LP Methods.S3 The Dual Linear Program

Transcription

1 LP Mthods.S3 Th Dual Linar Program Oprations Rsarch Modls and Mthods Paul A. Jnsn and Jonathan F. Bard Whn a solution is obtaind for a linar program with th rvisd simplx mthod, th solution to a scond modl, calld th dual problm, is radily availabl and provids usful information for snsitivity analysis as w hav just sn. Thr ar svral bnfits to b gaind from studying th dual problm, not th last of which is that it oftn has a practical intrprtation that nhancs th undrstanding of th original modl. Morovr, it is somtims asir to solv than th original modl, and likwis provids th optimal solution to th lattr at no xtra cost. Duality also has important implications for th thortical basis of mathmatical programming algorithms. In this sction, th dual problm is dfind, th proprtis that link it to th original (calld th primal) ar listd, and th procdur for idntifying th dual solution from th tablau is prsntd. Dfinition of th Dual LP Modl In discussing duality, it is common to dpart from th standard quality form of th LP givn in Sction 4.1 in ordr to highlight th symmtry of th primal-dual rlationships. Th dual modl is drivd by construction from th standard inquality form of linar programming modl as shown in Tabls 1 and 2. All constraints of th primal modl ar writtn as lss than or qual to, and right-hand-sid constants may b ithr positiv or ngativ. In th primal modl thr ar assumd to b n dcision variabls and m constraints, thus c and x ar n- dimnsional vctors. Th matrix of structural cofficints, A, has m rows and n columns. Th dual modl uss th sam arrays of cofficints but arrangd in a symmtric fashion. Th dual vctor has m componnts. Tabl 1. Matrix dfinition of primal and dual problms (P) Maximiz z P = cx (D) Minimiz z D = b subjct to Ax b x 0 subjct to A c 0 Tabl 2. Algbraic dfinition of primal and dual problms (P) Maximiz z P = c 1 x 1 + c 2 x c n x n (D) Minimiz z D = b 1 π 1 +b 2 π b m π m subjct to a 11 x 1 + a 12 x 2 + +a 1n x n b 1 subjct to a 11 π 1 +a 21 π a m1 π m c 1 a 21 x 1 + a 22 x a 2n x n b 2 a 12 π 1 +a 22 π a m2 π m c

2 Th Dual Linar Program 2 a m1 x 1 +a m2 x a mn x n b m a 1n π 1 +a 2n π a mn π m c n x 1 0, x 2 0,, x n 0 π 1 0, π 2 0,, π m 0 Exampl 4 (P)Maximiz z P = 2x 1 + 3x 2 subjct to x 1 + x 2 5 x 1 + 3x 2 35 x 1 20 (D) Minimiz z =5π D π π 3 subjct to π 1 + π 2 + π 3 2 π 1 + 3π 2 3 π 1 0, π 2 0, π 3 0 x 1 0, x 2 0 Th optimal solution to th primal including slacks is x * = (20, 5, 20, 0, 0) T with z P = 55. Th corrsponding dual solution including slacks is * = (0, 1, 1, 0, 0) with z D = 55. Not that z P = z D. This is always th cas as will b shown prsntly. From an algorithmic point of viw, solving th primal problm with th dual simplx mthod is quivalnt to solving th dual problm with th primal simplx mthod. Whn writtn in inquality form, th primal and dual modls ar rlatd in th following ways. a. Whn th primal has n variabls and m constraints, th dual has m variabls and n constraints. b. Th constraints for th primal ar all lss than or qual to, whil th constraints for th dual ar all gratr than or qual to. c. Th objctiv for th primal is to maximiz, whil th objctiv for th dual is to minimiz. d. All variabls for ithr problm ar rstrictd to b nonngativ. a. For vry primal constraint, thr is a dual variabl. Associatd with th ith primal constraint is dual variabl π i. Th dual objctiv function cofficint for π i is th right-hand sid of th ith primal constraint, b i. f. For vry primal variabl, thr is a dual constraint. Associatd with primal variabl x j is th jth dual constraint whos right-hand sid is th primal objctiv function cofficint c j.

