013

[9] WOgryczak ldquoInequalitymeasures and equitable approaches tolocation problemsrdquo European Journal of Operational Researchvol 122 no 2 pp 374ndash391 2000

[10] L Wang L Fang and K W Hipel ldquoBasin-wide cooperativewater resources allocationrdquo European Journal of OperationalResearch vol 190 no 3 pp 798ndash817 2008

[11] J Jaffe ldquoBottleneck ow controlrdquo IEEE Transactions on Commu-nications vol 7 pp 207ndash237 1980

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[14] D Bertsimas G Lulli and A Odoni ldquoAn integer optimizationapproach to large-scale air traffic flowmanagementrdquoOperationsResearch vol 59 no 1 pp 211ndash227 2011

[15] R S Klein and H Luss ldquoMinimax resource allocation with treestructured substitutable resourcesrdquoOperations Research vol 39pp 285ndash295 1991

[16] R S Klein H Luss and U G Rothblum ldquoMinimax resourceallocation problems with resource-substitutions represented bygraphsrdquo Operations Research vol 41 no 5 pp 959ndash971 1993

[17] J Pang and C S Yu ldquoA min-max resource allocation problemwith substitutionsrdquo European Journal of Operational Researchvol 41 no 2 pp 218ndash223 1989

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[20] W Ogryczak K Studzinski and K Zorychta ldquoA solver for themulti-objective transshipment problem with facility locationrdquoEuropean Journal of Operational Research vol 43 no 1 pp 53ndash64 1989

[21] M Koppen ldquoRelational optimization and its application frombottleneck flow control to wireless channel allocationrdquo Infor-matica vol 24 no 3 pp 413ndash433 2013

[22] W Ogryczak ldquoComments on properties of the minmax solu-tions in goal programmingrdquo European Journal of OperationalResearch vol 132 no 1 pp 17ndash21 2001

[23] W Ogryczak ldquoOn goal programming formulations of thereference point methodrdquo Journal of the Operational ResearchSociety vol 52 no 6 pp 691ndash698 2001

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[29] J A M Potters and S H Tijs ldquoThe nucleolus of a matrix gameand other nucleolirdquoMathematics of Operations Research vol 17no 1 pp 164ndash174 1992

[30] J R Rice ldquoTchebycheff approximation in a compact metricspacerdquo Bulletin of the American Mathematical Society vol 68pp 405ndash410 1962

[31] H Luss and D R Smith ldquoResource allocation among compet-ing activities a lexicographic minimax approachrdquo OperationsResearch Letters vol 5 no 5 pp 227ndash231 1986

[32] F A Behringer ldquoA simplex based algorithm for the lexicograph-ically extended linear maxmin problemrdquo European Journal ofOperational Research vol 7 no 3 pp 274ndash283 1981

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[39] Q CNguyen andR E Stone ldquoAmultiperiodminimax resourceallocation problem with substitutable resourcesrdquo ManagementScience vol 39 pp 964ndash974 1993

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[41] F Della Croce V T Paschos and A Tsoukias ldquoAn improvedgeneral procedure for lexicographic bottleneck problemsrdquoOperations Research Letters vol 24 no 4 pp 187ndash194 1999

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[44] W Ogryczak M Pioro and A Tomaszewski ldquoTelecommuni-cations network design and max-min optimization problemrdquoJournal of Telecommunications and Information Technology vol2005 no 3 pp 43ndash56 2005

[45] B Radunovic and J-Y Le Boudec ldquoA unified framework formax-min and min-max fairness with applicationsrdquo IEEEACMTransactions on Networking vol 15 no 5 pp 1073ndash1083 2007


[46] D Nace and M Pioro ldquoMax-min fairness and its applicationsto routing and load-balancing in communication networks atutorialrdquo IEEE Communications Surveys and Tutorials vol 10no 4 pp 5ndash17 2008

[47] R M Salles and J A Barria ldquoLexicographic maximin opti-misation for fair bandwidth allocation in computer networksrdquoEuropean Journal of Operational Research vol 185 no 2 pp778ndash794 2008

[48] W Ogryczak and T Sliwinski ldquoOn direct methods for lexico-graphic min-max optimizationrdquo in Computational Science andIts ApplicationsmdashICCSA 2006 vol 3982 of Lecture Notes inComputer Science pp 802ndash811 2006

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Economic Behavior and Organization vol 31 no 1 pp 13ndash351996

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[56] M Rothschild and J E Stiglitz ldquoSome further results on themeasurement of inequalityrdquo Journal of Economic Theory vol 6no 2 pp 188ndash204 1973

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[60] J A Mesa J Puerto and A Tamir ldquoImproved algorithms forseveral network location problems with equality measuresrdquoDiscrete AppliedMathematics vol 130 no 3 pp 437ndash448 2003

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[62] T Lan D Kao M Chiang and A Sabharwal ldquoAn axiomatictheory of fairness in network resource allocationrdquo in Pro-ceedings of the IEEE International Conference on ComputerCommunications (INFOCOM rsquo10) pp 1ndash9 IEEE San DiegoCalif USA March 2010

[63] M M Kostreva and W Ogryczak ldquoLinear optimization withmultiple equitable criteriardquoRAIROOperations Research vol 33no 3 pp 275ndash297 1999

[64] M M Kostreva W Ogryczak and A Wierzbicki ldquoEquitableaggregations and multiple criteria analysisrdquo European Journalof Operational Research vol 158 no 2 pp 362ndash377 2004

[65] W Ogryczak ldquoFair optimization methodological foundationsof fairness in network resource allocationrdquo in Proceedings of

the IEEE 38th Annual Computer Software and ApplicationsConference Workshops (COMPSACW rsquo14) pp 43ndash48 IEEE2014

[66] AWMarshall and I Olkin InequalitiesTheory ofMajorizationand its Applications vol 143 of Mathematics in Science andEngineering Academic Press New York NY USA 1979

[67] F P Kelly A K Maulloo and D Tan ldquoRate control for com-munication networks Shadow prices proportional fairness andstabilityrdquo Journal of the Operational Research Society vol 49 no3 pp 206ndash217 1997

[68] M Pioro G Malicsko and G Fodor ldquoOptimal link capacitydimensioning in proportionally fair networksrdquo in NETWORK-ING 2002 Networking Technologies Services and ProtocolsPerformance of Computer andCommunicationNetworksMobileand Wireless Communications vol 2345 of Lecture Notes inComputer Science pp 277ndash288 Springer Berlin Germany2002

[69] J Nash ldquoThe bargaining problemrdquo Econometrica vol 18 pp155ndash162 1950

[70] J Mo and J Walrand ldquoFair end-to-end window-based conges-tion controlrdquo IEEEACMTransactions on Networking vol 8 no5 pp 556ndash567 2000

[71] C Y Lee and H K Cho ldquoDiscrete bandwidth allocation con-sidering fairness and transmission load in multicast networksrdquoComputers and Operations Research vol 34 no 3 pp 884ndash8992007

[72] C Y Lee Y P Moon and Y J Cho ldquoA lexicographically fairallocation of discrete bandwidth formultiratemulticast trafficsrdquoComputers ampOperations Research vol 31 no 14 pp 2349ndash23632004

