optimal multiplicative generalized linear search plan for a

Hindawi Publishing CorporationJournal of OptimizationVolume 2013 Article ID 706176 13 pageshttpdxdoiorg1011552013706176

Research ArticleOptimal Multiplicative Generalized Linear Search Plan fora Discrete Random Walker

Abd-Elmoneim Anwar Mohamed and Mohamed Abd Allah El-Hadidy

Department of Mathematics Faculty of Science Tanta University Tanta 31527 Egypt

Correspondence should be addressed to Mohamed Abd Allah El-Hadidy melhadidieuneg

Received 18 February 2013 Accepted 14 June 2013

Academic Editor Bijaya Panigrahi

Copyright copy 2013 A-E A Mohamed and M A A El-Hadidy This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

This paper formulates a search model that gives the optimal search plan for the problem of finding a discrete random walk targetin minimum time The target moves through one of n-disjoint real lines in R119899 we have n-searchers starting the searching processfor the target from any point rather than the origin We find the conditions that make the expected value of the first meeting timebetween one of the searchers and the target finite Furthermore we show the existence of the optimal search plan that minimizesthe expected value of the first meeting time and find it The effectiveness of this model is illustrated using numerical example

1 Introduction

The search problem for a randomly moving target has aremarkable importance in our life due to its great applicabil-ity This problem is very interesting because it may arise inmany real world situations such as searching for randomlymoving persons or targets on roads That mathematicalanalysis collected all the researches derived from searchingapplications of the IIWorldWarThey solved with beauty twocomplementary objectives of the search find the target with(1) the smallest cost and (2) in minimum time Readers arereferred to Koopman [1] for the early works and to Benkoskiet al [2] and Frost and Stone [3] for a more recent survey

In the linear search problem the target moves on thereal line according to a known random process and itsinitial position is given by the value of a random variable1198830which has known probability distribution function A

searcher starts looking for the target at a point 1198670(|1198670| lt

infin) The searcher moves continuously along the line in bothdirections of the starting point119867

0until the target is met The

searcher would change its direction at suitable points manytimes before meeting its goalThus we consider that the pathlength which we represented is the cost of the searchThe aim

of the searcher is to minimize 119864120591120601 that is the expected value

of the first meeting time 120591120601between the searcher and the

target The problem is to find a search plan 120601(119905) such that119864120591120601lt infin in this case we call that 120601(119905) is a finite search plan

and if 119864120591120601lowastlt 119864120591120601

for all 120601(119905) isin Φ(119905) where Φ(119905) terms to theclass of all search plans then we call that 120601lowast(119905) is an optimalsearch plan

MC Cabe [4] found a finite search plan for a one-dimensional randomwalk target when the searcher starts thesearch from the origin and the initial position of the targethas a standard normal distribution Mohamed [5] discussedthe existence of a finite search plan for a one-dimensionalrandom walk target in general case which means that thesearch may start from any point on the real line and theinitial position of the target has any distribution El-Rayesand Mohamed [6] have shown the existence of a search planwhich minimizes the expected value of the first meetingtime between the searcher and the randomly moving targetwith imposed conditions Fristedt and Heath [7] derived theconditions for optimal search path which minimizes the costof effort of finding a randomly moving target on the real lineOhsumi [8] presented an optimal search plan for a target

2 Journal of Optimization

moving withMarkov process along one of119870-nonintersectingarcs to a safe destination within a time limit where the targetstarts at a safe base and tries to pass El-Rayes et al [9]illustrated this problemwhen the targetmoves on the real linewith a Brownian motion and the searcher starts the searchfrom the origin Recently Mohamed et al [10] discussed thisproblem for a Brownian targetmotion on one of 119899-intersectedreal lines in which any information of the target positionis not available to the searchers all the time Mohamed etal formulate a search model and find the conditions underwhich the expected value of the first meeting time betweenone of the searchers and the target is finite Furthermorethey showed the existence of the optimal search plan thatminimizes the expected value of the first meeting time andfound it

On the other hand when the target is located somewhereon the real line according to a known probability distributionthe searcher searches for it with known velocity and triesto find it in minimal expected time It is assumed that thesearcher can change the direction of its motion without anyloss of time The target can be detected only if the searcherreaches the target In an earlier work this problem has beenstudied extensively in many variations mostly by Beck et al[11ndash17] Franck [18] Rousseeuw [19] Reyniers [20 21] andBalkhi [22 23]

Furthermore Mohamed et al [24 25] have got moreinteresting results when they studied this problem to find arandomly located target in the plane The target has sym-metric or asymmetric distribution and with less informationabout this target available to the searchers More recentlyMohamed and El-Hadidy [26] studied this problemwhen thetarget moves with parabolic spiral in the plane and starts itsmotion from a randompoint AlsoMohamed and El-Hadidy[27] disscussed this problem in the plane when the targetmoves randomly with conditionally deterministic motion

Some problems of search may impose using more thanone searcher such aswhenwe search for a valuable target (egperson lost on one of 119899-disjoint roads) or search for a serioustarget (eg a car filledwith explosives whichmoves randomlyin one of 119899-disjoint roads) Thus the main contributions ofthis paper center around studying the problem of searchingfor a one-dimensional random walker target that is movingon one of a system of 119899-disjoint continuous real lines in R119899

(ie not intersected continuous real lines in the 119899-space)The problem that is studied here is very interesting where weargue to give the conditions on a strategy (or trajectories) of119899-searchers one on each line that make the expected valueof the first meeting time between one of the searchers and thetarget be finite and minimum In this problem there exists acomplication of such analysisThis complication is due to thefact that the searchers do not know the initial position of thetarget but only know its probability distribution Otherwisethe problem would be reduced to determine the strategy ofjust one searcher on the same targetrsquos line This problem isalready tackled in [6 7] This work focuses on the necessaryconditions for the existence of finite and optimal search planthat finds a random walker target

The optimal search plan that is proposed here shows thatthe special structure of the search problem can be exploited

Xnminus1

X3

Xn

X1

X2

L3 L2

L1

H32

H31

H33

H23H21H22H24

H14H12 H11

H13

Hn3

Hn1

Hn2

Hn4

12057520

12057530

120575n012057510

Lnminus1

Figure 1 The search plan 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) of the

searchers 119878119894 119894 = 1 2 119899

to obtain the efficient solution For example the search planfor a criminal drunk leaves its cache and walks up and downthrough one of 119899-disjoint streets totally disoriented

This paper is organized as follows In Section 2 weformulate the problem We display some properties that thesearch model should satisfy in Section 3 The search planand the conditions that make the expected value of the firstmeeting time between one of the searchers and the targetbe finite is discussed in Section 4 The existence of optimalsearch plan that minimizes the expected value of the firstmeeting time is presented in Section 5 The optimal searchplan is studied in Section 6 In Section 7 we illustrate theeffectiveness of this model using numerical example Finallythe paper concludes with a discussion of the results anddirections for future research

2 Problem Description

The problem under study can be formally described asfollows We have 119899-searchers 119878

119894 119894 = 1 2 119899 that start the

searching process from any point rather than the origin of theline 119871

119894 119894 = 1 2 119899 respectively as in Figure 1 Each of the

searchersmoves continuously along its line in both directionsof the starting pointThe searcher 119878

119894would conduct its search

in the following manner Start at 1205751198940and go to the left (right)

as far as 1198671198941 Then turn back to explore the right (left) part

of 1205751198940as far as119867

1198942 Retrace the steps again to explore the left

(right) part of 1198671198941as far as 119867

1198943and so forth In this paper

we need to determine 119867119894119895 119894 = 1 2 119899 119895 = 1 2 that

minimize the first meeting time between one of the searchersand the moving target

Journal of Optimization 3

Let 119868 be a set of integer numbers and 119868+ a nonnega-tive part of 119868 We also assume that 119883

119894119895119895ge1

are sequencesof independent identically distributed random variables in119871119894 119894 = 1 2 119899 respectively In addition we let a value

of ldquominus1rdquo indicate a step to the left and a value ldquo1rdquo a stepto the right so that for any 119895 ge 1 we have the 119899-dimensional probability vectors [119875(119883

1119895= 1) = 119901

1 119875(1198832119895=

1) = 1199012 119875(119883

119899119895= 1) = 119901

119899] and [119875(119883

1119895= minus1) =

1199021 119875(1198832119895= minus1) = 119902

2 119875(119883

119899119895= minus1) = 119902

119899] where

[1199011 1199012 119901

119899] + [119902

1 1199022 119902

119899] = [1 1 1]

Supposing that for 119905 gt 0 119905 isin 119868+ 119882(119905) =

sum119905

119895=1119883119894119895 119882(0) = 0 and 119883

0is a random variable that

represent the target initial position valued in 2119868 or (2119868 + 1)and independent of119882(119905) 119905 ge 0 if 120585

119894gt 0 and 119883

119894119895such that

0 le 1198701198941= (120585119894+ 119909119894)2 le 120585

119894 where 119870

1198941is an integer then

119875 (119882(120585119894) = 1198701198941)

=

(120585119894

1198701198941

)1199011198701198941

119894119902120585119894minus1198701198941

119894

0 if 1198701198941does not exist

(1)

The target is assumed to move randomly on one of 119899-disjoint real lines according to the process 119882(119905) 119905 isin 119868+where 119868+ is the set of positive integer numbers and119882(119905) is aone-dimensional randomwalkmotionThe initial position ofthe target is unknown but the searchers know its probabilitydistribution (ie the probability distribution of the target isgiven at time 0) and the process 119882(119905) 119905 isin 119868+which controlsthe targetrsquos motion

Figure 1 gives an illustration of the search plan that the 119899-searchers 119878

119894 119894 = 1 2 119899 follow it Moreover it is to be

noted that this search plan 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) is a

combination of continuous functions 120601119894(119905) with speed V

119894and

given by 120601119894(119905) 119868+ rarr 119868 119894 = 1 2 119899 such that

1003816100381610038161003816120601119894 (1199051) minus 120601119894 (1199052)1003816100381610038161003816 lt V119894

10038161003816100381610038161199051 minus 11990521003816100381610038161003816 forall119905

1 1199052isin 119868+

120601119894(0) = 0 119894 = 1 2 119899

(2)

The first meeting time 120591120601is a random variable valued in

119868+ and it is defined by

120591120601=

inf 119905 either one of 120601119894(119905) = 119883

0+119882(119905)

119894 = 1 2 119899

infin if the set is empty(3)

where1198830isin 119868 that is a randomvariable independent of119882(119905)

and it represents the initial position of the target We supposethat 119883

0= 119883119894if the target moves on 119871

119894 119894 = 1 2 119899 Also

assume that the set of all search plans of the searchers 119878119894with

speeds V119894 119894 = 1 2 119899 respectively satisfying condition

(2) be ΦV119894(119905) 119894 = 1 2 119899 The problem is to find a search

plan 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) for the 119899-searchers

such that 119864120591120601lt infin where 119864

120591120601is the expected value of

the first meeting time between one of the searchers and thetarget and Φ(119905) = (120601

1(119905) 1206012(119905) 120601

119899(119905)) 120601

119894(119905) isin ΦV119894

(119905) 119894 =

1 2 119899 is called a class of all sets of the search plans

Assuming that 120599119894 120582119894 119894 = 1 2 119899 are positive numbers

such that 120599119894gt 1 119862

119894= (120599119894minus 1)(120599

119894+ 1) gt |119901

119894minus 119902119894| 120582119894=

119887120603119894 119887 = 1 2 and 120603

119894(1 minus 119862

119894)2 is positive number also

In this problem we assume that V119894= 1 119894 = 1 2 119899 In

addition we define the sequences 119866119894119895 119889119894119895 and 119867

119894119895 119894 =

1 2 119899 119895 = 1 2 by 119866119894119895= 120582119894(120599119895

119894minus 1) 119889

119894119895=

(minus1)119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] and 119867119894119895= 119889119894119895+ 1205751198940 We also

use the notations 120595119894(119866119894(2119895+1)

) = 119882(119866119894(2119895+1)

) minus 1198701198941(119866119894(2119895+1)

)119894(119866119894(2119895)) = 119882(119866

119894(2119895)) + 119870

1198942(119866119894(2119895)) where 119870

1198941(119866119894(2119895+1)

) =

119862119894119866119894(2119895+1)

1198701198942(119866119894(2119895)) = 119862

119894119866119894(2119895)

which are positive functionsand the following remark

Remark 1 If 0 lt 120600sect lt 1 sect = 1 2 120581 then prod120581

sect=1120600sect lt

sum120581

sect=1 120600sect

There are known probability measures 120574119894 119894 = 1 2 119899

such that 1205741+1205742+ sdot sdot sdot + 120574

119899= 1 on 119871

1cup1198712cup sdot sdot sdot cup119871

119899 And they

describe the location of the target where 120574119894is induced by the

position of the target on 119871119894

The objective is to obtain the conditions that make thesearch plan 120601(119905) = (120601

1(119905) 1206012(119905) 120601

119899(119905)) be finite (ie 119864

120591120601lt

infin) We also need to show the existence of the optimal searchplan 120601lowast(119905) = (120601lowast

1(119905) 120601lowast2(119905) 120601lowast

119899(119905)) and find it that is to

give the minimum expected value of the first meeting time

3 Properties of the Search Model

In this section we start to discuss some properties that thesearch model for a one-dimensional random walk targetshould satisfy

The searcher 119878119894would conduct its search as in the above

manner that is detailed in Section 2 Consequently for anyline 119871

119894 119894 = 1 2 119899 119905 isin 119868+ we have the following

Case 1 If we consider thatQ is a set of positive even numberssuch thatQ = 2 4 6 119899 forF isin Q 119895 isin 119868+ thenwe have

sdot sdot sdot lt 119867119894(119895+2)

lt 0 lt 119867119894119895lt 119867119894(119895minus2)

lt sdot sdot sdot lt 1198671198942lt 1205751198940

lt 1198671198941lt 1198671198943lt sdot sdot sdot

(4)

for any 119905 isin 119868+ if 119866119894(2119895minus1)

le 119905 le 119866119894(2119895) 1 le 119895 le F2 then

120601119894(119905) = 119867

119894(2119895+1)minus 1205751198940minus [119905 minus 119866

119894(2119895minus1)] (5)

if 119866119894119895le 119905 le 119866

119894(119895+1) 119895 ge F + 1 then

120601119894(119905) = 119867

119894119895+ (minus1)

119895

[119905 minus 119866119894119895] (6)

if 119866119894(2119895)

le 119905 le 119866119894(2119895+1)

1 le 119895 le F2 then

120601119894(119905) = 120575

1198940minus 119867119894(2119895)+ [119905 minus 119866

119894(2119895)] (7)

Case 2 Also if 119874 is a set of positive odd numbers such that119874 = 1 3 5 forF isin 119874 119895 isin 119868

+ then we have

sdot sdot sdot lt 1198671198942lt 1205751198940lt 1198671198941lt 1198671198943lt sdot sdot sdot lt 119867

119894(119895minus2)lt 119867119894119895

lt 0 lt 119867119894(119895+2)

lt sdot sdot sdot (8)



le 119905 le 119866119894(2119895)

1 le 119895 le (F + 1)2 then

120601119894(119905) = 120575

1198940minus 119867119894(2119895minus1)

minus [119905 minus 119866119894(2119895minus1)

] (9)

if 119866119894119895le 119905 le 119866

119894(119895+1) 119895 ge F + 1 then

120601119894(119905) = 119867

119894119895+ (minus1)

119895

[119905 minus 119866119894119895] (10)

if 119866119894(2119895)

le 119905 le 119866119894(2119895+1)

1 le 119895 le (F minus 1)2 then

120601119894(119905) = 119867

119894(2119895)minus 1205751198940+ [119905 minus 119866

119894(2119895)] (11)

It is clear that the search path of 119878119894depends on 120582

119894 120599119894and

119895 isin 119868+ Since 1198830isin 119868 then the first meeting time is done at

119909119894isin 119868 where 119909

119894is an integer number For this reason we will

prove the conditions (properties) thatmake 119878119894meet the target

at 119909119894as in the following theorems

Theorem 2 If 119862119894is a rational number different from 1 to minus1

and for all 120585119894ge 1 119881(120585

119894) = (119882(120585

119894120599119894) minus 119862119894120585119894120599119894)2 then

(i) there exists a sequence 119884119894119895119895ge1

of independent identi-cally distributed random variables such that 119881(120585

119894) =

sumΓ

119895=1119884119894119895 and the distribution of 119884

119894119895is concentrated

on the integers 119864(119884119894119895) = (120599

1198942)[119864(119883

119894119895) minus 119862

119894] and the

probability vector [119901(1198841119895= F) gt 0 119901(119884

2119895= F) gt

0 119901(119884119899119895= F) gt 0] if and only if minus120599

119894(1 + 119862

119894)2 le

F le 120599119894(1 minus 119862

119894)2

(ii) The probability vector [119901(119881(1205851) = 1199091) gt 0 119901(119881(120585

2) =

1199092) gt 0 119901(119881(120585

119899) = 119909

119899) gt 0] holds if and only

if 119909119894is an integer such that minus120585

119894120599119894(1 + 119862

119894)2 le 119909

119894le

120585119894120599119894(1 minus 119862

119894)2

(iii) If 119862119894= 119901119894minus 119902119894then there exist constants 119903