3 Th Dual Linar Program 3 Modifications to Inquality Form g. Th numbr a ij is, in th primal, th cofficint of x j in th ith constraint, whil in th dual, a ij is th cofficint of π i in th jth constraint. It is rar that a linar program is givn in inquality form. This is spcially tru whn th modl has dfinitional constraints that ar introducd for convninc, or whn it has bn prpard for th tablau simplx mthod whr all RHS constants must b positiv. Nvrthlss, no mattr how th primal is statd, its dual can always b found by first convrting th primal to th inquality form in Tabl 1 and thn writing th dual accordingly. For xampl, givn an LP in standard quality form Maximiz z P = cx subjct to Ax = b, x 0 w can rplac th constraints Ax = b with two inqualitis: Ax b and A Ax b so th cofficint matrix bcoms A and th right-hand-sid vctor bcoms (b, b) T. Introducing a partitiond dual row vctor ( 1, 2 ) with 2m componnts, th corrsponding dual is Ltting Minimiz z D = 1 b 2 b subjct to 1 A 2 A c 1 0, 2 0 = 1 2 w may simplify th rprsntation of this problm to obtain th pair givn in Tabl 3. Tabl 3. Equality form of primal-dual modls (P) Maximiz z P =cx (D) Minimiz z D = b subjct to Ax = b subjct to A c x 0 This is th asymmtric form of th duality rlation. Similar transformations can b workd out for any linar program by first putting th primal into inquality form, constructing th dual, and thn simplifying th lattr to account for spcial structur. W say two LPs ar quivalnt

4 Th Dual Linar Program 4 if on can b transformd into anothr so that fasibl solutions, optimal solutions and corrsponding dual solutions ar prsrvd;.g., th inquality form in Tabl 1 and th quality form in Tabl 3 ar quivalnt primal-dual rprsntations. This suggsts th following rsult which can b provn by constructing th appropriat modls. Proposition 1: Duals of quivalnt problms ar quivalnt. Lt (P) rfr to an LP and lt (D) b its dual. Lt (^P) b an LP that is quivalnt to (P). Lt ( ^D) b th dual of (^P). Thn ( ^D) is quivalnt to (D), that is, thy hav th sam optimal objctiv function valus or thy ar both infasibl. Tabl 4 dscribs mor gnral rlations btwn th primal and dual that can b asily drivd from th standard dfinition. Thy rlat th sns of constraint i in th primal with th sign rstriction for π i in th dual, and sign rstriction of x j in th primal with th sns of constraint j in th dual. Not that whn ths altrnativ dfinitions ar allowd thr ar many ways to writ th primal and dual problms; howvr, thy ar all quivalnt. Tabl 4. Modifications in th primal-dual formulations Primal modl Dual modl Constraint i is π i 0 Constraint i is = π i is unrstrictd Constraint i is π i 0 x j 0 Constraint j is x j is unrstrictd Constraint j is = x j 0 Constraint j is Exampl 5 (P) Maximiz z P = 3x 1 2x 2 subjct to x 1 x 2 = 8 x 1 + 2x 2 13 x 1 0, x 2 unrstrictd (D) Minimiz z = 8π D π 2 subjct to π 1 + π 2 3 π 1 + 2π 2 = 2 π 1 unrstrictd, π 2 0

