research article a capacitated location-allocation model

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 507953 11 pageshttpdxdoiorg1011552013507953

Research ArticleA Capacitated Location-Allocation Model for FloodDisaster Service Operations with Border Crossing Passagesand Probabilistic Demand Locations

Seyed Ali Mirzapour1 Kuan Yew Wong1 and Kannan Govindan2

1 Department of Manufacturing and Industrial Engineering Faculty of Mechanical EngineeringUniversiti Teknologi Malaysia 81310 UTM Skudai Malaysia

2 Department of Business and Economics University of Southern Denmark 5230 Odense Denmark

Correspondence should be addressed to Kuan YewWong wongkyfkmutmmy

Received 16 August 2013 Revised 14 October 2013 Accepted 14 October 2013

Academic Editor Tsan-Ming Choi

Copyright copy 2013 Seyed Ali Mirzapour 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

Potential consequences of flood disasters including severe loss of life and property induce emergency managers to find theappropriate locations of relief rooms to evacuate people from the origin points to a safe place in order to lessen the possible impactof flood disasters In this research a p-center location problem is considered in order to determine the locations of some reliefrooms in a city and their corresponding allocation clustersThis study presents a mixed integer nonlinear programming model of acapacitated facility location-allocation problem which simultaneously considers the probabilistic distribution of demand locationsand a fixed line barrier in a region The proposed model aims at minimizing the maximum expected weighted distance from therelief rooms to all the demand regions in order to decrease the evacuation time of people from the affected areas before floodoccurrence A real-world case study has been carried out to examine the effectiveness and applicability of the proposed model

1 Introduction

Due to various occurrences of disasters such as floodshurricanes and earthquakes which lead to enormous prop-erty damages and human injuries disaster management hasrecently become a crucial issue Within disaster managementtasks finding the appropriate locations of relief rooms canhelp emergency managers to ensure public safety and well-being in tragic situations Thus transportation specialistsplay a prominent role in the emergency management processwhich focuses on people evacuation from disaster eventsin order to mitigate injury and loss of life In this regardthere has been an increasing interest among practitionersand scholars in the field of emergency facility locationproblems Compared to all disasters flood is more possibleto be predicted and prevented Therefore it is vital for flooddisaster managers to find the appropriate locations of reliefrooms or emergency medical services and allocate them topopulations in order to provide efficient services

A restricted region is a limitation in an area in whichthe geographic features obstruct the construction of reliefrooms Generally a restricted planar area is divided intothree categories forbidden regions congested regions andregions where neither placement nor travelling is permittedthrough them Lakes mountains highways and militaryzones are some practical examples of these restricted regionsHamacher andNickel [1] have performed an extensive reviewof facility location problems with restrictions These limi-tations were incorporated by Klamroth [2] into a model inwhich a fixed line barrier in a region divides it into two half-planes and two passages located along the line barrier providecommunication between these subregions The model pro-posed by Klamroth [2] is a specific formulation and it maynot be appropriate for cases that have regional probabilitydistributions of customers or different objective functionsIn the real world when we are dealing with establishingpermanent emergency service facilities for future disastersover a long term planning horizon it is essential to consider

2 Mathematical Problems in Engineering

the regional probability distribution of customers due to theuncertainties of events and partial information and dataTherefore a more applicable model which provides bettersolutions would be beneficial for overcoming this weakness

To the best of our knowledge few research activitieswhich consider barriers in emergency facility location prob-lems have been accomplished due to the computationalcomplexities associated with these problems Moreover fartoo little attention has been paid to developing a practicalmathematical model of emergency facility location thatconsiders probabilistic customer regions and a line barrierwith border crossings simultaneously In this research inorder to address the abovementioned problems we developa mixed integer nonlinear programming (MINLP) model ofa capacitated facility location-allocation problem (CFLAP)in an area in which a fixed line barrier and probabilisticcustomer regions are taken into account

The rest of this paper is organized as follows Section 2provides a comprehensive literature review on related topicsSection 3 gives an explanation of the problem and Section 4presents the proposed mathematical model In order to showthe applicability of the proposed model a suitable casestudy is illustrated in Section 5 and the model is validatedin Section 6 Finally Section 7 presents some concludingremarks and outlines a number of future research directions

2 Literature Review

21 Emergency Facility Location Problems The emergencyfacility location models are categorized into three differenttypes P-median P-center and maximal covering locationproblems Since the objective of this research is to design aneffective response strategy for disaster management organi-zations in order to reduce casualties a planar center locationproblem (PCLP) would best suit our purpose Actuallywhenever the average cost of servicing the customers is lessimportant than ensuring that no customer receives poorquality of services the minimax location problem is typicallyproposed [3] The procedure of the PCLP model which isin fact a minimax problem looks for the center in order tominimize the maximum service time cost or loss Researchin this area startedwith Sylvester [4] whichwas the first paperthat considered the Weber problem with the PCLP modelElzinga and Hearn [5] then introduced the Euclidean centerproblem with equal weights and offered an efficient solutiontechnique for their proposed model Francis et al [6] madeit possible to stress the importance of different customers byassigning unequal weights to them Mehrez et al [7] applieda single objective optimization to determine the location ofa new hospital The solutions of this problem were generatedby a qualitative approach based on the evaluation of expertsrsquojudgments After that Daskin et al [8] proposed an error-bound driven demand point aggregation for the minimaxproblemwith rectilinear distance Hurtado et al [9] came outwith some constraints on the minimax problem and used atime algorithm to solve it

In the last two decades the field of disaster manage-ment has gained increasing attention Many authors have

conducted several studies on predisaster preparedness andpostdisaster responsiveness including Altay and Green III[10] Simpson and Hancock [11] Ichoua [12] and Shisheboriand Jabalameli [13] Caunhye et al [14] andGalindo and Batta[15] performed an extensive review of optimization modelsin emergency logistics for disaster operations managementSherali et al [16] proposed a capacitated facility location-allocation model in order to minimize the congested evac-uation time by finding the optimal locations of evacuationshelters under hurricane or flood conditions Chang et al [17]studied the problem of locating relief rooms and allocatingrelief resources for flood emergency preparation They pro-posed two stochastic models under possible flood scenariosin which the demand locations are uncertain and depend onthe level of flood in different scenarios In addition Sheu[18] presented an earthquake relief model that forecasts thedemand of affected regions and coordinates the provisionof relief supplies Mete and Zabinsky [19] dealt with anoptimization approach for locating and distributing medicalsupplies under demand and transportation uncertainties fordisaster situations Some other recent relevant studies on thisissue include Jiang et al [20] Rawls and Turnquist [21] Yi etal [22] Halper and Raghavan [23] Wu et al [24] Xu et al[25] and Duran et al [26]

22 Facility Location Problems under Uncertainty Whenaddressing the strategic decision of establishing permanentfacilities to prepare for future disasters over a long termplanning horizon information is uncertain due to the timelag so considering the location of demands as a known pointwith certainty leads to poor model performance [27] Thefirst research in the probabilistic Weber problem was doneby Cooper [28] who focused on a single facility locationproblem After that Katz and Cooper [29] approximated theexpected distance of transportation via Euclidean distancein which the customer locations were assumed to have anormal distribution They proposed an algorithm to min-imize the total expected distance between the new facilityand existing facilities Later on for the case of rectilineardistance Wesolowsky [30] considered three different prob-ability distributions of bivariate normal bivariate symmetricexponential and bivariate uniform for customer locations insolving the probabilistic Weber problem

Although some studies have recently been done for thecase of multifacility minimax location problems much lessattention has been given to the probabilistic formulation ofminimax problems Stochastic demand locations were firstconsidered by Carbone and Mehrez [31] for the minimaxlocation problem where the locations of demand pointswere normally distributed with predetermined means andvariances An emergency location problem with a weightedminimax objective function has been presented by Bermanet al [32] in which the weights associated with each demandpoint were not given and each of themwas assumed to have auniform distributionThey proved that the objective functionis convex for parameters of the uniform distribution andthus the model can be solved by using standard optimizationtechniques The minimax regret location problem has been

Mathematical Problems in Engineering 3

investigated by Averbakh and Bereg [33] in the case ofuncertain weights and coordinates of demand points Theyproposed amodel with two objective functions of themedianand center problems by using rectilinear distance Foul [34]applied a mathematical model with the minimax objectivefunction to find the unique location where the demandpoints follow a bivariate uniform distributionThen Pelegrınet al [35] proposed a comprehensive framework for the1-center problem where the weights of demands have anarbitrary probability distributionThey have studied a varietyof problems using a number of objective functions

After that Durmaz et al [36] presented a model on theprobabilistic capacitated multifacility Weber problem in atwo-dimensional region by assuming that the locations ofcustomers have some multivariate probability distributionsLater on Canbolat and Von Massow [37] shared their workon locating emergency facilities with random demandsThey proposed an approach that explicitly addresses theuncertainty with respect to where an emergency will occurby minimizing the maximum risk through the minimizationof the maximum expected distance Then Hosseinijou andBashiri [38] formulated an appropriatemodel to represent theminimax single facility location problem in which demandareas are weighted and their coordinates have a bivariateuniform distributionThey proposed amodel which finds theoptimal location of the transfer point such that themaximumexpected weighted distance from the facility point to alldemand points through the transfer point is minimized