[73] D Bertsimas V F Farias and N Trichakis ldquoThe price offairnessrdquo Operations Research vol 59 no 1 pp 17ndash31 2011

[74] D Bertsimas V F Farias and N Trichakis ldquoOn the efficiency-fairness trade-offrdquo Management Science vol 58 no 12 pp2234ndash2250 2012

[75] A Tang J Wang and S H Low ldquoIs fair allocation alwaysinefficientrdquo in Proceedings of the Annual Joint Conference of theIEEEComputer andCommunications Societies (INFOCOM rsquo04)pp 35ndash45 Hong Kong China 2004

[76] H P Young Equity inTheory and Practice PrincetonUniversityPress Princeton NJ USA 1994

[77] H Moulin Axioms of Cooperative Decision Making CambridgeUniversity Press New York NY USA 1988

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[79] R R Yager and D P Filev Essentials of Fuzzy Modeling andControl Wiley New York NY USA 1994

[80] M Koppen K Yoshida R Verschae M Tsuru and Y OieldquoComparative study on meta-heuristics for achieving parabolicfairness in wireless channel allocationrdquo in Proceedings of theIEEEIPSJ 12th International Symposium onApplications and theInternet (SAINT rsquo12) pp 302ndash307 IEEE July 2012

[81] R R Yager ldquoOn the analytic representation of the Leximinordering and its application to flexible constraint propagationrdquoEuropean Journal of Operational Research vol 102 no 1 pp176ndash192 1997

[82] M Majdan and W Ogryczak ldquoDetermining OWA operatorweights by mean absolute deviation minimizationrdquo in ArtificialIntelligence and Soft Computing vol 7267 of Lecture Notes inComputer Science pp 283ndash291 Springer 2012


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[84] W Ogryczak ldquoTail mean and related robust solution conceptsrdquoInternational Journal of Systems Science Principles and Applica-tions of Systems and Integration vol 45 no 1 pp 29ndash38 2014

[85] WOgryczak andA Tamir ldquoMinimizing the sumof the 119896 largestfunctions in linear timerdquo Information Processing Letters vol 85no 3 pp 117ndash122 2003

[86] WOgryczak and T Sliwinski ldquoOn solving linear programswiththe ordered weighted averaging objectiverdquo European Journal ofOperational Research vol 148 no 1 pp 80ndash91 2003

[87] R R Yager ldquoConstrained OWA aggregationrdquo Fuzzy Sets andSystems vol 81 no 1 pp 89ndash101 1996

[88] W Ogryczak and A Ruszczynski ldquoDual stochastic dominanceand related mean-risk modelsrdquo SIAM Journal on Optimizationvol 13 no 1 pp 60ndash78 2002

[89] A Muller and D Stoyan Comparison Methods for StochasticModels and Risks John Wiley amp Sons Chichester UK 2002

[90] H Luss ldquoAn algorithm for separable nonlinear minimax prob-lemsrdquoOperations Research Letters vol 6 no 4 pp 159ndash162 1987

[91] H Luss ldquoA nonlinear minimax allocation problem with multi-ple knapsack constraintsrdquo Operations Research Letters vol 10no 4 pp 183ndash187 1991

[92] A Tomaszewski ldquoA polynomial algorithm for solving a generalmax-min fairness problemrdquo European Transactions on Telecom-munications vol 16 no 3 pp 233ndash240 2005

[93] W Ogryczak and T Sliwinski ldquoSequential algorithms for max-min fair bandwidth allocationrdquo in Proceedings of the EuropeanComputing Conference V Mastorakis V Mladenov and VT Kontargyri Eds vol 27 of Lecture Notes in ElectricalEngineering pp 511ndash522 Springer New York NY USA 2009

[94] W Ogryczak and T Sliwinski ldquoSequential algorithms for exactand approximate max-min fair bandwidth allocationrdquo in Pro-ceedings of the 15th International Telecommunications NetworkStrategy and Planning Symposium (NETWORKS rsquo12) pp 1ndash6Rome Italy October 2012

[95] W Ogryczak T Sliwinski and A Wierzbicki ldquoFair resourceallocation schemes and network dimensioning problemsrdquo Jour-nal of Telecommunications and Information Technology vol2003 no 3 pp 34ndash42 2003

[96] W Ogryczak AWierzbicki andMMilewski ldquoAmulti-criteriaapproach to fair and efficient bandwidth allocationrdquoOmega vol36 no 3 pp 451ndash463 2008

[97] D Nace and J B Orlin ldquoLexicographically minimum andmaximum load linear programming problemsrdquo OperationsResearch vol 55 no 1 pp 182ndash187 2007

[98] P Nilsson and M Pioro ldquoMax-min fair distribution of mod-ular network flows on fixed pathsrdquo in Proceedings of the 5thInternational IFIP-TC6 Networking Conference vol 3976 ofLecture Notes in Computer Science pp 916ndash927 Springer BerlinGermany 2006

[99] E Amaldi A Capone S Coniglio and L G Gianoli ldquoNetworkoptimization problems subject tomax-min fair flow allocationrdquoIEEE Communications Letters vol 17 no 7 pp 1463ndash1466 2013

[100] D Santos A de Sousa F Alvelos M Dzida and M PioroldquoOptimization of link load balancing in multiple spanning treerouting networksrdquo Telecommunication Systems vol 48 no 1-2pp 109ndash124 2011

[101] D Santos A de Sousa F Alvelos and M Pioro ldquoOptimizingnetwork load balancing an hybridization approach of meta-heuristics with column generationrdquo Telecommunication Sys-tems vol 52 no 2 pp 959ndash968 2013

[102] P A Nilsson andM Pioro ldquoSolving dimensioning tasks for pro-portionally fair networks carrying elastic trafficrdquo PerformanceEvaluation vol 49 no 1ndash4 pp 371ndash386 2002

[103] E Kubilinskas P Nilsson and M Pioro ldquoDesign modelsfor robust multi-layer next generation Internet core networkscarrying elastic trafficrdquo Journal of Network and Systems Man-agement vol 13 no 1 pp 57ndash76 2005

[104] M Pioro E Kubilinskas and P Nilsson ldquoResilient dimension-ing of proportionally fair networksrdquo European Transactions onTelecommunications vol 16 no 3 pp 241ndash251 2005

[105] M Pioro M Zotkiewicz B Staehle D Staehle and D YuanldquoOn max-min fair flow optimization in wireless mesh net-worksrdquo Ad Hoc Networks vol 13 pp 134ndash152 2014

[106] J Hurkala and T Sliwinski ldquoFair flow optimization withadvanced aggregation operators inWirelessMeshNetworksrdquo inProceedings of the Federated Conference on Computer Scienceand Information Systems (FedCSIS rsquo12) pp 415ndash421 IEEESeptember 2012

[107] A P Wierzbicki M Makowski and J Wessels Eds ModelBased Decision Support Methodology with Environmental Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 2000

[108] W Ogryczak and A Wierzbicki ldquoOn multi-criteria approachesto bandwidth allocationrdquo Control and Cybernetics vol 33 no 3pp 427ndash448 2004

[109] W Ogryczak M Milewski and A Wierzbicki ldquoOn fair andefficient bandwidth allocation by the multiple target approachrdquoin Proceedings of the 2nd Conference onNext Generation InternetDesign and Engineering (NGI rsquo06) pp 48ndash55 IEEE April 2006