1198941and 119903

1198942

depending on119862119894 119901119894such that for any 119909

119894isin R whereR

is the set of real numbers if 120585119894gt 1199031198941119909119894+1199031198942and if 119909

119894ge 0

we have [119901(0 le 119881(1205851+ 1) le 119909

1) 119901(0 le 119881(120585

2+ 1) le

1199092) 119901(0 le 119881(120585

119899+ 1) le 119909

119899)] le [119901(0 le 119881(120585

1) le

1199091) 119901(0 le 119881(120585

2) le 1199092) 119901(0 le 119881(120585

119899) le 119909119899)] and

if 119909119894lt 0 we have [119901(119909

1le 119881(120585

1+ 1) lt 0) 119901(119909

2le

119881(1205852+ 1) lt 0) 119901(119909

119899le 119881(120585

119899+ 1) lt 0)] le [119901(119909

1le

119881(1205851) lt 0) 119901(119909

2le 119881(120585

2) lt 0) 119901(119909

119899le 119881(120585

119899) lt

0)]

Proof (i) Define 119884119894119895= sum120599119894

F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2) 119895 ge 1

and [119901(1198841119895= 1199091) 119901(1198842119895= 1199092) 119901(119884

119899119895= 119909119899)] =

[119901(119882(1205991)) = 2119909

1+ 12059911198621 119901(119882(120599

2)) = 2119909

2+ 12059921198622

119901(119882(120599119899)) = 2119909

119899+ 120599119899119862119899]

Consequently if [119901(1198841119895= 1199091) gt 0 119901(119884

2119895= 1199092) gt

0 119901(119884119899119895= 119909119899) gt 0] then from (1) we have 119909

119894+ 120599119894(1 +

119862119894)2 that is an integer and since 119909

119894is an integer also then we

have 0 le 119909119894+ 120599119894(1 + 119862

119894)2 le 120599

119894 Therefore minus120599

119894(1 + 119862

119894)2 le

119909119894le 120599119894(1 minus 119862

119894)2 If 119909

119894= F then we have [119901(119884

1119895= F) gt

0 119901(1198842119895= F) gt 0 119901(119884

119899119895= F) gt 0] if and only if

minus120599119894(1 + 119862

119894)2 le F le 120599

119894(1 minus 119862

119894)2 In addition by using

(1) 119864(sum120599119894F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2)) = (120599

1198942)[119864(119883

119894119895) minus 119862119894] is

proved

(ii) Since 119881(120585119894) = (119882(120585

119894120599119894) minus 119862

119894120585119894120599119894)2 then we have

[119901(119881(1205851) = 119909

1) 119901(119881(120585

2) = 119909

2) 119901(119881(120585

119899) = 119909

119899)] =

[119901((119882(12058511205991) minus 119862112058511205991)2 = 119909

1) 119901((119882(120585

21205992) minus 119862212058521205992)2 =

1199092) 119901((119882(120585

119899120599119899) minus 119862

119899120585119899120599119899)2 = 119909

119899)] = [119901(119882(120585

11205991)) =

21199091+119862112058511205991) 119901(119882(120585

21205992)) = 2119909

2+119862212058521205992) 119901(119882(120585

119899120599119899)) =

2119909119899+ 119862119899120585119899120599119899)] and using (1) the prove is completed

(iii) By using (ii) if 119909119894ge 0 then we have [119901(0 le 119881(120585

1) le

1199091) 119901(0 le 119881(120585

2) le 119909

2) 119901(0 le 119881(120585

119899) le 119909

119899)] =

[sum[1199091]

119895=0119901(119881(120585

1) = 119895) sum

[1199092]

119895=0119901(119881(120585

2) = 119895) sum

[119909119899]

119895=0119901(119881(120585

119899) =

119895)] and if 119909119894lt 0we get [119901(119909

1le 119881(120585

1+1) lt 0) 119901(119909

2le 119881(120585

2+

1) lt 0) 119901(119909119899le 119881(120585

119899+ 1) lt 0)] = [sum

0

119895=[1199091]119901(119881(120585

1) =

119895) sum0

119895=[1199092]119901(119881(120585

2) = 119895) sum

0

119895=[119909119899]119901(119881(120585

119899) = 119895)] where

[119909119894] means the greatest integer less than or equal to 119909

119894 119894 =

1 2 119899 It is sufficient to show that

[119901 (119881 (1205851+ 1) = F) 119901 (119881 (120585

2+ 1) = F)

119901 (119881 (120585119899+ 1) = F)]

le [119901 (119881 (1205851) = F) 119901 (119881 (120585

2) = F)

119901 (119881 (120585119899) = F)] if 120585

119894gt 1199031198941119909119894+ 1199031198942

(12)

We have the following cases

Case (a) If 119862119894lt minus1 then from (ii) [119901(119881(120585

1) = F) gt

0 119901(119881(1205852) = F) gt 0 119901(119881(120585

119899) = F) gt 0] if and

only if 2F120599119894(1 minus 119862

119894) le 120585

119894le minus2F120599

119894(1 + 119862

119894) We take

1199031198941= minus2120599

119894(1 + 119862

119894) 1199031198942= 0 and then 120585

119894gt 1199031198941|F| rArr 120585

119894gt

minus2F120599119894(1 + 119862

119894) that leads to [119901(119881(120585

1) = F) = 0 119901(119881(120585

2) =

F) = 0 119901(119881(120585119899) = F) = 0] Consequently (12) holds

Case (b) If 119862119894gt 1 hence [119901(119881(120585

1) = F) gt 0 119901(119881(120585

2) =

F) gt 0 119901(119881(120585119899) = F) gt 0] if and only if minus2F120599

119894(1 +

119862119894) le 120585119894le 2F120599

119894(1 minus 119862

119894) From (ii) putting 119903

1198941= minus2120599

119894(1 minus

119862119894) 1199031198942= 0 then 120585

119894gt 1199031198941|F| rArr 120585

119894gt 2F120599

119894(1 minus 119862

119894) that leads

to [119901(119881(1205851) = F) = 0 119901(119881(120585

2) = F) = 0 119901(119881(120585

119899) =

F) = 0]Then (12) holds

Case (c) If minus1 lt 119862119894lt 1 then let 120572

119894= (1 minus 119862

119894)2 120573

119894=

1 minus 120572119894and Δ

119894= (120572119894119902119894)120572119894(120573119894119901119894)120573119894 where [119901

1 1199012 119901

119899] +

[1199021 1199022 119902

119899] = [1 1 1] In addition put 119903

1198942= 1(Δ

119894minus

1) 1199031198941= Δ119894max[(1120572

119894 1120573119894)120599119894(Δ119894minus1)] Since 119864(119883

119894119895) = 119862119894and

0 lt 120572119894lt 1 then 120572

119894= 119902119894and then Δ

119894gt 1 Consequently

1199031198941and 1199031198942are positive and well defined Assuming that 120585

119894gt

1199031198941|F| then minus120573

119894120585119894120599119894lt F lt 120572

119894120585119894120599119894 therefore from (ii) we

have [119901(119881(1205851) = F) gt 0 119901(119881(120585

2) = F) gt 0 119901(119881(120585

119899) =

F) gt 0]It remains to prove that if [119901(119881(120585

1) = F) gt 0 119901(119881(120585

2) =

F) gt 0 119901(119881(120585119899) = F) gt 0] and 120585

119894gt 1199031198941|F| + 119903

1198942then (2)

holdsConsidering that 119885

119894119895= (119883119894119895minus 119862119894)2 then [119901(119885

1119895= 1205721) =

1199011 119901(1198852119895= 1205722) = 119901

2 119901(119885

119899119895= 120572119899) = 119901

119899] [119901(119885

1119895=

minus1205731) = 119902

1 119901(1198852119895= minus120573

2) = 119902

2 119901(119885

119899119895= minus120573

119899) = 119902

119899]


and [119901(119881(1205851) = F) 119901(119881(120585

2) = F) 119901(119881(120585

119899) = F)] =

[119901(sum12058511205991

119895=11198851119895= F) 119901(sum

12058521205992

119895=11198852119895= F) 119901(sum

120585119899120599119899

119895=1119885119899119895=

F)] By using (1) we have

[119901 (119881 (120585

1+ 1) = F)

119901 (119881 (1205851) = F)

119901 (119881 (120585

2+ 1) = F)

119901 (119881 (1205852) = F)

119901 (119881 (120585119899+ 1) = F)

119901 (119881 (120585119899) = F)

]

= [

[

12059911205731

prod119895=1

12058511205991+ 119895

Δ1(12058511205991+ 119895 + (119895120572

1+F) 120573

1)

times

12059911205721

prod119895=1

12058511205991+ 119895 + 120599

11205731

Δ1(12058511205991+ 119895 + (119894120573

1minusF) 120572

1)

12059921205732

prod119895=1

12058521205992+ 119895

Δ2(12058521205992+ 119895 + (119895120572

2+F) 120573

2)

times

12059921205722

prod119895=1

12058521205992+ 119895 + 120599

21205732

Δ2(12058521205992+ 119895 + (119894120573

2minusF) 120572

2)

120599119899120573119899

prod119895=1

120585119899120599119899+ 119895

Δ119899(120585119899120599119899+ 119895 + (119895120572

119899+F) 120573

119899)

times

120599119899120572119899

prod119895=1

120585119899120599119899+ 119895 + 120599

119899120573119899

Δ119899(120585119899120599119899+ 119895 + (119894120573

119899minusF) 120572

119899)]

]

(13)

Since 120585119894gt 1199031198941|F| + 119903

1198942then every term is strictly less

than 1 Consequently [119901(119881(1205851+ 1) = F) 119901(119881(120585

2+ 1) =

F) 119901(119881(120585119899+ 1) = F)] lt [119901(119881(120585

1) = F) 119901(119881(120585

2) =

F) 119901(119881(120585119899) = F)]

Theorem 3 If 119864(119883119894119895) lt 119862

119894 119862119894isin R (the set of real

numbers) then there exists 120576119894 0 lt 120576

119894lt 1 such that

[119901(119882(1205851)) 119901(119882(120585

2)) 119901(119882(120585

119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for

all 120585119894 119894 = 1 2 119899

Proof For 120577119894gt 0 [119901(119882(120585

1) ge 119862

11205851) 119901(119882(120585

2) ge 119862

21205852)

119901(119882(120585119899) ge 119862

119899120585119899)] = [119901(exp120577

1(119882(1205851) minus 11986211205851) ge 1) 119901(exp

1205772(119882(1205852)minus11986221205852) ge 1) 119901(exp120577

119899(119882(120585119899)minus119862119899120585119899) ge 1)] le

[119864(exp1205771(119882(1205851) minus 119862

11205851)) 119864(exp120577

2(119882(1205852) minus 119862

21205852))

119864(exp120577119899(119882(120585119899) minus 119862

119899120585119899))] = [119891

1(1205771)1205851 1198912(1205772)1205852

119891119899(120577119899)120585119899] where [119891

1(1205771) 1198912(1205772) 119891

119899(120577119899)] = [119864(exp

1205771(1198831119895minus 1198621)) 119864(exp120577

2(1198832119895minus 1198622)) 119864(exp120577

119899(119883119899119895minus

119862119899))] = [119901

1exp1205771(1 minus 119862

1) + 119902

1exp1205771(minus1 minus 119862

1) 1199012

exp1205772(1 minus 119862

2) + 119902


2) 119901

119899exp

120577119899(1 minus 119862

119899) + 119902

119899exp120577119899(minus1 minus 119862

119899)] if 119864(119883

119894119895) lt 119862

119894 119894 =

1 2 119899 then [11989110158401(0) 1198911015840

2(0) 1198911015840

119899(0)] lt [0 0 0]

and since [1198911(0) 119891

2(0) 119891

119899(0)] lt [1 1 1]

then [min1205771gt01198911(1205771)min

1205772gt01198912(1205772) min

120577119899gt0119891119899(120577119899)]

= [1205761 1205762 120576

119899] lt [1 1 1]

By similar arguments if 119864(119883119894119895) gt 119862

119894 119894 = 1 2 119899

then there exist 120576119894lt 1 such that [119901(119882(120585

1)) 119901(119882(120585

2))

119901(119882(120585119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for all 120585

119894 119894 = 1 2 119899

4 Existence of a Finite Search Plan

In this section we find the conditions that make the searchplan be finite In addition we will discuss that under whatthese conditions are indeed finite This is a crucial issuerelated to the existence of a finite search plan So we willprovide useful theorems that help us to do it

Theorem 4 Let 120574119894be the measure defined on R by 119883

119894119895 119894 =

1 2 119899 and if 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) is a search

plan defined previously the expectation 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(14)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)

) gt minus119909119894)120574119894(119889119909119894) are finite when

1205751198940gt 0 And if 120575

1198940lt 0 then 119864

120591120601is finite if

intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(15)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite

Proof The hypotheses119883119894and 1205751198940are valued in 2119868 or (2119868 + 1)

then 119883119894+ 119882(119905) is greater than 120601

119894(119905) until the first meeting

also if 119883119894is smaller than 120575

1198940then 119883

119894+ 119882(119905) is smaller than

120601119894(119905) until the first meeting Since 120591

120601119894gt 119905 119894 = 1 2 119899 are

mutually exclusive events then 119901(120591120601119894gt 119905) = 119901(120591

1206011gt 119905 or

1205911206012gt 119905 or sdot sdot sdot or 120591

120601119899gt 119905) = sum

119899

119894=1119901(120591120601119894gt 119905) and for any 119895 ge 0

we have

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119883119894+119882(119866

119894(2119895)) lt 119867

119894(2119895)| 119883119894= 119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (119883119894+119882(119866

119894(2119895+1)) gt 119867

119894(2119895+1)| 119883119894= 119909119894) 120574119894(119889119909119894)

(16)

Using the notation 119894(119866119894(2119895)) = 119882(119866

119894(2119895)) + 119862119894119866119894(2119895)

weobtain 119882(119866

119894(2119895)) minus 119867

119894(2119895)lt minus119883

119894= 119909119894 then 119882(119866

119894(2119895)) minus

(minus1)2119895+1

119862119894[119866119894(2119895)

+ 1 + (minus1)2119895+1

] lt minus119909119894leads to 119882(119866

119894(2119895)) minus

(minus1)119862119894[119866119894(2119895)

+ 1 + (minus1)] = 119882(119866119894(2119895)) + 119862

119894[119866119894(2119895)] =


119894(119866119894(2119895)) lt minus119909

119894 Similarly by using the notation120595

119894(119866119894(2119895+1)

) =

119882(119866119894(2119895+1)

) minus 119862119894119866119894(2119895+1)

we get 120595119894(119866119894(2119895+1)

) gt minus119909119894 Conse-

quently

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894) 120574119894(119889119909119894)

(17)

also

119901 (120591120601119894gt 119866119894(2119895))

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895minus1)

) gt minus119909119894) 120574119894(119889119909119894)

(18)

From Remark 1 we obtain

119864120591120601= intinfin

0

119901 (120591120601gt 119905) 119889119905

le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 119905) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

(19)

where 1198661198940= 0 then

119864120591120601le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 1198661119895) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

=

119899

sum119894=1

infin

sum119895=0

(119866119894(119895+1)

minus 119866119894119895) 119901 (120591120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

[120582119894(120599119895+1

119894minus 1) minus 120582

119894(120599119895

119894minus 1)] 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

120582119894120599119895

119894(120599119894minus 1) 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

[

[

120582119894(120599119894minus 1)

infin

sum119895=0

120599119895

119894119901 (120591120601119894gt 119866119894119895)]

]

=

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 119863) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894119901 (120591120601119894gt 1198661198942) + 1205993

119894119901 (120591120601119894gt 1198661198943) + sdot sdot sdot )]

(20)

If 1205751198940lt 0 then we get

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205993

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205994

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198944) lt minus119909

119894) 120574119894(119889119909119894)

+ sdot sdot sdot

+ 120599F119894intinfin

1205751198940

119901 (120595119894(119866119894(Fminus1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+1119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+2119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+3119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+3)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ sdot sdot sdot )]

(21)


thus

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894) + sdot sdot sdot

+ 1205993

119894(120599119894+ 1)int

infin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+ 1205995

119894(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(1198661198945) gt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F+3119894

(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(119866119894(F+3)) gt minus119909119894) 120574119894 (119889119909119894) + sdot sdot sdot )]

(22)

Leads to

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times [1

119892119894+ (120599119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]]

=

119899

sum119894=1

120582119894(120599119894minus 1) (

1

119892119894)

+

119899

sum119894=1

[120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]

(23)

where

1

119892119894= 119901 (120591

120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

1

119908119894(119909119894) =

infin

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

1

119902119894(119909119894) =

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)

1

ℎ119894(119909119894) =

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(24)

Then 119864120591120601is finite if

intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(25)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite

By similar way if 1205751198940gt 0 then 119864


119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1) (

2

119892119894) + 120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

2

119908119894(119909119894) 120574119894(119889119909119894)

+ int1205751198940

minusinfin

2

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

2

ℎ119894(119909119894) 120574119894(119889119909119894))]

(26)


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)

) gt minus119909119894) are finite

Lemma5 (see El-Rayes et al [9]) If 119886ge 0 119886

+1le 119886 =

1 2 and 119889 = 1 2 are a strictly increasing se-

quence of integer numbers with 1198890= 0 then for any =

1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)

where suminfin=119896[119889+1

minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889

are vectors in formulas and they are taken to be rowvectors These vectors are defined as follows

infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)