23 Facility Location Problems with Barriers In many casesof facility location problems difficulties in solution method-ologies occur when barriers limit the establishment of newfacilities in a specified region The Weber problem with acircular barrier was first presented by Katz and Cooper [39]Since then researchers have studied a variety of locationproblems in the presence of barriers using different typesof objective function They showed that the objective func-tions of these problems are nonconvex and offered someheuristic methods for solving them Klamroth and Wiecek[40] focused on solving a multiobjective median problem(MOMP) with the minisum objective function by assuminga line barrier in an area and showed that their proposedmodel is nonconvex Many researchers have applied differentheuristic algorithms for theWeber problem that considers theconvex or nonconvex polyhedral barriers in a region (eg[41ndash43]) The location problem has also been extended byKlamroth [44] who suggested that the single circular barriercan be subdivided into some feasible convex regions Byusing this method the nonconvex objective function of theproblem has been transformed into some convex objectivefunctions in each region When the number of subdividedconvex regions increases this method is not suitable sincedividing the convex regions becomes awkward To overcomethis complexity a genetic algorithm has been proposed byBischoff and Klamroth [45] based on the Weiszfeld tech-nique to solve the model with a large number of demand

points Bischoff et al [46] presented two alternative location-allocation heuristics for themultifacilityWeber problemwithbarriers

The locations of optimum points for a given numberof passages in two cases of capacitated and incapacitatedfacility location problems have been discussed by Huang etal [48] their model tried to minimize the transportationcost Increasingly more authors have addressed the planarcenter location problem in the presence of barriers in whichthe objective function is to find a facility that minimizes themaximum distance (eg [49ndash51]) A different perspectiveof using a wave front approach was proposed by Frieszlig etal [52] for the minimax location problem in the presenceof barriers Canbolat and Wesolowsky [53] developed amodel by considering a probabilistic line barrier that occursrandomly on the horizontal route Their objective was tolocate a single new facility to minimize the sum of theexpected distances from all customer locations After thatCanbolat andWesolowsky [54] presented a new experimentalapproach for the Weber problem by using the Varignonframe Their proposed method tried to minimize the sum ofthe weighted distances from the facility to customer pointsby considering barriers in a region Then Canbolat andWesolowsky [47] presented a planar single facility locationproblem in the presence of border crossing They developedthe models for minisum and minimax problems in thepresence of a fixed line barrier which divides a region into twosubregions

In short although previous studies have proposed dif-ferent approaches to address the facility location-allocationproblem in the presence of a barrier they have somehowneglected the probabilistic nature of customer locationsMoreover very limited research activities have been carriedout on multifacility location problems albeit in many real-world applications more than a single facility is required tobe establishedTherefore an appropriatemathematicalmodelis proposed in this paper in order to address these issues

3 Problem Definition and Formulation

31 Facility Location Problem in the Presence of a Line BarrierIt is important to introduce some preliminary definitionsbefore formally defining the problem Based onKlamroth [2]the facility location problem in the presence of a line barrieris described as follows

Let 119871 be a line and 119875119896 = (119909119896 119910119896)120598119871 | 119896120598119870 = 1 119870

be a set of points on 119871 Then 119861119871 = 119871 1198751 119875119896 is calleda fixed line barrier with passages The feasible region 120579 isdefined as the union of two closed half-planes 1205791 and 1205792 onboth sides of 119861119871 119875119896 = (119909

119896 119910119896) is the coordinate of a border

crossing passage Since the connections are located on a fixedhorizontal line barrier we set 119910119896 = 120593 119896 = 1 119870 A setof existing demand regions with locations of 119883119895 119895 = 1 119869

and positiveweights of119908119895 is given in 120579Then119889119861119901(119883119894 119883119895) is the

p-norm barrier distance between 119883119894 and 119883119895 in the presenceof a fixed line barrier where communication between two


subplanes is allowed only through the 119870 passages The basicminimax model with barriers can be stated as follows

Min119865 (119883119894) = Max 119908119895 sdot 119889119861

119901(119883119894 119883119895)

119894 = 1 119868 119895 = 1 119869

(1)

Any demand region that is in the same half-plane willnot be affected by the presence of the barrier The otherdemand regions that are not in the same half-plane withthe new facility will connect to the new facility through justone of the passages along the line barrier For this problemthe rectilinear distance 119901 = 1 is considered to computethe distance between the new emergency facilities and alldemand regions

32 Stochastic Weighted Emergency Facility Location Problemin the Presence of a Line Barrier Assume that each demandregion 119895 has a coordinate of 119883119895 = (119880119895 119881119895) such that 119880119895 119881119895are independent random variables and each of them has abivariate uniform distribution This assumption is realisticin real-world cases wherein the geographical distributionof population in different regions is uniform in square orrectangular areas [38] Furthermore a uniform distributionprovides a ldquobuilding blockrdquo to deal with a situation which isnot precisely approximated [34]

The problem is to find the optimum location of the newfacility (119909119894

lowast 119910119894lowast) such that the maximum expected weighted

distance from the facility to all demand regions 119864119865(119883119894lowast) is

minimized

Min 119864119865 (119883119894lowast) = Max 119864 [119908119895 sdot 119889

119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

= Max 119908119895 sdot 119864 [119889119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

119894 = 1 119868 119895 = 1 119869

(2)

where 119864[sdot] represents the expected value The above men-tioned problem is illustrated in Figure 1 The weights ofemergency needs in each demand region are defined as afunction of two factors The first factor is related to thequalitative criterion of demand region 119895 which can be thequality of road construction from a demand region to thenew facility point whereas the next one is concerned withthe quantitative criterion of demand region 119895 which can bea function of population potential of emergency situationoccurrence and other issuesThe weight119876119895 of the qualitativefactor has an inverse relation with 119908119895 which means that119908119895 will reduce whenever the quality of road constructionincreases and this causes the new facility to be located fartherfrom demand region 119895 The weight 119875119895 of the quantitativefactor has a direct relation with 119908119895 for instance wheneverthe population of demand region 119895 increases119908119895 will increasetoo and the new emergency facility will tend to be locatednearer to this customer area In order to use both quantitativeand qualitative factors in the total weight of demand region119895 they must be defined in the same range to be comparableOne method of doing so is normalizing both types of factorsto be in the domain of zero to one Based on the experience

Y

X

BL

Pk

1205791

1205792

Figure 1 Demand locations with a fixed line barrier and twopassages on the plane

of emergency managers both quantitative and qualitativefactors have the same significance

In this paper the case of uniformly distributed demandregions in several rectangles is considered It uses the recti-linear distance concept for measuring distances Imagine anumber of demand regions 119895 in which119880119895119881119895 are independentrandom variables in [119886119895 119887119895] and [119888119895 119889119895] with probabilitydensity functions 119891119906(119880119895) = 1(119887119895 minus 119886119895) and 119891V(119881119895) = 1(119889119895 minus

119888119895) respectively Each area is a demand region and it isdesirable to find the optimum locations of emergency servicesamong all these regions The objective is to find the optimallocations of emergency facilities to provide efficient servicesfor people The mathematical expression for the expecteddistance between new facility 119894 and demand region 119895 ispresented as

119864 [119889 (119883119894 119883119895)] = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

= 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] + 119864 [

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

(3)

Let119864[|119880119895minus119909119894|] and119864[|119881119895minus119910119894|] be denoted by119867119894(119909) and119866119894(119910)respectively Then we have

119867119894 (119909) = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816sdot 119891119906 (119880119895) 119889119906

=1

119887119895 minus 119886119895

int

119887119895

119886119895

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816119889119906

(4)

119866119894 (119910) = 119864 [10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816sdot 119891V (119881119895) 119889V

=1

119889119895 minus 119888119895

int

119889119895

119888119895

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816119889V

(5)

The closed-form expression for the expected distance119864[119889(119883119894 119883119895)] was developed by Foul [34] which can bewritten as

119867119894 (119909) =

minus119909119894 +

119886119895 + 119887119895

2if 119909119894 le 119886119895

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

if 119886119895 le 119909119894 le 119887119895

119909119894 minus

119886119895 + 119887119895

2if 119909119894 ge 119887119895

(6)


119866119894(119910) can be defined like 119867119894(119909) by substituting 119909119894 119886119895 119887119895 with119910119894 119888119895 119889119895 respectively With this expression the problem isformulated as follows

Min 119864119865 (119883119894) = Max 119908119895 sdot [119867119894 (119909) + 119866119894 (119910)]

119894 = 1 119868 119895 = 1 119869

(7)

Equation (7) can be transformed into an equivalentconstrained nonlinear program which is presented here

Min 119885

Subject to 119885 ge 119908119895 sdot [119867119894 (119909) + 119866119894 (119910)]

119894 = 1 119868 119895 = 1 119869

(8)

4 Proposed Mathematical Model

In this section a MINLP model is proposed for CFLAPThe proposed MINLP model aims at finding the suitablelocations of some emergency facilities in the presence of afixed line barrier such that the expected weighted rectilineardistance between the emergency facilities and the farthestdemand regions is minimized The following assumptionsand notations are used in formulating the MINLP model forCFLAP

Assumptions (i) The demand regionsrsquo coordinates have abivariate uniform distribution because customer populations(residential areas) are normally dispersed or structured insquare or rectangular areas

(ii) There is a line barrier in the area which is fixed(iii) Two emergency service facilities are required to be

establishedThis will ensure amore balanced allocation of thedemand regions to the emergency service facilities since theentire region is divided into two half-planes by a line barrier

