Kleinc Mitteiliingcn 575

iimgewandelt werden :

7 , y ) ~ p ,cc<,:+B)? r-luo + - - + I-' /Ze(:mU-l U ) ( z - S ~ p - l ( r ~ ( 8 ) - Coo) O U ( S ) ds . (40)

0

Fiihrt man nun die Normen dcr in Gleichung (40) anftretenden GroBen als Summe der absoluten Werte der Komponenten ein, so kann (40) in die nachstehende Integralungleichnng umgeformt werden :

4 q l Bh2[2 :+<? R , - - R , ) z * ~ - ~ -

(43) sind. Die Konst.ant,e 7)1 ist um einen beliebig kleinen positiven \\'ert groBer als der gr6Bte positive Eigenwert der Matrix I,G -k H . Mit, Hilfe des &ONWALLschen Lemmas erhalten wir a m der Ungleichung (41) eine obere Abschat'znng fur den exak- ten LosnngsvekDor hinsichtlich der Norm :

niz i s I lC(6) -5rXIllh.

\\u\\ 5 M e " (44) Vergleicht man die Ergehnisse (36) und (40), so sieht man,

daO das zweite Glied von (40) gerade den gesuchten Fehler, d. h. die GroBe U - V darstellt. Geht man jetzt zn den Normen iiber:

1 1 r j _ _ j.'ll 5 M jc2~1( '-s) IT($) -- i m l II T W I ds , (45) 0

SO ergibt sich ails Ungleichiing (48) dnrch Anwendiing von (44) fur den F'ekler 11w -~flhPrungs?iaet?i.otle

I I U - ~ l l 5 J I N endl J IC(~) - iml e

S

146) x 5 15(i, -5ml(1t ds . " 0

Offensichtlich hiingt das Verha,lten des Fehlers (46) entschei- clend vom Vorzeichen der Konst,anten m, d. h. von den Vorzei. chenverhliltjnissen der Eigenwerte 1 ab. Anf Grnnd der Formel (3 1) sieht man leicht, daD nnter den Eigenwerten die folgenden Relationen bestjellen :

a) A2 .=A, < 0, wenn K < 0; 1)) &, < 0, Al = 0. wenn K = 0;

c) 1, < 0 <A,, wenn K > 0, falls die iibrigen auftretenden ProzeBparameter physikalisch sinnvolle Werte annehmen. Andererseits kann hinsichtlich des Verhaltens der Funktion - cml auf Grund von (1 1) nnd (14) sowie mitt>elr; physikalischer Uberlegnngen festgestellt werden, ( i d3

a) &o, -- Cool = 1 - j m < 1; b) lim -Cml = 0.

e++m

Beriicksichtigt man, daB die Ausgleichsprozessc fur Wiirme- iind Stofftransport, st,rt.s cxponent'iellcn Charalrter haben, erhalt. ninn

100

0 c) j IC(S) - Tml (1s < -$ . Wendet man das oben Gesagte an, so ergibt sich, daB der

Fehler (46) im Falle eines negativen K Null ist, wenn z = 0 gilt und im Grenzwert z = + 00 zumindest exponent'iell gegen Kull abninimt. Daher nimmt sein Wert in irgendeinem inneren Punkt des untersuchten Intervalls, den man bei bekannter Funktion c(?) feststellen kann, notwendig ein Maximum an.

1st K = 0, so ergibt die Formel (46) eine sehr schlechtr Gchatzung. Man sieht jedoch leicht, daB das zweite Glied von (40) folgende einfarhere Form annimmt :

(47)

denn in dieseni Fall ist I1 = 0. Beriicksichtigt man das uber die Eigenwerte A und uber das Verhalten der Funktion &) - Coo Gesagte und zieht die Beschranktheit des Losungsvektors U fiir K = 0 in Betracht, so folgt &us (47) sofort, daf3 sich der Fchler auch jetzt genauso verhiilt, wie im vorher angefuhrten Pall.

Im Falle eines positiven K nimmt der Fehler laut Forniel (46), voni anfanglichen Nnliwert ausgehend monoton zu und diver- giert im Grenzwert z = - 1 - 00. Diese Tatsacke hodeutet jedoch kein Hindernis dafiir, daB im Falle eines endlichen Reaktors der Fehler der Naherung abgeschstzt werden kann.

B e d e u t n n g d e r Xymbole Ak, A% - Reihenkoeffizienten; B - Proportionalitiitsfaktor; C - Gaskonzentration in der Flussigkeit; C* - Gaskonzentra- tion im Gasraum; c - Durchschnittskonzentration in der Dunnschicht; C$ - Eintrittskonzentration des Gases in den Gasraum; D - Diffusionskonstante des Gases in1 Flussigkeits- film; F , F* - LAPLACE-Transformierte von c bzw. C * ; G , f1, G , 17 - Koeffizientenmat,rizen bzw. deren Transformicrt,e; h -'Stoffaustauschfaktor; K - Konstante des Quellgliedes; k - Laufindex; ni, M , N - Konstanten; P, P-I - Transfor- mationsmatrix zur Diagonalisierung bzw. deren Inverse; p - Parameter dcr LAPLACE-Transformation; pe - Polstellen der LAPLACE-Transformierten; R, - Radius bis zur Innenseite der Dunnschicht; R, - Innenradius des Dunnschichtreaktors; 8 - Eigenvektor der Koeffizientenmatrix; s, t - Integrationsvari- able; U - Vektor der exakten Losung; U,, 8, - Anfangsnert bzw. dessen Transformierte; v - Durchschnittsge~chwindigkeit der Flussigkeit in der Diinnschicht ; v* - Diirchschnit,tsge- schwindigkeit des Gases im Gasraum; V , - Vektor der ap- proximativen Liisung bzw. dessen Transformierte; z - in der Diinnschichtdicke gemessene Ortskoordinate; z - Ortskoordi- nate entlang des Reaktors; 01 - kompiexer Parameter; i - Hilfsfunktion; coo - Grenzwert der Hilfsfunktion fur z = -1- m ; 1 - Eigenwerte dcr Koeffizientenmatrix; p - HENRYsche Kon- stante.

