Local Perturbation Analysis of Linear Programming with Functional relation Among Parameters

Transcription

1 4 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March Local Perturbaton Analyss o Lnear Prograng wth Functonal relaton Aong Paraeters Paya Hanaadeh, Allaeh Tabataba Unversty, Iran Abolal Ghae, Arabr Unversty o Technology, Iran Madd Tavana, La Salle Unversty, USA AbstrAct In ths paper, the authors study the senstvty analyss or a class o lnear prograng (LP) probles wth a unctonal relaton aong the obectve uncton paraeters or those o the rght-hand sde (RHS). The classcal ethods and standard senstvty analyss sotware pacages al to uncton when a unctonal relaton aong the LP paraeters preval. In order to overcoe ths decency, the authors derve a seres o senstvty analyss orulae and devse correspondng algorths or derent groups o hoogenous LP paraeters. The valdty o the derved orulae and devsed algorths s corroborated by open lterature exaples havng lnear as well as nonlnear unctonal relatons between ther vector b or vector c coponents. Keywords: Dependent Paraeters, Functonal Relaton, Lnear Prograng, Perturbaton Analyss, Senstvty Analyss IntroductIon The an purpose o classcal senstvty analyss s to exane the varatons o the obectve uncton s optal values and the soluton coponents as a result o the nntesal changes n one o the paraeters whle the other ones are ept xed (Bradley et al., 977). One o the probles conronted n classcal senstvty analyss s when several paraeters DOI:.48/ors. are changed sultaneously. In those cases, classcal ethods cannot obtan the eect o the perturbatons on the obectve uncton as well as on the optal soluton because when the sultaneous changes happen n the apltudes range o basc varables or bndng constrants, the order o basc soluton changes. However, there s a conservatve bound or the obectve uncton coecents or the rght-hand sde (RHS) paraeters on ther sultaneous changes. Ths bound s ntroduced under the percent rule o changes when the perturbed Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

2 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 4 coecents o obectve uncton are related to basc varables, or when the RHS paraeters are related to bndng constrants. The percent rule states that the su o the proper racton o the desred changes to the u possble change n that drecton s less than or equal to one then the current basc optal soluton wll rean unchanged (Bradley et al., 977). The general or o lnear prograng (LP) proble can be consdered as ollows: { c T x Ax b, x } () x where A s an n atrx wth ull ran and b s a colun vector called the RHS paraeters reseblng the aount o resources, and c s the coecent vector o the obectve uncton; they are collectvely called the paraeters o the LP proble. A splex algorth ay be used to solve the LP probles. A splex algorth, n each step, chooses a set o the ndependent coluns ro atrx A, provdes the correspondent basc easble solutons and checs or the optalty condtons. It s assued that a basc optal soluton s avalable or ths proble. Let us ntroduce the ollowng atheatcal notatons and dentons used throughout ths paper and or the splex table: S = { B, B,, B }:The set o basc varable B ndces S = { N, N,, N } :The set o non-basc N n varable ndces B:A sub-atrx o A whose coluns are assocated wth the basc varable o S B N:A sub-atrx o A whose coluns are assocated wth the non-basc varable o S N c B : The coecent vector o the obectve uncton whose eleents are related to the basc varable x B = B - b :The optal basc soluton o the LP proble T - = c B b :The optal value o the obectve B uncton Y = B N :The atrx wth entres o the nal table o the splex or the non-basc varables y : The entres o atrx Y In odelng and solvng LP probles, soe cases occur where the paraeters o the proble are unctonally related n a way that the change o a specc paraeter causes the sultaneous change o others. What separates the dscusson o the senstvty analyss n ths paper ro ts classcal ethods s the unctonal relaton aong the paraeters n LP. I the LP proble s n the or o (), the unctonal relaton aong the paraeters s dened as G(θ) =, where the unctonal relaton s consdered aong coponents o hoogenous paraeters o θ, (e.g. G( b, b,..., b ) = ). Ths relaton can be lnear or nonlnear whch heren s consdered to be contnuous, derentable, and ts range space s one densonal. The doan o G( ) = s dened on an e -neghborhood o θ, naely, N ε ( θ ) = θ : θ θ { }, where q s the ntal estaton o the perturbed paraeters and t s not epty. Two stuatons o unctonal relaton between proble paraeters are cted below: a. The nature o partcular probles or solvng ethods poses unctonal relaton on LP proble. For exaple, the transportaton splex s developed wth the prevalence o balance between supply and deand (naely the RHS paraeters o transportaton proble (Wendell, 98). b. Because o the envronental and physcal condtons nherent n the probles under the study, the unctonal relaton oten becoes necessary n the process o odelng (e.g., when the value o obectve uncton coecents coputed by the analytcal herarchy process (AHP); then there s a lnear uncton aong the coecents (Ghodsypour & O Bren, 998). Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