Notations

119868 Number of new facilities119869 Number of demand regions or customer regions119870 Number of passages119908119895 Positive weight of each existing demand region 119895

119905119894119895 Binary variable 1 if demand region 119895 is assigned tonew facility 119894 0 otherwise

ℎ119894119895119896 Binary variable 1 if demand region 119895 is served by newfacility 119894 through passage 119896 0 otherwise

119897119894119895 Binary variable 1 if demand region 119895 and new facility119894 are located in different half-planes 0 otherwise

119892119894 Binary variable 1 if new facility 119894 is located in theupper half-plane 0 otherwise

119902119895 Binary parameter 1 if demand region 119895 is located inthe upper half-plane 0 otherwise

119862119886119894 Maximum capacity of new facility 119894

1198781119894119895 Binary variable 1 if 119909119894 le 119886119895 0 otherwise1198782119894119895 Binary variable 1 if 119886119895 lt 119909119894 lt 119887119895 0 otherwise1198783119894119895 Binary variable 1 if 119909119894 ge 119887119895 0 otherwise

1198651119894119895 Binary variable 1 if 119910119894 le 119888119895 0 otherwise

1198652119894119895 Binary variable 1 if 119888119895 lt 119910119894 lt 119889119895 0 otherwise

1198653119894119895 Binary variable 1 if 119910119894 ge 119889119895 0 otherwise

1198781119895119896 Binary variable 1 if 119909119896le 119886119895 0 otherwise

1198782119895119896 Binary variable 1 if 119886119895 lt 119909119896lt 119887119895 0 otherwise

1198783119895119896 Binary variable 1 if 119909119896ge 119887119895 0 otherwise




Based on the above assumptions and notations theobjective function and constraints will be described

41 Objective Function The objective function tries to min-imize 119885 while satisfying constraint (9) Since 119885 needs tobe minimized the shortest possible path through one ofthe border crossings along the barrier will be selected Theobjective function can be formed as follows

Min 119885

where 119885ge119908119895 sdot 119905119894119895 sdot [[(

119870

sum

119896=1

[119879119896 (119909119896)+119877119896 (119910

119896) +

10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816

+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816] sdot ℎ119894119895119896) sdot 119897119894119895]

+ [[119867119894 (119909) + 119866119894 (119910)] sdot (1 minus 119897119894119895)] ]

(9)

Note that the expected distance from the passage coordi-nates along the line barrier to the demand regions is denotedby 119879119896(119909

119896) and 119877119896(119910

119896) which are equal to 119864[|119880119895 minus 119909

119896|] and

119864[|119881119895 minus 119910119896|] and they can be defined like 119867119894(119909) and 119866119894(119910) by

substituting 119909119894 with 119909119896 and 119910119894 with 119910

119896 respectively

42 Constraints The constraints of this problem are formu-lated as follows

421 Expected Distance Based on the obtained closed-formexpression in (6) the expected distance is computed indifferent ways depending on the location of the new facilityand the interval of demand regions Since this model will besolved by the GAMS software (full academic version 221)it is required to convert the formulas of 119867119894(119909) into a fewconstraints by introducing new binary variables of 1198781119894119895 1198782119894119895and 1198783119894119895 This makes it easier for the software to perform thecomputation process Likewise new binary variables of 11986511198941198951198652119894119895 and1198653119894119895 for119866119894(119910) are introduced119879119896(119909

119896) and119877119896(119910

119896) can

be manipulated like 119867119894(119909) and 119866119894(119910) by replacing 1198781119894119895 1198782119894119895


1198783119894119895 with 1198781119895119896 1198782119895119896 1198783119895119896 and 1198651119894119895 1198652119894119895 1198653119894119895 with 1198651119895 11986521198951198653119895 respectively These constraints can be written as follows

2119909119894 sdot 1198781119894119895 minus 2119886119895 sdot 1198781119894119895 le 119909119894 minus 119886119895 forall119894 119895 (10)

2119909119894 sdot 1198783119894119895 minus 2119887119895 sdot 1198783119894119895 ge 119909119894 minus 119887119895 forall119894 119895 (11)

1198781119894119895 + 1198782119894119895 + 1198783119894119895 = 1 forall119894 119895 (12)



1198651119894119895 + 1198652119894119895 + 1198653119894119895 = 1 forall119894 119895 (15)

2119909119896sdot 1198781119895119896 minus 2119886119895 sdot 1198781119895119896 le 119909

119896minus 119886119895 forall119895 119896 (16)

2119909119896sdot 1198783119895119896 minus 2119887119895 sdot 1198783119895119896 ge 119909

119896minus 119887119895 forall119895 119896 (17)

1198781119895119896 + 1198782119895119896 + 1198783119895119896 = 1 forall119895 119896 (18)

2120593 sdot 1198651119895 minus 2119888119895 sdot 1198651119895 le 120593 minus 119888119895 forall119895 (19)

2120593 sdot 1198653119895 minus 2119889119895 sdot 1198653119895 ge 120593 minus 119889119895 forall119895 (20)

1198651119895 + 1198652119895 + 1198653119895 = 1 forall119895 (21)

The first three constraints (constraints (10) (11) and(12)) are related to 119867119894(119909) Whenever the variable 119909119894 is lessthan 119886119895 (119909119894 le 119886119895) the binary variable 1198781119894119895 in constraint(10) will be equal to one otherwise it makes the binaryvariable to be equal to zero Constraint (11) ensures that theexpected distances between facilities and demand regions arecalculated by the third formula of (6) whenever the variable119909119894 is more than 119887119895 (119909119894 ge 119887119895) otherwise 1198783119894119895 will be equal tozero If neither 1198781119894119895 nor 1198783119894119895 is equal to one then the binaryvariable 1198782119894119895 must be equal to one due to constraint (12)Constraint (12) also guarantees that just one of these casescan occur for the same new facility and customer regionassignment of two or three binary variables to one will leadto infeasibility and is not allowed The rest of the constraints(constraints (13) to (21)) follow the same interpretation for119866119894(119910) 119879119896(119909

119896) and 119877119896(119910

119896) respectively

422 The Position of New Facilities and Customer RegionsConstraint (22) indicates whether the new facility is locatedin the upper half-plane or not

2119892119894 sdot 119910119894 minus 119910119894 ge 2119892119894 sdot 120593 minus 120593 forall119894 (22)

Constraint (23) shows whether the customer region andthe new facility are both located in the same half-plane or not

119897119894119895 =10038161003816100381610038161003816119902119895 minus 119892119894

10038161003816100381610038161003816 forall119894 119895 (23)

Constraints (22) and (23) help the objective function tocompute the expected distance between the new facilities andthe customer regions in different cases

423 Demand Allocation According to constraint (24) eachdemand region must be served just by one of the newfacilities

119868

sum

119894=1

119905119894119895 = 1 forall119895 (24)

The next constraint assures that the shortest path throughjust one of the passages will be selected to serve the allocatedcustomer region in a different half-plane

119870

sum

119896=1

ℎ119894119895119896 = 119897119894119895 forall119894 119895 (25)

424 Capacity Constraint (26) guarantees that each newfacility cannot serve more than its corresponding capacity

119869

sum

119895=1

119908119895 sdot 119905119894119895 le 119862119886119894 forall119894 (26)

The modified objective function with respect to con-straints (10) to (21) looks as followsMin 119885

where 119885 ge 119908119895 sdot 119905119894119895

sdot [

[

([

119870

sum

119896=1

[(minus119909119896+

119886119895 + 119887119895

2) sdot 1198781119895119896

+ (

(119909119896minus 119886119895)2

+ (119909119896minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119895119896

+ (119909119896minus

119886119895 + 119887119895

2) sdot 1198783119895119896

+ (minus119910119896+

119888119895 + 119889119895

2) sdot 1198651119895

+ (

(119910119896minus 119888119895)2

+ (119910119896minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119895

+ (119910119896minus

119888119895 + 119889119895

2) sdot 1198653119895

+10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816]sdot ℎ119894119895119896

]

]

sdot 119897119894119895)

+ ([(minus119909119894 +

119886119895 + 119887119895

2) sdot 1198781119894119895

+ (

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119894119895

+ (119909119894 minus

119886119895 + 119887119895

2) sdot 1198783119894119895

+ (minus119910119894 +

119888119895 + 119889119895

2) sdot 1198651119894119895

+ (

(119910119894 minus 119888119895)2

+ (119910119894 minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119894119895

+ (119910119894 minus

119888119895 + 119889119895

2) sdot 1198653119894119895] sdot (1 minus 119897119894119895))]

(27)


35

4

6

7

810

9

2

1

Figure 2 The illustration of the demand regions and passages

5 A Case Illustration

To examine the practicality and effectiveness of the proposedmodel for CFLAP a case is illustrated for the emergency facil-ity location problem One of the flood disaster managementactivities in the Amol city Mazandaran province northernpart of Iran was selected as a case study In recent yearssevere floods have occurred in the northern part of Iran In1999 floods in this area left 34 dead and 15 missing after morethan a week and presumed dead More than 100 people wereinjured and various areas were damagedwith estimated lossesof 15 million dollars [55] Recently the frequency of floodoccurrence in the Mazandaran province Iran has increasedand an appropriate model is required to find the suitablelocations of relief rooms in order to decrease the evacuationtime of people from the affected areas to a safe place beforeflood occurrence to reduce the possible impact of flooddisasters

