An Interactive Approach for Solving Multi-Objective Nonlinear Programming and its Application to Cooperative Continuous Static Games

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1 J. Appl. Res. Ind. Eng. Vol. 5, No. (08) Journal o Appled Researc on Industral Engneerng An Interactve Approac or Solvng Mult-Objectve Nonlnear Programmng and ts Applcaton to Cooperatve Contnuous Statc Games Hamden Abdelwaed Kala Department Operatons Researc, Insttute o Statstcal Studes and Researc, Caro Unversty, Gza, Egypt. P A P E R I N F O A B S T R A C T Croncle: Receved: 03 September 08 Revsed: November 08 Accepted: 3 August 08 Keywords: Mult-Objectve Nonlnear Programmng. Ecent Soluton. Reerence Drecton Metod. Attanable Reerence Pont. Interactve Approac. Satsactory Soluton. Cooperatve Contnuous Statc Games. Parametrc Study. In ts paper, an nteractve approac or solvng Mult-Objectve Nonlnear Programmng (MONLP) problem as ntroduced. Ts approac combnes wt te Reerence Drecton (RD) ntroduced by Narula et al. [0] and te Attanable Reerence Pont (ARP) metod ntroduced by Wang et al. [7]. In te nteractve approac, we stll start wt a weak ecent soluton as te rst step and use te correspondng objectve values to mprove te wegtng coecents o te augmented Lexcograpc Wegted Tcebyce Programmng (LWTP) problem; ence we mody te reerence pont n te case o an unsatsactory soluton or te Decson Maker (DM) e (se) wses. Te cooperatve contnuous statc game s ntroduced as an applcaton and ence te stablty set o te rst knd s determned correspondng to ts soluton troug te nteractve approac. Fnally, a numercal example as gven to te utlty o our nteractve approac.. Introducton Mult-objectve analyss assumes tat te objectves are generally n conlct. Wen modelng a Mult- Objectve Lnear Programmng (MOLP) problem, ow to calculate te exact values o te coecents s a probablstc task. Normally, te coecents are eter gven by a Decson Maker (DM) subjectvely or by statstcal nerence rom storcal data. Most mult-objectve metods are based on nteracton between a DM and te matematcal model o te problem under consderaton. A typcal nteractve metod exbts a erarccal structure composed o an analyss level, wc comprses te soluton o some auxlary sngle objectve optmzaton problem and decson level at wc te DM tres to nduce te analyss level to generate a soluton tat optmzes s (er) preerence uncton [5]. Mult-objectve optmzaton metods can be classed accordng to te DM nluence n te optmzaton process [5]: Metods were DM does not provde normaton (no-preerence metods). Correspondng autor E-mal address: amden_008@yaoo.com DOI: 0.05/jare

2 97 An nteractve approac or solvng mult-objectve nonlnear programmng and ts applcaton Metods were a posteror normaton s used (posteror metods). Metods were a pror normaton s used (pror metods). Metods were te progressve normaton s used (nteractve metods). Interactve approaces ave been nvented to combne advantages o bot posteror metods and pror metods and avod te dsadvantages. Snce te DM s nvolved n te entre soluton process, ts approac as ound better acceptance n practce. Among all te soluton approaces, nteractve metods ave become popular and are consdered promsng or Mult-Objectve Optmzaton Problems (MOPs). Altoug te numerous nteractve procedures ave been suggested, none as emerged as a clearly preerred approac [6]. Recently, researcers ave ntroduced te concept o te algortm wc lnks varous approaces n a way tat make use o te advantages [, 8]. A new algortm or solvng te MOP startng wt te utopan pont as been ntroduced by Sadrabad and Sadjad [5]. Te applcatons o game teory may be ound n economcs, engneerng, bology, and n many oter elds. Tree major classes o games are matrx games, contnuous statc games, and derental games. In contnuous statc games, te decson possbltes need not be dscrete and te decsons and costs are related n a contnuous rater tan a dscrete manner. Te game s statc n te sense tat no tme story s nvolved n te relatonsp between costs and decsons. Elsae [3] ntroduced an nteractve approac or solvng te Nas cooperatve contnuous statc games and also determned te stablty set o te rst knd correspondng to te obtaned compromse soluton. Kala and ZenElden [6] ntroduced an nteractve approac or solvng cooperatve contnuous statc games wt uzzy parameters n te objectve uncton coecents. Navd et al. [] presented a new game teoretcbased approac or mult response optmzaton problem. Osman et al. [] ntroduced a new procedure or contnuous tme open loop stackelberg derental game. Kala [7] proposed an nteractve approac or solvng mult-objectve nonlnear programmng problem. Te approac s based on te Reerence Drecton (RD) ntroduced by Narula et al. [0] and te Attanable Reerence Pont (ARP) metod ntroduced by Wang et al. [7]. Cruz and Smaan [] proposed te teory o ordnal games, were te players are able to rank-order ter decson coces aganst te coce by te oter players nstead o payo uncton objectve unctons. Muammed et al. [9] conducted a revew to nvestgate te varous actors aectng cooperaton n underwater acoustc sensor network. Tey studed te varous cooperaton tecnques used or underwater acoustc sensor network rom derent perspectves. Molnac and Earn [8] nvestgated a type o publc games played n groups o ndvduals wo coose ow muc to contrbute towards te product o a common good at te cost to temselves. Te stablty set gves a wde nsgt or te stablty o te soluton o parametrc nonlnear optmzaton problems due to a parametrc cange. Te essence o ts approac s n te denton, caracterzaton, and determnaton o a group o parameter sets, suc as te set o easble parameters, te solvablty set, and te stablty set o te rst and second knds. Tese sets ave been dened and caracterzed n te crsp envronment or parametrc nonlnear derentable programmng problems by Osman [] and Osman and Dauer [3]. Te rest o te paper s as ollows: In Secton, te Mult-Objectve Nonlnear Programmng (MONLP) problem s ormulated. In Secton 3, an nteractve approac or solvng te MONLP problem s presented. In Secton, te cooperatve contnuous statc game wt parameters n te rgt and sde o te constrants s presented. Fnally, some concludng remarks are reported n Secton 5.

