Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we performed the first itertion of the simple method for it, which resulted in the following Tbleu:
Tbleu Bsic vrible Eq. Coefficients of # F F -.. 6-6 Right side Add pivot row Add - pivot row Then when we tested for optimlity, we discovered tht we must do nother itertion becuse the coefficient of is negtive in the bove tbleu. I sked you to do this for homework. I will do it here, to provide you with the solution to the homework. So our only choice for the entering vrible is. To determine the leving vrible, we identify the constrints tht most limit the increse in, s shown in Tbleu below. Tbleu Bsic vrible Eq. Coefficients of # F F -.. 6 Right side 6-6 And so the constrint identifying the leving vrible is the lst one, since its rtio is smllest (<).
The leving vrible is therefore, since it is the vrible in the lst constrint which gets pushed to s increses. And so the tbleu is shown below with the pivot row, pivot column, nd pivot element identified. We hve lso modified the lst bsic vrible (column ) to be. Tbleu Bsic vrible Eq. Coefficients of # F F -.. 6-6 Right side Dividing the lst eqution by, we obtin: Tbleu Bsic vrible Eq. Coefficients of # F F -.. 6 -.. Right side Using the lst eqution to eliminte the - in the top eqution, we get: Tbleu Bsic vrible Eq. Coefficients of # F F. 6. 6 -.. Right side
Using the lst eqution to eliminte the from the second eqution, we get: Tbleu 6 Bsic vrible Eq. Coefficients of # F F. 6. -.. 6 -.. Right side The solution bove is optiml becuse ll coefficients in the objective function epression re positive.. Introduction to dulity Let s consider tht our liner progrmming problem is ctully resource lloction problem where the vrious constrints re ctully constrints on our resources. Reclling the originl form of the problem, nd giving it the nme Problem P : Problem P m F 8, we see tht our new interprettion indictes tht,, nd 8 represent the mimum mount of ech kind of resource tht we hve. ()
Now the question could very well rise: how might we gin the most, in terms of the vlue of our optimized objective function, by incresing one resource or nother? In order to provide bsis of comprison, let s llow ech resource to increse by unit. Wht will be the effect on the objective function? To nswer this question, I first solved the originl LP, (), in CPLEX. As we would epect, I got the nswer F *=6, consistent with Tbleu 6 bove. (The subscript on F, which is in this cse, indictes it is the solution to the LP defined by () bove). Then I used CPLEX to solve LP (). m F 8, where the only difference, reltive to (), is tht the upper bound on the first constrint ws incresed from to. CPLEX provides the nswer F *=6. Apprently, incresing resources on this constrint hs no effect. ()
Then I used CPLEX to solve LP (): m F 8, where the only difference, reltive to (), is tht the upper bound on the second constrint ws incresed from to. CPLEX provides the nswer F *=7.. Here, incresing the second resource by unit provides tht the objective function improves by n mount equl to.. Finlly, I used CPLEX to solve LP (): m F 9 (), where the only difference, reltive to (), is tht the upper bound on the third constrint ws incresed from 8 to 9. CPLEX provides the nswer F *=7. Here, incresing the third resource by unit provides tht the objective function improves by n mount equl to. () 6
And so incresing the resources, i.e., the right hnd sides of the first, second, nd third constrints, by unit, improve the optiml vlue of the objective function by,., nd, respectively. In other words, F * b F *. b () F * b This is quite useful informtion, in tht it could guide our future lloction (or relloction) or resources. In other words, ssume F* represents my optiml profits, nd I find I hve little etr money to spend. Then () tells me it is better to spend tht money to increse resource thn to increse resource, nd tht it will do me no good t ll to increse resource. It is of interest to inspect Tbleu 6 t this point. Tbleu 6 Bsic vrible Eq. Coefficients of # F F. 6. -.. 6 -.. Notice one very interesting thing: Right side 7
The coefficients of the slck vribles in this finl tbleu re ectly the sme s the right-hnd-sides of ():,., nd. These vlues re circled in Tbleu 6. This is no coincidence. In fct, it will lwys hppen. Tht is: The coefficients of the slck vribles in the objective function epression of the finl tbleu give the improvement in the objective for unit increse in the right-hnd-sides of the corresponding constrints. It is of interest to emine the units of these coefficients, n eercise most effectively ccomplished by returning to () where we cn see tht they hve units of (units of F)/(units of b). For emple, if F is mesured in dollrs, nd b is mesured in, sy, pipe fittings, then these coefficients would hve units of $/pipe fitting. These slck vrible coefficients hve nmes. They re clled the dul vribles. We will see why they re clled dul vribles lter. They re lso clled shdow prices. For now, let s give them the nomenclture λ i corresponding to the i th constrint. 8