5 Th Dual Linar Program 5 Rlations btwn Primal and Dual Objctiv Function Valus Thr ar a numbr of rlationships btwn solutions to th primal and dual problms that ar intrsting to thorticians, usful to algorithm dvloprs, and important to analysts for intrprting solutions. W prsnt ths rlationships as thorms and provn thm for th primal dual pair in Tabl 1; howvr, thy ar tru for all primal-dual formulations. In what follows, x rfrs to any fasibl solution of th primal and to any fasibl solution of th dual; x * and * ar th rspctiv optimal solutions if thy xist. Thorm 1 (Wak Duality) In a primal-dual pair of LPs, lt x b a primal fasibl solution and z P (x) th corrsponding valu of th primal objctiv function that is to b maximizd. Lt b a dual fasibl solution and z D ( ) th corrsponding dual objctiv function that is to b minimizd. Thn z P (x) z D ( ). This thorm shows that th objctiv valu for a fasibl solution to th dual will always b gratr than or qual to th objctiv function for a fasibl solution to th primal. Th following squnc dmonstrats this rsult. 1. Th primal solution is fasibl by hypothsis: Ax b 2. Prmultiply both sids by : Dx b 3. Th dual solution is fasibl by hypothsis: A c 4. Postmultiply both sids by x: Ax cx 5. Combin th rsults of 2 and 4: cx Ax b or z P (x) z D ( ) Thr ar a numbr of usful rlationships that can b drivd from Thorm 1. In particular, Th valu of z P (x) for any fasibl x is a lowr bound to z D ( * ). Th valu of z D ( ) for any fasibl is an uppr bound to z P (x * ). If thr xists a fasibl x and th primal problm is unboundd, thr is no fasibl.

6 Th Dual Linar Program 6 If thr xists a fasibl no fasibl x. and th dual problm is unboundd, thr is It is possibl that thr is no fasibl x and no fasibl. Th last point is dmonstratd by th following xampl. Maximiz z = x 1 + 3x 2 subjct to x 1 x 2 3 x 1 + x 2 5 x 1, x 2 unrstrictd Minimiz z D = 3π 1 5π 2 subjct to π 1 π 2 = 1 π 1 + π 2 = 3 π 1 0, π 2 0 Thorm 2 (Sufficint Optimality Critrion) In a primal-dual pair of LPs, lt z P (x) b th primal objctiv function and z D ( ) b th dual objctiv function. If (^x, ^) is a pair of primal and dual fasibl solutions satisfying z P (^x) = z D (^), thn ^x is an optimal solution of th primal and ^ is an optimal solution of th dual. Th proof can b sn in th following stps: 1. Dfinition of optimality for primal: z P (^x) z P (x * ) 2. Fasibl dual solution bound on z P : z P (x * ) z D ( * ) 3. Dfinition of optimality for dual: z D ( * ) z D (^) 4. Combin th rsults of 1, 2 and 3: z P (^x) z P (x * ) z D ( * ) z D (^) 5. Objctivs ar qual by hypothsis: z P (^x) = z D (^) 6. Combin 4 and 5: z P (^x) = z P (x * ) = z D ( * ) = z D (^) Thrfor, ^x and ^ ar optimal. This thorm stats that quality of objctiv valus implis optimality; morovr, w hav: h. Givn fasibl solutions x and for a primal-dual pair, if th objctiv valus ar qual, thy ar both optimal.

7 Th Dual Linar Program 7 i. If x * is an optimal solution to th primal, a finit optimal solution xists for th dual with objctiv valu z P (x * ). j. If * is an optimal solution to th dual, a finit optimal solution xists for th primal with objctiv valu z D ( * ). Taking ths rsults on stp farthr lad to th Fundamntal Duality Thorm. Thorm 3 (Strong Duality) In a primal-dual pair of LPs, if ithr th primal or th dual problm has an optimal fasibl solution, thn th othr dos also and th two optimal objctiv valus ar qual. Complmntary Solutions W will prov th rsult for th primal and dual problms givn in Tabl 3. Solving th primal problm by th simplx algorithm yilds an optimal solution x B = B 1 b, x N = 0 with - c = c B B 1 N c N 0, which can b writtn [c B, c N c B B 1 (B, N)] = c B B 1 A c 0. Now if w dfin = c B B 1 w hav A c and z P (x) = c B x B = c B B 1 b = b = z D ( ). By th sufficint optimality critrion, Thorm 2, is a dual optimal solution. This complts th proof whn th primal and dual ar as statd. In gnral, vry LP can b transformd into an quivalnt problm in standard quality form. This quivalnt problm is of th sam typ as th primal in Tabl 3, hnc th proof applis. Also, by Proposition 1, th dual of th quivalnt problm in standard form is quivalnt to th dual of th original problm. Thus th thorm must hold for it too. For purposs of this sction it is hlpful to rpat th dfinition of th primal and dual problms givn in Tabl 1 in a slightly diffrnt but quivalnt form. Tabl 5 contains th modifid rprsntation, whr I m is an m m idntity matrix and x s = (x s1,, x sm ) T an m-dimnsional vctor of slack variabls. Tabl 5. Equivalnt primal-dual pair (P) Maximiz z P = cx subjct to (A 1,, A n )x + I m x s = b x 0, x s 0 (D) Minimiz z D = b subjct to A j c j, j = 1,, n 0