Now we will discuss under what conditions of the chosensearch plan should be satisfied to make the previous integralsin Theorem 4 are indeed finite This is a crucial issue relatedto the existence of a finite search plan

Theorem 6 The chosen search plan should satisfy 2ℎ(119909) le119871(|119909|) 2119902(119909) le (|119909|) if 120575

1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and

(|119909|) are vectors of linear functions given by 1119908(119909) = [11199081

(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]

Proof We will prove this theorem for 2ℎ(119909) when 1205751198940gt 0

where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)

) gt minus1199091) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)

) gt minus119909119899)] and we obtain the follow-

ing cases(I) If 119909

119894gt 120575

1198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le minus12057510) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)

) le minus12057520) sum

infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575

1198940 then we get [

2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]

From (II) we have [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)

) le 0)] Consequently from Theorem 3 weget [2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)

) gt 0)] le [suminfin

119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1

Suppose the following holds

(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =

1 2 is a sequence of independent identicallydistributed random variables

(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)

(iv) is an integer such that 119889119894= 1199031198941|119909119894| + 1199031198942

(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)

Thus from Theorem 2 we have 119886119894(120585119894) is nonincreasing if

120585119894gt 119889119894120585119894 By applying Lemma 5 we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]

Since 119880119894(FF + 1) satisfies the conditions of renewal

theorem as in Feller [28] then 119880119894(FF + 1) is bounded

for all F 119894 = 1 2 119899 by a constant so 2ℎ(119909) =

[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)

then 2ℎ(119909) le 119871(|119909|) Similar to the previous analysis waywe can prove that 2119902(119909) le (|119909|) and also 1119908(119909) le 119872(|119909|)1ℎ(119909) le (|119909|) if 120575

1198940lt 0 Therefore the proof is

completed

Theorem 7 If there exists a finite search plan 120601(119905) =

(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite

Proof For 119864120591120601119894lt infin 119894 = 1 2 119899 we have 119901 (120591

120601is finite) =

1 and so 119901 (one of 120591120601119894is finite 119894 = 1 2 119899) = 1 Therefore

119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591

120601120573lt infin)) + sdot sdot sdot

+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)

Since 120591120601119894lt infin 119894 = 1 2 119899 are mutually exclusive

events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579

is finite) =1 for all 119894 = 120579 119894 120579 = 1 2 119899 If 119901 (120591

120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591

120601119894) with probability one and |119883

0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite

On the other hand if 119901 (120591120601120579

is finite) = 1 then 1198830=

120601120579(120591120601120579) minus 119882(120591

120601120579) with probability one and by the same

manner we can get 119864(|1198830|) is finite

Remark 8 A direct consequence of Theorems 4 6 and 7satisfies the existence of a finite search plan if and only if119864(|1198830|) is finite

5 Existence of an Optimal Search Plan

The goal of the searching strategy could minimize theexpected value of the first meeting time between one of thesearchers and the target Therefore the main problem here isto find a search paths 120601

119894(119905) 119894 = 1 2 119898 If such a search

paths exists we call it optimal search paths

Definition 9 Let 120601119894ℎ(119905)ℎge1

isin Φ1(119905) 119894 = 1 2 119898 be

sequences of search plans we say that 120601119894ℎ(119905) converges to 120601

119894(119905)

as ℎ tends toinfin if and only if for any 119905 isin 119868+ 120601119894ℎ(119905) converges

to 120601119894(119905) uniformly on every compact space (El-Rayes et al

[9])


Theorem 10 Let for any 119905 isin 119868+ and119882(119905) be a one-dimension-al random walk Then the mapping

(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)

is lower semicontinuous on Φ1(119905)

Proof Let 119861(120601119894 119905) be the indicator function of the set 120591

120601119894gt

119905 119894 = 1 2 119899 by the Fatou-Lesbesque theorem we get

119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)

for any sequences 120601119894ℎ

rarr 120601119894in Φ1(119905) where Φ

1(119905) is

sequentially compact see El-Rayes et al [9] Thus the map-ping (120601

1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864

120591120601is lower semicontinuous

mapping on Φ1(119905) and this completed the proof

6 Optimal Search Plan

In this section we will find the optimal search plan thatminimize the expected value of the first meeting time

Since 120591120601depends on infimum either one of the functions

120601119894(119905) 119894 = 1 2 119899 then we need to minimize 119864

120591120601 That

happens when we obtain the optimal search plan 120601lowast(119905) =120601lowast1(119905) 120601lowast2(119905) 120601lowast

119899(119905) where 120601

119894(119905) is a function on a dis-

tance 119867119894119895 119894 = 1 2 119899 1 le 119895 le F2 Therefore it can

be seen that the vector (1206011(119905) 1206012(119905) 120601

119899(119905)) are 119899-distinct

objective functions Since the speed of the 119899-searchers (searchteam) is V

119894= 1 119894 = 1 2 119899 then the objective functions

120601119894(119905) 119894 = 1 2 119899 are distinct set of constrains given from

Cases 1 and 2 In this problem we consider the constraint119866119894(2119895minus1)

le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then

the problem takes the form

min (1206011(119905) 1206012(119905) 120601

119899(119905))

subject to 119905 isin 119879 = 119905 isin 119868+

| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)

where 119905 is an 119899-dimensional vector of the decision variables1206011(119905) 1206012(119905) 120601

119899(119905) are 119899-distinct objective functions of the

decision vector of inequality constraints and 119879 is the feasibleset of constrained decisions

But V119894= 1 119894 = 1 2 119899 that leads to 119905 = 119867

119894119895

119894 = 1 2 119899 1 le 119895 le F2 then the above problem

is represented as the following multiobjective nonlinearprogramming problem (MONLP)

MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)

However on formulating the MONLP which closelydescribes and represents the real decision situation variousfactors of the real system should be reflected in the descrip-tion of the objective functions and the constraints Naturallythese objective functions and the constraints involve manyvariables and parameters whose possible values may beassigned by the experts In the conventional approach suchparameters are changed according to the variables in anexperimental andor subjective manner through the expertsunderstanding the nature of the variables

Substituting 119866119894119895= 119887120603119894(120599119895

119894minus 1) in the above MONLP we

have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940

minus [119905 minus 119887120603119894(1205992119895minus1

119894minus 1)]

(37)

where the variable 120599119894takes +ve numbers greater than 1 and

the parameter 120603119894is the least positive integer such that 120603

119894(1 minus

(120599119894minus 1)(120599

119894+ 1))2 is positive number 119867

119894119895ge 0 for all 119894 =

1 2 119899 1 le 119895 le F2 Furthermore 119867119894119895is a function of

the variable 120599119894and the parameter 120603

119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)


Then the previous MONLP(I) takes the form

MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895

119894minus 1) ge (ℎ

1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)

Choosing a certain weights 119894 119894 = 1 2 119899 such that

sum119899

119894=1119894= 1 and the certain parameters120603

119894= 1 119894 = 1 2 119899

then the previous MONLP(II) takes the form

MONLP(III) min 119894(ℎ3) (120599119894minus 1

120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

1 minus 120599119894lt 0 forall119887 = 1 2 1 le 119895 le

F

2

(40)

From the Kuhn-Tucker conditions we have

1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2

119894minus(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2

Figure 2 (a) and (b) give the optimal expected value of the first meeting time between 119878119894and the target that moves with discrete random

walk in one dimension

Solving (41)ndash(44) we found the optimal values of 120599119894that

give the optimal distances 119867119894119895for all 119887 = 1 2 1 le 119895 le

F2 and they are given by

120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)

7 An Illustrative Example

In this section we clarify the effectiveness of this model byconsidering the following example

Example 1 Suppose that a criminal drunk (target) leaves itscache and walks up and down through one of 119899-disjointstreets totally disoriented We model the streets as real lineswith cache at the point 119883

0 In addition assume that the

criminal drunk takes unit steps sowemay record his positionwith an integer Thus for example if he takes 5 steps tothe left he will be at a position 119883

0minus 5 If we consider

the previous conditions in Theorems 2 and 3 hold then theoptimal expected value of the first meeting time depends onthe optimal values of the distances 119867

119894119895that the searcher 119878

119894

should do it and the optimal values of 119866119894119895 119895 = 2 4 6 Using

the constraint 119866119894(2119895minus1)

le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le

F2 from Case 2 such that 119895 = 2 4 6 and for 119887 = 1 2in MONLP(III) and also using (38) we obtain the optimalvalues of 120599

119894that give the optimal values of119867

119894119895 119866119894119895in Table 1

It is clear that the optimal distances119867119894119895that 119878

119894should do

it depend on 119887 where 120582119894= 119887120603119894and 119866

119894119895= 120582119894(120599119895

119894minus 1)

Table 1 The optimal values of 120599119894 119867119894119895and 119866

119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940

From the previous numerical calculations and by consid-ering 120575

1198940= 10 we get Figure 2 that shows the projection

of the targetrsquos motion (one-dimensional random walk) onthe 119878119894path with 119887 = 1 2 where the current position on the

line represented in the vertical axis versus the time steps athorizontal axis

8 Discussion and Conclusions

A multiplicative generalized linear search plan for a one-dimensional random walk target on one of 119899-disjoint reallines has been presented where the target initial position isgiven by a random variable 119883

0 Therefore the target will be

meet if119864120591120601lt infin where120601(119905) = (120601

1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601

is the expected value of the first meeting time between oneof the searchers and the target We discuss some propertiesthat the search model should satisfy in Theorems 2 and 3We introduce the proof of conditions that make a search planfinite inTheorem 4 based on the continuity of the search planand the conditions inTheorems 2 and 3 to show that119864

120591120601lt infin

We providemore analysis by using Lemma 5Theorems 2 and3 in Theorem 6 to show that the search plan 120601(119905) is finite ifthe conditions 2ℎ(119909) le 119871(|119909|) 2119902(119909) le (|119909|) if 120575

1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)

(|119909|)119872(|119909|) and (|119909|) are vectors of linear functions Weuse Theorem 7 to show that if there exist a finite search planthen the expected value of the target initial position119864(|119883

0|) is


finite It will also be interesting to see a direct consequence ofTheorems 4 6 and 7 satisfying the existence of a finite searchplan if and only if119864(|119883

0|) is finiteThe existence of an optimal

search plan has been proved inTheorem 10To find the optimal distances 119867

119894119895 119894 = 1 119899 119895 =

1 2 that give the optimal search plan 120601lowast(119905) =

(120601lowast1(119905) 120601lowast2(119905) 120601lowast

119899(119905))we solve themultiobjective nonlinear

programming problem (MONLP) which contains thevariables 120599

119894and the parameters 120582

119894 119894 = 1 119899 We use

the Kuhn-Tucker conditions to obtain the optimal valuesof 120599119894that can give the optimal distances after substituting

(38) and then minimize the expected value of the firstmeeting time The effectiveness of this model is illustratedusing numerical example

In future research it seems that the proposed model willbe extendible to the multiple searchers case by consideringthe combinations of movement of multiple targets

References

[1] B O Koopman ldquoSearch and Screeningrdquo OEG Report 56 TheSummary Reports Group of the Columbia University DivisionofWarResearch (Available from theCenter forNavalAnalyses)1946

[2] S J Benkoski M G Monticino and J R Weisinger ldquoA surveyof the search theory literaturerdquoNaval Research Logistics vol 38pp 469ndash494 1991

[3] J R Frost and L D Stone ldquoReview of SearchTheory AdvancesandApplications to Search and RescueDecision Supportrdquo Finalreport CG-D-15-01 US Coast Guard Research and Develop-ment Center 2001

[4] B J MCCabe ldquoSearching for one dimensional randomwalkerrdquoJournal of Applied Probability vol 11 pp 86ndash93 1974

[5] A A Mohamed ldquoGeneralized search for one dimensionalrandom walkerrdquo International Journal of Pure and AppliedMathematics vol 19 pp 375ndash387 2005

[6] A B El-Rayes and A A Mohamed ldquoSearching for a randomlymoving targetrdquo in Proceedings of the 3rd ORMAConference vol2 pp 323ndash329 Egypt 1989

[7] B Fristedt and D Heath ldquoSearching for a particl and on the realLinerdquo Advances in Applied Probability vol 6 pp 79ndash102 1974

[8] A Ohsumi ldquoOptimal search for a markovian targetrdquo NavalResearch Logistics vol 38 pp 531ndash554 1991

[9] A B El-Rayes A E-M A Mohamed and H M Abou GaballdquoLinear search for a Brownian targetmotionrdquoActaMathematicaScientia vol 23 no 3 pp 321ndash327 2003

[10] A Mohamed M Kassem and M El-Hadidy ldquoMultiplicativelinear search for a brownian targetmotionrdquoAppliMathematicalModel vol 35 pp 4127ndash4139 2011

[11] A Beck ldquoOn the linear search problemrdquo Israel Journal ofMathematics vol 2 pp 221ndash228 1964

[12] A Beck ldquoMore on the linear search problemrdquo Israel Journal ofMathematics vol 3 pp 61ndash70 1965

[13] A Beck and M Beck ldquoSon of the linear search problemrdquo IsraelJournal of Mathematics vol 48 no 2-3 pp 109ndash122 1984

[14] A Beck and M Beck ldquoThe linear search problem rides againrdquoIsrael Journal of Mathematics vol 53 no 3 pp 365ndash372 1986

[15] A Beck and M Beck ldquoRevenge of the linear search problemrdquoSIAM Journal on Control and Optimization vol 30 no 1 pp112ndash122 1992

[16] A Beck and D Newman ldquoYet more on the linear searchproblemrdquo Israel Journal of Mathematics vol 8 pp 419ndash4291970

[17] A Beck and P Warren ldquoThe return of the linear searchproblemrdquo Israel Journal of Mathematics vol 10 pp 169ndash1831972

[18] W Franck ldquoOn an optimal search problemrdquo SIAM Review vol7 pp 503ndash512 1965

[19] P Rousseeuw ldquoOptimal search paths for random variablesrdquoJournal of Computational and Applied Mathematics vol 9 pp279ndash286 1983

[20] D J Reyniers ldquoCoordinated search for an object hidden on thelinerdquo European Journal of Operational Research vol 95 no 3pp 663ndash670 1996

[21] D J Reyniers ldquoCo-ordinating two searchers for an objecthidden on an intervalrdquo Journal of the Operational ResearchSociety vol 46 no 11 pp 1386ndash1392 1995

[22] Z T Balkhi ldquoThe generalized linear search problem existenceof optimal search pathsrdquo Journal of the Operations ResearchSociety of Japan vol 30 pp 399ndash420 1987

[23] Z T Balkhi ldquoGeneralized optimal search paths for continuousunivariate random variablerdquo Operations Research vol 23 pp67ndash96 1987

[24] A A Mohamed H M Abou Gabal and M A El-HadidyldquoCoordinated search for a randomly located target on the planerdquoEuropean Journal of Pure and Applied Mathematics vol 2 pp97ndash111 2009

[25] A A Mohamed H A Fergany and M A El-Hadidy ldquoOn thecoordinated search problem on the planerdquo Istanbul UniversityJournal of the School of Business Administration vol 41 pp 80ndash102 2012

[26] A Mohamed andM El-Hadidy ldquoExistence of a periodic searchstrategy for a parabolic spiral target motion in the planerdquoAfrikaMatematika vol 24 no 2 pp 145ndash160 2013

[27] AMohamed andM El-Hadidy ldquoCoordinated search for a con-ditionally deterministic target motion in the planerdquo EuropeanJournal ofMathematical Sciences vol 2 no 3 pp 272ndash295 2013

[28] W Feller An Introduction to Probability Theory and Its Applica-tions Wiley New York NY USA 2nd edition 1966

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013


Mathematical Problems in Engineering


Abstract and Applied Analysis

ISRN Applied Mathematics


Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics


Journal of Function Spaces and Applications

International Journal of Mathematics and Mathematical Sciences


ISRN Geometry



Discrete Dynamicsin Nature and Society


Volume 2013

Advances in

Mathematical Physics

ISRN Algebra


ProbabilityandStatistics

Journal of


ISRN Mathematical Analysis


Journal ofApplied Mathematics


Advances in

DecisionSciences



Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal


ISRN Discrete Mathematics


DifferentialEquations


Volume 2013


moving withMarkov process along one of119870-nonintersectingarcs to a safe destination within a time limit where the targetstarts at a safe base and tries to pass El-Rayes et al [9]illustrated this problemwhen the targetmoves on the real linewith a Brownian motion and the searcher starts the searchfrom the origin Recently Mohamed et al [10] discussed thisproblem for a Brownian targetmotion on one of 119899-intersectedreal lines in which any information of the target positionis not available to the searchers all the time Mohamed etal formulate a search model and find the conditions underwhich the expected value of the first meeting time betweenone of the searchers and the target is finite Furthermorethey showed the existence of the optimal search plan thatminimizes the expected value of the first meeting time andfound it

On the other hand when the target is located somewhereon the real line according to a known probability distributionthe searcher searches for it with known velocity and triesto find it in minimal expected time It is assumed that thesearcher can change the direction of its motion without anyloss of time The target can be detected only if the searcherreaches the target In an earlier work this problem has beenstudied extensively in many variations mostly by Beck et al[11ndash17] Franck [18] Rousseeuw [19] Reyniers [20 21] andBalkhi [22 23]