The Amol city is located at 36∘231015840N latitude and 52∘201015840Elongitude in northern Iran This city is situated on theHaraz river bank and it is less than 20 kilometers southof the Caspian sea and 10 kilometers north of the Alborzmountain Based on the 2010 census its population wasaround 224000 with 61085 families As shown in Figure 2the Haraz river divides the city into two subplanes andtwo bridges are located along the river to connect the twosubregions with each other With respect to the geographicinformation system (GIS) map of the city it has been dividedinto ten demand areas As we mentioned before the weightsof emergency needs in each demand region are a function ofboth qualitative and quantitative criteria The population ofeach area as a quantitative criterion was obtained from thestatistics center of Iran and is shown in Table 1 the data forquality of road construction were gained from the GIS mapof Amol which was prepared by the Department of Housingand Urban Development of Mazandaran Province Table 2presents the data for the qualitative factor

Table 1 Population data in different regions

119895 Population 119875119895

1 13464 00571322 26139 01109163 13984 00593394 25432 01079165 23916 01014836 8888 00377157 12972 00550448 35072 01488229 38392 016291010 37405 0158722

For the sake of normalizing the quantitative weight 119875119895

of each demand region the population number of each areais divided by the total population For the qualitative factorthe quality of road construction is categorized into 5 groupswhich are high quality (HQ) good quality (GQ) acceptablequality (AQ)minor repair required (MNR) andmajor repairrequired (MJR) Since this factor is inversely related with 119908119895the best quality construction receives the significance of 1 andtheworst quality construction gets themaximumsignificancescore which is equal to 5 The weight 119876119895 of the qualitativefactor is computed similarly as the quantitative criterionSince the significance of both criteria is the same the finalweight of each area is obtained by adding together half of theweights of the qualitative and quantitative factors

The coordinates of ten demand regions with their corre-sponding final weights are tabulated in Table 3 while Table 4gives the coordinates of the passages It is noted that theestablishment of two relief rooms with a maximum capacitypercentage of 0545 and 046 is required to cover all thedemand regions The proposed MINLP model contains 292integer variables and 4 noninteger variables in terms of itscomplexity Hence it has been implemented in the GAMS221 software using the BARON solver in order to identifythe optimal values of 119909lowast

119894and 119910

lowast

119894 BARON was created at the

University of Illinois which implements global optimizationalgorithms of a branch and bound type and solves nonconvexproblems to obtain global solutions [56] Computations wereperformed using a system with a Core 2 duo 21 GHz CPUand 2GB RAM and the computation time for solving themodel was about 11 minutes The time complexity growsexponentially with the increasing number of passages or newfacilities

The optimal locations of the emergency facilities and thevalue of the objective function are reported in Table 5 Inaddition Table 6 shows the optimal corresponding allocationclusters of the two new emergency facilities For a betterunderstanding Figure 3 illustrates the optimum locations ofthe new facilities and their corresponding allocation Basedon the results we recommend that demand regions 1 2 89 and 10 are served by the first new facility and the rest ofthe demand areas are served by the second new facility Theresults will be useful for emergency managers to decrease theevacuation time in affected areas before flood occurrence


Table 2 Data for quality of road construction

119895 HQ () GQ () AQ () MNR () MJR () Score () of each area 119876119895

1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744

Table 3 Parameter values for the demand regions

119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733

Table 4 The coordinates of passages

119896 119909119896

119910119896

1 69 852 101 85

Table 5 Optimal solutions for the locations of relief rooms

1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682

Table 6 Optimal allocation of demand areas to the two newfacilities

(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718

Figure 3 Illustration of the computation results

Table 7 Coordinates of randomly selected areas

119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264

6 Validation of the Model

Model validation is essential for evaluating the proposedmodel to make sure that the chosen new facility locationstruly minimize the maximum value of distance from thestochastic demand regions to the chosen new facility pointswithin the defined area Based on the validation procedureof Hosseinijou and Bashiri [38] ten demand points wererandomly generated within each demand region Then theobjective functions of the recommended emergency facilitylocations were compared with the objective function of theproposed model

Some efforts of validating the model are presented hereTen scenarios were created from five randomly selected areasin Table 3 A random data toolbox in the Minitab softwarewas used for generating random point data in each area Theweight of each point in the scenarios is equal to the weight of


Table 8 Validation results of the proposed model

Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]

times [137 182]119865(119883119894lowast) |Error|

1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023

Table 9 Analysis of validation results for the proposed model

119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387

its allocated area Table 7 presents the randomly selected areasfor validation of the proposedmodel In each of the scenariosthe actual value of the objective function is computed sincethe customer location is a point The maximum capacitypercentage of the two new facilities is set as 0425 and 058and the optimum expected value of the objective function forthe five selected areas is equal to 111 The validation resultsof the proposed model are tabulated in Table 8 The actualvalue of the objective function for each scenario is listed incolumn 7 column 8 shows the error of our model which isthe difference between the actual value in column 7 and theexpected value of the objective function for the five selectedregions

The final analysis of the model validation is reported inTable 9 The first row is related to the average of the objectivefunction values for the ten scenarioswhich are extracted fromcolumn 7 of Table 8 The second row indicates the averageerror between the expected objective function value of theproposedmodel and the obtained average amount in the firstrow The average error percentage of the model is presentedin the last row As can be seen in Table 9 the average errorpercentage is equal to 387 which shows that the proposedCFLAP model is a good estimator of the actual objectivefunction value and its solution is near to optimal Hence theproposed stochastic model is valid

In order to highlight the advantages of the proposedmodel as compared to the most relevant existing literaturea comparison has been made with the model developed byCanbolat and Wesolowsky [47] The results are summarizedin Table 10

As shown thework ofCanbolat andWesolowsky [47]wasdedicated to a single facility location problem where deter-ministic demand coordinates were utilized In addition theirmodel was tested using a numerical example In contrastthis study has developed a stochastic model that considersprobabilistic demand locations for a multifacility location-allocation problem The applicability of the proposed modelhas been evaluated using an actual case study thus justifyingits usefulness in the real-world context

7 Conclusions

In this study we have considered the planar center location-allocation problem for locating some emergency facilities inthe presence of a fixed line barrier in a region Although somestudies have been done on facility location problems withbarriers little attention has been devoted to the probabilisticformulation of these problemswhere demand coordinates aredistributed according to a bivariate probability distributionIn this paper we have proposed a MINLP model for CFLAPby considering probabilistic demand regions and a fixed linebarrier simultaneously thusmaking themodelmore realisticThis study aims at finding the optimum locations of two reliefrooms and their corresponding allocation which leads tominimizing the maximum expected weighted distance fromthe relief rooms to all the demand regions A real-world casestudy has been carried out in order to show the effectivenessof the proposed model and the problem has been solvedby the GAMS 221 software in order to find the optimumsolution

As an approximation uniform distributions have beenused to represent demand region locations Thus one pos-sible area for future research is to formulate the problem withthe exact probability distributions based on goodness-of-fittests In addition considering the minimax regret objectivefunction for hazardous location problems is another possibleextension that can be studied


Table 10 Comparison results

Canbolat and Wesolowsky [47] This studyDeterministic coordinates of demand locations Stochastic coordinates of demand locationsSingle facility location Multifacility locationNumerical example Real-world application

Conflict of Interests

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

References

[1] H W Hamacher and S Nickel ldquoRestricted planar locationproblems and applicationsrdquoNaval Research Logistics vol 42 no6 pp 967ndash992 1995

[2] K Klamroth ldquoPlanar weber location problems with line barri-ersrdquo Optimization vol 49 no 5-6 pp 517ndash527 2001

[3] A Krishna G S Dangayach and R Jain ldquoService recoveryliterature review and research issuesrdquo Journal of Service ScienceResearch vol 3 no 1 pp 71ndash121 2011

[4] J J Sylvester ldquoA question in the geometry of situationrdquo Quar-terly Journal of Pure and Applied Mathematics vol 1 no 1 pp79ndash80 1857

[5] J Elzinga and D W Hearn ldquoGeometrical solutions for someminimax location problemsrdquo Transportation Science vol 6 no4 pp 379ndash394 1972

[6] R L Francis L F McGinnis Jr and J A White FacilityLayout and Location An Analytical Approach Prentice-HallEnglewood Cliffs NJ USA 2nd edition 1992

[7] A Mehrez Z Sinuany-Stern T Arad-Geva and S BinyaminldquoOn the implementation of quantitative facility locationmodelsthe case of a hospital in a rural regionrdquo Journal of theOperationalResearch Society vol 47 no 5 pp 612ndash625 1996

[8] M S Daskin S M Hesse and C S Revelle ldquo120572-Reliable p-minimax regret a new model for strategic facility locationmodelingrdquo Location Science vol 5 no 4 pp 227ndash246 1997

[9] F Hurtado V Sacristan and G Toussaint ldquoSome constrainedminimax andmaximin location problemsrdquo Studies in LocationalAnalysis no 15 pp 17ndash35 2000

[10] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

[11] N C Simpson and P G Hancock ldquoFifty years of operationalresearch and emergency responserdquo Journal of the OperationalResearch Society vol 60 no 1 pp S126ndashS139 2009