8 Th Dual Linar Program 8 Each structural variabl x j is associatd with th dual constraint j, and ach slack variabl x si is associatd with dual variabl π i. Rcall that a basic solution is found slcting a st of basic variabls, constructing th basis matrix B, and stting th nonbasic variabls to zro. This givs th primal solution x B = B 1 b with z P = c B B 1 b. Th complmntary dual solution associatd with this basis is dfind to b = c B B 1 with z D = b = c B B 1 b. Evry basis dfins complmntary primal and dual solutions with idntical objctiv function valus. Thorm 4 (Optimality of Fasibl Complmntary Solutions) Givn th solution x B dtrmind from th basis B, whn x B is optimal to th primal, th complmntary solution = c B B 1 is optimal to th dual. Th proof of Thorm 4 can b sn in th following squnc. 1. Primal and dual objctiv valus ar qual by construction: z P (x B ) = c B x B = c B B 1 b z D ( ) = b = c B B 1 b 2. Primal objctiv valu for a basic solution whn nonbasic variabl x k is allowd to incras: z P = c B B 1 b ( A k c k )x k 3. Sinc th primal solution is optimal: - c k = A k c k 0 or A k c k 4. From 3, whn x k is a structural variabl, dual constraint k is satisfid: A k c k

9 Th Dual Linar Program 9 5. From 3, whn th nonbasic variabl is a slack variabl x si, π i is nonngativ: c si = 0 and A si = i so i 0 6. For basic variabls: c B B = c B c B B 1 B = 0 7. From 6, whn x k is a structural variabl and basic, th kth dual constraint is satisfid as an quality: c k A k = 0 8. From 6, whn x si is a slack variabl and basic, π i is zro: c si = 0 and A si = i so π i = 0 All constraints ar satisfid so optimal. is fasibl. By Thorm 2 it must b From Stp 3 of th proof, it can b infrrd that th rducd cost, c - k, for th primal variabl x k is quivalnt to th dual slack, π sk, for dual constraint k. Morovr, Stps 7 and 8 illustrat an important proprty known as complmntary slacknss. Givn th primal-dual pair in Tabl 1, w hav th following. Complmntary solutions proprty: For a givn basis, whn a primal structural variabl is basic, th corrsponding dual constraint is satisfid as an quality (th dual slack variabl is zro), and whn a primal slack variabl is basic (th primal constraint is loos), th corrsponding dual variabl is zro. This proprty holds whthr or not th primal and dual solutions ar fasibl. W hav alrady sn this in th simplx tablau. That is, whn a primal structural variabl is basic, its rduc cost (dual slack) is zro; whn a primal slack variabl is basic, th corrsponding structural dual variabls is zro (Stp 8 of proof). Illustration of Complmntary Solutions Tabls 6 and 7 rspctivly show th 10 basic solutions for th primal and dual problms givn in Exampl 4. In quality form, th primal problm has 5 variabls and 3 constraints, whil th dual has 5 variabls and 2 constraints. Thus thr ar n m = 5 3 = 5 2 = 10 potntial bass in ach cas. Th numbrs in th lftmost column of th tabls idntify th complmntary solutions;.g., No. 1 in Tabl 6 is complmntary to No. 1 in Tabl 7. Not that for No. 7, no solution xists bcaus th columns associatd with th variabls (x s1, x s2, x 2 ) as wll as th columns associatd with (π s1, π 3 ) do not form a basis.