Furthermore Mohamed et al [24 25] have got moreinteresting results when they studied this problem to find arandomly located target in the plane The target has sym-metric or asymmetric distribution and with less informationabout this target available to the searchers More recentlyMohamed and El-Hadidy [26] studied this problemwhen thetarget moves with parabolic spiral in the plane and starts itsmotion from a randompoint AlsoMohamed and El-Hadidy[27] disscussed this problem in the plane when the targetmoves randomly with conditionally deterministic motion

Some problems of search may impose using more thanone searcher such aswhenwe search for a valuable target (egperson lost on one of 119899-disjoint roads) or search for a serioustarget (eg a car filledwith explosives whichmoves randomlyin one of 119899-disjoint roads) Thus the main contributions ofthis paper center around studying the problem of searchingfor a one-dimensional random walker target that is movingon one of a system of 119899-disjoint continuous real lines in R119899

(ie not intersected continuous real lines in the 119899-space)The problem that is studied here is very interesting where weargue to give the conditions on a strategy (or trajectories) of119899-searchers one on each line that make the expected valueof the first meeting time between one of the searchers and thetarget be finite and minimum In this problem there exists acomplication of such analysisThis complication is due to thefact that the searchers do not know the initial position of thetarget but only know its probability distribution Otherwisethe problem would be reduced to determine the strategy ofjust one searcher on the same targetrsquos line This problem isalready tackled in [6 7] This work focuses on the necessaryconditions for the existence of finite and optimal search planthat finds a random walker target

The optimal search plan that is proposed here shows thatthe special structure of the search problem can be exploited

Xnminus1

X3

Xn

X1

X2

L3 L2

L1

H32

H31

H33

H23H21H22H24

H14H12 H11

H13

Hn3

Hn1

Hn2

Hn4

12057520

12057530

120575n012057510

Lnminus1

Figure 1 The search plan 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) of the

searchers 119878119894 119894 = 1 2 119899

to obtain the efficient solution For example the search planfor a criminal drunk leaves its cache and walks up and downthrough one of 119899-disjoint streets totally disoriented

This paper is organized as follows In Section 2 weformulate the problem We display some properties that thesearch model should satisfy in Section 3 The search planand the conditions that make the expected value of the firstmeeting time between one of the searchers and the targetbe finite is discussed in Section 4 The existence of optimalsearch plan that minimizes the expected value of the firstmeeting time is presented in Section 5 The optimal searchplan is studied in Section 6 In Section 7 we illustrate theeffectiveness of this model using numerical example Finallythe paper concludes with a discussion of the results anddirections for future research

2 Problem Description

The problem under study can be formally described asfollows We have 119899-searchers 119878

119894 119894 = 1 2 119899 that start the

searching process from any point rather than the origin of theline 119871

119894 119894 = 1 2 119899 respectively as in Figure 1 Each of the

searchersmoves continuously along its line in both directionsof the starting pointThe searcher 119878

119894would conduct its search

in the following manner Start at 1205751198940and go to the left (right)

as far as 1198671198941 Then turn back to explore the right (left) part

of 1205751198940as far as119867

1198942 Retrace the steps again to explore the left

(right) part of 1198671198941as far as 119867

1198943and so forth In this paper

we need to determine 119867119894119895 119894 = 1 2 119899 119895 = 1 2 that

minimize the first meeting time between one of the searchersand the moving target



119894119895119895ge1



1119895= 1) = 119901

1 119875(1198832119895=

1) = 1199012 119875(119883

119899119895= 1) = 119901

119899] and [119875(119883

1119895= minus1) =

1199021 119875(1198832119895= minus1) = 119902

2 119875(119883

119899119895= minus1) = 119902

119899] where

[1199011 1199012 119901

119899] + [119902

1 1199022 119902

119899] = [1 1 1]


sum119905

119895=1119883119894119895 119882(0) = 0 and 119883



119894gt 0 and 119883

119894119895such that

0 le 1198701198941= (120585119894+ 119909119894)2 le 120585

119894 where 119870


119875 (119882(120585119894) = 1198701198941)

=

(120585119894

1198701198941

)1199011198701198941

119894119902120585119894minus1198701198941

119894


(1)





119899(119905)) is a


119894and


1003816100381610038161003816120601119894 (1199051) minus 120601119894 (1199052)1003816100381610038161003816 lt V119894

10038161003816100381610038161199051 minus 11990521003816100381610038161003816 forall119905

1 1199052isin 119868+

120601119894(0) = 0 119894 = 1 2 119899

(2)



120591120601=

inf 119905 either one of 120601119894(119905) = 119883

0+119882(119905)

119894 = 1 2 119899





119894 119894 = 1 2 119899 Also




plan 120601(119905) = (1206011(119905) 1206012(119905) 120601





1(119905) 1206012(119905) 120601

119899(119905)) 120601

119894(119905) isin ΦV119894

(119905) 119894 =



such that 120599119894gt 1 119862

119894= (120599119894minus 1)(120599

119894+ 1) gt |119901

119894minus 119902119894| 120582119894=

119887120603119894 119887 = 1 2 and 120603

119894(1 minus 119862




119894119895 119894 =

1 2 119899 119895 = 1 2 by 119866119894119895= 120582119894(120599119895

119894minus 1) 119889

119894119895=

(minus1)119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] and 119867119894119895= 119889119894119895+ 1205751198940 We also


) = 119882(119866119894(2119895+1)

) minus 1198701198941(119866119894(2119895+1)

)119894(119866119894(2119895)) = 119882(119866

119894(2119895)) + 119870

1198942(119866119894(2119895)) where 119870

1198941(119866119894(2119895+1)

) =

119862119894119866119894(2119895+1)

1198701198942(119866119894(2119895)) = 119862

119894119866119894(2119895)



sect=1120600sect lt

sum120581

sect=1 120600sect



119899= 1 on 119871


119899 And they




1(119905) 1206012(119905) 120601

119899(119905)) be finite (ie 119864

120591120601lt


1(119905) 120601lowast2(119905) 120601lowast









sdot sdot sdot lt 119867119894(119895+2)

lt 0 lt 119867119894119895lt 119867119894(119895minus2)



(4)


le 119905 le 119866119894(2119895) 1 le 119895 le F2 then

120601119894(119905) = 119867


119894(2119895minus1)] (5)

if 119866119894119895le 119905 le 119866

119894(119895+1) 119895 ge F + 1 then

120601119894(119905) = 119867

119894119895+ (minus1)

119895

[119905 minus 119866119894119895] (6)

if 119866119894(2119895)

le 119905 le 119866119894(2119895+1)


120601119894(119905) = 120575

1198940minus 119867119894(2119895)+ [119905 minus 119866

119894(2119895)] (7)


+ then we have


119894(119895minus2)lt 119867119894119895

lt 0 lt 119867119894(119895+2)




le 119905 le 119866119894(2119895)

1 le 119895 le (F + 1)2 then

120601119894(119905) = 120575

1198940minus 119867119894(2119895minus1)


] (9)

if 119866119894119895le 119905 le 119866

119894(119895+1) 119895 ge F + 1 then

120601119894(119905) = 119867

119894119895+ (minus1)

119895

[119905 minus 119866119894119895] (10)

if 119866119894(2119895)

le 119905 le 119866119894(2119895+1)


120601119894(119905) = 119867

119894(2119895)minus 1205751198940+ [119905 minus 119866

119894(2119895)] (11)


119894 120599119894and


119909119894isin 119868 where 119909





and for all 120585119894ge 1 119881(120585

119894) = (119882(120585

119894120599119894) minus 119862119894120585119894120599119894)2 then



119894) =

sumΓ



on the integers 119864(119884119894119895) = (120599

1198942)[119864(119883

119894119895) minus 119862

119894] and the


2119895= F) gt


119894(1 + 119862

119894)2 le

F le 120599119894(1 minus 119862

119894)2


2) =

1199092) gt 0 119901(119881(120585

119899) = 119909



119894120599119894(1 + 119862

119894)2 le 119909

119894le

120585119894120599119894(1 minus 119862

119894)2


1198941and 119903

1198942


119894isin R whereR


119894ge 0

we have [119901(0 le 119881(1205851+ 1) le 119909

1) 119901(0 le 119881(120585

2+ 1) le

1199092) 119901(0 le 119881(120585

119899+ 1) le 119909

119899)] le [119901(0 le 119881(120585

1) le

1199091) 119901(0 le 119881(120585

2) le 1199092) 119901(0 le 119881(120585

119899) le 119909119899)] and

if 119909119894lt 0 we have [119901(119909

1le 119881(120585

1+ 1) lt 0) 119901(119909

2le

119881(1205852+ 1) lt 0) 119901(119909

119899le 119881(120585

119899+ 1) lt 0)] le [119901(119909

1le

119881(1205851) lt 0) 119901(119909

2le 119881(120585

2) lt 0) 119901(119909

119899le 119881(120585

119899) lt

0)]


F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2) 119895 ge 1

and [119901(1198841119895= 1199091) 119901(1198842119895= 1199092) 119901(119884

119899119895= 119909119899)] =

[119901(119882(1205991)) = 2119909

1+ 12059911198621 119901(119882(120599

2)) = 2119909

2+ 12059921198622

119901(119882(120599119899)) = 2119909

119899+ 120599119899119862119899]


2119895= 1199092) gt


119894+ 120599119894(1 +



have 0 le 119909119894+ 120599119894(1 + 119862

119894)2 le 120599


119894(1 + 119862

119894)2 le

119909119894le 120599119894(1 minus 119862

119894)2 If 119909

119894= F then we have [119901(119884

1119895= F) gt

0 119901(1198842119895= F) gt 0 119901(119884

119899119895= F) gt 0] if and only if

minus120599119894(1 + 119862

119894)2 le F le 120599

119894(1 minus 119862


(1) 119864(sum120599119894F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2)) = (120599

1198942)[119864(119883

119894119895) minus 119862119894] is

proved

(ii) Since 119881(120585119894) = (119882(120585

119894120599119894) minus 119862

119894120585119894120599119894)2 then we have

[119901(119881(1205851) = 119909

1) 119901(119881(120585

2) = 119909

2) 119901(119881(120585

119899) = 119909

119899)] =

[119901((119882(12058511205991) minus 119862112058511205991)2 = 119909

1) 119901((119882(120585

21205992) minus 119862212058521205992)2 =

1199092) 119901((119882(120585

119899120599119899) minus 119862

119899120585119899120599119899)2 = 119909

119899)] = [119901(119882(120585

11205991)) =

21199091+119862112058511205991) 119901(119882(120585

21205992)) = 2119909

2+119862212058521205992) 119901(119882(120585

119899120599119899)) =



1) le

1199091) 119901(0 le 119881(120585

2) le 119909

2) 119901(0 le 119881(120585

119899) le 119909

119899)] =

[sum[1199091]

119895=0119901(119881(120585

1) = 119895) sum

[1199092]

119895=0119901(119881(120585

2) = 119895) sum

[119909119899]

119895=0119901(119881(120585

119899) =

119895)] and if 119909119894lt 0we get [119901(119909

1le 119881(120585

1+1) lt 0) 119901(119909

2le 119881(120585

2+

1) lt 0) 119901(119909119899le 119881(120585

119899+ 1) lt 0)] = [sum

0

119895=[1199091]119901(119881(120585

1) =

119895) sum0

119895=[1199092]119901(119881(120585

2) = 119895) sum

0

119895=[119909119899]119901(119881(120585

119899) = 119895)] where


119894 119894 =


[119901 (119881 (1205851+ 1) = F) 119901 (119881 (120585

2+ 1) = F)

119901 (119881 (120585119899+ 1) = F)]

le [119901 (119881 (1205851) = F) 119901 (119881 (120585

2) = F)

119901 (119881 (120585119899) = F)] if 120585

119894gt 1199031198941119909119894+ 1199031198942

(12)



1) = F) gt

0 119901(119881(1205852) = F) gt 0 119901(119881(120585

119899) = F) gt 0] if and

only if 2F120599119894(1 minus 119862

119894) le 120585

119894le minus2F120599

119894(1 + 119862

119894) We take

1199031198941= minus2120599

119894(1 + 119862

119894) 1199031198942= 0 and then 120585

119894gt 1199031198941|F| rArr 120585

119894gt

minus2F120599119894(1 + 119862

119894) that leads to [119901(119881(120585

1) = F) = 0 119901(119881(120585

2) =


Case (b) If 119862119894gt 1 hence [119901(119881(120585

1) = F) gt 0 119901(119881(120585

2) =


119894(1 +

119862119894) le 120585119894le 2F120599

119894(1 minus 119862


1198941= minus2120599

119894(1 minus

119862119894) 1199031198942= 0 then 120585

119894gt 1199031198941|F| rArr 120585

119894gt 2F120599

119894(1 minus 119862

119894) that leads

to [119901(119881(1205851) = F) = 0 119901(119881(120585

2) = F) = 0 119901(119881(120585

119899) =



119894= (1 minus 119862

119894)2 120573

119894=

1 minus 120572119894and Δ

119894= (120572119894119902119894)120572119894(120573119894119901119894)120573119894 where [119901

1 1199012 119901

119899] +

[1199021 1199022 119902

119899] = [1 1 1] In addition put 119903

1198942= 1(Δ

119894minus

1) 1199031198941= Δ119894max[(1120572

119894 1120573119894)120599119894(Δ119894minus1)] Since 119864(119883

119894119895) = 119862119894and

0 lt 120572119894lt 1 then 120572

119894= 119902119894and then Δ



119894gt

1199031198941|F| then minus120573

119894120585119894120599119894lt F lt 120572


have [119901(119881(1205851) = F) gt 0 119901(119881(120585

2) = F) gt 0 119901(119881(120585

119899) =


1) = F) gt 0 119901(119881(120585

2) =

F) gt 0 119901(119881(120585119899) = F) gt 0] and 120585

119894gt 1199031198941|F| + 119903

1198942then (2)


119894119895= (119883119894119895minus 119862119894)2 then [119901(119885

1119895= 1205721) =

1199011 119901(1198852119895= 1205722) = 119901

2 119901(119885

119899119895= 120572119899) = 119901

119899] [119901(119885

1119895=

minus1205731) = 119902

1 119901(1198852119895= minus120573

2) = 119902

2 119901(119885

119899119895= minus120573

119899) = 119902

119899]


and [119901(119881(1205851) = F) 119901(119881(120585

2) = F) 119901(119881(120585

119899) = F)] =

[119901(sum12058511205991

119895=11198851119895= F) 119901(sum

12058521205992

119895=11198852119895= F) 119901(sum

120585119899120599119899

119895=1119885119899119895=


[119901 (119881 (120585

1+ 1) = F)

119901 (119881 (1205851) = F)

119901 (119881 (120585

2+ 1) = F)

119901 (119881 (1205852) = F)

119901 (119881 (120585119899+ 1) = F)

119901 (119881 (120585119899) = F)

]

= [

[

12059911205731

prod119895=1

12058511205991+ 119895

Δ1(12058511205991+ 119895 + (119895120572

1+F) 120573

1)

times

12059911205721

prod119895=1

12058511205991+ 119895 + 120599

11205731

Δ1(12058511205991+ 119895 + (119894120573

1minusF) 120572

1)

12059921205732

prod119895=1

12058521205992+ 119895

Δ2(12058521205992+ 119895 + (119895120572

2+F) 120573

2)

times

12059921205722

prod119895=1

12058521205992+ 119895 + 120599

21205732

Δ2(12058521205992+ 119895 + (119894120573

2minusF) 120572

2)

120599119899120573119899

prod119895=1

120585119899120599119899+ 119895

Δ119899(120585119899120599119899+ 119895 + (119895120572

119899+F) 120573

119899)

times

120599119899120572119899

prod119895=1

120585119899120599119899+ 119895 + 120599

119899120573119899

Δ119899(120585119899120599119899+ 119895 + (119894120573

119899minusF) 120572

119899)]

]

(13)

Since 120585119894gt 1199031198941|F| + 119903



2+ 1) =

F) 119901(119881(120585119899+ 1) = F)] lt [119901(119881(120585

1) = F) 119901(119881(120585

2) =

F) 119901(119881(120585119899) = F)]

Theorem 3 If 119864(119883119894119895) lt 119862




[119901(119882(1205851)) 119901(119882(120585

2)) 119901(119882(120585

119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for

all 120585119894 119894 = 1 2 119899

Proof For 120577119894gt 0 [119901(119882(120585

1) ge 119862

11205851) 119901(119882(120585

2) ge 119862

21205852)

119901(119882(120585119899) ge 119862

119899120585119899)] = [119901(exp120577

1(119882(1205851) minus 11986211205851) ge 1) 119901(exp

1205772(119882(1205852)minus11986221205852) ge 1) 119901(exp120577

119899(119882(120585119899)minus119862119899120585119899) ge 1)] le

[119864(exp1205771(119882(1205851) minus 119862

11205851)) 119864(exp120577

2(119882(1205852) minus 119862

21205852))