[12] S Ichoua ldquoRelief distribution networks design and opera-tionsrdquo in Supply Chain Optimization Design and ManagementAdvances and Intelligent Methods I Minis V Zeimpekis GDounias and N Ampazis Eds pp 171ndash186 IGI GlobalHershey Pa USA 2011

[13] D Shishebori and M S Jabalameli ldquoImproving the efficiencyof medical services systems a new integrated mathematicalmodeling approachrdquo Mathematical Problems in Engineeringvol 2013 Article ID 649397 13 pages 2013

[14] A M Caunhye X Nie and S Pokharel ldquoOptimization modelsin emergency logistics a literature reviewrdquo Socio-EconomicPlanning Sciences vol 46 no 1 pp 4ndash13 2012

[15] G Galindo and R Batta ldquoReview of recent developments inORMS research in disaster operationsmanagementrdquo EuropeanJournal of Operational Research vol 230 no 2 pp 201ndash211 2013

[16] H D Sherali T B Carter and A G Hobeika ldquoA location-allocation model and algorithm for evacuation planning underhurricaneflood conditionsrdquo Transportation Research Part Bvol 25 no 6 pp 439ndash452 1991

[17] M-S Chang Y-L Tseng and J-W Chen ldquoA scenario planningapproach for the flood emergency logistics preparation problemunder uncertaintyrdquo Transportation Research Part E vol 43 no6 pp 737ndash754 2007

[18] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research Part E vol 43 no 6 pp 687ndash709 2007

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[21] C G Rawls and M A Turnquist ldquoPre-positioning of emer-gency supplies for disaster responserdquo Transportation ResearchPart B vol 44 no 4 pp 521ndash534 2010

[22] P Yi S K George J A Paul and L Lin ldquoHospital capacityplanning for disaster emergency managementrdquo Socio-EconomicPlanning Sciences vol 44 no 3 pp 151ndash160 2010

[23] R Halper and S Raghavan ldquoThe mobile facility routing prob-lemrdquo Transportation Science vol 45 no 3 pp 413ndash434 2011

[24] L Wu Q Tan and Y Zhang ldquoDelivery time reliability modelof logistic networkrdquoMathematical Problems in Engineering vol2013 Article ID 879472 5 pages 2013

[25] J Xu J Gang and X Lei ldquoHazmats transportation networkdesign model with emergency response under complex fuzzyenvironmentrdquoMathematical Problems in Engineering vol 2013Article ID 517372 22 pages 2013

[26] S Duran M A Gutierrez and P Keskinocak ldquoPre-positioningof emergency items for CARE internationalrdquo Interfaces vol 41no 3 pp 223ndash237 2011

[27] W Klibi S Ichoua and A Martel ldquoPrepositioning emer-gency supplies to support disaster relief a stochastic program-ming approachrdquo Centre interuniversitaire de recherche sur lesreseaux drsquoentreprise la logistique et le transport (CIRRELT)Canada 2013

[28] L Cooper ldquoA random locational equilibrium problemrdquo Journalof Regional Sciences vol 14 no 1 pp 47ndash54 1974

[29] I N Katz and L Cooper ldquoAn always-convergent numericalscheme for a random locational equilibrium problemrdquo SIAMJournal of Numerical Analysis vol 11 no 4 pp 683ndash692 1974

[30] G O Wesolowsky ldquoThe weber problem with rectangulardistances and randomly distributed destinationsrdquo Journal ofRegional Sciences vol 17 no 1 pp 53ndash60 1977


[31] R Carbone and A Mehrez ldquoThe single facility minimaxdistance problem under stochastic location of demandrdquo Man-agement Science vol 26 no 1 pp 113ndash115 1980

[32] O Berman J Wang Z Drezner and G O Wesolowsky ldquoAprobabilistic minimax location problem on the planerdquo Annalsof Operations Research vol 122 no 1ndash4 pp 59ndash70 2003

[33] I Averbakh and S Bereg ldquoFacility location problems withuncertainty on the planerdquo Discrete Optimization vol 2 no 1pp 3ndash34 2005

[34] A Foul ldquoA 1-center problem on the plane with uniformlydistributed demand pointsrdquoOperations Research Letters vol 34no 3 pp 264ndash268 2006

[35] B Pelegrın J Fernandez and B Toth ldquoThe 1-center problemin the plane with independent random weightsrdquo Computers ampOperations Research vol 35 no 3 pp 737ndash749 2008

[36] E Durmaz N Aras and I K Altınel ldquoDiscrete approximationheuristics for the capacitated continuous locationmdashallocationproblem with probabilistic customer locationsrdquo Computers ampOperations Research vol 36 no 7 pp 2139ndash2148 2009

[37] M S Canbolat and M Von Massow ldquoLocating emergencyfacilities with random demand for risk minimizationrdquo ExpertSystems with Applications vol 38 no 8 pp 10099ndash10106 2011

[38] SAHosseinijou andMBashiri ldquoStochasticmodels for transferpoint location problemrdquo International Journal of AdvancedManufacturing Technology vol 58 no 1ndash4 pp 211ndash225 2012

[39] I N Katz and L Cooper ldquoFacility location in the presenceof forbidden regions I formulation and the case of Euclideandistance with one forbidden circlerdquo European Journal of Opera-tional Research vol 6 no 2 pp 166ndash173 1981

[40] KKlamroth andMMWiecek ldquoAbi-objectivemedian locationproblem with a line barrierrdquo Operations Research vol 50 no 4pp 670ndash679 2002

[41] Y P Aneja andM Parlar ldquoAlgorithms for weber facility locationin the presence of forbidden regions andor barriers to travelrdquoLocation Science vol 3 no 2 pp 134ndash134 1995

[42] S E Butt and T M Cavalier ldquoAn efficient algorithm for facilitylocation in the presence of forbidden regionsrdquo European Journalof Operational Research vol 90 no 1 pp 56ndash70 1996

[43] R G McGarvey and T M Cavalier ldquoA global optimal approachto facility location in the presence of forbidden regionsrdquoComputers and Industrial Engineering vol 45 no 1 pp 1ndash152003

[44] K Klamroth ldquoAlgebraic properties of location problems withone circular barrierrdquo European Journal of Operational Researchvol 154 no 1 pp 20ndash35 2004

[45] M Bischoff and K Klamroth ldquoAn efficient solution methodfor weber problems with barriers based on genetic algorithmsrdquoEuropean Journal of Operational Research vol 177 no 1 pp 22ndash41 2007

[46] M Bischoff T Fleischmann and K Klamroth ldquoThe multi-facility location-allocation problem with polyhedral barriersrdquoComputers amp Operations Research vol 36 no 5 pp 1376ndash13922009

[47] M S Canbolat and G O Wesolowsky ldquoA planar single facilitylocation andborder crossing problemrdquoComputersampOperationsResearch vol 39 no 12 pp 3156ndash3165 2012

[48] S Huang R Batta K Klamroth and R Nagi ldquoThe K-connection location problem in a planerdquo Annals of OperationsResearch vol 136 no 1 pp 193ndash209 2005

[49] P Nandikonda R Batta and R Nagi ldquoLocating a 1-center on aManhattan plane with ldquoarbitrarilyrdquo shaped barriersrdquo Annals ofOperations Research vol 123 no 1ndash4 pp 157ndash172 2003

[50] P M Dearing K Klamroth and R Segars Jr ldquoPlanar locationproblems with block distance and barriersrdquo Annals of Opera-tions Research vol 136 no 1 pp 117ndash143 2005

[51] A Sarkar R Batta and R Nagi ldquoPlacing a finite size facilitywith a center objective on a rectangular plane with barriersrdquoEuropean Journal of Operational Research vol 179 no 3 pp1160ndash1176 2007

[52] L Friebszlig K Klamroth and M Sprau ldquoA wavefront approachto center location problems with barriersrdquoAnnals of OperationsResearch vol 136 no 1 pp 35ndash48 2005

[53] M S Canbolat and G OWesolowsky ldquoThe rectilinear distanceweber problem in the presence of a probabilistic line barrierrdquoEuropean Journal of Operational Research vol 202 no 1 pp114ndash121 2010

[54] M S Canbolat and G O Wesolowsky ldquoOn the use of theVarignon frame for single facility weber problems in thepresence of convex barriersrdquo European Journal of OperationalResearch vol 217 no 2 pp 241ndash247 2012

[55] A R Gharagozlou A R A Ardalan andM H Diarjan ldquoEnvi-ronmental planning for disaster management by using GIS(a case study about flood in Mazandaran)rdquo The InternationalArchives of the Photogrammetry Remote Sensing and SpatialInformation Sciences vol 37 B8 pp 323ndash326 2008

[56] N Sahinidis and M Tawarmalani ldquoBaronrdquo in GAMSmdashTheSolver Manuals pp 19ndash32 GAMS Development CorporationWashington DC USA 2008

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

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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


the regional probability distribution of customers due to theuncertainties of events and partial information and dataTherefore a more applicable model which provides bettersolutions would be beneficial for overcoming this weakness

To the best of our knowledge few research activitieswhich consider barriers in emergency facility location prob-lems have been accomplished due to the computationalcomplexities associated with these problems Moreover fartoo little attention has been paid to developing a practicalmathematical model of emergency facility location thatconsiders probabilistic customer regions and a line barrierwith border crossings simultaneously In this research inorder to address the abovementioned problems we developa mixed integer nonlinear programming (MINLP) model ofa capacitated facility location-allocation problem (CFLAP)in an area in which a fixed line barrier and probabilisticcustomer regions are taken into account