10 Th Dual Linar Program 10 Tabl 6. Basic solutions for primal problm Basic Nonbasic Primal No. variabls variabls x 1 x 2 x s1 x s2 x s3 z P status 1 x s1, x s2, x s3 x 1, x Fasibl 2 x 1, x s2, x s3 x 2, x s Infasibl 3 x 2, x s2, x s3 x 1, x s Fasibl 4 x s1, x 1, x s3 x 2, x s Infasibl 5 x s1, x 2, x s3 x 1, x s Infasibl 6 x s1, x s2, x 1 x 2, x s Fasibl 7 x s1, x s2, x 2 x 1, x s3 No solution 8 x 1, x 2, x s3 x s 1, x s Fasibl 9 x 1, x s2, x 2 x s1, x s Infasibl 10 x s1, x 1, x 2 x s2, x s Fasibl

11 Th Dual Linar Program 11 Tabl 7. Basic solutions for dual problm Basic Nonbasic Dual No. variabls variabls π s1 π s2 π 1 π 2 π 3 z D status 1 π s1, π s2 π 1, π 2, π Infasibl 2 π s2, π 1 π s1, π 2, π Infasibl 3 π s1, π 1 π s2, π 2, π Infasibl 4 π s2, π 2 π 1, π s1, π Fasibl 5 π s1, π 2 π 1, π s2, π Infasibl 6 π s2, π 3 π 1, π 2, π s Infasibl 7 π s1, π 3 π 1, π 2, π s2 No solution 8 π 1, π 2 π s1, π s2, π Infasibl 9 π 1, π 3 π s1, π 2, π s Fasibl 10 π 2, π 3 π 1, π s1, π s Fasibl Th conditions drivd in this sction ar illustratd by th data in th tabls. Fig. 1 shows th objctiv function valus for th solutions that ar fasibl for th primal and dual problms. As can b sn, th objctiv valu for vry primal fasibl solution provids a lowr bound for th optimal dual objctiv (z D = 55). Th objctiv valu for vry fasibl dual solution provids an uppr bound for th optimal primal objctiv (z P = 55). Ths bounds convrg to th optimum as might b xpctd; howvr, z P = z D for all points. Th complmntary solutions proprty is similarly xhibitd by all points. For xampl, considr No. 9. Hr, x 1 is basic in th primal and π s1 is nonbasic in th dual so th first dual constraint is satisfid as an quality. Also, x s2 is basic and π 2, th corrsponding dual variabl, is nonbasic. Ths two obsrvations can b writtn mathmatically as x 1 π s1 = 0 and x s2 π 2 = 0. In th nxt sction, w provid a gnral statmnt of this rsult for all complmntary pairs.

12 Th Dual Linar Program 12 Fasibl for primal Fasibl for dual -10 #1 #3 #6 & #8 #10 #4 # Fasibl for primal Fasibl for dual z Figur 1. Objctiv valus of primal and dual solutions Finding Complmntary Solutions for Standard Inquality Forms Th simplx algorithm solvs th primal and dual problms simultanously. This is obvious for th rvisd simplx mthod which uss th complmntary dual solution dirctly in th computations. Whn th primal and dual problms ar in th standard inquality form givn in Tabl 1, th tablau mthod provids all dual valus in row 0. Fig. 2 shows row 0 of th gnral tablau. Not that th x's ar labls but th ntris in row 0 ar numbrs corrsponding th valus of th dual variabls. Th dual slacks appar undr th primal structural variabl labls, and th dual structural variabls appar undr th primal slack variabl labls. Row Basic Cofficints no. variabls z x 1 x 2... x n x s1 x s2... x sm RHS 0 z 1 π s1 π s2... π sn π 1 π 2... π m z D = z P Figur 2. Dual variabls shown in th simplx tablau To illustrat, solution No. 3 from Tabls 6 and 7 is displayd in th tablau blow. Th primal solution is shown as th RHS vctor. Th complmntary dual solution is givn in row 0. In particular, π s1 = 5, π 1 = 3 and z D = 15, whil all othr dual variabls ar zro.