119864(exp120577119899(119882(120585119899) minus 119862

119899120585119899))] = [119891

1(1205771)1205851 1198912(1205772)1205852

119891119899(120577119899)120585119899] where [119891

1(1205771) 1198912(1205772) 119891

119899(120577119899)] = [119864(exp

1205771(1198831119895minus 1198621)) 119864(exp120577

2(1198832119895minus 1198622)) 119864(exp120577

119899(119883119899119895minus

119862119899))] = [119901

1exp1205771(1 minus 119862

1) + 119902


1) 1199012

exp1205772(1 minus 119862

2) + 119902


2) 119901

119899exp

120577119899(1 minus 119862

119899) + 119902

119899exp120577119899(minus1 minus 119862

119899)] if 119864(119883

119894119895) lt 119862

119894 119894 =

1 2 119899 then [11989110158401(0) 1198911015840

2(0) 1198911015840

119899(0)] lt [0 0 0]

and since [1198911(0) 119891

2(0) 119891

119899(0)] lt [1 1 1]

then [min1205771gt01198911(1205771)min

1205772gt01198912(1205772) min

120577119899gt0119891119899(120577119899)]

= [1205761 1205762 120576

119899] lt [1 1 1]


119894 119894 = 1 2 119899


1)) 119901(119882(120585

2))

119901(119882(120585119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for all 120585

119894 119894 = 1 2 119899




119894119895 119894 =

1 2 119899 and if 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) is a search


int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(14)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)


1205751198940gt 0 And if 120575

1198940lt 0 then 119864


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(15)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite





1198940then 119883

119894+ 119882(119905) is smaller than


120601119894gt 119905 119894 = 1 2 119899 are


1206011gt 119905 or


120601119899gt 119905) = sum

119899

119894=1119901(120591120601119894gt 119905) and for any 119895 ge 0

we have

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119883119894+119882(119866

119894(2119895)) lt 119867

119894(2119895)| 119883119894= 119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (119883119894+119882(119866

119894(2119895+1)) gt 119867

119894(2119895+1)| 119883119894= 119909119894) 120574119894(119889119909119894)

(16)


119894(2119895)) + 119862119894119866119894(2119895)

weobtain 119882(119866

119894(2119895)) minus 119867

119894(2119895)lt minus119883

119894= 119909119894 then 119882(119866

119894(2119895)) minus

(minus1)2119895+1

119862119894[119866119894(2119895)

+ 1 + (minus1)2119895+1

] lt minus119909119894leads to 119882(119866

119894(2119895)) minus

(minus1)119862119894[119866119894(2119895)

+ 1 + (minus1)] = 119882(119866119894(2119895)) + 119862

119894[119866119894(2119895)] =


119894(119866119894(2119895)) lt minus119909


119894(119866119894(2119895+1)

) =

119882(119866119894(2119895+1)

) minus 119862119894119866119894(2119895+1)

we get 120595119894(119866119894(2119895+1)


quently

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894) 120574119894(119889119909119894)

(17)

also

119901 (120591120601119894gt 119866119894(2119895))

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895minus1)

) gt minus119909119894) 120574119894(119889119909119894)

(18)


119864120591120601= intinfin

0

119901 (120591120601gt 119905) 119889119905

le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 119905) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

(19)

where 1198661198940= 0 then

119864120591120601le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 1198661119895) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

=

119899

sum119894=1

infin

sum119895=0

(119866119894(119895+1)

minus 119866119894119895) 119901 (120591120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

[120582119894(120599119895+1

119894minus 1) minus 120582

119894(120599119895

119894minus 1)] 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

120582119894120599119895

119894(120599119894minus 1) 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

[

[

120582119894(120599119894minus 1)

infin

sum119895=0

120599119895

119894119901 (120591120601119894gt 119866119894119895)]

]

=

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 119863) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894119901 (120591120601119894gt 1198661198942) + 1205993

119894119901 (120591120601119894gt 1198661198943) + sdot sdot sdot )]

(20)


119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205993

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205994

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198944) lt minus119909

119894) 120574119894(119889119909119894)

+ sdot sdot sdot

+ 120599F119894intinfin

1205751198940

119901 (120595119894(119866119894(Fminus1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+1119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+2119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+3119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+3)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ sdot sdot sdot )]

(21)


thus

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F119894(120599119894+ 1)

times int1205751198940

minusinfin


+ 1205993

119894(120599119894+ 1)int

infin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+ 1205995

119894(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(1198661198945) gt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F+3119894

(120599119894+ 1)

times intinfin

1205751198940


(22)

Leads to

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times [1

119892119894+ (120599119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]]

=

119899

sum119894=1

120582119894(120599119894minus 1) (

1

119892119894)

+

119899

sum119894=1

[120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]

(23)

where

1

119892119894= 119901 (120591

120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

1

119908119894(119909119894) =

infin

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

1

119902119894(119909119894) =

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)

1

ℎ119894(119909119894) =

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(24)


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(25)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite



119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1) (

2

119892119894) + 120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

2

119908119894(119909119894) 120574119894(119889119909119894)

+ int1205751198940

minusinfin

2

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

2

ℎ119894(119909119894) 120574119894(119889119909119894))]

(26)


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)



+1le 119886 =



1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)


minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889


infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)



1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and


(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]


where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)



119894gt 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013



119894119895119895ge1



1119895= 1) = 119901

1 119875(1198832119895=

1) = 1199012 119875(119883

119899119895= 1) = 119901

119899] and [119875(119883

1119895= minus1) =

1199021 119875(1198832119895= minus1) = 119902

2 119875(119883

119899119895= minus1) = 119902

119899] where

[1199011 1199012 119901

119899] + [119902

1 1199022 119902

119899] = [1 1 1]


sum119905

119895=1119883119894119895 119882(0) = 0 and 119883



119894gt 0 and 119883

119894119895such that

0 le 1198701198941= (120585119894+ 119909119894)2 le 120585

119894 where 119870


119875 (119882(120585119894) = 1198701198941)

=

(120585119894

1198701198941

)1199011198701198941

119894119902120585119894minus1198701198941

119894


(1)





119899(119905)) is a


119894and


1003816100381610038161003816120601119894 (1199051) minus 120601119894 (1199052)1003816100381610038161003816 lt V119894

10038161003816100381610038161199051 minus 11990521003816100381610038161003816 forall119905

1 1199052isin 119868+

120601119894(0) = 0 119894 = 1 2 119899

(2)



120591120601=

inf 119905 either one of 120601119894(119905) = 119883

0+119882(119905)

119894 = 1 2 119899





119894 119894 = 1 2 119899 Also




plan 120601(119905) = (1206011(119905) 1206012(119905) 120601





1(119905) 1206012(119905) 120601

119899(119905)) 120601

119894(119905) isin ΦV119894

(119905) 119894 =



such that 120599119894gt 1 119862

119894= (120599119894minus 1)(120599

119894+ 1) gt |119901

119894minus 119902119894| 120582119894=

119887120603119894 119887 = 1 2 and 120603

119894(1 minus 119862




119894119895 119894 =

1 2 119899 119895 = 1 2 by 119866119894119895= 120582119894(120599119895

119894minus 1) 119889

119894119895=

(minus1)119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] and 119867119894119895= 119889119894119895+ 1205751198940 We also


) = 119882(119866119894(2119895+1)

) minus 1198701198941(119866119894(2119895+1)

)119894(119866119894(2119895)) = 119882(119866

119894(2119895)) + 119870

1198942(119866119894(2119895)) where 119870

1198941(119866119894(2119895+1)

) =

119862119894119866119894(2119895+1)

1198701198942(119866119894(2119895)) = 119862

119894119866119894(2119895)



sect=1120600sect lt

sum120581

sect=1 120600sect



119899= 1 on 119871


119899 And they




1(119905) 1206012(119905) 120601

119899(119905)) be finite (ie 119864

120591120601lt


1(119905) 120601lowast2(119905) 120601lowast









sdot sdot sdot lt 119867119894(119895+2)

lt 0 lt 119867119894119895lt 119867119894(119895minus2)



(4)


le 119905 le 119866119894(2119895) 1 le 119895 le F2 then

120601119894(119905) = 119867


119894(2119895minus1)] (5)

if 119866119894119895le 119905 le 119866

119894(119895+1) 119895 ge F + 1 then

120601119894(119905) = 119867

119894119895+ (minus1)

119895

[119905 minus 119866119894119895] (6)

if 119866119894(2119895)

le 119905 le 119866119894(2119895+1)


120601119894(119905) = 120575

1198940minus 119867119894(2119895)+ [119905 minus 119866

119894(2119895)] (7)


+ then we have


119894(119895minus2)lt 119867119894119895

lt 0 lt 119867119894(119895+2)




le 119905 le 119866119894(2119895)

1 le 119895 le (F + 1)2 then

120601119894(119905) = 120575

1198940minus 119867119894(2119895minus1)


] (9)

if 119866119894119895le 119905 le 119866

119894(119895+1) 119895 ge F + 1 then

120601119894(119905) = 119867

119894119895+ (minus1)

119895

[119905 minus 119866119894119895] (10)

if 119866119894(2119895)

le 119905 le 119866119894(2119895+1)


120601119894(119905) = 119867

119894(2119895)minus 1205751198940+ [119905 minus 119866

119894(2119895)] (11)


119894 120599119894and


119909119894isin 119868 where 119909





and for all 120585119894ge 1 119881(120585

119894) = (119882(120585

119894120599119894) minus 119862119894120585119894120599119894)2 then



119894) =

sumΓ



on the integers 119864(119884119894119895) = (120599

1198942)[119864(119883

119894119895) minus 119862

119894] and the


2119895= F) gt


119894(1 + 119862

119894)2 le

F le 120599119894(1 minus 119862

119894)2


2) =

1199092) gt 0 119901(119881(120585

119899) = 119909



119894120599119894(1 + 119862

119894)2 le 119909

119894le

120585119894120599119894(1 minus 119862

119894)2


1198941and 119903

1198942


119894isin R whereR


119894ge 0

we have [119901(0 le 119881(1205851+ 1) le 119909

1) 119901(0 le 119881(120585

2+ 1) le

1199092) 119901(0 le 119881(120585

119899+ 1) le 119909

119899)] le [119901(0 le 119881(120585

1) le

1199091) 119901(0 le 119881(120585

2) le 1199092) 119901(0 le 119881(120585

119899) le 119909119899)] and

if 119909119894lt 0 we have [119901(119909

1le 119881(120585

1+ 1) lt 0) 119901(119909

2le

119881(1205852+ 1) lt 0) 119901(119909

119899le 119881(120585

119899+ 1) lt 0)] le [119901(119909

1le

119881(1205851) lt 0) 119901(119909

2le 119881(120585

2) lt 0) 119901(119909

119899le 119881(120585

119899) lt

0)]


F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2) 119895 ge 1

and [119901(1198841119895= 1199091) 119901(1198842119895= 1199092) 119901(119884

119899119895= 119909119899)] =

[119901(119882(1205991)) = 2119909

1+ 12059911198621 119901(119882(120599

2)) = 2119909

2+ 12059921198622

119901(119882(120599119899)) = 2119909

119899+ 120599119899119862119899]


2119895= 1199092) gt


119894+ 120599119894(1 +



have 0 le 119909119894+ 120599119894(1 + 119862

119894)2 le 120599


119894(1 + 119862

119894)2 le

119909119894le 120599119894(1 minus 119862

119894)2 If 119909

119894= F then we have [119901(119884

1119895= F) gt

0 119901(1198842119895= F) gt 0 119901(119884

119899119895= F) gt 0] if and only if

minus120599119894(1 + 119862

119894)2 le F le 120599

119894(1 minus 119862


(1) 119864(sum120599119894F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2)) = (120599

1198942)[119864(119883

119894119895) minus 119862119894] is

proved

(ii) Since 119881(120585119894) = (119882(120585

119894120599119894) minus 119862

119894120585119894120599119894)2 then we have

[119901(119881(1205851) = 119909

1) 119901(119881(120585

2) = 119909

2) 119901(119881(120585

119899) = 119909

119899)] =

[119901((119882(12058511205991) minus 119862112058511205991)2 = 119909

1) 119901((119882(120585

21205992) minus 119862212058521205992)2 =

1199092) 119901((119882(120585

119899120599119899) minus 119862

119899120585119899120599119899)2 = 119909

119899)] = [119901(119882(120585

11205991)) =

21199091+119862112058511205991) 119901(119882(120585

21205992)) = 2119909

2+119862212058521205992) 119901(119882(120585

119899120599119899)) =



1) le

1199091) 119901(0 le 119881(120585

2) le 119909

2) 119901(0 le 119881(120585

119899) le 119909

119899)] =

[sum[1199091]

119895=0119901(119881(120585

1) = 119895) sum

[1199092]

119895=0119901(119881(120585

2) = 119895) sum

[119909119899]

119895=0119901(119881(120585

119899) =

119895)] and if 119909119894lt 0we get [119901(119909

1le 119881(120585

1+1) lt 0) 119901(119909

2le 119881(120585

2+

1) lt 0) 119901(119909119899le 119881(120585

119899+ 1) lt 0)] = [sum

0

119895=[1199091]119901(119881(120585

1) =

119895) sum0

119895=[1199092]119901(119881(120585

2) = 119895) sum

0

119895=[119909119899]119901(119881(120585

119899) = 119895)] where


119894 119894 =


[119901 (119881 (1205851+ 1) = F) 119901 (119881 (120585

2+ 1) = F)

119901 (119881 (120585119899+ 1) = F)]

le [119901 (119881 (1205851) = F) 119901 (119881 (120585

2) = F)

119901 (119881 (120585119899) = F)] if 120585

119894gt 1199031198941119909119894+ 1199031198942

(12)



1) = F) gt

0 119901(119881(1205852) = F) gt 0 119901(119881(120585

119899) = F) gt 0] if and

only if 2F120599119894(1 minus 119862

119894) le 120585

119894le minus2F120599

119894(1 + 119862

119894) We take

1199031198941= minus2120599

119894(1 + 119862

119894) 1199031198942= 0 and then 120585

119894gt 1199031198941|F| rArr 120585

119894gt

minus2F120599119894(1 + 119862

119894) that leads to [119901(119881(120585

1) = F) = 0 119901(119881(120585

2) =


Case (b) If 119862119894gt 1 hence [119901(119881(120585

1) = F) gt 0 119901(119881(120585

2) =


119894(1 +

119862119894) le 120585119894le 2F120599

119894(1 minus 119862


1198941= minus2120599

119894(1 minus

119862119894) 1199031198942= 0 then 120585

119894gt 1199031198941|F| rArr 120585

119894gt 2F120599

119894(1 minus 119862

119894) that leads

to [119901(119881(1205851) = F) = 0 119901(119881(120585

2) = F) = 0 119901(119881(120585

119899) =



119894= (1 minus 119862

119894)2 120573

119894=

1 minus 120572119894and Δ

119894= (120572119894119902119894)120572119894(120573119894119901119894)120573119894 where [119901

1 1199012 119901

119899] +

[1199021 1199022 119902

119899] = [1 1 1] In addition put 119903

1198942= 1(Δ

119894minus

1) 1199031198941= Δ119894max[(1120572

119894 1120573119894)120599119894(Δ119894minus1)] Since 119864(119883

119894119895) = 119862119894and

0 lt 120572119894lt 1 then 120572

119894= 119902119894and then Δ



119894gt

1199031198941|F| then minus120573

119894120585119894120599119894lt F lt 120572


have [119901(119881(1205851) = F) gt 0 119901(119881(120585

2) = F) gt 0 119901(119881(120585

119899) =


1) = F) gt 0 119901(119881(120585

2) =

F) gt 0 119901(119881(120585119899) = F) gt 0] and 120585

119894gt 1199031198941|F| + 119903

1198942then (2)


119894119895= (119883119894119895minus 119862119894)2 then [119901(119885

1119895= 1205721) =

1199011 119901(1198852119895= 1205722) = 119901

2 119901(119885

119899119895= 120572119899) = 119901

119899] [119901(119885

1119895=

minus1205731) = 119902

1 119901(1198852119895= minus120573

2) = 119902

2 119901(119885

119899119895= minus120573

119899) = 119902

119899]


and [119901(119881(1205851) = F) 119901(119881(120585

2) = F) 119901(119881(120585

119899) = F)] =

[119901(sum12058511205991

119895=11198851119895= F) 119901(sum

12058521205992

119895=11198852119895= F) 119901(sum

120585119899120599119899

119895=1119885119899119895=


[119901 (119881 (120585

1+ 1) = F)

119901 (119881 (1205851) = F)

119901 (119881 (120585

2+ 1) = F)

119901 (119881 (1205852) = F)

119901 (119881 (120585119899+ 1) = F)

119901 (119881 (120585119899) = F)

]

= [

[

12059911205731

prod119895=1

12058511205991+ 119895

Δ1(12058511205991+ 119895 + (119895120572

1+F) 120573

1)

times

12059911205721

prod119895=1

12058511205991+ 119895 + 120599

11205731

Δ1(12058511205991+ 119895 + (119894120573

1minusF) 120572

1)

12059921205732

prod119895=1

12058521205992+ 119895

Δ2(12058521205992+ 119895 + (119895120572

2+F) 120573

2)

times

12059921205722

prod119895=1

12058521205992+ 119895 + 120599

21205732

Δ2(12058521205992+ 119895 + (119894120573

2minusF) 120572

2)

120599119899120573119899

prod119895=1

120585119899120599119899+ 119895

Δ119899(120585119899120599119899+ 119895 + (119895120572

119899+F) 120573

119899)

times

120599119899120572119899

prod119895=1

120585119899120599119899+ 119895 + 120599

119899120573119899

Δ119899(120585119899120599119899+ 119895 + (119894120573

119899minusF) 120572

119899)]