3 44 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March The rest o the paper s organed as ollows. We present a revew o the relevant lterature on senstvty analyss n the next secton. We ollow ths revew wth a dscusson o the senstvty analyss or the LP probles wth unctonal relatons between the hoogeneous paraeters. We then provde our nuercal exaples llustratng the ecency o the proposed ethods n coparson to the classcal ethods. Fnally, we conclude wth our conclusons and uture research drectons. the LIterAture review The portance o senstvty analyss n lnear prograng has been wdely stressed n the anageent scence lterature. However, the research on senstvty analyss wth unctonal relaton has been sporadc and scattered. Usually varaton occurs n the RHS o the constrants and/or the obectve uncton coecents. A coprehensve survey o senstvty analyss o LP probles can be ound n Gal and Greenberg (997) and Roos et al. (997) and the reerences theren, but no-sultaneous perturbaton o both the RHS and the obectve uncton data s consdered. Greenberg () studed sultaneous perturbaton o the RHS and the obectve uncton data when the pral and dual lnear optaton probles are n canoncal or. However, or lnear optaton probles n canoncal or t s necessary to dene the optal partton to separate the actve and nactve varables as well as the actve and nactve constrants or all optal solutons. In ths secton, we brely revew the state o the art n local perturbaton (senstvty) analyss. Ths revew s conned to paraetrc analyss, range analyss and tolerance approach. The post optalty analyss o the LP probles was ntroduced by Arsha (99). Ths was the rst eort to study senstvty analyss on a specc structure o LP probles wth unctonal relaton aong ts paraeters. Ths outcoe steed ro the act that the balance between supply and deand posed a unctonal relaton on the RHS paraeters o the transportaton proble (Arsha, 99). Arsha and Kahan (989) used the requred noraton or conductng senstvty analyss ro the transportaton splex. Ghodsypour and O Bren (998) ntroduced a ethod or suppler selecton and optal purchase by ntegratng AHP wth LP where the obectve uncton coecent values were obtaned on the bass o anagers expert udgent n the process o usng AHP. The obectve uncton coecents o the suppler selecton proble ollowed a lnear unctonal relaton. Prevalence o ths relatonshp prevented usng the coon ethods o senstvty analyss or the sad proble. Ghodsypour and O Bren (998) studed ths shortcong and called or the need to develop conventonal ethods or senstvty analyss when unctonal relatons preval. Jansen et al. (997) warned researchers and practcng anagers aganst the conusng results when degeneracy o the optal basc soluton s gnored and proved that the brea ponts n the optal obectve value (as a uncton o b and c ) occur where the optal partton as well as the representaton o the optal uncton changes. The ntervals between two consequent brea ponts are the unons o the ranges obtaned ro ndvdual optal bases by a splex ethod wth the coon property o dentcal representaton or the optal value uncton, a act whch was prevously proven by Adler and Montero (99). Kolta and Terlay () ntroduced three ponts o vew wth respect to senstvty analyss o LP probles. The rst pont dened a space or the changes o the proble s paraeters, n a way that the optal bass reaned optal. The second pont dened a space or the paraeter varatons where the ndces set o the optal soluton (not necessarly basc varables) dd not change. The thrd pont deterned the bounds or the paraeter changes where representaton o the optal value uncton (n unparaetrc case, ts slope) reaned unchanged. These derences were all due to the degeneracy o the proble. For the ultparaetrc case, ater revsng the tolerance approach, Flpp () obtaned the u Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

4 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 4 pertted varaton o the paraeters usng a geoetrcal ethod. He ound unque ranges whch deterned the u and nu sultaneous changes or ndependent paraeters. Ghaar-Hadgheh and Terlay (6a) urther analyed the propertes o the optal value uncton when sultaneous perturbaton o the RHS and the obectve uncton happen n convex quadratc optaton. Recently, senstvty analyss n b-paraetrc lnear optaton probles has been the ocus o several studes (Ghaar-Hadgheh & Terlay, 6a, 6b; Ghaar-Hadgheh et al., 6; Ghaar-Hadgheh et al., 7; Dehghan et al., 7; Mrna & Ghaar-Hadgheh, 7). Ghaar-Hadgheh et al. (8) have proved that nvarancy regons are separated by vertcal and horontal lnes and generate a esh-le area. They have also showed that the boundares o these regons can be dented n polynoal te. sensitivity AnALysIs with FunctIonAL relation In ths secton, we ocus on local perturbaton (senstvty) analyss and not the paraetrc one whch s well explaned n Facco (98). As a result, the range o varaton o paraeters s very sall and alls wthn an e -neghborhood o the estated paraeters. We devse senstvty analyss ethodologes or probles wth unctonal relaton aong the hoogenous paraeters. Ths unctonal relaton can be lnear or nonlnear or the RHS paraeter values or or the obectve uncton coecent values. We should reterate that when the unctonal relaton s a nonlnear uncton, very sall perturbaton or paraeters values are consdered. It s assued that varatons all wthn an e -neghborhood o the estated paraeters. It s also assued that a basc optal soluton s avalable. However, the underlyng optal basc soluton s degenerate, then, the ethod ght al to gve a proper result (See Jansen et al., 997). Moreover, the ethod s not applcable the optal soluton s not a basc one. varatons n the rhs values In ths secton, rst the slope changes o the optal values o the obectve uncton and the basc varables are analyed wth respect to the changes n RHS paraeter values regardless o the prevalence o a unctonal relaton. Then, usng the total derentaton operator, the ethod or calculatng the slope changes n case unctonal relatons aong the RHS paraeters are provded. A lowchart o the senstvty analyss wth respect to the changes n the RHS values s presented n Fgure. Let us consder solvng proble () on the bass o the prary estaton o b and obtan the optal soluton: T - = c B b () B x - B = B b () When there s no relatonshp between the coponents o vector b, then we have: d T d = c B - e or = v =,,..., B (4) where v s the value o dual varable correspondng to the shadow prce o the th constrant. Also, dx = y =,,..., (), where y, s the (,) th coponent o the B - splex optal tableau. Relatons (4) and () show the senstvty o the obectve uncton and the value o basc varable to the changes n paraeter b. When there s a unctonal relaton between the coponents o vector b, we have: G( b,..., ) = (6) b Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