[110] A Tomaszewski ldquoDesign of optical wireless networks with fairtraffic flowsrdquo Journal of Applied Mathematics vol 2014 ArticleID 938483 5 pages 2014

[111] H Luss ldquoEquitable bandwidth allocation in content distribu-tion networksrdquo Naval Research Logistics vol 57 no 3 pp 266ndash278 2010

[112] S Sarkar and L Tassiulas ldquoDistributed algorithms for compu-tation of fair rates in multirate multicast treesrdquo in Proceedingsof the 19th Annual Joint Conference of the IEEE Computer andCommunications Societies (IEEE INFOCOM rsquo00) pp 52ndash61March 2000

[113] H Luss ldquoA distributed algorithm for equitable bandwidthallocation for content distribution in a tree networkrdquo Journalof the Operational Research Society vol 63 no 4 pp 460ndash4692012

[114] H Luss ldquoAn equitable bandwidth allocation model for video-on-demand networksrdquo Networks and Spatial Economics vol 8no 1 pp 23ndash41 2008

[115] S Sarkar andL Tassiulas ldquoFair allocation of utilities inmultiratemulticast networks a framework for unifying diverse fairnessobjectivesrdquo IEEE Transactions on Automatic Control vol 47 no6 pp 931ndash944 2002

[116] S Sarkar and L Tassiulas ldquoFair bandwidth allocation for multi-casting in networkswith discrete feasible setrdquo IEEETransactionson Computers vol 53 no 7 pp 785ndash797 2004

[117] D Chiu and R Jain ldquoAnalysis of the increase and decreasealgorithms for congestion avoidance in computer networksrdquoComputer Networks and ISDN Systems vol 17 no 1 pp 1ndash141989


[118] L Massoulie and J W Roberts ldquoBandwidth sharing and admis-sion control for elastic trafficrdquo Telecommunication Systems vol15 no 1-2 pp 185ndash201 2000

[119] DNaceNDoan E Gourdin andB Liau ldquoComputing optimalmax-min fair resource allocation for elastic flowsrdquo IEEEACMTransactions on Networking vol 14 no 6 pp 1272ndash1281 2006

[120] D Nace L Nhat Doan O Klopfenstein and A BashllarildquoMax-min fairness in multi-commodity flowsrdquo Computers andOperations Research vol 35 no 2 pp 557ndash573 2008

[121] L Tassiulas and S Sarkar ldquoMaxmin fair scheduling in wirelessad hoc networksrdquo IEEE Journal on Selected Areas in Communi-cations vol 23 no 1 pp 163ndash173 2005

[122] KWakuda S Kasahara Y Takahashi Y Kure andE Itakura ldquoApacket scheduling algorithm for max-min fairness in multihopwireless LANsrdquo Computer Communications vol 32 no 13-14pp 1437ndash1444 2009

[123] T Bonald and J Roberts ldquoScheduling network traffic SIGMET-RICS performrdquo Evaluation Review vol 34 no 4 pp 29ndash352007

[124] S Lai and B Ravindran ldquoAchieving max-min lifetime andfairness with rate allocation for data aggregation in sensornetworksrdquo Ad Hoc Networks vol 9 no 5 pp 821ndash834 2011

[125] A Gogu D Nace S Chatterjea andA Dilo ldquoMax-min fair linkquality inWSNbased on SINRrdquo Journal of AppliedMathematicsvol 2014 Article ID 693212 11 pages 2014

[126] P B Coulter ldquoMeasuring the inequality of urbanpublic servicesa methodological discussion with applicationsrdquo Policy StudiesJournal vol 8 pp 693ndash698 1980

[127] L D Mayhew and G Leonardi ldquoEquity efficiency and accessi-bility in urban and regional health-care systemsrdquo Environmentand Planning A vol 14 pp 1479ndash1507 1982

[128] M Labbe D Peeters and J-F Thisse ldquoLocation on networksrdquoin Handbook in Operations Research and Management ScienceNetwork Routing M O Ball T L Magnanti C L Monma andG L Nemhauser Eds pp 551ndash624 North-Holland Amster-dam The Netherlands 1996

[129] J Malczewski andW Ogryczak ldquoAn interactive approach to thecentral facility location problem locating pediatric hospitals inWarsawrdquoGeographical Analysis vol 22 no 3 pp 244ndash258 1990

[130] W Ogryczak ldquoOn the distribution approach to location prob-lemsrdquo Computers and Industrial Engineering vol 37 no 3 pp595ndash612 1999

[131] M M Kostreva W Ogryczak and D W Tonkyn ldquoRelocationproblems arising in conservation biologyrdquo Computers andMathematics withApplications vol 37 no 4-5 pp 135ndash150 1999

[132] O Berman ldquoMean-variance location problemsrdquo TransportationScience vol 24 no 4 pp 287ndash293 1990

[133] O Maimon ldquoThe variance equity measure in locational deci-sion theoryrdquo Annals of Operations Research vol 6 no 5 pp147ndash160 1986

[134] E Erkut ldquoInequality measures for location problemsrdquo LocationScience vol 1 pp 199ndash217 1993

[135] OMaimon ldquoAn algorithm for the Lorenzmeasure in locationaldecisions on treesrdquo Journal of Algorithms vol 9 no 4 pp 583ndash596 1988

[136] M B Mandell ldquoModelling effectiveness-equity trade-offs inpublic service delivery systemsrdquo Management Science vol 37no 4 pp 467ndash482 1991

[137] W Ogryczak ldquoInequality measures and equitable locationsrdquoAnnals of Operations Research vol 167 pp 61ndash86 2009

[138] M M Kostreva and W Ogryczak ldquoEquitable approaches tolocation problemsrdquo in SpatialMulticriteriaDecisionMaking andAnalysis A Geographic Informatio Sciences Approach J CThillEd pp 103ndash126 Ashgate London UK 1999

[139] W Ogryczak ldquoOn cent-dians of general networksrdquo LocationScience vol 5 no 1 pp 15ndash28 1997

[140] P J Slater ldquoCenters to centroids in graphsrdquo Journal of GraphTheory vol 2 no 3 pp 209ndash222 1978

[141] A Tamir ldquoThe 119896-centrum multi-facility location problemrdquoDiscrete Applied Mathematics vol 109 no 3 pp 293ndash307 2001

[142] W Ogryczak and M Zawadzki ldquoConditional median a para-metric solution concept for location problemsrdquo Annals ofOperations Research vol 110 pp 167ndash181 2002

[143] W Ogryczak and T Sliwinski ldquoOn equitable approaches toresource allocation problems the conditional minimax solu-tionrdquo Journal of Telecommunications and Information Technol-ogy vol 2002 no 3 pp 40ndash48 2002

[144] S Nickel and J Puerto Location Theory A Unified ApproachSpringer Berlin Germany 2005

[145] A Neidhardt H Luss and K R Krishnan ldquoData fusion andoptimal placement of fixed and mobile sensorsrdquo in Proceedingsof the 3rd IEEE Sensors Applications Symposium (SAS rsquo08) pp128ndash133 Atlanta Ga USA February 2008

[146] N Megiddo ldquoOptimal flows in networks with multiple sourcesand sinksrdquoMathematical Programming vol 7 no 1 pp 97ndash1071974

