Appled Mathematcal Scences Vol 5 0 no 65 3 33 Interactve B-Level Mult-Objectve Integer Non-lnear Programmng Problem O E Emam Department of Informaton Systems aculty of Computer Scence and nformaton Helwan Unversty PO Bo 795 Egypt emam_o_e@yahoocom Abstract Ths paper presents a b-level mult-objectve nteger non-lnear programmng (BLMINP problem wth lnear or non-lnear constrants and an nteractve algorthm for solvng such model At the frst phase of the soluton algorthm to avod the complety of non convety of ths problem we begn by fndng the conve hull of ts orgnal set of constrants usng the cuttng-plane algorthm to convert the BLMINP problem to an equvalent b-level mult-objectve non-lnear programmng (BLMNP problem At the second phase the algorthm smplfes an equvalent (BLMNP problem by transformng t nto separate mult-objectve decson-makng problems wth herarchcal structure and solvng t by usng ε - constrant method to avod the dffculty assocated wth non-conve mathematcal programmng In addton the author put forward the satsfactorness concept as the frst-level decson-maker preference nally an llustratve numercal eample s gven to demonstrate the obtaned results Mathematcs Subject Classfcaton: 90C9; 90C30; 4A58; 90C0 Keywords: B-level programmng; Nonlnear programmng; Management decson makng; nteger programmng ( Introducton B-level programmng (BLP s a subset of the mult-level programmng problem whch dentfed as a mathematcal programmng problem that solves decentralzed plannng problems wth two decson makers (DMs n a two- level or herarchcal organzaton ([3] [4] [5] [6] [7] [9] An algorthm for the nteractve mult-level non-lnear mult-objectve decson-makng problem s presented n many searches ( Osman et al [7] and Sh and Xa [0]
3 O E Emam The nteractve algorthm uses the concepts of satsfactorness to mult-objectve optmzaton at every level untl a preferred soluton s reach Based on (Sh and Xa [0] satsfactory soluton concepts the proposed soluton method proceed from the frst-level decson-maker (LDM to the second-level decson-maker (SLDM The LDM gets the preferred or satsfactory solutons that are acceptable n rank order to the SLDM The SLDM wll search for the preferred soluton of the LDM untl the preferred soluton s reached Integer multobjectve programmng has attracted the attenton of many researchers n the past The man reason for nterest n lnear or nonlnear programmng stems from the fact that programmng models could better ft the real problems f we consder optmzaton of economc quanttes ([] [8] Ths paper s organzed as follows: we start n Secton by formulatng the model of b-level mult-objectve nteger non-lnear programmng problem wth the soluton concept s ntroduced In Secton 3 Defntons and Theorems s carred out In Secton 4 an nteractve model for BLMINP problem s presented In Secton 5 an nteractve algorthm for BLMINP problem s presented In Secton 6 an eample s provded to llustrate the developed results nally n Secton 7 some open ponts are stated for future research work n the area of nteractve mult-level nteger programmng optmzaton problems ( Problem ormulaton and Soluton Concept n Let R ( be a vector varables ndcatng the frst decson level s choce the second decson level s choce Let the LDM and SLDM have N and N objectve functons respectvely And M s the set of feasble choces {( } So the BLMINP problem may be formulated as follows: Ma Ma f f ( [ st Level] where solves Ma N [ nd Level] ( Ma f ( f ( N ( Subject to M {( g ( 0 m j 0 and nteger j } (3 Where M s a non-conve constrant set and are non-lnear functons The decson mechansm of BLMINP problem s that the LDM and SLDM adopt the two-planner Stackelberg game Accordng to the two-planner Stackelberg game and mathematcal programmng the defntons of soluton for the model of BLMINP problem are gven as follows Defnton or any ( M { ( M } decson-makng varable ( M { ( M } soluton of the SLDM then ( s a feasble soluton of BLMINP problem gven by LDM f the s the non-nferor