5 46 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March Fgure. Flowchart o the senstvty analyss wth respect to the changes n the RHS values Then, the classcal ethods o the senstvty analyss addressed above wll not yeld the correct results. Under such unctonal prevalence, one ay use total derentaton as ollows: d dx = b + b b xb x = + b x b (7) (8) Consderng that the rate o changes o and x wth respect to b s s not a sooth (contnuously derentable) uncton, the above relatons are vald only or the sooth ranges and are not dened n brea ponts. For the unctonal relaton G, we have: G + G + + G... b b b = (9) Heren, senstvty analyss o and x wth respect to b s wll be derved or the paraeter ranges where the basc soluton reans optal. In ths case, we consder the ollowng two ors: Only two coponents o vector b are perturbed sultaneously. eleents o vector b are perturbed sultaneously ( ). Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

6 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 47 The senstvty analyses or both o the above cases are dscussed below: Perturbaton o two rhs Paraeters Suppose that the eect o perturbaton b on and x s studed n a way that ths perturbaton s absorbed by a second paraeter, b, so that the prevalent unctonal relaton s satsed. In such a case, we wll have: G + G b b d d dx = b = G/ G/ b () Also, ro (7): = b + b = b + b b G/ b G/ Slarly, ro (8), we have: x = b x B B + b () G/ b G/ b () ollows: (I uncton G s lnear, the eects o changes are calculated accurately). = + G/ b v v G/ b () = + G/ b x y y B,, G/ b (4) These two relatons are vald when G s a lnear uncton. When G s a nonlnear one, these relatons ay be vald or very sall perturbaton o paraeters. As the devaton gets larger, the lnear approxaton o the nonlnear uncton G wll no longer be satsed. The optalty soluton pont under sultaneous perturbaton o the RHS paraeters s studed n two cases: A. General varatons on rhs Paraeters In ths case, t s sucent to eep the easblty o the soluton as: x + x =,,..., G/ b x + y + y B,, G/ b Thereore, b Then, we obtan the u pertted changes o paraeter b n order that the current optal soluton reans easble. Usng () and (), the obvous eect o paraeter b can be respectvely estated on and x, as x dx + B I dx <, then Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

7 48 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March x dx / and n the general case, we have: G/ b G/ b + (7) x b n B S B dx / dx < () Then, Db s the u ncrease o b ro ts ntal value and Db s related to the two ollowng cases: b. varatons n rhs Paraeters related to the bndng constrants In ths case, the basc soluton wll rean unchanged, provded that the percent rule o varatons s granted (Bradley et al., 977), that s: + b and Db and Db and be nte, G/ b b b G/ + (6) ust have the sae sgn G/ b G/ b + b and the upper and lower bounds are dened as ollows accordng to the sgn o Db : B.) b > B..) I G/ b G/ b, Db s non-postve and Db s the u decrease o b ro ts ntal value. B..) I G/ b G/ b <, Db s postve and Db s the u ncrease o b ro ts ntal value. B.) I b < G/ b G/ b + (8) Then, Db s the u decrease o b ro ts ntal value and Db s related to the two ollowng cases: B..) I G/ b G/ b, Db s non-negatve and Db s the u ncrease o b ro ts ntal value. B..) I G/ b G/ b <, Db s negatve and Db s the u decrease o b ro ts ntal value. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

8 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 49 Perturbaton o coponents o vector b (General case) The senstvty analyss o and x should be studed when coponents o vector b change and others rean xed. Ths happens when one wants to study the eect o the change o b on and x so that the other ( -) paraeters change. In ths case, the varatons o ( -) paraeters Db are arbtrarly changed and th paraeter s obtaned ro unctonal relatong. d = = + b b b b (9) Here, or splcty, the rst paraeters are taen or the above analyss wthout loss o generalty. Slarly, we have: dx = = + x x b + + x x B B B B... B b b b b Now, by replacng d we have: = v and dx = + b + + b v v... v b b = x y + y, () = y y, b b B, In above equaton, we replace, wth Db. The easblty o the soluton due to the Db sultaneous changes o the RHS paraeters s studed n two cases: A) General varatons on the rhs Paraeters In ths case, n order to rean the soluton easble, t s sucent that: x + x =,,..., B B Thereore, x dx + B I dx x dx / <, then and n general, we have x b n B S B dx / dx < b) varatons o the rhs Paraeters related to the bndng constrants () In ths case, the basc soluton reans optal, provded that the percent rule o changes s satsed (Bradley et al., 977), that s: = b where Db and Db have the sae sgn and are nte. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