]

(13)

Since 120585119894gt 1199031198941|F| + 119903



2+ 1) =

F) 119901(119881(120585119899+ 1) = F)] lt [119901(119881(120585

1) = F) 119901(119881(120585

2) =

F) 119901(119881(120585119899) = F)]

Theorem 3 If 119864(119883119894119895) lt 119862




[119901(119882(1205851)) 119901(119882(120585

2)) 119901(119882(120585

119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for

all 120585119894 119894 = 1 2 119899

Proof For 120577119894gt 0 [119901(119882(120585

1) ge 119862

11205851) 119901(119882(120585

2) ge 119862

21205852)

119901(119882(120585119899) ge 119862

119899120585119899)] = [119901(exp120577

1(119882(1205851) minus 11986211205851) ge 1) 119901(exp

1205772(119882(1205852)minus11986221205852) ge 1) 119901(exp120577

119899(119882(120585119899)minus119862119899120585119899) ge 1)] le

[119864(exp1205771(119882(1205851) minus 119862

11205851)) 119864(exp120577

2(119882(1205852) minus 119862

21205852))

119864(exp120577119899(119882(120585119899) minus 119862

119899120585119899))] = [119891

1(1205771)1205851 1198912(1205772)1205852

119891119899(120577119899)120585119899] where [119891

1(1205771) 1198912(1205772) 119891

119899(120577119899)] = [119864(exp

1205771(1198831119895minus 1198621)) 119864(exp120577

2(1198832119895minus 1198622)) 119864(exp120577

119899(119883119899119895minus

119862119899))] = [119901

1exp1205771(1 minus 119862

1) + 119902


1) 1199012

exp1205772(1 minus 119862

2) + 119902


2) 119901

119899exp

120577119899(1 minus 119862

119899) + 119902

119899exp120577119899(minus1 minus 119862

119899)] if 119864(119883

119894119895) lt 119862

119894 119894 =

1 2 119899 then [11989110158401(0) 1198911015840

2(0) 1198911015840

119899(0)] lt [0 0 0]

and since [1198911(0) 119891

2(0) 119891

119899(0)] lt [1 1 1]

then [min1205771gt01198911(1205771)min

1205772gt01198912(1205772) min

120577119899gt0119891119899(120577119899)]

= [1205761 1205762 120576

119899] lt [1 1 1]


119894 119894 = 1 2 119899


1)) 119901(119882(120585

2))

119901(119882(120585119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for all 120585

119894 119894 = 1 2 119899




119894119895 119894 =

1 2 119899 and if 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) is a search


int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(14)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)


1205751198940gt 0 And if 120575

1198940lt 0 then 119864


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(15)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite





1198940then 119883

119894+ 119882(119905) is smaller than


120601119894gt 119905 119894 = 1 2 119899 are


1206011gt 119905 or


120601119899gt 119905) = sum

119899

119894=1119901(120591120601119894gt 119905) and for any 119895 ge 0

we have

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119883119894+119882(119866

119894(2119895)) lt 119867

119894(2119895)| 119883119894= 119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (119883119894+119882(119866

119894(2119895+1)) gt 119867

119894(2119895+1)| 119883119894= 119909119894) 120574119894(119889119909119894)

(16)


119894(2119895)) + 119862119894119866119894(2119895)

weobtain 119882(119866

119894(2119895)) minus 119867

119894(2119895)lt minus119883

119894= 119909119894 then 119882(119866

119894(2119895)) minus

(minus1)2119895+1

119862119894[119866119894(2119895)

+ 1 + (minus1)2119895+1

] lt minus119909119894leads to 119882(119866

119894(2119895)) minus

(minus1)119862119894[119866119894(2119895)

+ 1 + (minus1)] = 119882(119866119894(2119895)) + 119862

119894[119866119894(2119895)] =


119894(119866119894(2119895)) lt minus119909


119894(119866119894(2119895+1)

) =

119882(119866119894(2119895+1)

) minus 119862119894119866119894(2119895+1)

we get 120595119894(119866119894(2119895+1)


quently

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894) 120574119894(119889119909119894)

(17)

also

119901 (120591120601119894gt 119866119894(2119895))

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895minus1)

) gt minus119909119894) 120574119894(119889119909119894)

(18)


119864120591120601= intinfin

0

119901 (120591120601gt 119905) 119889119905

le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 119905) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

(19)

where 1198661198940= 0 then

119864120591120601le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 1198661119895) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

=

119899

sum119894=1

infin

sum119895=0

(119866119894(119895+1)

minus 119866119894119895) 119901 (120591120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

[120582119894(120599119895+1

119894minus 1) minus 120582

119894(120599119895

119894minus 1)] 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

120582119894120599119895

119894(120599119894minus 1) 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

[

[

120582119894(120599119894minus 1)

infin

sum119895=0

120599119895

119894119901 (120591120601119894gt 119866119894119895)]

]

=

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 119863) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894119901 (120591120601119894gt 1198661198942) + 1205993

119894119901 (120591120601119894gt 1198661198943) + sdot sdot sdot )]

(20)


119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205993

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205994

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198944) lt minus119909

119894) 120574119894(119889119909119894)

+ sdot sdot sdot

+ 120599F119894intinfin

1205751198940

119901 (120595119894(119866119894(Fminus1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+1119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+2119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+3119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+3)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ sdot sdot sdot )]

(21)


thus

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F119894(120599119894+ 1)

times int1205751198940

minusinfin


+ 1205993

119894(120599119894+ 1)int

infin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+ 1205995

119894(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(1198661198945) gt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F+3119894

(120599119894+ 1)

times intinfin

1205751198940


(22)

Leads to

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times [1

119892119894+ (120599119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]]

=

119899

sum119894=1

120582119894(120599119894minus 1) (

1

119892119894)

+

119899

sum119894=1

[120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]

(23)

where

1

119892119894= 119901 (120591

120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

1

119908119894(119909119894) =

infin

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

1

119902119894(119909119894) =

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)

1

ℎ119894(119909119894) =

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(24)


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(25)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite



119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1) (

2

119892119894) + 120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

2

119908119894(119909119894) 120574119894(119889119909119894)

+ int1205751198940

minusinfin

2

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

2

ℎ119894(119909119894) 120574119894(119889119909119894))]

(26)


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)



+1le 119886 =



1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)


minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889


infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)



1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and


(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]


where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)



119894gt 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013



le 119905 le 119866119894(2119895)

1 le 119895 le (F + 1)2 then

120601119894(119905) = 120575

1198940minus 119867119894(2119895minus1)


] (9)

if 119866119894119895le 119905 le 119866

119894(119895+1) 119895 ge F + 1 then

120601119894(119905) = 119867

119894119895+ (minus1)

119895

[119905 minus 119866119894119895] (10)

if 119866119894(2119895)

le 119905 le 119866119894(2119895+1)


120601119894(119905) = 119867

119894(2119895)minus 1205751198940+ [119905 minus 119866

119894(2119895)] (11)


119894 120599119894and


119909119894isin 119868 where 119909





and for all 120585119894ge 1 119881(120585

119894) = (119882(120585

119894120599119894) minus 119862119894120585119894120599119894)2 then



119894) =

sumΓ



on the integers 119864(119884119894119895) = (120599

1198942)[119864(119883

119894119895) minus 119862

119894] and the


2119895= F) gt


119894(1 + 119862

119894)2 le

F le 120599119894(1 minus 119862

119894)2


2) =

1199092) gt 0 119901(119881(120585

119899) = 119909



119894120599119894(1 + 119862

119894)2 le 119909

119894le

120585119894120599119894(1 minus 119862

119894)2


1198941and 119903

1198942


119894isin R whereR


119894ge 0

we have [119901(0 le 119881(1205851+ 1) le 119909

1) 119901(0 le 119881(120585

2+ 1) le

1199092) 119901(0 le 119881(120585

119899+ 1) le 119909

119899)] le [119901(0 le 119881(120585

1) le

1199091) 119901(0 le 119881(120585

2) le 1199092) 119901(0 le 119881(120585

119899) le 119909119899)] and

if 119909119894lt 0 we have [119901(119909

1le 119881(120585

1+ 1) lt 0) 119901(119909

2le

119881(1205852+ 1) lt 0) 119901(119909

119899le 119881(120585

119899+ 1) lt 0)] le [119901(119909

1le

119881(1205851) lt 0) 119901(119909

2le 119881(120585

2) lt 0) 119901(119909

119899le 119881(120585

119899) lt

0)]


F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2) 119895 ge 1

and [119901(1198841119895= 1199091) 119901(1198842119895= 1199092) 119901(119884

119899119895= 119909119899)] =

[119901(119882(1205991)) = 2119909

1+ 12059911198621 119901(119882(120599

2)) = 2119909

2+ 12059921198622

119901(119882(120599119899)) = 2119909

119899+ 120599119899119862119899]


2119895= 1199092) gt


119894+ 120599119894(1 +



have 0 le 119909119894+ 120599119894(1 + 119862

119894)2 le 120599


119894(1 + 119862

119894)2 le

119909119894le 120599119894(1 minus 119862

119894)2 If 119909

119894= F then we have [119901(119884

1119895= F) gt

0 119901(1198842119895= F) gt 0 119901(119884

119899119895= F) gt 0] if and only if

minus120599119894(1 + 119862

119894)2 le F le 120599

119894(1 minus 119862


(1) 119864(sum120599119894F=1((119883119894(F+(119895minus1)120599119894)

minus 119862119894)2)) = (120599

1198942)[119864(119883

119894119895) minus 119862119894] is

proved

(ii) Since 119881(120585119894) = (119882(120585

119894120599119894) minus 119862

119894120585119894120599119894)2 then we have

[119901(119881(1205851) = 119909

1) 119901(119881(120585

2) = 119909

2) 119901(119881(120585

119899) = 119909

119899)] =

[119901((119882(12058511205991) minus 119862112058511205991)2 = 119909

1) 119901((119882(120585

21205992) minus 119862212058521205992)2 =

1199092) 119901((119882(120585

119899120599119899) minus 119862

119899120585119899120599119899)2 = 119909

119899)] = [119901(119882(120585

11205991)) =

21199091+119862112058511205991) 119901(119882(120585

21205992)) = 2119909

2+119862212058521205992) 119901(119882(120585

119899120599119899)) =



1) le

1199091) 119901(0 le 119881(120585

2) le 119909

2) 119901(0 le 119881(120585

119899) le 119909

119899)] =

[sum[1199091]

119895=0119901(119881(120585

1) = 119895) sum

[1199092]

119895=0119901(119881(120585

2) = 119895) sum

[119909119899]

119895=0119901(119881(120585

119899) =

119895)] and if 119909119894lt 0we get [119901(119909

1le 119881(120585

1+1) lt 0) 119901(119909

2le 119881(120585

2+

1) lt 0) 119901(119909119899le 119881(120585

119899+ 1) lt 0)] = [sum

0

119895=[1199091]119901(119881(120585

1) =

119895) sum0

119895=[1199092]119901(119881(120585

2) = 119895) sum

0

119895=[119909119899]119901(119881(120585

119899) = 119895)] where


119894 119894 =


[119901 (119881 (1205851+ 1) = F) 119901 (119881 (120585

2+ 1) = F)

119901 (119881 (120585119899+ 1) = F)]

le [119901 (119881 (1205851) = F) 119901 (119881 (120585

2) = F)

119901 (119881 (120585119899) = F)] if 120585

119894gt 1199031198941119909119894+ 1199031198942

(12)



1) = F) gt

0 119901(119881(1205852) = F) gt 0 119901(119881(120585

119899) = F) gt 0] if and

only if 2F120599119894(1 minus 119862

119894) le 120585

119894le minus2F120599

119894(1 + 119862

119894) We take

1199031198941= minus2120599

119894(1 + 119862

119894) 1199031198942= 0 and then 120585

119894gt 1199031198941|F| rArr 120585

119894gt

minus2F120599119894(1 + 119862

119894) that leads to [119901(119881(120585

1) = F) = 0 119901(119881(120585

2) =


Case (b) If 119862119894gt 1 hence [119901(119881(120585

1) = F) gt 0 119901(119881(120585

2) =


119894(1 +

119862119894) le 120585119894le 2F120599

119894(1 minus 119862


1198941= minus2120599

119894(1 minus

119862119894) 1199031198942= 0 then 120585

119894gt 1199031198941|F| rArr 120585

119894gt 2F120599

119894(1 minus 119862

119894) that leads

to [119901(119881(1205851) = F) = 0 119901(119881(120585

2) = F) = 0 119901(119881(120585

119899) =



119894= (1 minus 119862

119894)2 120573

119894=

1 minus 120572119894and Δ

119894= (120572119894119902119894)120572119894(120573119894119901119894)120573119894 where [119901

1 1199012 119901

119899] +

[1199021 1199022 119902

119899] = [1 1 1] In addition put 119903

1198942= 1(Δ

119894minus

1) 1199031198941= Δ119894max[(1120572

119894 1120573119894)120599119894(Δ119894minus1)] Since 119864(119883

119894119895) = 119862119894and

0 lt 120572119894lt 1 then 120572

119894= 119902119894and then Δ



119894gt

1199031198941|F| then minus120573

119894120585119894120599119894lt F lt 120572


have [119901(119881(1205851) = F) gt 0 119901(119881(120585

2) = F) gt 0 119901(119881(120585

119899) =


1) = F) gt 0 119901(119881(120585

2) =

F) gt 0 119901(119881(120585119899) = F) gt 0] and 120585

119894gt 1199031198941|F| + 119903

1198942then (2)


119894119895= (119883119894119895minus 119862119894)2 then [119901(119885

1119895= 1205721) =

1199011 119901(1198852119895= 1205722) = 119901

2 119901(119885

119899119895= 120572119899) = 119901

119899] [119901(119885

1119895=

minus1205731) = 119902

1 119901(1198852119895= minus120573

2) = 119902

2 119901(119885

119899119895= minus120573

119899) = 119902

119899]


and [119901(119881(1205851) = F) 119901(119881(120585

2) = F) 119901(119881(120585

119899) = F)] =

[119901(sum12058511205991

119895=11198851119895= F) 119901(sum

12058521205992

119895=11198852119895= F) 119901(sum

120585119899120599119899

119895=1119885119899119895=


[119901 (119881 (120585

1+ 1) = F)

119901 (119881 (1205851) = F)

119901 (119881 (120585

2+ 1) = F)

119901 (119881 (1205852) = F)

119901 (119881 (120585119899+ 1) = F)

119901 (119881 (120585119899) = F)

]

= [

[

12059911205731

prod119895=1

12058511205991+ 119895

Δ1(12058511205991+ 119895 + (119895120572

1+F) 120573

1)

times

12059911205721

prod119895=1

12058511205991+ 119895 + 120599

11205731

Δ1(12058511205991+ 119895 + (119894120573

1minusF) 120572

1)

12059921205732

prod119895=1

12058521205992+ 119895

Δ2(12058521205992+ 119895 + (119895120572

2+F) 120573

2)

times

12059921205722

prod119895=1

12058521205992+ 119895 + 120599

21205732

Δ2(12058521205992+ 119895 + (119894120573

2minusF) 120572

2)

120599119899120573119899

prod119895=1

120585119899120599119899+ 119895

Δ119899(120585119899120599119899+ 119895 + (119895120572

119899+F) 120573

119899)

times

120599119899120572119899

prod119895=1

120585119899120599119899+ 119895 + 120599

119899120573119899

Δ119899(120585119899120599119899+ 119895 + (119894120573

119899minusF) 120572

119899)]

]

(13)

Since 120585119894gt 1199031198941|F| + 119903



2+ 1) =

F) 119901(119881(120585119899+ 1) = F)] lt [119901(119881(120585

1) = F) 119901(119881(120585

2) =

F) 119901(119881(120585119899) = F)]

Theorem 3 If 119864(119883119894119895) lt 119862




[119901(119882(1205851)) 119901(119882(120585

2)) 119901(119882(120585

119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for

all 120585119894 119894 = 1 2 119899

Proof For 120577119894gt 0 [119901(119882(120585

1) ge 119862

11205851) 119901(119882(120585

2) ge 119862

21205852)

119901(119882(120585119899) ge 119862

119899120585119899)] = [119901(exp120577

1(119882(1205851) minus 11986211205851) ge 1) 119901(exp

1205772(119882(1205852)minus11986221205852) ge 1) 119901(exp120577

119899(119882(120585119899)minus119862119899120585119899) ge 1)] le

[119864(exp1205771(119882(1205851) minus 119862

11205851)) 119864(exp120577

2(119882(1205852) minus 119862

21205852))

119864(exp120577119899(119882(120585119899) minus 119862

119899120585119899))] = [119891

1(1205771)1205851 1198912(1205772)1205852

119891119899(120577119899)120585119899] where [119891

1(1205771) 1198912(1205772) 119891

119899(120577119899)] = [119864(exp

1205771(1198831119895minus 1198621)) 119864(exp120577

2(1198832119895minus 1198622)) 119864(exp120577

119899(119883119899119895minus

119862119899))] = [119901

1exp1205771(1 minus 119862

1) + 119902


1) 1199012

exp1205772(1 minus 119862

2) + 119902


2) 119901

119899exp

120577119899(1 minus 119862

119899) + 119902

119899exp120577119899(minus1 minus 119862

119899)] if 119864(119883

119894119895) lt 119862

119894 119894 =

1 2 119899 then [11989110158401(0) 1198911015840

2(0) 1198911015840

119899(0)] lt [0 0 0]

and since [1198911(0) 119891

2(0) 119891

119899(0)] lt [1 1 1]

then [min1205771gt01198911(1205771)min

1205772gt01198912(1205772) min

120577119899gt0119891119899(120577119899)]