[147] L Georgiadis P Georgatsos K Floros and S Sartzetakis ldquoLex-icographically optimal balanced networksrdquo in Proceedings of the20th Annual Joint Conference of the IEEE Computer and Com-munications Societies (INFOCOM rsquo01) vol 2 pp 689ndash698Anchorage AK USA April 2001

[148] J Kleinberg Y Rabani and E Tardos ldquoFairness in routing andload balancingrdquo Journal of Computer and System Sciences vol63 no 1 pp 2ndash21 2001

[149] P Nilsson Fairness in communication and computer networkdesign [PhD thesis] Lund University Lund Sweden 2006

[150] E Amaldi S Coniglio L G Gianoli and C U Ileri ldquoOn single-path network routing subject to max-min fair flow allocationrdquoElectronic Notes in Discrete Mathematics vol 41 no 5 pp 543ndash550 2013

[151] M Koppen K Yoshida K Ohnishi and M Tsuru ldquoMeta-heuristic approach to proportional fairnessrdquo Evolutionary Intel-ligence vol 5 no 4 pp 231ndash244 2012

[152] J Kalcsics ldquoImproved complexity results for several multifacil-ity location problems on treesrdquo Annals of Operations Researchvol 191 pp 23ndash36 2011

[153] J Kalcsics S Nickel J Puerto and A Tamir ldquoAlgorithmicresults for ordered median problemsrdquo Operations ResearchLetters vol 30 no 3 pp 149ndash158 2002

[154] P Brucker S A Kravchenko and Y N Sotskov ldquoOn thecomplexity of two machine job-shop scheduling with regularobjective functionsrdquo OR Spektrum vol 19 no 1 pp 5ndash10 1997

[155] P Brucker Y N Sotskov and F Werner ldquoComplexity of shop-scheduling problems with fixed number of jobs a surveyrdquoMathematical Methods of Operations Research vol 65 no 3 pp461ndash481 2007

[156] M R Garey D S Johnson and R Sethi ldquoThe complexity offlowshop and jobshop schedulingrdquo Mathematics of OperationsResearch vol 1 no 2 pp 117ndash129 1976

[157] N V Shakhlevich Y N Sotskov and F Werner ldquoComplexity ofmixed shop scheduling problems a surveyrdquo European Journalof Operational Research vol 120 no 2 pp 343ndash351 2000


[158] Y Mati S Dauzere-Peres and C Lahlou ldquoA general approachfor optimizing regular criteria in the job-shop schedulingproblemrdquoEuropean Journal ofOperational Research vol 212 no1 pp 33ndash42 2011

[159] M L Balinski and H P Young Fair Representation Meetingthe Ideal of One Man Yale University Press New Haven ConnUSA 1982

[160] H Groeflin H Luss M B Rosenwein and E T WahlsldquoFinal assembly sequencing for just-in-time manufacturingrdquoInternational Journal of Production Research vol 27 no 2 pp199ndash213 1989

[161] F Dugardin F Yalaoui and L Amodeo ldquoNew multi-objectivemethod to solve reentrant hybrid flow shop scheduling prob-lemrdquo European Journal of Operational Research vol 203 no 1pp 22ndash31 2010

[162] MC Lopez-de-los-Mozos J AMesa and J Puerto ldquoA general-izedmodel of equality measures in network location problemsrdquoComputers and Operations Research vol 35 no 3 pp 651ndash6602008

[163] Z Chen ldquoA modified resource distribution fairness measurerdquoJournal of Mathematics vol 2013 Article ID 956039 6 pages2013

[164] W Ogryczak ldquoOn principles of fair resource allocation forimportance weighted agentsrdquo in Proceedings of the InternationalWorkshop on Social Informatics (SOCINFO rsquo09) pp 57ndash62Warsaw Poland June 2009

[165] V Torra ldquoThe weighted OWA operatorrdquo International Journalof Intelligent Systems vol 12 no 2 pp 153ndash166 1997

[166] V Torra and Y Narukawa Modeling Decisions InformationFusion and Aggregation Operators Springer Berlin Germany2007

[167] W Ogryczak and T Sliwinski ldquoOn Optimization of theimportance weighted OWA aggregation of multiple criteriardquo inComputational Science and Its ApplicationsmdashICCSA 2007 vol4705 of Lecture Notes in Computer Science pp 804ndash817 2007

[168] W Ogryczak and T Sliwinski ldquoOn efficient WOWA optimiza-tion for decision support under riskrdquo International Journal ofApproximate Reasoning vol 50 no 6 pp 915ndash928 2009

[169] J Hurkała and T Sliwinski ldquoThreshold accepting heuristic forfair flow optimization in wireless mesh networksrdquo Journal ofAppliedMathematics vol 2014 Article ID 108673 11 pages 2014

[170] Lrsquo Buzna M Kohani and J Janacek ldquoAn approximationaLgorithm for the facility location problem with lexicographicminimax objectiverdquo Journal of Applied Mathematics vol 2014Article ID 562373 12 pages 2014

[171] A Ghodsi M Zaharia B Hindman A Konwinski S Shenkerand I Stoica ldquoDominant resource fairness fair allocation ofmultiple resource typesrdquo in Proceedings of the 8th USENIXConference on Networked Systems Design and Implementationpp 24ndash37 USENIX Association 2011

[172] M Koppen K Ohnishi and M Tsuru ldquoMulti-Jain fairnessindex of per-entity allocation features for fair and efficient allo-cation of network resourcesrdquo in Proceedings of the 5th Inter-national Conference on Intelligent Networking and CollaborativeSystems (INCoS rsquo13) pp 841ndash846 Xirsquoan China 2013

[173] C Joe-Wong S Sen T Lan and M Chiang ldquoMulti-resourceallocation fairness-efficiency tradeoffs in a unifying frame-workrdquo in Proceedings of the IEEE Conference on ComputerCommunications (INFOCOM rsquo12) pp 1206ndash1214 March 2012

[174] A Wierzbicki Trust and Fairness in Open Distributed SystemsSpringer Berlin Germany 2010

[175] S Hsu and A S Tsai ldquoA game-theoretic analysis of bandwidthallocation under a user-grouping constraintrdquo Journal of AppliedMathematics vol 2013 Article ID 480962 9 pages 2013

[176] N Nisam T Roughgarden E Tardos and V V Vazirani EdsAlgorithmic Game Theory Cambridge University Press Cam-bridge UK 2007

[177] K Rzadca D Trystram and A Wierzbicki ldquoFair game-theo-retic resource management in dedicated gridsrdquo in Proceedingsof the 7th IEEE International Symposium on Cluster Computingand the Grid (CCGrid rsquo07) pp 343ndash350 May 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


developed a particular example is models of resource allo-cation that try to achieve some form of fairness in resourceallocation patterns [5] In general the models relate to theoptimization of systems which serve many users and thequality of service provided to every individual user definesthe optimization criteria That pattern applies among othersto telecommunication and Internet networks in those net-works it is important to allocate network resources such asavailable bandwidth so as to provide equitable performanceto all services and all origin-destination pairs of nodes [6 7]Still there aremany other pressing examples of systemswherefair distribution of resources is required Problems of efficientand fair resource allocation arise in complex systems of sys-tems when the system combines a number of component sys-tems such as resource supply systems utilization systems atdemand sites or users stakeholders and their coalitionsystems economic and governmental institutions policysystems and environmental systems Actually addressingfairness in particular types of systems of systems has become agreat challenge of the 21 century [8] as fairly dividing limitednatural resources (such as the fossil fuels the clean water andthe environments capacity to absorb greenhouse gases) isperceived as being of utmost importance