9 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March varatons n the obectve Functon coecents An ndrect approach or analyng the senstvty values wth respect to the coecents o the obectve uncton s to use the dual proble because the obectve uncton coecents n the pral proble play the role o the RHS paraeters n the dual one. Here, a drect approach s used to do the senstvty analyss wth respect to the perturbatons n the obectve uncton coecents. The sequence o the steps necessary or conductng the senstvty analyss wth respect to perturbatons o two obectve uncton coecents s shown n Fgure. Let us consder solvng proble () on the bass o the ntal value o the coponent o vector c and obtan the optal soluton: T - = c B b B x - B = B b c = c y c = c y c B B, I there s no unctonal relaton aong the coponents o c, we have: () Fgure. Flowchart o the senstvty analyss wth respect to the changes n the obectve uncton coecent values Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

10 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March The senstvty analyses or both o the above cases are dscussed below: Perturbaton o only two coecents o the obectve Functon () Equatons () and () respectvely show the senstvtes o the obectve uncton and the shadow prce o each varable wth respect to the perturbaton o c. In cases wth unctonal relaton aong the coponents o vector c : H( c, c,, ) = (4) c n Slar to the prevous secton, usng the concept o total derentaton, we have: d = c + c + + c n n () Suppose that one wants to study the eect o perturbaton o c on and - c such that another coponent o vector c le c s also perturbed and the unctonal relaton between coponents o c s satsed. In ths case we have: H c + H c = c = H/ and also: d (8) c c H/ c = + = + c c (9) and also: H + H + + H c c c = n n (6) (7) where equaton (9) shows the senstvty o wth respect to the perturbaton o c. These two relatons are vald when H s a lnear uncton. I H s a nonlnear one, these relatons ay be vald or very sall perturbaton o paraeters. As the devatons ncrease n se, lnear approxaton o the nonlnear uncton H s no longer satsed. For replacng the equaton () wth (9), we consder the ollowng states: Agan, we consder the ollowng two ors: Perturbaton o only two coecents o the obectve uncton Varatons n coponents o the obectve uncton coecents A) I both paraeters are related to non-basc varables. B) I one o the paraeters s related to a basc varable and the other one corresponds to a non-basc one. C) I both paraeters are related to basc varables. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

11 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March For each state, we calculate equaton (9) as ollows: A) d = - I c relates to basc varables and c to non-basc varables then, d = x + B - I c relates to basc varables and c to non-basc varables then, d H/ c = + x B H/ c C) d = x + x B B Slarly, ro (6): ( c ) ( c ) H/ c d ( c ) = + c c () As entoned above, when H s a nonlnear uncton, () wll be vald or very sall perturbatons o perturbed paraeters. For replacng (4) wth (), we consder the ollowng states: A) I both paraeters are related to non-basc varables and -, ¹ - At least one o the corresponds to B) I one o the paraeters s related to a basc varable and the other one s assocated wth a non-basc one and - Non-basc varables do not correspond to - Non-basc varables correspond to C) I both paraeters are related to basc varables. For each o the above states, we calculate equaton () as ollows: A.) d ( c ) = A..) c = c d ( c ) = + A..) c = c H/ c d ( c ) ( ) = + B.) c reers to basc varables and c reers to non-basc varables B..) ¹ d ( c ) y = +, B..) = H/ c d ( c ) y ( ) = +, B.)c reers to non-basc varables and c reers to basc varables B..) ¹ H/ c d ( c ) y = +, B..) = H/ c d ( c ) y = +, C.), ¹ H/ c d ( c ) y y = +,, C.) = d ( c ) y = +, C.) = H/ c d ( c ) y = +, Next, we derve the upper and lower bounds or the value o the paraeter c (sultaneous change o c ) so that the current optal soluton reans optal. Keepng optalty o the soluton under sultaneous changes o the Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

12 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March obectve uncton coecents s studed n two cases as ollows: A) General varatons n obectve Functon coecents In ths case, n order to eep the soluton optal, t s sucent that: =,,, n c + ( c ) Consderng (), we have: d( c ) ( c ) = c Thus d( c ) c + c Now, d ( c ) <, then ( c c ) d( c ) Generally, we have: ( c ) d c n < SB d( c ) ( ) () b) varatons n coecents related to basc varables I the percent rule or varatons s satsed, the basc soluton wll rean optal, that s: + c H/ c H/ c + c But upper and lower bounds are dened accordng to the sgn o Dc as ollows: I c > H/ c + () where Dc s the u ncrease o c ro ts ntal value and Dc reers to the two ollowng cases: A.) I H/ c, Dc s non-postve and Dc s the u decrease o c ro ts ntal value. A.) I H/ c <, Dc s postve and Dc s the u ncrease o c ro ts ntal value. B) I c < Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