One note of cution, here. Reclling tht λ = in our problem, we understnd tht if we increse b by, from 8 to 9, we will improve our objective function vlue (t the optimum) by, from 6 to 7, nd tht is indeed the cse. But wht if we increse b by 6, mking it? Will we see n increse in F* to? Use of CPLEX indictes this does turn out to be the cse. However, if we increse b to 6, we still obtin F*=, result which suggests tht somewhere between b = nd b =6, the third constrint becme no longer binding (it ws no longer one of the two constrints tht defined the corner point). This is result of the fct tht whenever one hs multiple resources, ech of which is constrined, it will be the cse tht only subset of the resources ctully limit the objective. For emple, you my hve lbor hours, trucks, & pipe fittings s your resources, ech of which re individully constrined. You hve lrge truck fleet nd whole wrehouse of pipe fittings, but you don t hve enough lbor. And so you increse lbor until you hit your limit on pipe fitting inventory. You then begin incresing pipe fitting inventory, but soon you hit the limit on trucks. 9
. Motivting the dul problem Consider gin our originl problem., 8 m F (6) Let s epress liner combintions of multiples of the constrints, where the multipliers on the constrints re denoted λ i on the i th constrint. The below problem illustrtes (the λ i must be nonnegtive): 8 ) ( ) ( 8) ( ) ( ) ( m F (7) The lst reltion cn be rewritten s follows: 8 ) ( ) ( 8) ( ) ( ) ( m F (8) COMPOSITE INEQUALITY
The left-hnd-side of the composite inequlity, being liner combintion of our originl inequlities, must hold t ny fesible solution (, ), nd in * * prticulr, t the optiml solution (, ). Tht is, it is necessry condition for stisfying the inequlities from which it cme. Write the objective function epression together with the composite inequlity: F ( ) ( ) 8 (9) Let s develop criteri for selecting λ, λ, λ. Consider the following concepts wtch closely: Concept : Mke sure tht our choices of λ, λ, λ re such tht ech coefficient of i in the composite inequlity is t lest s gret s the corresponding coefficient in the objective function epression, i.e., () This gurntees tht ny solution (, ) results in vlue F which is less thn or equl to the left-hndside of the composite inequlity. Although necessry, it is not sufficient. A simple emple will show this. Choose λ = λ = λ =, nd the combined inequlity is then +. All vlues of (, ) tht stisfy the originl inequlities must stisfy this one, but there will be some tht stisfy this one tht do not stisfy one or more of the originl inequlities, for emple, (,) results here in, but the third inequlity of the originl ones results in ()+()= which is greter thn its right-hnd-side of 8.
Concept : Becuse the left-hnd side of the composite inequlity is less thn or equl to the right hnd side of the composite inequlity, we cn lso sy tht ny solution (, ) results in vlue F which is less thn or equl to the right-hnd-side of the composite inequlity. Concept : Concept implies the right-hnd-side of the composite inequlity is n upper bound on the vlue tht F my tke. This is true for ny vlue of F, even the mimum vlue F*. In other words, if we look t the vlue F* nd the right-hnd-side of the composite inequlity on the rel number line, they pper s below, with F* to the left, nd there would be some difference Δ between them. Δ F* Right-hnd-side of composite inequlity Concept : Now choose λ, λ, λ to minimize the right-hnd-side of the composite inequlity, subject to constrints (). This cretes lest upper bound to F*, i.e., it pushes the right-hnd-side of the composite inequlity s fr left s possible, while gurnteeing right-hnd-side remins greter thn or equl to F* (due to enforcement of constrints ()). Concept : Given tht the right-hnd-side of the composite inequlity is n upper bound to F*, then finding its minimum, subject to (), implies Δ=.
Concept tells us tht if we solve the following problem, cll it Problem D, Problem D min G subject to 8,, tht the vlue of the obtined objective function, t the optimum, will be the sme s the vlue of the originl objective function t its optimum, i.e., F*=G*. In other words, solving Problem D is equivlent to solving Problem P. Problem P Problem D m F min G 8 subject to 8, PrimlProblem,, Dul Problem Problem P is clled the priml problem. Problem D is clled the dul problem. They re precisely equivlent. To show this, let s use CPLEX to solve Problem D.
The CPLEX code to do it is below (note I m using y insted of λ): minimize y + y + 8 y subject to y + y >= y + y >= y >= y >= y >= end The solution gives G*=6 λ = λ =. λ = It is of interest to inspect Tbleu 6 of the priml. Tbleu 6 Bsic vrible Eq. Coefficients of # F F. 6. -.. 6 -.. Right side We note the vlues of the decision vribles obtined for the dul problem re ectly the coefficients of the slck vribles in the priml problem.
We lso note tht G*=F*=6. One lst thing tht is very interesting here. Using CPLEX, following solution of the dul, we cn lso obtin the coefficients for the dul problem slck vribles (the slck vribles to problem D), in the objective function row, using the following commnd, disply solution dul - nd they re: λ. λ 6. which is precisely the solution of the priml problem, =, =6, s cn be red off from Tbleu 6 bove. This commnd ws mde fter solution of the bove dul problem in CPLEX, therefore it is giving the solution to the dul of the dul, which is the priml! Cution to void confusion: The bove vlues for λ nd λ re not the vlues of the slck vribles. They re the coefficients of the slck vribles of the objective function epression in the lst tbleu of the dul problem. This suggests tht there is certin circulr reltionship here, which cn be stted s The dul of the dul to priml is the priml. Tht is, if you clled Problem D our priml problem, nd took its dul, you would get our originl priml problem bck, s illustrted below.