= [1205761 1205762 120576

119899] lt [1 1 1]


119894 119894 = 1 2 119899


1)) 119901(119882(120585

2))

119901(119882(120585119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for all 120585

119894 119894 = 1 2 119899




119894119895 119894 =

1 2 119899 and if 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) is a search


int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(14)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)


1205751198940gt 0 And if 120575

1198940lt 0 then 119864


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(15)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite





1198940then 119883

119894+ 119882(119905) is smaller than


120601119894gt 119905 119894 = 1 2 119899 are


1206011gt 119905 or


120601119899gt 119905) = sum

119899

119894=1119901(120591120601119894gt 119905) and for any 119895 ge 0

we have

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119883119894+119882(119866

119894(2119895)) lt 119867

119894(2119895)| 119883119894= 119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (119883119894+119882(119866

119894(2119895+1)) gt 119867

119894(2119895+1)| 119883119894= 119909119894) 120574119894(119889119909119894)

(16)


119894(2119895)) + 119862119894119866119894(2119895)

weobtain 119882(119866

119894(2119895)) minus 119867

119894(2119895)lt minus119883

119894= 119909119894 then 119882(119866

119894(2119895)) minus

(minus1)2119895+1

119862119894[119866119894(2119895)

+ 1 + (minus1)2119895+1

] lt minus119909119894leads to 119882(119866

119894(2119895)) minus

(minus1)119862119894[119866119894(2119895)

+ 1 + (minus1)] = 119882(119866119894(2119895)) + 119862

119894[119866119894(2119895)] =


119894(119866119894(2119895)) lt minus119909


119894(119866119894(2119895+1)

) =

119882(119866119894(2119895+1)

) minus 119862119894119866119894(2119895+1)

we get 120595119894(119866119894(2119895+1)


quently

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894) 120574119894(119889119909119894)

(17)

also

119901 (120591120601119894gt 119866119894(2119895))

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895minus1)

) gt minus119909119894) 120574119894(119889119909119894)

(18)


119864120591120601= intinfin

0

119901 (120591120601gt 119905) 119889119905

le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 119905) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

(19)

where 1198661198940= 0 then

119864120591120601le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 1198661119895) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

=

119899

sum119894=1

infin

sum119895=0

(119866119894(119895+1)

minus 119866119894119895) 119901 (120591120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

[120582119894(120599119895+1

119894minus 1) minus 120582

119894(120599119895

119894minus 1)] 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

120582119894120599119895

119894(120599119894minus 1) 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

[

[

120582119894(120599119894minus 1)

infin

sum119895=0

120599119895

119894119901 (120591120601119894gt 119866119894119895)]

]

=

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 119863) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894119901 (120591120601119894gt 1198661198942) + 1205993

119894119901 (120591120601119894gt 1198661198943) + sdot sdot sdot )]

(20)


119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205993

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205994

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198944) lt minus119909

119894) 120574119894(119889119909119894)

+ sdot sdot sdot

+ 120599F119894intinfin

1205751198940

119901 (120595119894(119866119894(Fminus1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+1119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+2119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+3119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+3)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ sdot sdot sdot )]

(21)


thus

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F119894(120599119894+ 1)

times int1205751198940

minusinfin


+ 1205993

119894(120599119894+ 1)int

infin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+ 1205995

119894(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(1198661198945) gt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F+3119894

(120599119894+ 1)

times intinfin

1205751198940


(22)

Leads to

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times [1

119892119894+ (120599119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]]

=

119899

sum119894=1

120582119894(120599119894minus 1) (

1

119892119894)

+

119899

sum119894=1

[120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]

(23)

where

1

119892119894= 119901 (120591

120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

1

119908119894(119909119894) =

infin

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

1

119902119894(119909119894) =

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)

1

ℎ119894(119909119894) =

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(24)


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(25)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite



119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1) (

2

119892119894) + 120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

2

119908119894(119909119894) 120574119894(119889119909119894)

+ int1205751198940

minusinfin

2

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

2

ℎ119894(119909119894) 120574119894(119889119909119894))]

(26)


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)



+1le 119886 =



1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)


minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889


infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)



1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and


(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]


where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)



119894gt 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


and [119901(119881(1205851) = F) 119901(119881(120585

2) = F) 119901(119881(120585

119899) = F)] =

[119901(sum12058511205991

119895=11198851119895= F) 119901(sum

12058521205992

119895=11198852119895= F) 119901(sum

120585119899120599119899

119895=1119885119899119895=


[119901 (119881 (120585

1+ 1) = F)

119901 (119881 (1205851) = F)

119901 (119881 (120585

2+ 1) = F)

119901 (119881 (1205852) = F)

119901 (119881 (120585119899+ 1) = F)

119901 (119881 (120585119899) = F)

]

= [

[

12059911205731

prod119895=1

12058511205991+ 119895

Δ1(12058511205991+ 119895 + (119895120572

1+F) 120573

1)

times

12059911205721

prod119895=1

12058511205991+ 119895 + 120599

11205731

Δ1(12058511205991+ 119895 + (119894120573

1minusF) 120572

1)

12059921205732

prod119895=1

12058521205992+ 119895

Δ2(12058521205992+ 119895 + (119895120572

2+F) 120573

2)

times

12059921205722

prod119895=1

12058521205992+ 119895 + 120599

21205732

Δ2(12058521205992+ 119895 + (119894120573

2minusF) 120572

2)

120599119899120573119899

prod119895=1

120585119899120599119899+ 119895

Δ119899(120585119899120599119899+ 119895 + (119895120572

119899+F) 120573

119899)

times

120599119899120572119899

prod119895=1

120585119899120599119899+ 119895 + 120599

119899120573119899

Δ119899(120585119899120599119899+ 119895 + (119894120573

119899minusF) 120572

119899)]

]

(13)

Since 120585119894gt 1199031198941|F| + 119903



2+ 1) =

F) 119901(119881(120585119899+ 1) = F)] lt [119901(119881(120585

1) = F) 119901(119881(120585

2) =

F) 119901(119881(120585119899) = F)]

Theorem 3 If 119864(119883119894119895) lt 119862




[119901(119882(1205851)) 119901(119882(120585

2)) 119901(119882(120585

119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for

all 120585119894 119894 = 1 2 119899

Proof For 120577119894gt 0 [119901(119882(120585

1) ge 119862

11205851) 119901(119882(120585

2) ge 119862

21205852)

119901(119882(120585119899) ge 119862

119899120585119899)] = [119901(exp120577

1(119882(1205851) minus 11986211205851) ge 1) 119901(exp

1205772(119882(1205852)minus11986221205852) ge 1) 119901(exp120577

119899(119882(120585119899)minus119862119899120585119899) ge 1)] le

[119864(exp1205771(119882(1205851) minus 119862

11205851)) 119864(exp120577

2(119882(1205852) minus 119862

21205852))

119864(exp120577119899(119882(120585119899) minus 119862

119899120585119899))] = [119891

1(1205771)1205851 1198912(1205772)1205852

119891119899(120577119899)120585119899] where [119891

1(1205771) 1198912(1205772) 119891

119899(120577119899)] = [119864(exp

1205771(1198831119895minus 1198621)) 119864(exp120577

2(1198832119895minus 1198622)) 119864(exp120577

119899(119883119899119895minus

119862119899))] = [119901

1exp1205771(1 minus 119862

1) + 119902


1) 1199012

exp1205772(1 minus 119862

2) + 119902


2) 119901

119899exp

120577119899(1 minus 119862

119899) + 119902

119899exp120577119899(minus1 minus 119862

119899)] if 119864(119883

119894119895) lt 119862

119894 119894 =

1 2 119899 then [11989110158401(0) 1198911015840

2(0) 1198911015840

119899(0)] lt [0 0 0]

and since [1198911(0) 119891

2(0) 119891

119899(0)] lt [1 1 1]

then [min1205771gt01198911(1205771)min

1205772gt01198912(1205772) min

120577119899gt0119891119899(120577119899)]

= [1205761 1205762 120576

119899] lt [1 1 1]


119894 119894 = 1 2 119899


1)) 119901(119882(120585

2))

119901(119882(120585119899))] le [120576

1205851

1 1205761205852

2 120576120585119899

119899] for all 120585

119894 119894 = 1 2 119899




119894119895 119894 =

1 2 119899 and if 120601(119905) = (1206011(119905) 1206012(119905) 120601

119899(119905)) is a search


int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(14)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)


1205751198940gt 0 And if 120575

1198940lt 0 then 119864


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(15)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite





1198940then 119883

119894+ 119882(119905) is smaller than


120601119894gt 119905 119894 = 1 2 119899 are


1206011gt 119905 or


120601119899gt 119905) = sum

119899

119894=1119901(120591120601119894gt 119905) and for any 119895 ge 0

we have

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119883119894+119882(119866

119894(2119895)) lt 119867

119894(2119895)| 119883119894= 119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (119883119894+119882(119866

119894(2119895+1)) gt 119867

119894(2119895+1)| 119883119894= 119909119894) 120574119894(119889119909119894)

(16)


119894(2119895)) + 119862119894119866119894(2119895)

weobtain 119882(119866

119894(2119895)) minus 119867

119894(2119895)lt minus119883

119894= 119909119894 then 119882(119866

119894(2119895)) minus

(minus1)2119895+1

119862119894[119866119894(2119895)

+ 1 + (minus1)2119895+1

] lt minus119909119894leads to 119882(119866

119894(2119895)) minus

(minus1)119862119894[119866119894(2119895)

+ 1 + (minus1)] = 119882(119866119894(2119895)) + 119862

119894[119866119894(2119895)] =


119894(119866119894(2119895)) lt minus119909


119894(119866119894(2119895+1)

) =

119882(119866119894(2119895+1)

) minus 119862119894119866119894(2119895+1)

we get 120595119894(119866119894(2119895+1)


quently

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894) 120574119894(119889119909119894)

(17)

also

119901 (120591120601119894gt 119866119894(2119895))

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895minus1)

) gt minus119909119894) 120574119894(119889119909119894)

(18)


119864120591120601= intinfin

0

119901 (120591120601gt 119905) 119889119905

le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 119905) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

(19)

where 1198661198940= 0 then

119864120591120601le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 1198661119895) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

=

119899

sum119894=1

infin

sum119895=0

(119866119894(119895+1)

minus 119866119894119895) 119901 (120591120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

[120582119894(120599119895+1

119894minus 1) minus 120582

119894(120599119895

119894minus 1)] 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

120582119894120599119895

119894(120599119894minus 1) 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

[

[

120582119894(120599119894minus 1)

infin

sum119895=0

120599119895

119894119901 (120591120601119894gt 119866119894119895)]

]

=

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 119863) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894119901 (120591120601119894gt 1198661198942) + 1205993

119894119901 (120591120601119894gt 1198661198943) + sdot sdot sdot )]

(20)


119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205993

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205994

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198944) lt minus119909

119894) 120574119894(119889119909119894)

+ sdot sdot sdot

+ 120599F119894intinfin

1205751198940

119901 (120595119894(119866119894(Fminus1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+1119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+2119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+3119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+3)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ sdot sdot sdot )]

(21)


thus

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F119894(120599119894+ 1)

times int1205751198940

minusinfin


+ 1205993

119894(120599119894+ 1)int

infin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+ 1205995

119894(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(1198661198945) gt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F+3119894

(120599119894+ 1)

times intinfin

1205751198940


(22)

Leads to

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times [1

119892119894+ (120599119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]]

=

119899

sum119894=1

120582119894(120599119894minus 1) (

1

119892119894)

+

119899

sum119894=1

[120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]

(23)

where

1

119892119894= 119901 (120591

120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

1

119908119894(119909119894) =

infin

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

1

119902119894(119909119894) =

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)

1

ℎ119894(119909119894) =

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(24)


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(25)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite



119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1) (

2

119892119894) + 120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

2

119908119894(119909119894) 120574119894(119889119909119894)

+ int1205751198940

minusinfin

2

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

2

ℎ119894(119909119894) 120574119894(119889119909119894))]

(26)


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)



+1le 119886 =



1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)


minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889


infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)



1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and


(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]


where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)



119894gt 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


119894(119866119894(2119895)) lt minus119909


119894(119866119894(2119895+1)

) =

119882(119866119894(2119895+1)

) minus 119862119894119866119894(2119895+1)

we get 120595119894(119866119894(2119895+1)


quently

119901 (120591120601119894gt 119866119894(2119895+1)

)

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894) 120574119894(119889119909119894)

(17)

also

119901 (120591120601119894gt 119866119894(2119895))

le int1205751198940

minusinfin

119901 (119894(119866119894(2119895)) lt minus119909

119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

119901 (120595119894(119866119894(2119895minus1)

) gt minus119909119894) 120574119894(119889119909119894)

(18)


119864120591120601= intinfin

0

119901 (120591120601gt 119905) 119889119905

le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 119905) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

(19)

where 1198661198940= 0 then

119864120591120601le

infin

sum119895=0

int1198661(119895+1)

1198661119895

119901 (1205911206011gt 1198661119895) 119889119905 + sdot sdot sdot

+

infin

sum119895=0

int119866119899(119895+1)

119866119899119895

119901 (120591120601119899gt 119905) 119889119905

=

119899

sum119894=1

infin

sum119895=0

(119866119894(119895+1)

minus 119866119894119895) 119901 (120591120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

[120582119894(120599119895+1

119894minus 1) minus 120582

119894(120599119895

119894minus 1)] 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

infin

sum119895=0

120582119894120599119895

119894(120599119894minus 1) 119901 (120591

120601119894gt 119866119894119895)

=

119899

sum119894=1

[

[

120582119894(120599119894minus 1)

infin

sum119895=0

120599119895

119894119901 (120591120601119894gt 119866119894119895)]

]

=

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 119863) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894119901 (120591120601119894gt 1198661198942) + 1205993

119894119901 (120591120601119894gt 1198661198943) + sdot sdot sdot )]

(20)


119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205993

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894)

+ 1205994

119894intinfin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+int1205751198940

minusinfin

119901 (119894(1198661198944) lt minus119909

119894) 120574119894(119889119909119894)

+ sdot sdot sdot

+ 120599F119894intinfin

1205751198940

119901 (120595119894(119866119894(Fminus1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+1119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894F) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+2119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+1)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ 120599F+3119894

intinfin

1205751198940

119901 (120595119894(119866119894(F+3)) gt minus119909119894) 120574119894 (119889119909119894)

+int1205751198940

minusinfin

119901 (119894(119866119894(F+2)) lt minus119909119894) 120574119894 (119889119909119894)

+ sdot sdot sdot )]

(21)


thus

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F119894(120599119894+ 1)

times int1205751198940

minusinfin


+ 1205993

119894(120599119894+ 1)int

infin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+ 1205995

119894(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(1198661198945) gt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F+3119894

(120599119894+ 1)

times intinfin

1205751198940


(22)

Leads to

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times [1

119892119894+ (120599119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]]

=

119899

sum119894=1

120582119894(120599119894minus 1) (

1

119892119894)

+

119899

sum119894=1

[120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]

(23)

where

1

119892119894= 119901 (120591

120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

1

119908119894(119909119894) =

infin

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

1

119902119894(119909119894) =

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)

1

ℎ119894(119909119894) =

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(24)


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(25)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite



119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1) (

2

119892119894) + 120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

2

119908119894(119909119894) 120574119894(119889119909119894)

+ int1205751198940

minusinfin

2

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

2

ℎ119894(119909119894) 120574119894(119889119909119894))]

(26)


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)



+1le 119886 =



1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)


minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889


infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)



1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and


(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]


where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)



119894gt 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


thus

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times (119901 (120591120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894(120599119894+ 1)

times int1205751198940

minusinfin

119901 (119894(1198661198942) lt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F119894(120599119894+ 1)

times int1205751198940

minusinfin


+ 1205993

119894(120599119894+ 1)int

infin

1205751198940

119901 (120595119894(1198661198943) gt minus119909

119894) 120574119894(119889119909119894)

+ 1205995

119894(120599119894+ 1)

times intinfin

1205751198940

119901 (120595119894(1198661198945) gt minus119909

119894) 120574119894(119889119909119894) + sdot sdot sdot

+ 120599F+3119894

(120599119894+ 1)

times intinfin

1205751198940


(22)

Leads to

119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1)

times [1

119892119894+ (120599119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]]

=

119899

sum119894=1

120582119894(120599119894minus 1) (

1

119892119894)

+

119899

sum119894=1

[120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

1

119908119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

119902119894(119909119894) 120574119894(119889119909119894)

+ intinfin

1205751198940

1

ℎ119894(119909119894) 120574119894(119889119909119894))]

(23)

where

1

119892119894= 119901 (120591

120601119894gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

1

119908119894(119909119894) =

infin

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

1

119902119894(119909119894) =

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)

1

ℎ119894(119909119894) =

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(24)


intinfin

1205751198940

[

[

infin

sum119895=(F+1)2

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

+

(Fminus1)2

sum119895=1

1205992119895+1

119894119901 (119894(119866119894(2119895+1)