The rest of this paper is organized as follows Section 2provides a comprehensive literature review on related topicsSection 3 gives an explanation of the problem and Section 4presents the proposed mathematical model In order to showthe applicability of the proposed model a suitable casestudy is illustrated in Section 5 and the model is validatedin Section 6 Finally Section 7 presents some concludingremarks and outlines a number of future research directions

2 Literature Review

21 Emergency Facility Location Problems The emergencyfacility location models are categorized into three differenttypes P-median P-center and maximal covering locationproblems Since the objective of this research is to design aneffective response strategy for disaster management organi-zations in order to reduce casualties a planar center locationproblem (PCLP) would best suit our purpose Actuallywhenever the average cost of servicing the customers is lessimportant than ensuring that no customer receives poorquality of services the minimax location problem is typicallyproposed [3] The procedure of the PCLP model which isin fact a minimax problem looks for the center in order tominimize the maximum service time cost or loss Researchin this area startedwith Sylvester [4] whichwas the first paperthat considered the Weber problem with the PCLP modelElzinga and Hearn [5] then introduced the Euclidean centerproblem with equal weights and offered an efficient solutiontechnique for their proposed model Francis et al [6] madeit possible to stress the importance of different customers byassigning unequal weights to them Mehrez et al [7] applieda single objective optimization to determine the location ofa new hospital The solutions of this problem were generatedby a qualitative approach based on the evaluation of expertsrsquojudgments After that Daskin et al [8] proposed an error-bound driven demand point aggregation for the minimaxproblemwith rectilinear distance Hurtado et al [9] came outwith some constraints on the minimax problem and used atime algorithm to solve it

In the last two decades the field of disaster manage-ment has gained increasing attention Many authors have

conducted several studies on predisaster preparedness andpostdisaster responsiveness including Altay and Green III[10] Simpson and Hancock [11] Ichoua [12] and Shisheboriand Jabalameli [13] Caunhye et al [14] andGalindo and Batta[15] performed an extensive review of optimization modelsin emergency logistics for disaster operations managementSherali et al [16] proposed a capacitated facility location-allocation model in order to minimize the congested evac-uation time by finding the optimal locations of evacuationshelters under hurricane or flood conditions Chang et al [17]studied the problem of locating relief rooms and allocatingrelief resources for flood emergency preparation They pro-posed two stochastic models under possible flood scenariosin which the demand locations are uncertain and depend onthe level of flood in different scenarios In addition Sheu[18] presented an earthquake relief model that forecasts thedemand of affected regions and coordinates the provisionof relief supplies Mete and Zabinsky [19] dealt with anoptimization approach for locating and distributing medicalsupplies under demand and transportation uncertainties fordisaster situations Some other recent relevant studies on thisissue include Jiang et al [20] Rawls and Turnquist [21] Yi etal [22] Halper and Raghavan [23] Wu et al [24] Xu et al[25] and Duran et al [26]

22 Facility Location Problems under Uncertainty Whenaddressing the strategic decision of establishing permanentfacilities to prepare for future disasters over a long termplanning horizon information is uncertain due to the timelag so considering the location of demands as a known pointwith certainty leads to poor model performance [27] Thefirst research in the probabilistic Weber problem was doneby Cooper [28] who focused on a single facility locationproblem After that Katz and Cooper [29] approximated theexpected distance of transportation via Euclidean distancein which the customer locations were assumed to have anormal distribution They proposed an algorithm to min-imize the total expected distance between the new facilityand existing facilities Later on for the case of rectilineardistance Wesolowsky [30] considered three different prob-ability distributions of bivariate normal bivariate symmetricexponential and bivariate uniform for customer locations insolving the probabilistic Weber problem

Although some studies have recently been done for thecase of multifacility minimax location problems much lessattention has been given to the probabilistic formulation ofminimax problems Stochastic demand locations were firstconsidered by Carbone and Mehrez [31] for the minimaxlocation problem where the locations of demand pointswere normally distributed with predetermined means andvariances An emergency location problem with a weightedminimax objective function has been presented by Bermanet al [32] in which the weights associated with each demandpoint were not given and each of themwas assumed to have auniform distributionThey proved that the objective functionis convex for parameters of the uniform distribution andthus the model can be solved by using standard optimizationtechniques The minimax regret location problem has been










Let 119871 be a line and 119875119896 = (119909119896 119910119896)120598119871 | 119896120598119870 = 1 119870








Min119865 (119883119894) = Max 119908119895 sdot 119889119861

119901(119883119894 119883119895)

119894 = 1 119868 119895 = 1 119869

(1)






minimized


119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

= Max 119908119895 sdot 119864 [119889119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

119894 = 1 119868 119895 = 1 119869

(2)


Y

X

BL

Pk

1205791

1205792





119864 [119889 (119883119894 119883119895)] = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

= 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] + 119864 [

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

(3)


119867119894 (119909) = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816sdot 119891119906 (119880119895) 119889119906

=1

119887119895 minus 119886119895

int

119887119895

119886119895

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816119889119906

(4)

119866119894 (119910) = 119864 [10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816sdot 119891V (119881119895) 119889V

=1

119889119895 minus 119888119895

int

119889119895

119888119895

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816119889V

(5)


119867119894 (119909) =

minus119909119894 +

119886119895 + 119887119895

2if 119909119894 le 119886119895

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

if 119886119895 le 119909119894 le 119887119895

119909119894 minus

119886119895 + 119887119895

2if 119909119894 ge 119887119895

(6)



Min 119864119865 (119883119894) = Max 119908119895 sdot [119867119894 (119909) + 119866119894 (119910)]

119894 = 1 119868 119895 = 1 119869

(7)


Min 119885


119894 = 1 119868 119895 = 1 119869

(8)






Notations




















Min 119885


119870

sum

119896=1

[119879119896 (119909119896)+119877119896 (119910

119896) +

10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816

+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816] sdot ℎ119894119895119896) sdot 119897119894119895]

+ [[119867119894 (119909) + 119866119894 (119910)] sdot (1 minus 119897119894119895)] ]

(9)


119896) and 119877119896(119910


119896|] and



119896 respectively



119896) and119877119896(119910

119896) can






1198781119894119895 + 1198782119894119895 + 1198783119894119895 = 1 forall119894 119895 (12)



1198651119894119895 + 1198652119894119895 + 1198653119894119895 = 1 forall119894 119895 (15)


119896minus 119886119895 forall119895 119896 (16)


119896minus 119887119895 forall119895 119896 (17)

1198781119895119896 + 1198782119895119896 + 1198783119895119896 = 1 forall119895 119896 (18)



1198651119895 + 1198652119895 + 1198653119895 = 1 forall119895 (21)


119896) and 119877119896(119910





119897119894119895 =10038161003816100381610038161003816119902119895 minus 119892119894

10038161003816100381610038161003816 forall119894 119895 (23)



119868

sum

119894=1

119905119894119895 = 1 forall119895 (24)


119870

sum

119896=1

ℎ119894119895119896 = 119897119894119895 forall119894 119895 (25)


119869

sum

119895=1

119908119895 sdot 119905119894119895 le 119862119886119894 forall119894 (26)


where 119885 ge 119908119895 sdot 119905119894119895

sdot [

[

([

119870

sum

119896=1

[(minus119909119896+

119886119895 + 119887119895

2) sdot 1198781119895119896

+ (

(119909119896minus 119886119895)2

+ (119909119896minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119895119896

+ (119909119896minus

119886119895 + 119887119895

2) sdot 1198783119895119896

+ (minus119910119896+

119888119895 + 119889119895

2) sdot 1198651119895

+ (

(119910119896minus 119888119895)2

+ (119910119896minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119895

+ (119910119896minus

119888119895 + 119889119895

2) sdot 1198653119895

+10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816]sdot ℎ119894119895119896

]

]

sdot 119897119894119895)

+ ([(minus119909119894 +

119886119895 + 119887119895

2) sdot 1198781119894119895

+ (

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119894119895

+ (119909119894 minus

119886119895 + 119887119895

2) sdot 1198783119894119895

+ (minus119910119894 +

119888119895 + 119889119895

2) sdot 1198651119894119895

+ (

(119910119894 minus 119888119895)2

+ (119910119894 minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119894119895

+ (119910119894 minus

119888119895 + 119889119895

2) sdot 1198653119894119895] sdot (1 minus 119897119894119895))]

(27)


35

4

6

7

810

9

2

1






119895 Population 119875119895

1 13464 00571322 26139 01109163 13984 00593394 25432 01079165 23916 01014836 8888 00377157 12972 00550448 35072 01488229 38392 016291010 37405 0158722




119894and 119910

lowast







1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744


119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733


119896 119909119896

119910119896

1 69 852 101 85


1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682


(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718



119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264






Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Let 119871 be a line and 119875119896 = (119909119896 119910119896)120598119871 | 119896120598119870 = 1 119870








Min119865 (119883119894) = Max 119908119895 sdot 119889119861

119901(119883119894 119883119895)

119894 = 1 119868 119895 = 1 119869

(1)






minimized


119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

= Max 119908119895 sdot 119864 [119889119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

119894 = 1 119868 119895 = 1 119869

(2)


Y

X

BL

Pk

1205791

1205792





119864 [119889 (119883119894 119883119895)] = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