Non-lnear programmng problem 33 * * Defnton If ( * * feasble soluton ( M ests such that f ( f j ( * * least one j ( j N ; so problem s a feasble soluton of the BLMINP problem; no other j wth at s the preferred soluton of the BLMINP In what follows an equvalent b-level mult-objectve nonlnear programmng (BLMNP problem assocated wth problem (-(3 can be stated wth the help of cuttng-plane technque ([] [] [8] together wth Balnsk algorthm [] Ths equvalent BLMNP problem can be wrtten n the followng form: (LDM Ma ( Ma f ( f ( where solves N (4 (SLDM Ma ( Ma f ( f ( N (5 Subject to [M ] (6 where [M ] s the conve hull of the feasble regon M defned by (3 earler Ths conve hull s defned by: ( s n ( s ( s [ M ] M R { R A b 0} (7 and n addton A ( s A a as and b ( s b b c c m s (8 are the orgnal constrant matr A and the rght-hand sde vector b respectvely wth s-addtonal constrants each correspondng to an effcent cut n the form a By an effcent cut we mean that a cut whch s not redundant c
34 O E Emam (3 Defntons and Theorems We wll obtan the soluton of the equvalent BLMNP problem of the BLMINP problem by solvng LDM and SLDM problems each one separately In ths way we can quanttatvely present satsfactorness and the preferred soluton n vew of sngular-level mult-objectve decson-makng problem and ntroduce several theorems wth the help of the qualty of ε -constrant method to provde a theoretcal bass for upper-level mult-objectve decson-makng Consder a mult-objectve decson-makng problem as follows: Ma ( f( f n ( (9 h j 0 j denotes the decson-makng varable and f denotes the objectve functon of the mult-objectve decson- q n + n Where ( R ( ( makng problem Let Ω { h j ( 0 j q } and a Mn f ( Ω b Ma f ( On u [ a b ] defne A f A Ω ( ( f μ meet ( and ( as below : u whose membershp functon ( When the objectve value f ( approaches or equals the decson-maker s deal value μ A ( f ( approaches or equals Otherwse 0 ( If f ( > f ( then μ ( f ( μ ( f ( n Defnton 3 If satsfactorness of A A s a non-nferor soluton then A f ( to objectve ( f Defnton 4 μ Mn μ ( f ( n A μ s defned as the ( s defned as the satsfactorness of non- nferor soluton to all the objectves Defnton 5 Wth a certan value s 0 gven n advance by the decson-maker f non-nferor soluton satsfes μ ( s0 then s the preferred soluton correspondng to the satsfactorness s 0 We gve membershp functon A ( f ( f ( a ( f ( μ as below: μ A (0 b a It s decded accordng to the decson-maker s requrements Obvously (0 μ meets the two requrements ( and ( for ( The ε -constrant method s effectve for solvng mult-objectve decson-makng problems The formalzaton of P ( ε s as follows: Ma f ( A f
Non-lnear programmng problem 35 f ( ε n Ω Assume ε ε ( ε n { f ( n Ω } X ( ε ε and Ε { ε ( ε φ ( empty set } Theorem If ε ( ε ε 3 ε n Ε then the optmal soluton to P ests and ncludes the non-nferor soluton of (9 (see Sh and Xa [0] P then s the non- Corollary If nferor soluton of (9 s the only optmal soluton to ε ε The ε -constrant problem ncludng satsfactorness s as follows: Ma f ( ( f ( δ n Ω Theorem If P ( ε ( s has no soluton or has the non-nferor soluton and f ( δ then no non-nferor soluton ests such that μ ( s Proof: If s a non-nferor soluton of (9 such that μ ( s