) le minus119909119894)]

]

120574119894(119889119909119894)

(25)

int1205751198940

minusinfin

suminfin

119895=11205992119895

119894119901(119894(119866119894(2119895)) lt minus119909

119894)120574119894(119889119909119894) are finite



119864120591120601le

119899

sum119894=1

[120582119894(120599119894minus 1) (

2

119892119894) + 120582119894(120599119894minus 1) (120599

119894+ 1)

times (int1205751198940

minusinfin

2

119908119894(119909119894) 120574119894(119889119909119894)

+ int1205751198940

minusinfin

2

119902119894(119909119894) 120574119894(119889119909119894)

+intinfin

1205751198940

2

ℎ119894(119909119894) 120574119894(119889119909119894))]

(26)


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)



+1le 119886 =



1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)


minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889


infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)



1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and


(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]


where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)



119894gt 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


where2

119892119894= 119901 (120591

1206011gt 0) + 120599

119894119901 (120591120601119894gt 1198661198941)

+ 1205992

119894intinfin

1205751198940

119901 (120595119894(1198661198941) gt minus119909

119894) 120574119894(119889119909119894)

2

119908119894(119909119894) =

F2

sum119895=1

1205992119895

119894119901 (120595119894(119866119894(2119895)) lt minus119909

119894)

2

119902119894(119909119894) =

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)

2

ℎ119894(119909119894) =

infin

sum119895=1

1205992119895+1

119894119901 (120595119894(119866119894(2119895+1)

) gt minus119909119894)

(27)

And 119864120591120601is finite if

int1205751198940

minusinfin

[

[

F2

sum119895=1

1205992119895

119894119901 (119894(119866119894(2119895)) lt minus119909

119894)

+

infin

sum119895=F2+1

1205992119895

119894119901 (119894(119866119894(2119895)) le minus119909

119894)]

]

120574119894(119889119909119894)

(28)

intinfin

1205751198940

suminfin

119895=11205992119895+1

119894119901(120595119894(119866119894(2119895+1)



+1le 119886 =



1 2 infin

sum

=119896

[119889+1minus 119889] 119886119889+1

le

infin

sum

=119889

119886

le

infin

sum

=119896

[119889+1minus 119889] 119886119889

(29)


minus 119889]119886119889+1 suminfin

=119889

119886 and suminfin

=119896[119889+1

minus

119889]119886119889


infin

sum

=119896

[119889+1minus 119889] 119886119889+1

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891(+1)

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892(+1)

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899(+1)

]

]

infin

sum

=119889

119886= [

[

infin

sum

=1198891

1198861

infin

sum

=1198892

1198862

infin

sum

=119889119899

119886119899

]

]

infin

sum

=119896

[119889+1minus 119889] 119886119889

= [

[

infin

sum

=119896

[1198891(+1)

minus 1198891] 11988611198891

infin

sum

=119896

[1198892(+1)

minus 1198892] 11988621198892

infin

sum

=119896

[119889119899(+1)

minus 119889119899] 119886119899119889119899

]

]

(30)



1198940gt 0 and 1119908(119909) le 119872(|119909|) 1ℎ(119909)

le (|119909|) if 1205751198940lt 0 where 119871(|119909|) (|119909|) 119872(|119909|) and


(1199091)1 1199082(1199092) 1 119908

119899(119909119899)] 1ℎ(119909) = [

1

ℎ1(1199091)1 ℎ2(1199092)

1ℎ119899(119909119899)] 2ℎ(119909) = [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] 2119902(119909) =

[2

1199021(1199091)2 1199022(1199092) 2 119902

119899(119909119899)] 119871(|119909|) = [119871

1(|1199091|) 1198712(|1199092|)

119871119899(|119909119899|)] (|119909|) = [

1(|1199091|) 2(|1199092|)

119899(|119909119899|)] 119872

(|119909|) = [1198721(|1199091|)1198722(|1199092|) 119872

119899(|119909119899|)] and (|119909|) =

[1(|1199091|) 2(|1199092|)

119899(|119909119899|)]


where 2ℎ(119909) = [2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [sum

infin

119895=11205992119895+1

1119901

(1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt minus1199092)

suminfin

119895=11205992119895+1119899119901(120595119899(119866119899(2119895+1)



119894gt 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)]+[sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt 1205952

(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)](II) If 0 le 119909

119894le 1205751198940 then we have [

2

ℎ1(1199091)2 ℎ2(1199092)

2

ℎ119899(119909119899)] = [

2

ℎ1(0)2

ℎ2(0)

2

ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1

119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le

0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)]


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


(III) If 119909119894le 120575


2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] = [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)]minus[sum

infin

119895=11205992119895+1

1119901(0 lt

1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(0 lt 120595

2(1198662(2119895+1)

) le minus1199092)

suminfin

119895=11205992119895+1119899119901(0 lt 120595

119899(119866119899(2119895+1)

) le minus119909119899)]


ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] le [

2

ℎ1

(0)2 ℎ2(0) 2 ℎ

119899(0)] and for 119909

119894ge 120575

1198940[2

ℎ1(1199091)2 ℎ2

(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899(1205751198990)] +

[suminfin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)


119895=11205992119895+1

2119901(minus1199092lt

1205952(1198662(2119895+1)


infin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le

minus1205751198990)] but [

2

ℎ1(1199091)2 ℎ2(1199092) 2 ℎ

119899(119909119899)] = [

2

ℎ1(0)2ℎ

2(0)

2ℎ119899(0)] + [sum

infin

119895=11205992119895+1

1119901(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=1

1205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

) le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899

(119866119899(2119895+1)


ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901(1205951(1198661(2119895+1)

) gt

0) suminfin

119895=11205992119895+1

2119901(1205952(1198662(2119895+1)

) gt 0) suminfin

119895=11205992119895+1119899119901(120595119899

(119866119899(2119895+1)


119895=11205992119895+1

11205761198661(2119895+1)

1 suminfin

119895=11205992119895+1

21205761198662(2119895+1)

2

suminfin

119895=11205992119895+1119899120576119866119899(2119895+1)

119899] le [1205993

1(12059921minus1) 1205993

2(12059922minus1) 1205993

119899(1205992119899minus

1)] 0 lt 120576119894lt 1


(i) 119889119894120585119894= 119866119894(2120585119894+1)

120590119894= 120583119894(1205992120585119894+1

119894minus 1)

(ii) 120583119894(120585119894) = 120595

119894(120585119894120590119894)2 = sum

120585119894

119895=1119883119894119895 where 119883

119894119895 119895 =


(iii) 119886119894(120585119894) = 119901(minus119909

1198942 lt 120583

119894(120585119894) le 0) = sum

|119909119894|2

F=0119901(minus(F + 1) lt

120583119894(120585119894) le minusF)


(v) 119894= 1205992119894120583119894(120599119894minus 1)

(vi) 119880119894(FF + 1) = sum

infin

120585119894=0119901(minus(F + 1) lt 120583

119894(120585119894) le minusF)



2

ℎ1(1199091)2 ℎ2(1199092)

2 ℎ119899(119909119899)] minus [

2

ℎ1(0)2 ℎ

2(0) 2 ℎ

119899(0)] = [sum

infin

119895=11205992119895+1

1119901

(minus1199091lt 1205951(1198661(2119895+1)

) le 0) suminfin

119895=11205992119895+1

2119901(minus1199092lt 1205952(1198662(2119895+1)

)

le 0) suminfin

119895=11205992119895+1119899119901(minus119909119899lt 120595119899(119866119899(2119895+1)

) le 0)] = [sum

1205851=1

12059921205851+1

11198861(11988911205851) sum

1205852=112059921205852+1

21198862(11988921205852) sum

120585119899=11205992120585119899+1119899

119886119899(119889119899120585119899)]

+ [suminfin1205851=+1

12059921205851+1

11198861(11988911205851) suminfin

1205852=+112059921205852+1

21198862(11988921205852) sum

infin

120585119899=+1

1205992120585119899+1119899

119886119899(119889119899120585119899)] le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

]

+ [1suminfin

1205851=+1(11988911205851

minus 1198891(1205851minus1)

)1198861(11988911205851) 2suminfin

1205852=+1(11988921205852

minus

1198892(1205852minus1)

)1198862(11988921205852)

119899suminfin

120585119899=+1(119889119899120585119899

minus 119889119899(120585119899minus1)

)119886119899(119889119899120585119899)]

le [sum

1205851=112059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1suminfin

1205851=1198891

1198861(1205851) 2suminfin

1205852=11988921198862(1205852)

119899suminfin

120585119899=119889119899119886119899(120585119899)] le [sum

1205851=1

12059921205851+1

1 sum

1205852=112059921205852+1

2 sum

120585119899=11205992120585119899+1119899

] + [1sum|1199091|2

F=01198801(FF +

1) 2sum|1199092|2

F=01198802(FF + 1)

119899sum|119909119899|2

F=0119880119899(FF + 1)]




[1199021(1199091) 1199022(1199092) 119902

119899(119909119899)] le [

2

ℎ1(12057510)2 ℎ2(12057520) 2 ℎ

119899

(1205751198990)] + [

2

Λ112 Λ21 2 Λ

1198991] + [

2

Λ12|1199091|2 Λ22|1199092|

2 Λ1198992|119909119899|] = [119871

1(|1199091|) 1198712(|1199092|) 119871

119899(|119909119899|)] = 119871(|119909|)



completed


(1206011(119905) 1206012(119905) 120601

119899(119905)) isin Φ(119905) then 119864(|119883

0|) is finite


120601is finite) =


119901(

119899

⋃119894=1

(120591120601119894lt infin)) =

119899

sum119894=1

119901 (120591120601119894lt infin)

minussum

119894120573

119901 ((120591120601119894lt infin) cap (120591


+ (minus1)119899

119901(

119899

⋂119894=1

(120591120601119894lt infin))

(31)


events then119899

sum119894=1

119901 (120591120601119894lt infin) = 1 (32)

Consequently 119901 (120591120601119894

is finite) = 1 or 119901 (120591120601120579


120601119894is finite) = 1 then

1198830= 120601119894(120591120601119894) minus 119882(120591


0| le

|120601119894(120591120601119894)| + |119882(120591

120601119894)| le 120591

120601119894+ |119882(120591

120601119894)| that leads to 119864(|119883

0|) le

119864120591120601119894+ 119864(|119882(120591

120601119894)|) But 119882(120591

120601119894) lt 120591120601119894 then 119864(|119882(120591

120601119894)|) lt 119864

120591120601119894

and 119864(|1198830|) is finite



120601120579(120591120601120579) minus 119882(120591






119894(119905) 119894 = 1 2 119898 If such a search



isin Φ1(119905) 119894 = 1 2 119898 be


119894(119905)



[9])



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013



(1206011(119905) 1206012(119905) 120601

119899(119905)) 997888rarr 119864

120591120601isin 119868+

(33)



120601119894gt


119864120591120601= 119864[

infin

sum119905=0

119861 (120601119894 119905)]

= 119864[

infin

sum119905=0

limℎrarrinfin

inf 119861 (120601119894ℎ(119905) 119905)]

le

infin

sum119905=0

limℎrarrinfin

inf 119864(1120591120601ℎgt119905)

(34)


rarr 120601119894in Φ1(119905) where Φ

1(119905) is


1(119905) 1206012(119905) 120601

119899(119905)) rarr 119864







120591120601 That


119899(119905) where 120601




119899(119905)) are 119899-distinct





le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le F2 Then


min (1206011(119905) 1206012(119905) 120601

119899(119905))


| 119866119894(2119895minus1)

le 119905 le 119866119894(2119895)

119894 = 1 2 119899 1 le 119895 leF

2

(35)





119894119895



MONLP

min (Ξ1(1198671119895) Ξ2(1198672119895) Ξ

119899(119867119899119895))

subject to 119866119894(2119895minus1)

le 119867119894119895le 119866119894(2119895)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867


119894(2119895minus1)]

(36)


Substituting 119866119894119895= 119887120603119894(120599119895


have

MONLP(I)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1) le 119867

119894119895 119867119894119895le 119887120603119894(1205992119895

119894minus 1)

119867119894119895isin 119868+

119894 = 1 2 119899 1 le 119895 leF

2

where Ξ119894(119867119894119895) = 119867

119894(2119895+1)minus 1205751198940


119894minus 1)]

(37)



119894(1 minus

(120599119894minus 1)(120599


119894119895ge 0 for all 119894 =



119894 where

119867119894119895= (minus1)

119895+1

119862119894[119866119894119895+ 1 + (minus1)

119895+1

] + 1205751198940

= (minus1)119895+1

(120599119894minus 1

120599119894+ 1) [119887120603

119894(120579119895

119894minus 1) + 1 + (minus1)

119895+1

]

+ 1205751198940

(38)



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013



MONLP(II)

min (Ξ1(1205991 1206031) Ξ2(1205992 1206032) Ξ

119899(120599119899 120603119899))

subject to 119887120603119894(1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1] + 1205751198940

119887120603119894(1205992119895


1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

where Ξ119894(120599119894 120603119894) = (ℎ

3) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887120603119894(120599119895

119894minus 1) + ℎ

5]

+ 119887120603119894(1205992119895minus1

119894minus 1)

ℎ1= 1 + (minus1)

119895+1

ℎ2= (minus1)

119895+2

ℎ3= (minus1)

2119895+1

ℎ4= 1 + (minus1)

2119895

ℎ5= 1 + (minus1)

119895

120599119894gt 1

120603119894gt 0 forall119894 = 1 2 119899 119887 = 1 2 1 le 119895 le

F

2

(39)


sum119899


119894= 1 119894 = 1 2 119899



120599119894+ 1)

times [119887 (1205992119895minus1

119894minus 1) + ℎ

4]

minus 21205751198940+ (ℎ2) (120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

5]

+ 119887 (1205992119895minus1

119894minus 1)

subject to 119887 (1205992119895minus1

119894minus 1)

le (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940

119887 (1205992119895

119894minus 1)

ge (ℎ1minus 1) (

120599119894minus 1

120599119894+ 1)

times [119887 (120599119895

119894minus 1) + ℎ

1]

+ 1205751198940


F

2

(40)


1

2

2

(120599119894+ 1)2[(ℎ3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

+ (120599119894minus 1

120599119894+ 1) 119887 [(ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894+ (ℎ2minus 1) 119895120599

119895minus1

119894]

+ 119887 (2119895 minus 1) 1205992119895minus2

119894

+ 1199061119887 (2119895 minus 1) 120599

2119895minus2


120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

+ 1199062minus119887 (2119895 minus 1) 120599

2119895minus2

119894+(ℎ1minus 1)

120599119894+ 1

times [2

120599119894+ 1[119887 (120599119895

119894minus 1) + ℎ

1] + (120599

119894minus 1) 119895119887120599

119895minus1

119894]

minus 1199063= 0

(41)

1199061119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) minus (ℎ

1minus 1) (120599

119894minus 1)

times [119887 (120599119895minus1

119894minus 1) + ℎ

1] = 0

(42)

1199062(ℎ1minus 1) (120599

119894minus 1) [119887 (120599

119895

119894minus 1) + ℎ

1]

minus119887 (1205992119895minus1

119894minus 1) (120599

119894+ 1) = 0

(43)

11990631 minus 120599

119894 = 0 (44)


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

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Volume 2013


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Volume 2013

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Volume 2013


1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(a) 119887 = 1

1 2 3 4 5 6 7 8 109

15

10

5

0

minus5

minus10

(b) 119887 = 2






120599119894= radic

Υ + Λ

119887 (1 minus 2119895) 1205992119895minus2

119894

minus 1 120599119894gt 1 (45)

whereΥ = 2 [(ℎ

3minus 1) [119887 (120599

2119895minus1

119894minus 1) + ℎ

4]

+ (ℎ2minus 1) [119887 (120599

119895

119894minus 1) + ℎ

5]]

Λ = 119887 (ℎ3minus 1) (2119895 minus 1) 120599

2119895

119894minus 119887 (ℎ

3minus 1) (2119895 minus 1) 120599

2119895minus2

119894

+ 119887 (ℎ2minus 1) 119895120599

119895+1

119894minus 119887 (ℎ

2minus 1) 119895120599

119895minus1

119894

(46)









119894



le 119905 le 119866119894(2119895) 119894 = 1 2 119899 1 le 119895 le



119894119895 119866119894119895in Table 1


119894should do


119894119895= 120582119894(120599119895

119894minus 1)


119894119895 119895 = 2 4 6 119887 = 1 2

119887 119895 120599119894

119866119894119895

119867119894119895

12 18836 08836 027075 + 120575

1198940

4 25265 15265 06607 minus 1205751198940

6 26982 16982 07798 + 1205751198940

22 25226 30452 13162 + 120575

1198940

4 27897 35794 22963 minus 1205751198940

6 28652 37304 18 + 1205751198940









1(119905) 1206012(119905) 120601

119899(119905)) and119864

120591120601


120591120601lt infin


1198940gt 0 and

1119908(119909) le 119872(|119909|) 1ℎ(119909) le (|119909|) if 1205751198940lt 0 where 119871(|119909|)


0|) is





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013





119894119895 119894 = 1 119899 119895 =






119894 119894 = 1 119899 We use




References






























OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013

optimal multiplicative generalized linear search plan for a

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