= 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] + 119864 [

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

(3)


119867119894 (119909) = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816sdot 119891119906 (119880119895) 119889119906

=1

119887119895 minus 119886119895

int

119887119895

119886119895

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816119889119906

(4)

119866119894 (119910) = 119864 [10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816sdot 119891V (119881119895) 119889V

=1

119889119895 minus 119888119895

int

119889119895

119888119895

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816119889V

(5)


119867119894 (119909) =

minus119909119894 +

119886119895 + 119887119895

2if 119909119894 le 119886119895

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

if 119886119895 le 119909119894 le 119887119895

119909119894 minus

119886119895 + 119887119895

2if 119909119894 ge 119887119895

(6)



Min 119864119865 (119883119894) = Max 119908119895 sdot [119867119894 (119909) + 119866119894 (119910)]

119894 = 1 119868 119895 = 1 119869

(7)


Min 119885


119894 = 1 119868 119895 = 1 119869

(8)






Notations




















Min 119885


119870

sum

119896=1

[119879119896 (119909119896)+119877119896 (119910

119896) +

10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816

+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816] sdot ℎ119894119895119896) sdot 119897119894119895]

+ [[119867119894 (119909) + 119866119894 (119910)] sdot (1 minus 119897119894119895)] ]

(9)


119896) and 119877119896(119910


119896|] and



119896 respectively



119896) and119877119896(119910

119896) can






1198781119894119895 + 1198782119894119895 + 1198783119894119895 = 1 forall119894 119895 (12)



1198651119894119895 + 1198652119894119895 + 1198653119894119895 = 1 forall119894 119895 (15)


119896minus 119886119895 forall119895 119896 (16)


119896minus 119887119895 forall119895 119896 (17)

1198781119895119896 + 1198782119895119896 + 1198783119895119896 = 1 forall119895 119896 (18)



1198651119895 + 1198652119895 + 1198653119895 = 1 forall119895 (21)


119896) and 119877119896(119910





119897119894119895 =10038161003816100381610038161003816119902119895 minus 119892119894

10038161003816100381610038161003816 forall119894 119895 (23)



119868

sum

119894=1

119905119894119895 = 1 forall119895 (24)


119870

sum

119896=1

ℎ119894119895119896 = 119897119894119895 forall119894 119895 (25)


119869

sum

119895=1

119908119895 sdot 119905119894119895 le 119862119886119894 forall119894 (26)


where 119885 ge 119908119895 sdot 119905119894119895

sdot [

[

([

119870

sum

119896=1

[(minus119909119896+

119886119895 + 119887119895

2) sdot 1198781119895119896

+ (

(119909119896minus 119886119895)2

+ (119909119896minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119895119896

+ (119909119896minus

119886119895 + 119887119895

2) sdot 1198783119895119896

+ (minus119910119896+

119888119895 + 119889119895

2) sdot 1198651119895

+ (

(119910119896minus 119888119895)2

+ (119910119896minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119895

+ (119910119896minus

119888119895 + 119889119895

2) sdot 1198653119895

+10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816]sdot ℎ119894119895119896

]

]

sdot 119897119894119895)

+ ([(minus119909119894 +

119886119895 + 119887119895

2) sdot 1198781119894119895

+ (

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119894119895

+ (119909119894 minus

119886119895 + 119887119895

2) sdot 1198783119894119895

+ (minus119910119894 +

119888119895 + 119889119895

2) sdot 1198651119894119895

+ (

(119910119894 minus 119888119895)2

+ (119910119894 minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119894119895

+ (119910119894 minus

119888119895 + 119889119895

2) sdot 1198653119894119895] sdot (1 minus 119897119894119895))]

(27)


35

4

6

7

810

9

2

1






119895 Population 119875119895

1 13464 00571322 26139 01109163 13984 00593394 25432 01079165 23916 01014836 8888 00377157 12972 00550448 35072 01488229 38392 016291010 37405 0158722




119894and 119910

lowast







1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744


119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733


119896 119909119896

119910119896

1 69 852 101 85


1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682


(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718



119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264






Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Min119865 (119883119894) = Max 119908119895 sdot 119889119861

119901(119883119894 119883119895)

119894 = 1 119868 119895 = 1 119869

(1)






minimized


119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

= Max 119908119895 sdot 119864 [119889119861

1((119909119894lowast 119910119894lowast) (119880119895 119881119895))]

119894 = 1 119868 119895 = 1 119869

(2)


Y

X

BL

Pk

1205791

1205792





119864 [119889 (119883119894 119883119895)] = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

= 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] + 119864 [

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816]

(3)


119867119894 (119909) = 119864 [10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816sdot 119891119906 (119880119895) 119889119906

=1

119887119895 minus 119886119895

int

119887119895

119886119895

10038161003816100381610038161003816119880119895 minus 119909119894

10038161003816100381610038161003816119889119906

(4)

119866119894 (119910) = 119864 [10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816] = int

infin

minusinfin

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816sdot 119891V (119881119895) 119889V

=1

119889119895 minus 119888119895

int

119889119895

119888119895

10038161003816100381610038161003816119881119895 minus 119910119894

10038161003816100381610038161003816119889V

(5)


119867119894 (119909) =

minus119909119894 +

119886119895 + 119887119895

2if 119909119894 le 119886119895

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

if 119886119895 le 119909119894 le 119887119895

119909119894 minus

119886119895 + 119887119895

2if 119909119894 ge 119887119895

(6)



Min 119864119865 (119883119894) = Max 119908119895 sdot [119867119894 (119909) + 119866119894 (119910)]

119894 = 1 119868 119895 = 1 119869

(7)


Min 119885


119894 = 1 119868 119895 = 1 119869

(8)






Notations




















Min 119885


119870

sum

119896=1

[119879119896 (119909119896)+119877119896 (119910

119896) +

10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816

+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816] sdot ℎ119894119895119896) sdot 119897119894119895]

+ [[119867119894 (119909) + 119866119894 (119910)] sdot (1 minus 119897119894119895)] ]

(9)


119896) and 119877119896(119910


119896|] and



119896 respectively



119896) and119877119896(119910

119896) can






1198781119894119895 + 1198782119894119895 + 1198783119894119895 = 1 forall119894 119895 (12)



1198651119894119895 + 1198652119894119895 + 1198653119894119895 = 1 forall119894 119895 (15)


119896minus 119886119895 forall119895 119896 (16)


119896minus 119887119895 forall119895 119896 (17)

1198781119895119896 + 1198782119895119896 + 1198783119895119896 = 1 forall119895 119896 (18)



1198651119895 + 1198652119895 + 1198653119895 = 1 forall119895 (21)


119896) and 119877119896(119910





119897119894119895 =10038161003816100381610038161003816119902119895 minus 119892119894

10038161003816100381610038161003816 forall119894 119895 (23)



119868

sum

119894=1

119905119894119895 = 1 forall119895 (24)


119870

sum

119896=1

ℎ119894119895119896 = 119897119894119895 forall119894 119895 (25)


119869

sum

119895=1

119908119895 sdot 119905119894119895 le 119862119886119894 forall119894 (26)


where 119885 ge 119908119895 sdot 119905119894119895

sdot [

[

([

119870

sum

119896=1

[(minus119909119896+

119886119895 + 119887119895

2) sdot 1198781119895119896

+ (

(119909119896minus 119886119895)2

+ (119909119896minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119895119896

+ (119909119896minus

119886119895 + 119887119895

2) sdot 1198783119895119896

+ (minus119910119896+

119888119895 + 119889119895

2) sdot 1198651119895

+ (

(119910119896minus 119888119895)2

+ (119910119896minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119895

+ (119910119896minus

119888119895 + 119889119895

2) sdot 1198653119895

+10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816]sdot ℎ119894119895119896

]

]

sdot 119897119894119895)

+ ([(minus119909119894 +

119886119895 + 119887119895

2) sdot 1198781119894119895

+ (

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119894119895

+ (119909119894 minus

119886119895 + 119887119895

2) sdot 1198783119894119895

+ (minus119910119894 +

119888119895 + 119889119895

2) sdot 1198651119894119895

+ (

(119910119894 minus 119888119895)2

+ (119910119894 minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119894119895

+ (119910119894 minus

119888119895 + 119889119895

2) sdot 1198653119894119895] sdot (1 minus 119897119894119895))]

(27)


35

4

6

7

810

9

2

1






119895 Population 119875119895

1 13464 00571322 26139 01109163 13984 00593394 25432 01079165 23916 01014836 8888 00377157 12972 00550448 35072 01488229 38392 016291010 37405 0158722




119894and 119910

lowast







1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744


119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733


119896 119909119896

119910119896

1 69 852 101 85


1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682


(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718



119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264






Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Min 119864119865 (119883119894) = Max 119908119895 sdot [119867119894 (119909) + 119866119894 (119910)]

119894 = 1 119868 119895 = 1 119869

(7)


Min 119885


119894 = 1 119868 119895 = 1 119869

(8)






Notations




















Min 119885


119870

sum

119896=1

[119879119896 (119909119896)+119877119896 (119910

119896) +

10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816

+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816] sdot ℎ119894119895119896) sdot 119897119894119895]

+ [[119867119894 (119909) + 119866119894 (119910)] sdot (1 minus 119897119894119895)] ]

(9)