namely μ ( ( s ( n Then s a feasble soluton of ( ε ( s f δ A f therefore ( ( s and f ( f( δ P P ε has a non-nferor soluton such that whch s n contradcton to the hypothess Theorem 3 Assume s < s f there s no preferred soluton to s then go to s [0] Theorem 4 Assume s a non-nferor soluton of P ( ε ( s and f n f ε n and let δ ( Let ( ε ( ε ε 3 ε n Then s stll an optmal soluton of ( ε If s the only optmal soluton of P ( ε If other optmal soluton of P ( ε ests and L { n} f L ( ε L then s nferor soluton P then s a non-nferor soluton ests such that
36 O E Emam Proof (a ε δ ( n namely ε ( ε ε 3 ε n ( δ δ 3 δ n ; ( ε ( ε ( s let to be a non-nferor soluton of P ( ε ( s and ( ε then f ( Ma f( Ma f ( ( s ( ( ε and (a s proven by Corollary (b f( f( and f ( f ( ε whch f holds when L therefore s nferor soluton Therefore s a non-nferor soluton of P (4 An nteractve model for BLMINP problem To solve the BLMINP problem by adoptng the two-planner Stackelberg game frst we have to retransfer set of constrants M to ts equvalent [M] so we wll obtan an equvalent BLMNP problem then the LDM gves the preferred or satsfactory solutons that are acceptable n rank order to the SLDM and then the SLDM wll seek the solutons by ε -constrant method and to arrve at the soluton that gradually approaches the preferred soluton or satsfactory soluton to the LDM nally the LDM decde the preferred soluton of the BLMINP problem accordng to the satsfactorness 4-The rst-level Decson-Maker (LDM Problem The frst-level decson-maker problem of the (BLMINP problem s as follows: Ma Ma f f (3 N [M ] To obtan the preferred soluton of the LDM problem; we transform (3 nto the followng mult-objectve decson-makng problem: Ma f ( (4 f δ j N (5 j j Ω n + n ( R So the algorthm steps for solvng (4-(5 are as follows: 4-The Algorthm for LDM Problem Step : (a Use Balnsk ' s algorthm to fnd all the vertces of the feasble regon M (bselect one of the non-nteger vertces ( n of the soluton space In the tableau of ths verte choose the row vector
Non-lnear programmng problem 37 where the basc varable has the largest fractonal value and construct ts correspondng Gomory ' s fractonal cut n the form a c (c Add the frst cut a c to the orgnal set of the constrants M Ths wll yeld a new feasble regon M (d Repeat agan the steps (a (c untl at some step r the obtaned vertces of the soluton space all are ntegers (e Elmnate (drop all the redundant constrants of the appled cuts (f Add all the constrants of appled s-effcent cuts to the orgnal set of constrants M to get [M] Step: ormulate the equvalent lnear fractonal program wth the constrants [M] Step3: Set the satsfactorness Let s s0 at the begnnng and let s s s respectvely Step 4: Set the ε -constrant problem P ( ε ( s f P ( ε ( s has no soluton or has a non-nferor soluton makng f ( < δ then go to step 3 to adjust s s j+ < s j Otherwse go to step 5 Step 5: Assumng that s a non-nferor soluton of P ( ε ( s judge by theorem 4 whether or not s a non-nferor soluton of (4-(5 If s a non-nferor soluton turn to step 6 f s nferor f f and at least soluton there must be a such that one ">" ; Repeat step 5 wth Step 6: If the decson-maker s satsfed wth then s a preferred soluton Otherwse go to step 7 Step 7: Adjust the satsfactorness Let s s j f s and go to step 4 4-The Second-Level Decson-Maker (SLDM Problem Secondly accordng to the nteractve mechansm of the BLMINP problem the LDM varables should be gven to the SLDM; hence the SLDM problem can be wrtten as follows: Ma Ma f f (6 [ ( N ( M ] The SLDM wll convert (6 nto the followng sngle objectve functon as follows: Ma f ( + (7 f δ j N (8 ( j ( Ω j j