3 Kala / J. Appl. Res. Ind. Eng. 5() (08) Problem Formulaton and Soluton Concepts Consder te ollowng MONLP problem mn Z ( x ) ( ( x ), ( x ),..., ( x )) t s s.t. () x M. Were n x R and x),,,3,..., s ( are real valued unctons. Is assumed tat: Te easble regon n M R s non-empty and compact. ( x),,,3,..., s, are contnuous. Denton. (Wang et al. [7]). Let ) t (,,..., s be an attanable reerence pont,.e. *,,,3,..., s and may be easble or neasble or te Eq. (), and x M. * x s sad to be reerence non-domnated soluton o MONLP problem MONLP Eq. (), and ( x * ). * A pont x s sad to be reerence (non-domnated) satsactory soluton o Eq. () ecent soluton and ( x * ) s satsactory or te DM. * x s an ecent soluton o * x s reerence 3. An Interactve Approac Te steps o te nteractve approac Step. Fnd as ollows: max,,... m y, s.t. () y ( x);, x M, y R. Step. Gven an ntal reerence pont s 0 R suc tat 0 0. Let I {,,..., s}, I I, 0. Step 3. Calculate te wegtng vector rom te ollowng relaton w,,,3,..., s, (3) and ten solve te LWTP problem.

4 99 An nteractve approac or solvng mult-objectve nonlnear programmng and ts applcaton s Lex mn, ( x),,..., s s.t. () w ( x) x M, 0 R. ;,,3,..., s, Let x be te obtaned optmal soluton. Step : Determne te termnaton. Wen nal soluton. Wen s not satsactory and reerence soluton o Eq. (). Oterwse, go to Step 5. Step 5: Mody te reerence pont. () DM selects any objectve n J J J : s satsactory to te DM, stop wt x x as te or s, tere s no satsactory ecent e n J suc tat : at. Let J J /{ e }. Separate e s an unsatsactory J nto two parts: at, J J J. () For and DM wses to release te value o / J te DM provdes, te amount to be relaxed or at suc tat,. For J, let. For or all J { e }, return to (), to separate / te amount to be relaxed or some J ) at ( tere s no satsactory ecent soluton. In te case tat J / J, let. 0. Let () In te case J once agan or return to () to ncrease, te DM wses to do so. Oterwse, stop and Let e e,, e, and solve te ollowng auxlary problem: mn e (x) or some J { e }, go to (v) / s.t. (5) ( x),,,..., s, e, x M. To get te soluton x, wen ( ) ( e x e x ) or ( ) e x or objectve DM, return to () to ncrease te amount to be relaxed or some J ) at ( e s not satsactory to te, te DM wses to do so. Oterwse, stop tere s no satsactory ecent soluton. Wen ( ) ( e x e x ) and e ( x ) or objectve mproved or ( e e x wen x e s satsactory to te DM, e/ se provdes, te largest amount to be e at ( x ), ) suc tat 0, ( x ) ( x ). Let ( x ). (v) I e, return to () wt e e. Oterwse, let x x, s an unque optmal soluton o te Eq. (5) or let x e e e e and return to (v) be an optmal soluton o te ollowng:

5 Kala / J. Appl. Res. Ind. Eng. 5() (08) mn,,..., s s.t. (6) w ( x) x M, 0 R.,,,..., s, and return to step (). I ( x ), Let Example. Consder te ollowng problem: e e let and return to Step 3. s.t. mn Z ( x) ( ( x), ( x) ) ( x 3) ( x ) 0, x x 0. T Were ( x) x and ( x) x. Let us apply te steps o te nteractve approac as: Step. Startng wt x by solvng te ollowng problem: s.t. max, x, x, ( x 3) ( x ) 0, xx 0, 0 R,,. Let te soluton be x (3,009,.9967) and te correspondng objectve values are (3,009,,9967). Step. DM speces an ntal attanable reerence pont, suc tat s ? 5.000, and ?.000. Ten, ) T (5.000,.000. Let J {,,}, J O J. or every, tat Set te teraton counter 0, and go to Step 3.

6 30 An nteractve approac or solvng mult-objectve nonlnear programmng and ts applcaton Te ollowng steps o te nteractve approac as llustrated as n te ollowng lowcart. Fg.. Steps o te nteractve approac.

7 Kala / J. Appl. Res. Ind. Eng. 5() (08) Step 3. Calculate te wegtng vector rom te ollowng relaton: w 0.500, and w Step. Solve te problem troug te LWTP Eq. () wt te calculatng wegts as: s.t. mn {, x x.9967 } x 3.009, 0.99 x.9967, ( x x x, x 3) x ( x 0, 0,0 ) R. 0, Step 5. Case: * * x (3.009,0), (3.009,0), (5.000,,000), and (3.009,.9967) s te DM satsed Y/N: Y, stop wt x * (3.009,0) and * (3.009,0) as te nal soluton.. An Applcaton n Cooperatve Contnuous Statc Games Consder te ollowng Cooperatve Contnuous Statc Game (CCSG), wt parameters n te rgt and sde o te constrants, and wt m players. Tese players respectvely ave te costs s.t. a,, F a,,..., F a,, mn F m a, 0, j,, n j..., (8) k R gl a l l r : (, ),,,...,. (9) (7) Were n k R R, l n k F,,,..., m are convex unctons on R R, g, l,,..., r are concave unctons on are any real numbers. Assume tat tere exsts a uncton a ( ). I te unctons j ( a, ) j ( a, ), j,,..., n are derentable ten te Jacobn 0, j, k,,..., n a negborood o a soluton pont ( a, ) to Eq. (8), a ( ) s te soluton to Eq. (8) generated by. Derentablty assumpton are not needed or te unctons F ( a, ) and j ( a, ). Also, ( ) s a regular and compact set. Ten te correspondng mult-objectve optmzaton problem wt parameters n te rgt and sde o constrants s: l k n a

8 303 An nteractve approac or solvng mult-objectve nonlnear programmng and ts applcaton mn ( ), F ( ),..., F t m ( F ) s.t. (0) gl ( ) l, l,,..., r. k s k Were F : R R,,,..., m are convex unctons on R, gl : R R, l,,..., r are concave k unctons on R, F ( ) F ( ( ), ), and gl ( ) gl ( ( ), ). Here, we assume tat Eq. (0) s stable []. Te proposed nteractve procedure ntroduced n Secton 3 s appled to te Eq. (0). Let r be te nal soluton correspondng to R obtan by solvng Eq. (0) troug te use o te proposed nteractve procedure. To determne te stablty set o te rst knd correspondng to denoted by S ( ), applyng te ollowng condtons: u ( g ( )) 0, l,,...,r l l l u 0, l,,...,r. l Example. Consder te ollowng two player games wt F ( ) ( ) ( ) F ( ) ( ) ( ) Were player controls R and player controls R wt,,,,,,, R. 3 3 By applyng te proposed nteractve approac to ts problem we obtan te soluton 0 (0.6,.85) T corresponds to (,,0,0) to determne S (0.6,.85) ; we apply te ollowng condtons:,. We ave,,3, I. u ( 3. ) 0, u (.6 )0, u ( 0.6 ) 0, 3 u (.85 ) 0. 3 For I, u 0, u 0, u 0, u 0. Ten 3