13 4 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March H/ c + () Dc s the u decrease o c ro ts ntal value and Dc s related to the two ollowng cases: B.) I H/ c, Dc s non-negatve and Dc s the u pertted ncrease o c ro ts ntal value. B.) I H/ c <, Dc s negatve and Dc s the u pertted decrease o c ro ts ntal value. varatons n coponents o the obectve Functon coecents Suppose that the eect o change o c on and - c s to be studed when other ( -) paraeters change and the unctonal relaton reans satsed. Slar to the prevous secton, usng the concept o total derentaton, we have: d = = + c c c + + c (4) Here, wthout loss o generalty, or the sae o splcty the rst paraeters are assued as the subect paraeters. Slarly, we have: Equatons () and () deterne the values o and ( c ) respectvely n c c equatons (4) and (). Here, because the specc values o, c and ( c ) are derent and dependent on c ther correspondng basc varables, we only recall ths general equaton (see the secton on the perturbaton o only two coecents o the obectve uncton). Retanng the soluton optalty n the stuaton o sultaneous changes o the obectve uncton coecents (change o paraeters) s studed n two ollowng cases: A) General varatons n obectve Functon s coecents In ths case, the basc soluton reans optal, ( ) d c c + c =,,, n ( ) < I d c c c d c ( ) ( ) In the general case, we have: ( c ) d c c n SB d ( c ) ( ) < (6) () Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

14 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March b) varatons n obectve Functon s coecents correspondng to basc varables I the percent rule or varatons s satsed, the basc soluton wll rean optal, that s: = c B (7) nuerical exaples In ths secton, we corroborate the valdty o our proposed algorths by applyng the orulae derved n ths study to our nuercal exaples and coparng our results wth those obtaned by Ghodsypour and O Bren (998). exaple : suppler selecton Proble In ths exaple, we consder a suppler selecton proble. Suppler selecton s a ultcrtera proble wth both qualtatve and quanttatve actors. In order to select the best supplers t s necessary to ae a trade-o between these qualtatve and quanttatve actors soe o whch ay conlct. Ghodsypour and O Bren (998) have devsed a procedure that ntegrates AHP and LP and consders both qualtatve and quanttatve actors n choosng the best supplers and placng the optu order quanttes aong the such that the total value o purchasng s ed. Ther ethod calculates the attracton o every suppler by consderng desrablty and anageent crtera. The aount o purchase ro each suppler s the decson varable n ther LP odel and the attracton rates are used as the coecents o the obectve uncton along wth constrants such as budget, aount o deand and product qualty. Ghodsypour and O Bren (998) suggest exanng the senstvty o the estated paraeters n ther odel because the attracton rate o each suppler s extracted on the bass o expert udgent. However, Ghodsypour and O Bren (998) gnore the unctonal relatons aong the coecents o ther obectve uncton. Ths unctonal relaton s one o the characterstcs o AHP where the su o the obectve uncton coecents ust equal. The prevalence o ths lnear relatonshp n the odel prevents one to use the classcal ethods or dong the senstvty analyss. Ths shortcong could be overcoe by usng the proposed ethods n ths paper. To clary the above asserton, we present the ollowng proble orulaton proposed by Ghodsypour and O Bren (998): Maxe =. 97x +. x +. 68x +. 8x Subect to :. x +. x +. x +. 6x 4 x + x + x + x = 4 x 4 x 7 x 6 x 4 x =,,, 4 4 (8) The nal (optal) soluton tableau reported by Ghodsypour and O Bren (998) s presented n Table. I the obectve uncton coecent related to varable s denoted by c, the subect unctonal relaton ay be dened as: H( c, c, c, c ) = c + c + c + c = 4 4 The senstvty analyss results reported by Ghodsypour and O Bren (998) are presented n Table. The changes n paraeters o the obectve uncton coecents are studed usng the optal soluton and the senstvty analyss results are presented n Tables and, respectvely, regardless o the unctonal relatons shown n Table (output o Lndo ). In order to conduct senstvty analyss, the steps ndcated n the lowchart presented n Fgure are taen. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

15 6 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March Table. The nal tableau or exaple RHS x x 9 x 8 x 7 x 6 x x 4 x x x x 6 - x 4 x 7 x x 9 x Table. The senstvty analyss results or the suppler selecton proble Suppler ratng x 4 x x x Area () 6 c > c > c > c 4 () 4 6 c > c > c > c 4 () 4 6 c > c > c > c 4 (4) 4 c > c > c > c 4 () 4 c > c > c > c 4 For the purpose o usng all steps o the lowchart, rst the varatons o c and c are consdered (n ths case, c s related to a nonbasc varable and step () s not used). Then, the varaton o c and c s studed (n ths case, both paraeters are related to basc varables and step (4) s not used). Step : deterne the desred value o change n the rst paraeter: c =.. Step : calculate the value o change n the second paraeter usng the uncton H : c + c = c =. A) Varatons o c and c Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