Problem D min G subject to 8,, Priml Problem Problem P m F, DulProblem. Obtining the dul from the priml 8 It is useful to mke the following observtions:. Number of decision vribles nd constrints: Number of dul decision vribles is number of priml constrints. Number of dul constrints is number of priml decision vribles.. Coefficients of decision vribles in dul objective re right-hnd-sides of priml constrints. Problem P Problem D m F min G 8 subject to 8, PrimlProblem,, Dul Problem 6
. Coefficients of decision vribles in priml objective re right-hnd-sides of dul constrints. Problem P m F, PrimlProblem Problem D min G subject to 8,, Dul Problem 8. Coefficients of one vrible cross multiple priml constrints re coefficients of multiple vribles in one dul constrint. Problem P m F, PrimlProblem PrimlProblem Problem D min G 8 subject to 8,, Dul Problem Problem P Problem D m F min G 8 subject to 8,,, Dul Problem 7
8. Coefficients of one vrible cross multiple dul constrints re coefficients of multiple vribles in one priml constrint. PrimlProblem, 8 m Problem P F Problem Dul to subject G Problem D,, 8 min PrimlProblem, 8 m Problem P F Problem Dul to subject G Problem D,, 8 min PrimlProblem, 8 m Problem P F Problem Dul to subject G Problem D,, 8 min
6. If priml objective is mimiztion, then dul objective is minimiztion. 7. If priml constrints re, dul constrints re. From the bove, we should be ble to immeditely write down the dul given the priml. Emple: Let s return to the emple we used to illustrte use of CPLEX in the notes clled Intro_CPLEX. m F Subject to,, The dul problem cn be written down by inspection. min G 8 8 Subject to,, 9
We cn use CPLEX to check. First, we solve the priml problem using the below code: mimize + + subject to + + <= + + <= + + <= 8 >= >= >= end The solution is (,, )=(,,), F*=. The dul vribles re: (λ, λ, λ )=(,,) Now, solve the dul problem using the below code: minimize y + y + 8 y subject to y + y + y >= y + y + y >= y + y + y >= y >= y >= y >= end The solution is (λ, λ, λ )=(,,), G*=. The dul vribles re: (,, )=(,,).
. Viewing the priml-dul reltionship Another wy to view the reltionship between the priml nd the dul is vi use of the priml-dul tble. Although this offers no new informtion reltive to wht we hve lredy lerned, you might find it helpful in remembering the structurl spects to the reltionship. Let s consider the following generlized priml-dul problems.,...,,.............. m n m n mn m m n n n n n n b b b s t c c c F,...,,.............. min n n m mn n n m m m m m m c c c s t b b b G The priml-dul tble is shown below.
Dul Problem Coefficients of Coefficients for dul objective function Right side Priml Problem Coefficients of n Right side λ n b λ n b λm m m mn b m c c c n Coefficients for priml objective function For emple, our previous emple problem hs priml-dul tble s shown below.
Dul Problem Coefficients of λ Coefficients for dul objective function Priml Problem Coefficients of Right side λ Right side λ 8 Coefficients for priml objective function
6. The dulity theorem We hve lredy been using this theorem, nd so now we merely formlize it. Dulity theorem: If the priml problem hs n optiml solution *, then the dul problem hs n optiml solution λ* such tht G( *) F( *) The proof is given in [, pp. 8-9]. The dulity theorem rises n interesting question. Wht if the priml does not hve n optiml solution? Then wht hppens in the dul? To nswer this, we must first consider wht re the lterntives for finding n optiml solution to the priml? There re two:. The priml is unbounded.. The priml is infesible.
7. Unbounded priml We hve lredy seen n emple of n unbounded priml, illustrted by the problem below nd its corresponding fesible region. m F, 8
Recll tht the objective function for the dul estblishes n upper bound for the objective function of the priml, i.e., G( ) F( ) for ny sets of fesible solutions λ nd. If F() is unbounded, then the only possibility for G(λ) is tht it must be infesible. Let s write down the dul of the bove priml to see. min G 8 s. t.,, nd we immeditely see tht the second constrint cnnot be stisfied, nd so the dul is infesible. Likewise, we cn show tht if the dul is unbounded, the priml must be infesible. However, it is not necessrily true tht if the priml (or dul) is infesible, tht the dul (or priml) is unbounded. It is possible for n infesible priml to hve n infesible dul nd vice-vers, tht is, both the priml nd the dul my be both be infesible. Reference [, p. 6] provides such cse. [] V. Chvtl, Liner Progrmming, Freemn & Compny, NY, 98. 6