9 Kala / J. Appl. Res. Ind. Eng. 5() (08) I 3. S (0.6,.85) R : 3.,.6, 0.6, v.85 I,, u 0, u 0, u 0, u 0. Ten For I 3 3. S (0.6,.85) R : 3.,.6, 0.6, v.85. For I u 0, l,,3,. Ten I 3 3, 3. S (0.6,.85) R : 3.,.6, 0.6, v.85. I,,3,, u 0, l,,3,. Ten For I 3. S (0.6,.85) R : 3.,.6, 0.6, v.85. Te stablty set correspondng to te oter proper subsets I q o,,3, can be computed n te same way. Hence, S (0.6,.85) s gven by: 5. Concluson 6 S(0.6,.85) S q I q (0.6,.85). In ts paper, an nteractve approac or solvng MONLP problems was proposed. Ts approac combned wt te RD metod ntroduced by Narula et al. [0] and te ARP metod ntroduced by Wang et al. [7]. Te advantages are te reducton o te number o teratons; ence te computatonal eort requred to obtan te nal soluton was reduced. Also, te cooperatve contnuous statc game was ntroduced as an applcaton. Acknowledgements Te autors are very grateul to te anonymous revewers or s/er nsgtul and constructve comments and suggestons tat ave led to an mproved verson o ts paper. Reerences [] Cruz, J. B., & Smaan, M. A. (000). Ordnal games and generalzed Nas and Stackelberg solutons. Journal o optmzaton teory and applcatons, 07(), 05-. [] Cankong, V., & Hames, Y. Y. (008). Mult-objectve decson makng: teory and metodology. Courer Dover Publcatons. [3] Elsae, M. M. K. (007). An nteractve approac or solvng Nas cooperatve contnuous statc games. Internatonal journal o contemporary matematcal scences, (), 7-6. [] Gardner, L. R., & Steuer, R. E. (99). Uned nteractve multple objectve programmng. European journal o operatonal researc, 7(3), [5] Hwang, C. L., & Masud, A. S. M. (0). Multple objectve decson makng metods and applcatons: a state-o-te-art survey (Vol. 6). Sprnger Scence & Busness Meda.

10 305 An nteractve approac or solvng mult-objectve nonlnear programmng and ts applcaton [6] Kala, H. A., & Zeneldn, R. A. (05). An nteractve approac or solvng uzzy cooperatve contnuous statc games. Internatonal journal o computer applcatons, 3(). [7] Kala, H. A. (06). An nteractve approac or solvng uzzy mult-objectve non-lnear programmng problems. Te journal o uzzy matematcs, 3(), [8] Molna, C., & Earn, D. J. (07). Evolutonary stablty n contnuous nonlnear publc goods games. Journal o matematcal bology, 7(-), [9] Muammed, D., Ans, M., Zaree, M., Vargas-Rosales, C., & Kan, A. (08). Game teory-based cooperaton or underwater acoustc sensor networks: Taxonomy, revew, researc callenges and drectons. Sensors, 8(), 5. [0] Narula, S. C., Krlov, L., & Vasslev, V. (99). An nteractve algortm or solvng multple objectve nonlnear programmng problems. In G. H. Tzeng H. F. Wang U. P. Wen & P. L. Yu (Eds.). Multple crtera decson makng (pp. 9-7). Sprnger, New York, NY. [] Navd, H., Amr, A., & Kamranrad, R. (0). Mult-responses optmzaton troug game teory approac. Internatonal journal o ndustral engneerng and producton researc, 5(3), 5-. [] Osman, M. S. A. (977). Qualtatve analyss o basc notons n parametrc convex programmng. I. Parameters n te constrants. Aplkace matematky, (5), [3] Osman, M., & Dauer, J. (983). Caracterzaton o basc notons n mult-objectve convex programmng problems. Retreved rom Lncoln, Unversty o Nebraska database. [] Osman, M. S., El-Koly, N. A., & Solman, E. I. (05). A recent approac to contnuous tme open loop stackelberg dynamc game wt mn- max cooperatve and non-cooperatve ollowers. European scentc journal, (3), [5] Sadrabad, M. R., & Sadjad, S. J. (009). A new nteractve metod to solve multobjectve lnear programmng problems. JSEA, (), [6] Sn, W. S., & Ravndran, A. (99). A comparatve study o nteractve tradeo cuttng plane metods or MOMP. European journal o operatonal researc, 56(3), [7] Wang, X. M., Qn, Z. L., & Hu, Y. D. (00). An nteractve algortm or multcrtera decson makng: te attanable reerence pont metod. IEEE transactons on systems, man, and cybernetcs-part A: systems and umans, 3(3), [8] Wendell, R. E., & Lee, D. N. (977). Ecency n mult-objectve optmzaton problems. Matematcal programmng, (), 06-.