16 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 7 Table. The allowable varaton bounds o the obectve uncton coecents Var. OFC INC DEC x.97.9 x..6 x x Step : are the changed paraeters related to the basc varables? No, so go to step 4. Step 4: does the change o rst paraeter satsy ()? In order to answer ths queston, the slope o -c s s calculated wth respect to the perturbaton o c. d( c ) ( c ) ( c ) H/ c = + c c d( c ) d( c ) = [ y ] (), = [ y ] ( ),. <. 8 Ths eans that, applyng these changes, the basc soluton reans optal and vald. Step 6: calculate the changes o and the -c s. Accordng to () and (): ( c ) =. c =. 6 > ( c ) =. c =. 8 > ( c ) =. c =. 4 > d = c + c c H/ = [ + x ( )] B () and () result n: The changes are appled or testng: = c x + ( c + c ) x + ( c + c ) x + c x n{, } =. 8 Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

17 8 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March Accordng to step 4, the u pertted value o c that can be ncreased under the condton o eepng the basc soluton unchanged and optal s:. 8. Now let us assue that c =. ; then, the change o the bass should be consdered. For ths purpose, the values o - c n step 4 are updated usng the resulted slopes. c + c = c =. Step : are the changed paraeters related to basc varables? No Step 4: do the changed paraeters satsy the percent rule? ( c ) =. c =. 4 < ( c ) =. c =. 7 > ( c ) =. c =. 9 > H/ c + Ths noraton shows that x enters nto the bass because t contans a negatve - c but x exts ro the bass. Then, the new basc vector s: x = 4, x =, x =, x = 4 In order to copare ths result wth the results presented n Table, we rst need to apply the changes n the coecents and arrange the as: The values o Dc and Dc are extracted ro Table. = <. 4 That eans the optalty o the basc soluton s not ept. I the changes o step and are appled, the new bass ay be as ollows: x = ( x, x, x, x, x, x ) = (, 6, 7, ) B 8 7 Thereore: c > c > c > c whch ths order 4 s n lne wth the 4 th row o Table and thus the results are the sae. B) Varatons o c and c Step : deterne the desred value o change n c =. Step : calculate the value o Dc usng H. The new bass only ders n the auxlary varable x 7 whch was replaced by x 9 n the prevous bass. Ths eans that the values o the decson varables x and x are not changed. Arrangng the updated obectve uncton coecent values, we have: c > c > c > c 4 Ths order s n lne wth the nd row o Table, so the obtaned results are the sae. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

18 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 9 exaple : consder the Followng LP Proble: e Subectto : = x + x x + x b x + x b x x b x + x b4 x, x Suppose that aong the entres o vector b, the ollowng unctonal relaton s dened: G( b, b, b, b ) = 4 7b + 6b + b + 4b = 4 The ntal values o coponents o b are b T = ( 8, 9, 6, ). The nal (optal) tableau or ths proble s presented n Table 4. The range or the pertted changes o the RHS paraeters s on the bass o the coon senstvty analyss and s presented n Table regardless o the unctonal relaton (Output o Lndo ). Now assue that b has changed and b has to be changed n order to satsy the unctonal relaton. The ollowng steps are dened n the lowchart shown n Fgure : Table 4. The nal tableau or exaple RHS x 6 x x 4 x x x / 7/ - x x x 8 x 6 Table. The range or pertted changes n the rght-hand sde paraeters the unctonal relaton between coponents o b s gnored RHS INC DEC b = 8 b = 9.6 b = 6 b 4 = Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

19 6 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March Step : deterne the desred value o change n the rst paraeter: b = Step : calculate the value o change n the second paraeter usng G : 7 b + 6 b = b = Step : are the changed paraeters related to the bndng constrants? Yes, accordng to the splex table, the rst and second constrants are bndng ones. Step 4: s the percent rule o changes satsed? Yes. Because o b <, the equaton (8) s used. Thus, because o the decrease n b and the ncrease n b, Db s the u value o the pertted decrease o b, and, Db s the u value o the pertted ncrease o b ; thereore, b = and b = (accordng to Table ); and. 6 7/ 6 x x = ( ) 7, ( ) = 6 = 7 ( ) ( ) =, 6 = 7 + ( ) ( ) = 6 The valdty o the results s exaned by studyng the change n the optal soluton through the proble constrants. The rst and second constrants denty the optal soluton, so that the changed equatons are (see Box). Accordng to the splex table, t s clear that x = and x = whch are n accordance wth the obtaned result ro the lowchart. exaple : consder the LP Proble Presented n exaple wth the Followng Functonal relaton Suppose b has changed and b has to be changed n order to satsy the above unctonal relaton. Steps and : =, then: b But snce >. 6, the desred changes do not ae the optal soluton neasble. Step 6: calculate the changes o and x. Accordng to equatons () and (4): The changed paraeters are related to bndng constrants, but the change o b s toward unlted ncrease, because b s able Box. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