119896) and 119877119896(119910


119896|] and



119896 respectively



119896) and119877119896(119910

119896) can






1198781119894119895 + 1198782119894119895 + 1198783119894119895 = 1 forall119894 119895 (12)



1198651119894119895 + 1198652119894119895 + 1198653119894119895 = 1 forall119894 119895 (15)


119896minus 119886119895 forall119895 119896 (16)


119896minus 119887119895 forall119895 119896 (17)

1198781119895119896 + 1198782119895119896 + 1198783119895119896 = 1 forall119895 119896 (18)



1198651119895 + 1198652119895 + 1198653119895 = 1 forall119895 (21)


119896) and 119877119896(119910





119897119894119895 =10038161003816100381610038161003816119902119895 minus 119892119894

10038161003816100381610038161003816 forall119894 119895 (23)



119868

sum

119894=1

119905119894119895 = 1 forall119895 (24)


119870

sum

119896=1

ℎ119894119895119896 = 119897119894119895 forall119894 119895 (25)


119869

sum

119895=1

119908119895 sdot 119905119894119895 le 119862119886119894 forall119894 (26)


where 119885 ge 119908119895 sdot 119905119894119895

sdot [

[

([

119870

sum

119896=1

[(minus119909119896+

119886119895 + 119887119895

2) sdot 1198781119895119896

+ (

(119909119896minus 119886119895)2

+ (119909119896minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119895119896

+ (119909119896minus

119886119895 + 119887119895

2) sdot 1198783119895119896

+ (minus119910119896+

119888119895 + 119889119895

2) sdot 1198651119895

+ (

(119910119896minus 119888119895)2

+ (119910119896minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119895

+ (119910119896minus

119888119895 + 119889119895

2) sdot 1198653119895

+10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816]sdot ℎ119894119895119896

]

]

sdot 119897119894119895)

+ ([(minus119909119894 +

119886119895 + 119887119895

2) sdot 1198781119894119895

+ (

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119894119895

+ (119909119894 minus

119886119895 + 119887119895

2) sdot 1198783119894119895

+ (minus119910119894 +

119888119895 + 119889119895

2) sdot 1198651119894119895

+ (

(119910119894 minus 119888119895)2

+ (119910119894 minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119894119895

+ (119910119894 minus

119888119895 + 119889119895

2) sdot 1198653119894119895] sdot (1 minus 119897119894119895))]

(27)


35

4

6

7

810

9

2

1






119895 Population 119875119895

1 13464 00571322 26139 01109163 13984 00593394 25432 01079165 23916 01014836 8888 00377157 12972 00550448 35072 01488229 38392 016291010 37405 0158722




119894and 119910

lowast







1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744


119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733


119896 119909119896

119910119896

1 69 852 101 85


1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682


(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718



119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264






Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1198781119894119895 + 1198782119894119895 + 1198783119894119895 = 1 forall119894 119895 (12)



1198651119894119895 + 1198652119894119895 + 1198653119894119895 = 1 forall119894 119895 (15)


119896minus 119886119895 forall119895 119896 (16)


119896minus 119887119895 forall119895 119896 (17)

1198781119895119896 + 1198782119895119896 + 1198783119895119896 = 1 forall119895 119896 (18)



1198651119895 + 1198652119895 + 1198653119895 = 1 forall119895 (21)


119896) and 119877119896(119910





119897119894119895 =10038161003816100381610038161003816119902119895 minus 119892119894

10038161003816100381610038161003816 forall119894 119895 (23)



119868

sum

119894=1

119905119894119895 = 1 forall119895 (24)


119870

sum

119896=1

ℎ119894119895119896 = 119897119894119895 forall119894 119895 (25)


119869

sum

119895=1

119908119895 sdot 119905119894119895 le 119862119886119894 forall119894 (26)


where 119885 ge 119908119895 sdot 119905119894119895

sdot [

[

([

119870

sum

119896=1

[(minus119909119896+

119886119895 + 119887119895

2) sdot 1198781119895119896

+ (

(119909119896minus 119886119895)2

+ (119909119896minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119895119896

+ (119909119896minus

119886119895 + 119887119895

2) sdot 1198783119895119896

+ (minus119910119896+

119888119895 + 119889119895

2) sdot 1198651119895

+ (

(119910119896minus 119888119895)2

+ (119910119896minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119895

+ (119910119896minus

119888119895 + 119889119895

2) sdot 1198653119895

+10038161003816100381610038161003816119909119896minus 119909119894

10038161003816100381610038161003816+10038161003816100381610038161003816119910119896minus 119910119894

10038161003816100381610038161003816]sdot ℎ119894119895119896

]

]

sdot 119897119894119895)

+ ([(minus119909119894 +

119886119895 + 119887119895

2) sdot 1198781119894119895

+ (

(119909119894 minus 119886119895)2

+ (119909119894 minus 119887119895)2

2 sdot (119887119895 minus 119886119895)

) sdot 1198782119894119895

+ (119909119894 minus

119886119895 + 119887119895

2) sdot 1198783119894119895

+ (minus119910119894 +

119888119895 + 119889119895

2) sdot 1198651119894119895

+ (

(119910119894 minus 119888119895)2

+ (119910119894 minus 119889119895)2

2 sdot (119889119895 minus 119888119895)

) sdot 1198652119894119895

+ (119910119894 minus

119888119895 + 119889119895

2) sdot 1198653119894119895] sdot (1 minus 119897119894119895))]

(27)


35

4

6

7

810

9

2

1






119895 Population 119875119895

1 13464 00571322 26139 01109163 13984 00593394 25432 01079165 23916 01014836 8888 00377157 12972 00550448 35072 01488229 38392 016291010 37405 0158722




119894and 119910

lowast







1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744


119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733


119896 119909119896

119910119896

1 69 852 101 85


1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682


(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718



119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264






Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










35

4

6

7

810

9

2

1






119895 Population 119875119895

1 13464 00571322 26139 01109163 13984 00593394 25432 01079165 23916 01014836 8888 00377157 12972 00550448 35072 01488229 38392 016291010 37405 0158722




119894and 119910

lowast







1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744


119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733


119896 119909119896

119910119896

1 69 852 101 85


1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682


(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718



119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264






Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 038 4091 375 2019 0 27546 00895122 165 3976 3576 1153 941 28162 00915143 0 1176 2843 2549 3431 38232 01242364 137 665 2329 3933 2935 38861 01262805 296 20 2889 2444 237 34589 01123986 072 1115 4066 2475 227 3575 01161717 278 5945 3087 645 0 24009 00780188 597 681 2071 336 137 22459 00729819 154 7051 2615 179 0 22817 007414510 1579 1404 1404 1053 45 35311 0114744


119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 00733222 [114 145] [89 15] 01012153 [92 182] [59 8] 009178754 [92 154] [38 58] 01170985 [47 91] [47 8] 010694056 [47 91] [03 46] 00769437 [19 84] [89 114] 00665318 [05 84] [115 136] 011090159 [62 113] [137 182] 0118527510 [86 113] [94 136] 0136733


119896 119909119896

119910119896

1 69 852 101 85


1199091lowast

1199101lowast

1199092lowast

1199102lowast

119865(119883119894lowast)

98 13312 86 6139 0682


(119894 119895) 119905119894119895 (119894 119895) 119905119894119895

1 1 1 2 1 01 2 1 2 2 01 3 0 2 3 11 4 0 2 4 11 5 0 2 5 11 6 0 2 6 11 7 0 2 7 11 8 1 2 8 01 9 1 2 9 01 10 1 2 10 0

9

810

1

27

5 3

4

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7654321

891011

151412

16

13

1718



119895 [119886119895 119887119895] [119888119895 119889119895] 119908119895

1 [146 191] [114 171] 01593 [92 182] [59 8] 01985 [47 91] [47 8] 02347 [19 84] [89 114] 01459 [62 113] [137 182] 0264






Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Scenario1198831= (119880

1 1198811)

[146 191]

times [114 171]

1198833= (119880

3 1198813)

[92 182]

times [59 8]

1198835= (119880

5 1198815)

[47 91]

times [47 8]

1198837= (119880

7 1198817)

[19 84]

times [89 114]

1198839= (119880

9 1198819)

[62 113]


1 (1843 1678) (1205 774) (796 707) (526 1126) (795 1557) 1160 005

2 (1910 1344) (1365 613) (878 762) (375 969) (1096 1686) 1147 0037

3 (1657 1669) (1067 755) (486 587) (705 927) (769 1457) 1092 0018

4 (1742 1274) (1348 78) (546 65) (345 1045) (993 1623) 1189 0079

5 (1667 1529) (1071 757) (503 744) (235 989) (746 1393) 1049 0061

6 (1497 1232) (1595 695) (497 484) (266 1068) (1047 1796) 1425 0315

7 (1566 1460) (1209 7) (516 7) (685 1015) (75 1714) 1062 0048

8 (1655 1311) (1351 706) (899 7) (217 981) (1069 1683) 1179 0069

9 (1630 1464) (1665 687) (655 744) (818 953) (1057 1798) 1144 0034

10 (1828 1350) (1447 654) (603 633) (425 931) (1027 1564) 1087 0023


119865 (119883119894lowast) 1153

|119865 (119883119894lowast) minus 119864119865 (119883119894

lowast) | 0043

|119864119865 (119883119894lowast) minus 119865 (119883119894

lowast)|

119864119865 (119883119894lowast)

387





7 Conclusions








References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a capacitated location-allocation model

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