Essentially fairness is an abstract sociopolitical conceptthat implies impartiality justice and equity In order to ensurefairness in a given system all system entities have to beequally well provided with the systemrsquos services For examplethe issue of equity iswidely recognized in the analysis of locat-ing public services where the clients of a system are entitledto fair treatment according to community regulations In thatcontext the decisions often concern the placement of servicecenters or other facilities at such positions that all users aretreated in an equitable way with respect to certain criteria[9] In particular location of the facilities pertaining to publicservices such as police and fire departments and emergencymedical facilities should provide fair response time to alldemand locationswithin ametropolitan area Similarly waterresources should be allocated fairly [10]

As far as technical systems are concerned the importanceof fairness was early recognized with respect to problems ofallocation of bandwidth in telecommunication networks [1112] (resulting in many models and methods of fair optimiza-tion [7]) flight scheduling [13] and allocation of takeoff andlanding ldquoslotsrdquo at airports [14] In such areas as allocation ofresources in high-tech manufacturing and optimal allocationof water and energy resources the context of fair resourceallocation was additionally enriched by considering possiblesubstitutions among the resources models with such substi-tutions are presented in [5 Ch 4] and [15ndash17]

In general complex systems require mathematical pro-gramming models in order to describe the dependencies andto enable system optimization Many such models are basedon some kind of network of connections and dependenciesIn particular wide range of systemmodels are related to somekind of network flows that express realizations of competingactivities [18] This applies to telecommunication systemspower distribution systems transportation systems logisticssystems and so forthThe discrete location problems can alsobe viewed in terms of such network system [19 20]

The general purpose of this paper is to review fairoptimizationmodels and algorithms supporting efficient andfair resource allocation in problems related to such networkmodels The particular focus is on location-allocation prob-lems and allocation problems related to communication net-works since in those areas the fair optimization concepts havebeen extensively developed and widely applied

The paper is organized as follows In the next sectionwe present methodological foundations of fair optimizationmodels In Section 3 the most important models and meth-ods of fair optimization in communication networks arereviewed Section 4 aims at reviewing applications of fair-ness optimization in location and allocation problems Thecomputational complexity issues are addressed in Section 5The paper is concluded by addressing the most importantdirections of the development of fair optimization method-ology for network systems

2 Fairness Equity and Fair Optimization

21 Efficiency and Equity The generic allocation problemdeals with a system comprising a set 119868 of119898 services (activitiesagents) and a given set 119876 of allocation patterns (allocationdecisions) For each service 119894 isin 119868 a function 119891

119894(x) of the allo-

cation pattern x isin 119876 is defined This function measures theoutcome (effect) 119910

119894= 119891119894(x) of allocation pattern x for service

119894 In applications we consider this measure that usuallyexpresses the service quality In general outcomes can bemeasured (modeled) as service time service costs andservice delays as well as in a more subjective way In typicalformulations a larger value of the outcome means a bettereffect (higher service quality or client satisfaction) Other-wise the outcomes can be replacedwith their complements tosome large number Therefore without loss of generality wecan assume that each individual outcome 119910

119894is to be maxi-

mized which allows us to view the generic resource allocationproblem as a vector maximization model Consider

max f (x) x isin 119876 (1)

where f(x) is a vector-function that maps the decision space119883 = 119877

119899 into the criterion space 119884 = 119877119898 and 119876 sub 119883 denotes

the feasible set We consider complex systems represented bymathematical programming models and specifically modelsbased on some network of connections and dependencies

An outcome vector y is attainable if it expresses outcomesof a feasible solution x isin 119876 (ie y = f(x)) The set of all theattainable outcome vectors is denoted by 119860 Note that ingeneral convexity of the feasible set 119876 and concavity of theoutcome function f do not guarantee convexity of the corre-sponding attainable set 119860 Nevertheless the multiple criteriamaximization model (1) can be rewritten in the equivalentform

max y 119910119894le 119891119894 (x) forall119894 x isin 119876 (2)

where the attainable set 119860 is convex whenever 119876 is convexand functions 119891

119894are concave

Model (1) only specifies that we are interested in maxi-mization of all objective functions 119891

119894for 119894 isin 119868 = 1 2 119898




max 119892 (f (x)) x isin 119876 (3)



119898


outcome 120583(y) = 119879(y)119898 = (1119898)sum119898



119910119894 The mean


max min119894=1119898

119891119894 (x) x isin 119876 (4)


max min119894=1119898

119910119894

119898

sum

119894=1

119886119894119910119894le 119887 (5)


119898





1= 1199102= sdot sdot sdot = 119910













(119910120587(1)

119910120587(2)

119910120587(119898)

) cong (1199101 1199102 119910

119898)


(6)


1199101198941015840 gt 11991011989410158401015840 997904rArr y minus 120576e

1198941015840 + 120576e11989410158401015840 ≻ y

for 0 lt 120576 lt 1199101198941015840 minus 11991011989410158401015840

(7)




119910120587(2)

119910120587(119898)

) = 119892(1199101 1199102 119910



1198941015840 minus

120576 11991011989410158401015840 + 120576 119910

119898) gt 119892(119910

1 1199102 119910





119898



max

119898

sum

119894=1

119906 (119891119894 (x)) x isin 119876 (8)


max

119898

sum

119894=1

log (119910119894)

119898

sum

119894=1

119886119894119910119894le 119887 (9)





119906 (119910119894 120572) =

1199101minus120572

119894

(1 minus 120572) if 120572 = 1

log (119910119894) if 120572 = 1

(10)


W( )

y

u( ) = u( )

B( )

y2 = y1

y2

y1

y

yy

y










POF (119865 119860) =

(sum119898

119894=1119910119879

119894minus sum119898

119894=1119910119865

119894)

sum119898

119894=1119910119879

119894

(11)





119898 (12)


POF (MMF 119860) le 1 minus4119898

(119898 + 1)2 (13)



1205781+1120572

+ 119898 minus 120578

1205781+1120572 + (119898 minus 120578) 120578

cong 1 minus 119874 (119898minus120572(120572+1)

)

(14)



119898rarr 119877

119898 such thatΘ(y) = (120579

1(y) 1205792(y) 120579

119898(y)) where 120579

1(y) le 120579



120579119894(y) = 119910



120579119894(y) =

119894

sum

119895=1

120579119895(y) for 119894 = 1 119898 (15)



119894(y1015840) ge 120579

119894(y10158401015840) for all 119894 isin 119868



max (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (16)






(17)



119894=1119904119894120579119894(y) which can


119895=119894119904119895(119894 = 1 119898) allo-


max

119898

sum

119894=1

120596119894120579119894 (f (x)) x isin 119876 (18)


1gt

1205962


gt 120596119898





W( )

B( )

OOWA

y2 = y1

y2

y1

y

y

y


y2 = y1

y2

y1

1205962 = 1205961

1205961 ≫ 1205962

1205961 gt 1205962

1205961 gt 1205962



andness (120596) =sum119898

119894=1((119898 minus 119894) (119898 minus 1)) 120596119894

sum119898

119894=1120596119894

(19)