20 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 6 B to be lted when dx < (see equaton ()) [step & 4], or ths reason, t s observed that: dxb b y y G/ =, +, ( ) G/ b dx 4 = ( ) = 4 dx 4 = + ( ) = 4 dx 4 = + ( ) = 4 dx = + ( ) = Thereore, the basc soluton reans easble n the drecton o the deterned changes. It s because all the slopes are postve. Step 6: accordng to equatons () and (4): x x b = 7 + = = =, =, = Exanng the valdty o these results: new = 8 + = 9 b = 9 + = new x + x = 9 x x + x = x = = exaple 4: consder the Followng LP Proble e = c x + c x + c x + c x Subect to : x + 4x x 4 x + x x 4 6 x + x + x + x4 x, x, x, x Assue that the unctonal relaton between coponents o c s as ollows: G( c, c, c, c ) = c + c + c c = 4 4 We nd the optal table presented n Table 6 assung an ntal value o c = (,, 4, ) T or vector c. Furtherore, Table 7 presents the range o changes n the obectve uncton coecents (output: Lndo ). Now suppose that one wants to exane the eect o c on and the -c s, so that paraeters c and c can change (the change o three paraeters). The ollowng steps are dened n the lowchart shown n Fgure. The desred change o the two paraeters s appled n the unctonal relaton and the value o change n the thrd paraeter s obtaned. Step : c = c = Step : Step : paraeters are related to basc and nonbasc varables. Step 4: to satsy relaton (6), we have, Accordng to the splex table, t s observed that x = and x = whch conr the obtaned results ro the lowchart. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

21 6 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March Table 6. The nal tableau or exaple 4 RHS x 7 x 6 x x 4 x x x x x 6 - x d ( c ) = + + c = + ( ) = / d ( c ) c = + c + c = + + ( ) = / snce <, the basc soluton reans optal. Step : by usng (), we have: 7 4 c n,, = Also, by usng (4), we have: = + + ( ) c = / Table 7. The pertted bound o changes n the obectve uncton coecents the unctonal relaton between coponents o c s gnored Var. OFC INC DEC x x - 7 x 4 x 4-6 Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

22 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 6 conclusion And Future research directions The portance o senstvty analyss n lnear prograng has been wdely stressed n the anageent scence lterature. However, the research on senstvty analyss wth unctonal relaton has been sporadc and scattered. Usually varaton occurs n the RHS o the constrants and/or the obectve uncton coecents. In ths paper, we consdered the senstvty analyss or a class o LP probles wth unctonal relaton aong the obectve uncton paraeters or those o the RHS. The classcal ethods and standard senstvty analyss sotware pacages al to uncton when a unctonal relaton aong the LP paraeters prevals. In order to overcoe ths decency, we derved a seres o senstvty analyss orulae and devsed correspondng algorths or derent group o hoogenous LP paraeters. The valdty o the derved orulae and devsed algorths was corroborated by open lterature exaples havng lnear as well as nonlnear unctonal relatons between ther vector b or vector c coponents. The derved orulae and the devsed procedures developed n ths study are not autoated. There are no theoretcal obstacles to the developent and pleentaton o coputer progras or perorng senstvty analyss n LP probles wth unctonal relaton. We plan to autoate these orulae and procedures. Further research s also needed to develop explct orulae that descrbe the behavor o the optal value under lnear and nonlnear perturbatons o the constrant coecent atrx. The research on senstvty analyss could also benet ro the extenson o the current ethod to ult-densonal unctonal relatons. Fnally, the proposed approach can be extended to cases wth probablstc noraton on hoogenous paraeters (e.g., when the paraeters are correlated then the change o a specc paraeter causes the sultaneous change o others). AcnowLedGent The authors would le to than the anonyous revewers and the edtor or ther nsghtul coents and suggestons. references Adler, I., & Montero, R. (99). A geoetrc vew o paraetrc lnear prograng. Algorthca, 8(), do:.7/bf7884 Arsha, H. (99). Post optalty analyss o the transportaton proble. The Journal o the Operatonal Research Socety, 4(), 9. Arsha, H., & Kahan, A. B. (989). A splex-type algorth or general transportaton probles: An alternatve to steppng-stone. The Journal o the Operatonal Research Socety, 4(6), 8 9. Bradley, S. P., Hax, A. C., & Mangnant, T. L. (977). Appled Matheatcal prograng. Readng, MA: Addson-Wesley. Dehghan, M., Ghaar-Hadgheh, A. R., & Mrna, K. (7). Support set nvarancy senstvty analyss n b-paraetrc lnear optaton. Advanced Modelng and Optaton, 9(), Facco, A. (98). Introducton to senstvty analyss n nonlnear prograng. New Yor: Acadec Press. Flpp, C. (). A resh vew on the tolerance approach to senstvty analyss n lnear prograng. European Journal o Operatonal Research, 67(), 9. do:.6/.eor.4.. Gal, T., & Greenberg, H. J. (997). Advances n senstvty analyss and paraetrc prograng. Boston: Kluwer. Ghaar-Hadgheh, A. R., Ghaar-Hadgheh, H., & Terlay, T. (8). -paraetrc optal partton nvarancy senstvty analyss n lnear optaton. Central European Journal o Operatons Research, 6(), 8. do:.7/s Ghaar-Hadgheh, A. R., Mrna, K., & Terlay, T. (6). Senstvty analyss n lnear and convex quadratc optaton: nvarant actve constrant set and nvarant partton ntervals. Inoraton Systes and Operatonal Research, 44(), 9. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