13 Th Dual Linar Program 13 Cofficints Row Basic z x 1 x 2 x s1 x s2 x s3 RHS 0 z x x s x s Th optimal tablau for th xampl is shown nxt. From this tablau w can rad both th primal and dual solutions. For th dual problm th optimum is π * 2 = π* 3 = 1, and π* s1 = π* s2 = π* s3 = 0. Cofficints Row Basic z x 1 x 2 x s1 x s2 x s3 RHS 0 z x /3 1/3 5 2 x x s /3 4/3 20 Finding Complmntary Solutions for Nonstandard Forms Whn th primal linar programming problm is in a nonstandard form with quality or gratr than or qual to constraints, th dual variabls do not appar dirctly in th tablau. For an quality in th primal, thr is no slack variabl in th tablau; howvr, if a unit vctor is insrtd in phas 1 to rprsnt th artificial variabl for that constraint in th initial tablau, th dual variabl will appar in row 0 undr that artificial variabl (assuming th artificial is givn a zro objctiv cofficint in phas 2). For a gratr than or qual to constraint th dual variabl associatd with that constraint is th ngativ of th valu apparing in row 0 of th column of th slack variabl for th constraint. Finally, bcaus th primal simplx mthod rquirs that all variabls b rstrictd to b nonngativ, nonstandard forms that contain unrstrictd variabls or variabls constraind to b nonpositiv ar not allowd. Th forgoing dvlopmnts ar natly summarizd in th following thorm.

14 Th Dual Linar Program 14 Thorm 5 (Ncssary and Sufficint Optimality Conditions) Considr a primaldual pair of LPs. Lt x and b th primal and dual variabls and lt z P (x) and z D ( ) b th corrsponding objctiv functions. If x is a primal fasibl solution, it is optimal iff thr xists a dual fasibl solution satisfying z P (x) = z D ( ). Complmntary Slacknss Th if (sufficint) part of th proof follows dirctly from th sufficint optimality conditions of Thorm 2. Th only (ncssary) part follows from th optimality of complmntary solutions statd in Thorm 4. It also follows from th Fundamntal Duality Thorm. W hav obsrvd th complmntary slacknss proprty of complmntary basic solutions which holds for all bass whthr optimal or not. To prsnt this proprty in mathmatical trms, w first rcast th primal and dual problms givn in Tabl 1 by introducing slack variabls. Tabl 8 dfins th rvisd modls. Tabl 8. Primal and dual problms with slack variabls addd (P) Maximiz z P = cx (D) Minimiz z D = b subjct to Ax + I m x s = b subjct to A s I n = c x 0, x s 0 0, s 0 Th vctor of slacks for th primal is x s = (x s1, x s2,, x sm ), with x si th slack variabl for th ith constraint. Th vctor of slacks for th dual is s = (π s1, s2,, π sn ), with π sj th slack variabl for th jth constraint. I m and I n ar idntity matrics of siz m and n, rspctivly. Both problms in quality form hav n + m variabls. Th primal and dual variabls ar linkd by idntifying n + m pairs with on variabl in th pair from ach problm. Pair (x j, π sj ) for j = 1 to n: Th primal variabl x j is paird with th slack variabl π sj associatd with th jth dual constraint. Pair (x si, π i ) for i = 1 to m: Th dual variabl π i is paird with th slack variabl x si associatd with th ith primal constraint. Complmntary slacknss is th condition that at last on mmbr of ach pair is zro. For a particular pair (x j, π sj ), th proprty implis that