119896(y) = max

119905(119896119905minussum

119898

119894=1max119905minus119910

119894 0)where 119905 is an


max (1205781 1205782 120578

119898) (20a)

st x isin 119876 (20b)

120578119896= 119896119905119896minus

119898

sum

119894=1

119889119894119896

119896 = 1 119898

(20c)

119905119896minus 119889119894119896

le 119891119894 (x) 119889

119894119896ge 0

119894 119896 = 1 119898

(20d)


120575120591(y) =

1

119898

119898

sum

119894=1

(120591 minus 119910119894)+ (21)


120591(y1015840) le









120591119895

(f(x)) Consider

min (1205751205911

(f (x)) 1205751205912

(f (x)) 120575120591119903

(f (x))) x isin 119876

(22)








lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942


(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)






3

2

4

1

5

2

34

11

2


3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1




119894(119909119894) in (23a)






e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












max 119892 (f (x)) x isin 119876 (3)



119898


outcome 120583(y) = 119879(y)119898 = (1119898)sum119898



119910119894 The mean


max min119894=1119898

119891119894 (x) x isin 119876 (4)


max min119894=1119898

119910119894

119898

sum

119894=1

119886119894119910119894le 119887 (5)


119898





1= 1199102= sdot sdot sdot = 119910













(119910120587(1)

119910120587(2)

119910120587(119898)

) cong (1199101 1199102 119910

119898)


(6)


1199101198941015840 gt 11991011989410158401015840 997904rArr y minus 120576e

1198941015840 + 120576e11989410158401015840 ≻ y

for 0 lt 120576 lt 1199101198941015840 minus 11991011989410158401015840

(7)




119910120587(2)

119910120587(119898)

) = 119892(1199101 1199102 119910



1198941015840 minus

120576 11991011989410158401015840 + 120576 119910

119898) gt 119892(119910

1 1199102 119910





119898



max

119898

sum

119894=1

119906 (119891119894 (x)) x isin 119876 (8)


max

119898

sum

119894=1

log (119910119894)

119898

sum

119894=1

119886119894119910119894le 119887 (9)





119906 (119910119894 120572) =

1199101minus120572

119894

(1 minus 120572) if 120572 = 1

log (119910119894) if 120572 = 1

(10)


W( )

y

u( ) = u( )

B( )

y2 = y1

y2

y1

y

yy

y










POF (119865 119860) =

(sum119898

119894=1119910119879

119894minus sum119898

119894=1119910119865

119894)

sum119898

119894=1119910119879

119894

(11)





119898 (12)


POF (MMF 119860) le 1 minus4119898

(119898 + 1)2 (13)



1205781+1120572

+ 119898 minus 120578

1205781+1120572 + (119898 minus 120578) 120578

cong 1 minus 119874 (119898minus120572(120572+1)

)

(14)



119898rarr 119877

119898 such thatΘ(y) = (120579

1(y) 1205792(y) 120579

119898(y)) where 120579

1(y) le 120579



120579119894(y) = 119910



120579119894(y) =

119894

sum

119895=1

120579119895(y) for 119894 = 1 119898 (15)



119894(y1015840) ge 120579

119894(y10158401015840) for all 119894 isin 119868



max (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (16)






(17)



119894=1119904119894120579119894(y) which can


119895=119894119904119895(119894 = 1 119898) allo-


max

119898

sum

119894=1

120596119894120579119894 (f (x)) x isin 119876 (18)


1gt

1205962


gt 120596119898





W( )

B( )

OOWA

y2 = y1

y2

y1

y

y

y


y2 = y1

y2

y1

1205962 = 1205961

1205961 ≫ 1205962

1205961 gt 1205962

1205961 gt 1205962



andness (120596) =sum119898

119894=1((119898 minus 119894) (119898 minus 1)) 120596119894

sum119898

119894=1120596119894

(19)





119896(y) = max

119905(119896119905minussum

119898

119894=1max119905minus119910

119894 0)where 119905 is an


max (1205781 1205782 120578

119898) (20a)

st x isin 119876 (20b)

120578119896= 119896119905119896minus

119898

sum

119894=1

119889119894119896

119896 = 1 119898

(20c)

119905119896minus 119889119894119896

le 119891119894 (x) 119889

119894119896ge 0

119894 119896 = 1 119898

(20d)


120575120591(y) =

1

119898

119898

sum

119894=1

(120591 minus 119910119894)+ (21)


120591(y1015840) le









120591119895

(f(x)) Consider

min (1205751205911

(f (x)) 1205751205912

(f (x)) 120575120591119903

(f (x))) x isin 119876

(22)








lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942


(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)






3

2

4

1

5

2

34

11

2


3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1




119894(119909119894) in (23a)






e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















(119910120587(1)

119910120587(2)

119910120587(119898)

) cong (1199101 1199102 119910

119898)


(6)


1199101198941015840 gt 11991011989410158401015840 997904rArr y minus 120576e

1198941015840 + 120576e11989410158401015840 ≻ y

for 0 lt 120576 lt 1199101198941015840 minus 11991011989410158401015840

(7)




119910120587(2)

119910120587(119898)

) = 119892(1199101 1199102 119910



1198941015840 minus

120576 11991011989410158401015840 + 120576 119910

119898) gt 119892(119910

1 1199102 119910





119898



max

119898

sum

119894=1

119906 (119891119894 (x)) x isin 119876 (8)


max

119898

sum

119894=1

log (119910119894)

119898

sum

119894=1

119886119894119910119894le 119887 (9)





119906 (119910119894 120572) =

1199101minus120572

119894

(1 minus 120572) if 120572 = 1

log (119910119894) if 120572 = 1

(10)


W( )

y

u( ) = u( )

B( )

y2 = y1

y2

y1

y

yy

y










POF (119865 119860) =

(sum119898

119894=1119910119879

119894minus sum119898

119894=1119910119865

119894)

sum119898

119894=1119910119879

119894

(11)





119898 (12)


POF (MMF 119860) le 1 minus4119898

(119898 + 1)2 (13)



1205781+1120572

+ 119898 minus 120578

1205781+1120572 + (119898 minus 120578) 120578

cong 1 minus 119874 (119898minus120572(120572+1)

)

(14)



119898rarr 119877

119898 such thatΘ(y) = (120579

1(y) 1205792(y) 120579

119898(y)) where 120579

1(y) le 120579



120579119894(y) = 119910



120579119894(y) =

119894

sum

119895=1

120579119895(y) for 119894 = 1 119898 (15)



119894(y1015840) ge 120579

119894(y10158401015840) for all 119894 isin 119868



max (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (16)






(17)



119894=1119904119894120579119894(y) which can


119895=119894119904119895(119894 = 1 119898) allo-


max

119898

sum

119894=1

120596119894120579119894 (f (x)) x isin 119876 (18)


1gt

1205962


gt 120596119898





W( )

B( )

OOWA

y2 = y1

y2

y1

y

y

y


y2 = y1

y2

y1

1205962 = 1205961

1205961 ≫ 1205962

1205961 gt 1205962

1205961 gt 1205962



andness (120596) =sum119898

119894=1((119898 minus 119894) (119898 minus 1)) 120596119894

sum119898

119894=1120596119894

(19)