23 64 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March Ghaar-Hadgheh, A. R., Mrna, K., & Terlay, T. (7). Actve constrant set nvarancy senstvty analyss n lnear optaton. Journal o Optaton Theory and Applcatons, (),. do:.7/s Ghaar-Hadgheh, A. R., & Terlay, T. (6a). Senstvty analyss n lnear optaton: Invarant support set nterval. European Journal o Operatonal Research, 69(), 8 7. do:.6/. eor Ghaar-Hadgheh, A. R., & Terlay, T. (6b). Generaled support set nvarancy senstvty analyss n lnear optaton. Industral and Manageent Optaton, (), 8. Ghodsypour, S. H., & O Bren, C. (998). A decson support syste or suppler selecton usng an ntegrated analytc herarchy process and lnear prograng. Internatonal Journal o Producton Econocs, 6-7(), 99. do:.6/s9-7(97)9- Greenberg, H. J. (). Sultaneous pral-dual rght-hand-sde Senstvty Analyss ro a Strctly Copleentary Soluton o a lnear prograng. SIAM Journal on Optaton, (), do:.7/s6496 Jansen, B., De Jong, J. J., Ross, C., & Terlay, T. (997). Senstvty Analyss n Lnear Prograng: Just be Careul! European Journal o Operatonal Research, (), 8. do:.6/s77-7(96)7- Kolta, T., & Terlay, T. (). The derence between anageral and atheatcal nterpretaton o senstvty analyss results n lnear prograng. Internatonal Journal o Producton Econocs, 6(), do:.6/s9-7(99)6- Mrna, K., & Ghaar-Hadgheh, A. R. (7). Support set expanson senstvty analyss n convex quadratc optaton. Optaton Methods & Sotware, (4), do:.8/ Roos, C., Terlay, T., & Val, J.-Ph. (997). Theory and algorths or lnear optaton: An nteror pont approach. New Yor: John Wley & Sons. Wendell, R. E. (98). A New Perspectve on Senstvty Analyss n Lnear prograng: a Tolerance Approach (Tech. Rep. 448). Bradord, PA: Unversty o Pttsburg, Graduate school o Busness. Paya Hanaadeh s an Assstant Proessor o Industral Manageent at Allaeh Tabataba' Unversty n Tehran, Iran and a eber o the Desgn Optaton under Uncertanty Group at the Unversty o Waterloo, Canada. He receved hs MSc and PhD n Industral Engneerng ro Tehran Polytechnc Unversty and pursues hs research n Inoraton Systes and Decson-ang under Uncertanty. He has publshed n such ournals as the Inoraton Socety, Journal o Global Inoraton Manageent, Telecouncatons Polcy, Matheatcal and Coputer Modelng, Expert Systes wth Applcatons, Internatonal Journal o Inoraton Manageent, aong others. Abolal Ghae s an Adunct Assstant Proessor o Industral Engneerng at Arabr Unversty o Technology n Tehran. He receved hs MSc, ASD and PhD n Operatons Research ro George Washngton Unversty. Hs acadec area o nterest s atheatcal prograng wth ephass on senstvty and paraetrc analyss on nonlnear progras. As the charan o Tehran Energy Consultant, he has conducted extensve research on Upstrea Petroleu Engneerng studes and supervsed several theses n optaton o petroleu engneerng related topcs. He has publshed n such ournals as Operatons Research and Coputers and Operatons Research. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global

24 Internatonal Journal o Operatons Research and Inoraton Systes, (), 4-6, January-March 6 Madd Tavana s a Proessor o Manageent Inoraton Systes and Decson Scences and the Lnac Dstngushed Char o Inoraton Systes at La Salle Unversty where he served as Charan o the Manageent Departent and Drector o the Center or Technology and Manageent. He has been a dstngushed research ellow at NASA's Kennedy Space Center, NASA s Johnson Space Center, Naval Research Laboratory - Stenns Space Center, and Ar Force Research Laboratory. He was awarded the prestgous Space Act Award by NASA n. He holds an MBA, a PMIS, and a PhD n Manageent Inoraton Systes. Dr. Tavana receved hs post-doctoral dploa n strategc noraton systes ro the Wharton School o the Unversty o Pennsylvana. He s the Edtor-n-Che or the Internatonal Journal o Strategc Decson Scences, the Internatonal Journal o Enterprse Inoraton Systes, and the Internatonal Journal o Appled Decson Scences. He has publshed n ournals such as Decson Scences, Interaces, Inoraton Systes, Annals o Operatons Research, Inoraton and Manageent, Journal o the Operatonal Research Socety, Coputers and Operatons Research, and Advances n Engneerng Sotware, aong others. Copyrght, IGI Global. Copyng or dstrbutng n prnt or electronc ors wthout wrtten persson o IGI Global