15 Th Dual Linar Program 15 ithr x j is zro or th corrsponding dual constraint is satisfid as an quality. For th pair (x si, π i ), it implis that ithr primal constraint i is satisfid as an quality, or th corrsponding dual variabl is zro. Thorm 6 (Complmntary Slacknss) Th pairs (x, x s ) and (, s ) of primal and dual fasibl solutions ar optimal to thir rspctiv problms iff whnvr a slack variabl in on problm is strictly positiv, th valu of th associatd nonngativ variabl of th othr problm is zro. For th primal-dual pair in Tabl 8, th thorm has th following intrprtation. Whnvr Altrnativly, w hav n x si = b i a ij x j > 0 w hav π i = 0 (3) j=1 m π sj = a ij π i c j > 0 w hav x j = 0 (4) i=1 π i x si = π i x j π sj = x j n b i a ij x j = 0, i = 1,..., m (5) m i=1 j=1 a ij π i c j = 0, j = 1,..., n (6) Th proof of th thorm is lft as an xrcis. In vctor notation (5) and (6) can b writtn collctivly as x s = 0 and x s = 0, rspctivly. Conditions (3) or (5) only rquir that if x si > 0, thn π i = 0. Thy do not rquir that if x si = 0, thn π i must b > 0; that is, both x si and π i could b zro and th conditions of th thorm would b satisfid. Morovr, conditions (3) or (5) automatically imply that if π i > 0, thn x si = 0. Th sam is tru for (4) or (6). For instanc, if x j > 0, thn π sj = 0. Th complmntary slacknss thorm dos not say anything about th valus of unrstrictd variabls (corrsponding to quality constraints

16 Th Dual Linar Program 16 Economic Intrprtation in th othr problm) in a pair of optimal fasibl solutions. This is th situation, for xampl, whn th primal is writtn in quality form as in Tabl 3. It is concrnd only with nonngativ variabls of on problm and th slack variabls corrsponding to th associatd inquality constraints in th othr problm. Considr th primal-dual pair givn in Tabl 1. Assum that th primal problm rprsnts a chmical manufacturr that is undr dirction to limit its output of toxic wasts. Suppos that ovr a givn priod of tim it producs n diffrnt typs of chmicals that rturn a unit profit of c j ach for j = 1,..., n, and that no mor than b i units of toxic wast i can rsult from th manufacturing procss, i = 1,..., m. Lt a ij b th amount of byproduct i gnratd by th manufactur of on unit of chmical j. Th problm is to dcid how many units of j to produc, dnotd by x j, so that no toxic wast lvls ar xcdd. Ths constraints can b writtn as Σ n j=1 a ij x j b i.for all i. To driv an quivalnt dual problm, lt π i 0 b th unit contribution to profit associatd with byproduct i. Thus π i can b intrprtd as th amount th company should b willing to pay to b allowd to gnrat on unit of toxic wast i. Consquntly, th trm Σ m i=1 π i a ij rprsnts th implid contribution to profit associatd with th currnt mix of toxic wasts whn on unit of chmical j is producd. Bcaus th sam mix of wasts could probably b gnratd in othr ways as wll, no altrnativ us should b considrd if it is lss profitabl than chmical j. This lads to th constraint Σ m i=1 π i a ij c j for all j; howvr, if a strict inquality holds for som j giving Σ m i=1 π i a ij > c j, bttr us of th prmissibl toxic wast lvls can b found so it is optimal to st x j = 0. If x j > 0, th implid valu of th toxic wast mix should b just qual to th unit profit for chmical j, giving Σ m i=1 π i a ij = c j. Similarly, if Σ n j=1 a ij x j < b i for som i, th marginal contribution to profit associatd with th ith toxic wast limit is zro so w should st π i = 0. If π i > 0 th manufacturr should b gnrating as much byproduct i as prmissibl, implying Σ n j=1 a ij x j = b i. Ths conditions ar nothing mor than complmntary slacknss in Eqs. (3) and (4). Whn thy ar satisfid, thr is no incntiv for th manufacturr to altr its production

17 Th Dual Linar Program 17 plan or to chang th implid pric structur. Th objctiv of th dual problm is to minimiz pollution costs which can b intrprtd as minimizing th total implicit valu of toxic wasts gnratd in th manufacturing procss. At optimality, th minimum cost incurrd is xactly th maximum rvnu ralizd in th primal. This rsults in an conomic quilibrium whr cost = rvnu, or Σ m i=1 π i b i = Σn j=1 c j x j.