119896(y) = max

119905(119896119905minussum

119898

119894=1max119905minus119910

119894 0)where 119905 is an


max (1205781 1205782 120578

119898) (20a)

st x isin 119876 (20b)

120578119896= 119896119905119896minus

119898

sum

119894=1

119889119894119896

119896 = 1 119898

(20c)

119905119896minus 119889119894119896

le 119891119894 (x) 119889

119894119896ge 0

119894 119896 = 1 119898

(20d)


120575120591(y) =

1

119898

119898

sum

119894=1

(120591 minus 119910119894)+ (21)


120591(y1015840) le









120591119895

(f(x)) Consider

min (1205751205911

(f (x)) 1205751205912

(f (x)) 120575120591119903

(f (x))) x isin 119876

(22)








lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942


(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)






3

2

4

1

5

2

34

11

2


3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1




119894(119909119894) in (23a)






e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119910120587(2)

119910120587(119898)

) = 119892(1199101 1199102 119910



1198941015840 minus

120576 11991011989410158401015840 + 120576 119910

119898) gt 119892(119910

1 1199102 119910





119898



max

119898

sum

119894=1

119906 (119891119894 (x)) x isin 119876 (8)


max

119898

sum

119894=1

log (119910119894)

119898

sum

119894=1

119886119894119910119894le 119887 (9)





119906 (119910119894 120572) =

1199101minus120572

119894

(1 minus 120572) if 120572 = 1

log (119910119894) if 120572 = 1

(10)


W( )

y

u( ) = u( )

B( )

y2 = y1

y2

y1

y

yy

y










POF (119865 119860) =

(sum119898

119894=1119910119879

119894minus sum119898

119894=1119910119865

119894)

sum119898

119894=1119910119879

119894

(11)





119898 (12)


POF (MMF 119860) le 1 minus4119898

(119898 + 1)2 (13)



1205781+1120572

+ 119898 minus 120578

1205781+1120572 + (119898 minus 120578) 120578

cong 1 minus 119874 (119898minus120572(120572+1)

)

(14)



119898rarr 119877

119898 such thatΘ(y) = (120579

1(y) 1205792(y) 120579

119898(y)) where 120579

1(y) le 120579



120579119894(y) = 119910



120579119894(y) =

119894

sum

119895=1

120579119895(y) for 119894 = 1 119898 (15)



119894(y1015840) ge 120579

119894(y10158401015840) for all 119894 isin 119868



max (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (16)






(17)



119894=1119904119894120579119894(y) which can


119895=119894119904119895(119894 = 1 119898) allo-


max

119898

sum

119894=1

120596119894120579119894 (f (x)) x isin 119876 (18)


1gt

1205962


gt 120596119898





W( )

B( )

OOWA

y2 = y1

y2

y1

y

y

y


y2 = y1

y2

y1

1205962 = 1205961

1205961 ≫ 1205962

1205961 gt 1205962

1205961 gt 1205962



andness (120596) =sum119898

119894=1((119898 minus 119894) (119898 minus 1)) 120596119894

sum119898

119894=1120596119894

(19)





119896(y) = max

119905(119896119905minussum

119898

119894=1max119905minus119910

119894 0)where 119905 is an


max (1205781 1205782 120578

119898) (20a)

st x isin 119876 (20b)

120578119896= 119896119905119896minus

119898

sum

119894=1

119889119894119896

119896 = 1 119898

(20c)

119905119896minus 119889119894119896

le 119891119894 (x) 119889

119894119896ge 0

119894 119896 = 1 119898

(20d)


120575120591(y) =

1

119898

119898

sum

119894=1

(120591 minus 119910119894)+ (21)


120591(y1015840) le









120591119895

(f(x)) Consider

min (1205751205911

(f (x)) 1205751205912

(f (x)) 120575120591119903

(f (x))) x isin 119876

(22)








lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942


(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)






3

2

4

1

5

2

34

11

2


3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1




119894(119909119894) in (23a)






e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119898 (12)


POF (MMF 119860) le 1 minus4119898

(119898 + 1)2 (13)



1205781+1120572

+ 119898 minus 120578

1205781+1120572 + (119898 minus 120578) 120578

cong 1 minus 119874 (119898minus120572(120572+1)

)

(14)



119898rarr 119877

119898 such thatΘ(y) = (120579

1(y) 1205792(y) 120579

119898(y)) where 120579

1(y) le 120579



120579119894(y) = 119910



120579119894(y) =

119894

sum

119895=1

120579119895(y) for 119894 = 1 119898 (15)



119894(y1015840) ge 120579

119894(y10158401015840) for all 119894 isin 119868



max (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (16)






(17)



119894=1119904119894120579119894(y) which can


119895=119894119904119895(119894 = 1 119898) allo-


max

119898

sum

119894=1

120596119894120579119894 (f (x)) x isin 119876 (18)


1gt

1205962


gt 120596119898





W( )

B( )

OOWA

y2 = y1

y2

y1

y

y

y


y2 = y1

y2

y1

1205962 = 1205961

1205961 ≫ 1205962

1205961 gt 1205962

1205961 gt 1205962



andness (120596) =sum119898

119894=1((119898 minus 119894) (119898 minus 1)) 120596119894

sum119898

119894=1120596119894

(19)





119896(y) = max

119905(119896119905minussum

119898

119894=1max119905minus119910

119894 0)where 119905 is an


max (1205781 1205782 120578

119898) (20a)

st x isin 119876 (20b)

120578119896= 119896119905119896minus

119898

sum

119894=1

119889119894119896

119896 = 1 119898

(20c)

119905119896minus 119889119894119896

le 119891119894 (x) 119889

119894119896ge 0

119894 119896 = 1 119898

(20d)


120575120591(y) =

1

119898

119898

sum

119894=1

(120591 minus 119910119894)+ (21)


120591(y1015840) le









120591119895

(f(x)) Consider

min (1205751205911

(f (x)) 1205751205912

(f (x)) 120575120591119903

(f (x))) x isin 119876

(22)








lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942


(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)






3

2

4

1

5

2

34

11

2


3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1




119894(119909119894) in (23a)






e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










W( )

B( )

OOWA

y2 = y1

y2

y1

y

y

y


y2 = y1

y2

y1

1205962 = 1205961

1205961 ≫ 1205962

1205961 gt 1205962

1205961 gt 1205962



andness (120596) =sum119898

119894=1((119898 minus 119894) (119898 minus 1)) 120596119894

sum119898

119894=1120596119894

(19)





119896(y) = max

119905(119896119905minussum

119898

119894=1max119905minus119910

119894 0)where 119905 is an


max (1205781 1205782 120578

119898) (20a)

st x isin 119876 (20b)

120578119896= 119896119905119896minus

119898

sum

119894=1

119889119894119896

119896 = 1 119898

(20c)

119905119896minus 119889119894119896

le 119891119894 (x) 119889

119894119896ge 0

119894 119896 = 1 119898

(20d)


120575120591(y) =

1

119898

119898

sum

119894=1

(120591 minus 119910119894)+ (21)


120591(y1015840) le









120591119895

(f(x)) Consider

min (1205751205911

(f (x)) 1205751205912

(f (x)) 120575120591119903

(f (x))) x isin 119876

(22)








lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942


(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)