38 O E Emam Our basc though on solvng (7-(8 s to fnd the second-level nonnferor soluton ( S that s closest to the LDM preferred soluton ( S Now we wll test whether ( s preferred soluton to the LDM or t may be changed by the followng test: If S ( ( <δ (9 S ( So ( S s a preferred soluton to the LDM where postve constant gven by the LDM whch means ( S soluton of the BLMINP problem (5 Interactve Algorthm for BLMINP problem δ s a small s a preferred Step : -Set k 0 ; solve the st level decson-makng problem to obtan a set of preferred solutons that are acceptable to the LDM The LDM puts the solutons n order n the format as follows: k k k + p k + p Preferred soluton ( ( Preferred rankng (satsfactory rankng k k k + k + k + p k + p f ( f f Step : -Gven Step 3: -If to the SLDM solve the SLDM problem to obtan S <δ S Where δ s a farly small postve number gven by the LDM then go to step 4 Otherwse go to step 5 S Step 4: - ( s the preferred soluton to the BLMINP problem Step 5: - Set k k + then go to step (6 Numercal Eample To demonstrate the soluton for nteractve BLMINP problem let us consder the followng eample: Ma Ma + 3 + [ st level] [ ] where solves [ nd level] Ma ( Ma + ( + [ + ] + 7
Non-lnear programmng problem 39 + 5 0 0 and ntegers rst the gven b-level nteger mult-objectve non-lnear programmng problem can be converted nto ts equvalent b-level mult-objectve non-lnear programmng problem as follows: Ma Ma + 3 + [ st level] ( [ ] [ nd level] Ma ( Ma + ( + + [ ] Subject to + 7 + 5 0 + 4 + 3 0 rst the LDM solves hs/her problem as follows: - nd ndvdual optmal soluton by solvng (3 we get ( b b ( 7 4 ( a a ( 00 - Usng the soluton of LDM problem we can formulate (4-(5 as follows: Ma + Subject to + 7 + 5 0 + 4 + 3 + 0 δ b a s + a Where So the LDM soluton s ( gven by LDM and ( s 03 δ 0 are Secondly the SLDM solves hs/her problem as follows: - nd the ndvdual optmal solutons by solvng (6 we get: ( b b ( 46 ( a a ( 0 - Usng the results from SLDM problem we can formulate (7-(8 as follows: Ma + Subject to + 7
330 O E Emam + 5 0 + 4 + 3 ( + + 85 0 Where ( b a s + a 8 5 δ So the SLDM soluton s ( S ( and( s 0 5 nally by usng (9 we wll fnd that ( S ( s a preferred soluton to the LDM from the followng test: ( ( 0 p 0 ( So ( S ( s the preferred soluton to the BLMINP problem (7 Summary and Concludng Remarks Ths paper has proposed an nteractve algorthm for solvng a b-level mult-objectve nteger non-lnear programmng (BLMINP problem wth lnear or non-lnear constrants We start by fndng the conve hull of ts orgnal set of constrants usng the cuttng-plane algorthm to convert the BLMINP problem to an equvalent (BLMNP problem Then the algorthm smplfes an equvalent (BLMNP problem by transformng t nto separate mult-objectve decsonmakng problems wth herarchcal structure and solvng t by usng ε -constrant method to avod the dffculty assocated wth non-conve mathematcal programmng s ntroduced However there are many other aspects whch should be eplored and studed n the area of mult-level optmzaton such as: Interactve b-level and mult-level nteger fractonal mult-objectve decson-makng problems Interactve b-level and mult-level nteger stochastc non-lnear multobjectve decson-makng problems 3 Interactve b-level and mult-level nteger large-scale non-lnear multobjectve decson-makng problems
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33 O E Emam [0]X Sh and H Xa Model and nteractve algorthm of b-level multobjectve decson-makng wth multple nterconnected decson- makers Journal of Mult-Crtera Decson Analyss 0 (00 7-34 Receved: Aprl 0