3

2

4

1

5

2

34

11

2


3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1




119894(119909119894) in (23a)






e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















lexmaxx

Θ (f (x)) = (1198911198941

(1199091198941

) 1198911198942

(1199091198942

) 119891119894119898

(119909119894119898

))

(23a)

st 1198911198941

(1199091198941

) le 1198911198942

(1199091198942


(119909119894119898

) (23b)

sum

119894isin119868

119886119894119895119909119894le 119887119895 forall119895 isin 119869 (23c)






3

2

4

1

5

2

34

11

2


3

2

4

1

5

2

3 4

11

2Path 3

Path 2

Path 1




119894(119909119894) in (23a)






e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










e1 e21205921 1205922 1205923



1lt 1205912



119896(y) = sum

119896

119897=1ℎ119897(y) with ℎ

119896(y)





lex min (

119898

sum

119894=1

1199111119894

119898

sum

119894=1

1199112119894

119898

sum

119894=1

119911119903minus1119894

) (24a)


120591119896+1

minus 119891119894 (x) le 119872119911

119896119894 119911119896119894

isin 0 1

119894 isin 119868 119896 lt 119903

(24c)






(V1 V2 V3) two links (119890


1=

V1 V2 1198892

= V2 V3 1198893



1= 1198882

=







1=

1198832= 1198833= 075 (119879(119883) = 225)


2+ log119883

3)

1198831= 1198832= 1 119883

3= 05 (119879(119883) = 25) and


1198832= 15 119883

3= 0 (119879(119883) = 3)




3






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















1 V2 V3) two links


1= V1 V2 1198892

= V2 V3

1198893



1= 1198882

= 15) Let 119883 = (1198831 1198832 1198833) be



Let 119883119889




119890=



sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (25a)

119883119889ge ℎ (119889) 119889 isin D (25b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (25c)

119884119890le 119888 (119890) 119890 isin E (25d)

119909119901isin X 119901 isin P (25e)


+and







119889isinD119883119889 119879(119883) =

sum119889isinD 119883








sum

119901isinP(119889)

119906119901= 1 119889 isin D (26a)

119909119901le 119872119906

119901 119901 isin P (26b)

119906119901isin 0 1 119901 isin P (26c)







119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901 119901 isin



min 119903 (27a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (27b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903 119890 isin E (27c)



119890


lex min Θ (119877) (28a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (28b)

sum

119901isinQ(119890)

119909119901le 119888 (119890) 119903119890 119890 isin E (28c)

119903119890isin R 119890 isin E

119909119901isin X 119901 isin P

(28d)



119890


lex max Θ(119884) (29a)

sum

119901isinP(119889)

119909119901= ℎ (119889) 119889 isin D (29b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (29c)

119884119890le 119888 (119890) 119890 isin E (29d)

119884119890= 119888 (119890) minus 119884

119890 119890 isin E (29e)

119909119901isin X 119901 isin P (29f)


lex max Θ (119883) (30a)

sum

119901isinP(119889)

119909119901= 119883119889 119889 isin D (30b)

sum

119901isinQ(119890)

119909119901= 119884119890 119890 isin E (30c)

sum

119890isinE

120585 (119890) 119884119890 le 119861 119890 isin E (30d)

119909119901isin X 119901 isin P (30e)




the cost 120581(119889) = sum119890isin119901(119889)

120585(119890) Clearly

119883lowast=

119861

sum119889isinD 120581 (119889)

(31)


min sum

119889isinD

log 119883119889 (32)


119883lowast

119889=

119861

120581 (119889) |D| 119889 isin D (33)








1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1= V1 V2 1198892

= V2 V3 1198893

=


1 1198832 1198833) are



1= 15 119883

2= 15 119883



1= 075 119883

2= 075 119883

3= 075 which is the

MMF solution
















sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (34)











119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119909119904= min 119903 (119904) 119910119890 119890 isin E 119904 isin S (119890) (35)




119890



119909119904= min119903 (119904) min

119890isinE(119904)119910119890 119904 isin S (36)




119904cannot




119890isinE(119904)119910119890




lex max Θ (119909) (37a)

119909119904le 119903 (119904) 119904 isin S (37b)

sum

119904isinS(119890)

119909119904119897 (119904) le 119888 (119890) 119890 isin E (37c)

119909119904isin R+ 119904 isin S (37d)





1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2

3 4 5

6 7 8 9 10 11

1 2 3 4 5 6

(1 2) (2 3 5) (1 5) (3 4 6) (1 2 4 5) (4 6)

(2 6)(3 4)(1 5)

100

100100

100

100 200

200150

8080 80

150











119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119890119862119890)1minus120572 and (120572 minus 1)


119890
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119899

sum

119895=1

119909119895= 119901 119909

119895isin 0 1 for 119895 = 1 119899 (38)


1015840

119894119895(119894 = 1 2 119898 119895 = 1 2 119899) equal


119899

sum

119895=1

1199091015840

119894119895= 1 for 119894 = 1 119898

1199091015840

119894119895le 119909119895 for 119894 = 1 119898 119895 = 1 119899

1199091015840

119894119895isin 0 1 for 119894 = 1 119898 119895 = 1 119899

(39)


Let 119889119894119895


119891119894 (x) =

119899

sum

119895=1

1198891198941198951199091015840

119894119895 for 119894 = 1 119898 (40)


119894119895= 119889 minus 119889

119894119895



119894119895= exp (minus120573119889

119894119895)



119894gt 0 are


119894= 119908119894sum119898

119894=1119908119894for 119894 =



are given as 119908119894= 1119898



119863(y) =1

2

119898

sum

119894=1

119898

sum

119895=1

10038161003816100381610038161003816119910119894minus 119910119895

10038161003816100381610038161003816119908119894119908119895 (41)


MAD (y) =

119898

sum

119894=1

1003816100381610038161003816119910119894 minus 120583 (y)1003816100381610038161003816 119908119894 (42)



2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

100100

1205921

1205922 1205923





the other 11988912


13= 11988923

=








1) = (1 minus 120576)(1 + 99120576) and in





3) =








119872120572119863

(y) = 120583 (y) + 120572119863 (y)

= (1 minus 120572)

119898

sum

119894=1

119910119894119908119894

+ 120572

119898

sum

119894=1

119898

sum

119895=1

max 119910119894 119910119895119908119894119908119895

(43)



min (1205791 (f (x)) 1205792 (f (x)) 120579119898 (f (x))) x isin 119876 (44)

where 120579119894(y) = sum

119898






119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119898

119894=1120596119894120579119894(y)


1lt 1205962






(45)



119895be the





1

3

4

5

2

3

4

5


Sensitive locations

M N

2

2

2

1



lex max Θ (119883) = (1198831198951

1198831198952

119883119895119899

) (46a)

st 1198831198951

le 1198831198952


(46b)

119883119895= sum

119894isin119868119895

119909119894 forall119895 isin 119873 (46c)

sum

119894isin119872

119909119894= 119870 (46d)



119894119895gt 0 for 119894 isin 119868





5 Complexity Issues







5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














5)





4) for the
















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Acknowledgment


References
































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









review article fair optimization and networks: a surveyof fairness was early recognized with respect...

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