I

L i t e r a t 11 r 1 PoLrHszKr, K. ct nl., Aetn Chimica Acnd. Sci. IInng.ar. 86, 1F1 - (1975) Z I'OLINSZRT, K. rt xi., Hiingar. J. Industr. Chem. 4, 151 - (1076).

Eingereicht: 27. 1. 1978, endgultige Fassung: 7. 3. 1979

Anschrift: Dr. T i ~ h VAJDA, Forschungsinstitut fiir Tech- nische Chemie der Ungarischen Akademie der Wissen- Hehaften, P. 0. B. 98, H-1502 Budapest,, Ungarn

SURESH CHANDRA M. CHANDRAMOHAN

An Improved Branch and Bound Method for Mixed Integer Linear Fractional Programs

1 . I n t r o d u c t ' i o n Branch and bound methods for solving integer linear fractional programs have already been proposed by AGRAWAL [I], [2]. Of t,hese, the former is based on the branch and bound method of LAND and Dorc [7) and the latter on that of DAKIN [53. The computational efficiency of DAKIN'S method, when applied to integer linear programs, has been improved considerably by BEALE and SMALL 131 and TOYLIN [9]. But, AQRAWAL has not put forth any such proposals. In fact, the improvements in [3] and [9] were dependent mainly upon the monotonicity of the objectfive function values a t the successive dual feasible solutions generated by the dual simplex method and hence such improve- ments in tmhe case of integer linear fractional programs do not appear to he straightforward.

Using the CHARNES and COOPER transformation we develop here a branch and bound method for solving mixed integer linear fract.ional programs. The method is similar to tha t of DAKIN [51 wit'h iniprovenients similar to those of BEALE and SMALL [ 31. A detailed study of t,his is necessitated from t,hc fact that the, variables of the equivalent program are n o t con- strained to be integers and hence t,he branching strategy is to be modified properly. However, t h e subproblems occurring here are linear programs and so, addit'ional constraints can easily be augmented by employing the dual simplex method. Moreover, i,hc penalty calculations proposed here do not require much of rompiitational effort. It is also shown that t'he procedure is finite. Becaiise of these advantages the method can claim superiority over those in [l] and [Z]. A small numerical example is also given t o illustrate the algorit)hm.

2. K o t a t i o n Let A be given real ?TI. >/ n matrix and Rn dcnote the n-di- mensional real veceor space. Let C, D E Rs, b E Rm, a, @ 6 R be all given. Let M = {1, 2 , ... , w}, N = { I , 2, ... , n } and N, be a given srthnet of N . Furt,her, let x, y E 3 7 2 and t E R be variable vectors with components of z and y donotfed by xf, j E N , and y,, j E 3, respectively. For convenience t,hc ( n -1- 1) component vector [y, tIT ivill be written as (y, t ) . The superscript T denotes transposition. Let S = {z I B z 5 b, x 2 0) be non- empty and bounded and let

It is assunied that DTz -+- /3 > 0 for all z E 8. 1 ' - { ( y , t ) I Ay - bt 5 0 , UTy + / I t 1 ~ ?/ 2 0 , t ;< 0) .

3. E q u i v a l e n t p r o g r a m CHARXES and COOPER [4], while' establishing the re,lationsliips hetween n linear fractional program and a n equivalent linear program, proved that every (y, t ) E V satisfies t > 0. Therefore for every (y, t ) E V , there exists a unique 2 E S where X I = yr/t for all j E A'.

Now, we consider the prohlem (P) given by: CTX 4- a

Maximize DTz + P

subject to Az 5 b, z 2 0, xj integer, ,j E ATl.

solved by solving (Q), where (Q) is given by

((4)

GRANOT and GRAKOT [6] observed that prohlem (P) mn he

Maximize z = cTy + nt

siil>jects to ~y - bt 5 0 ,

y ;: 0,

uTy + fit = I , t 2 0, yr/t integer for j E iVl .

The following t,heorcni cst'ablishes the rela,t,ions brtwrrn ('Pj

T h e o r e m 1 : (i) If there exists a n opfimc~l solution x* fo (P), tAen (y*, t * ) is

mi oplimnl solution to (Q) &ere y* = x*t* and t* = l/(UTz*+j). (ii) Conversely, if there exists nn optimal solution (y*, t * ) to ( Q ) ,

t l m t* > 0 an.d x* is an optimal solution to (P) where x* = ?/* I t * . Part (ii) of t'he above theorem is the same as theorcm I in [6].

130th of these parts can be proved easily by following the proof of the Theorem 1 in [4]. So, the proof is omitted here.

Using Theorem 1 we give below a branch and bonnd method similar t'o that of DAKIS [5 ] wit,h penalty calcnlnt.ion~ similar to thmc of BEALE and SarALL [s].

and (Q).

3. Branch a,nd b o u n d m r t h o d

4.1. ~)~z~?~op?tf,,ltlen.t of tke n t p l h o t l Considcr the problem:

(Q,,): Masimize z = CTy + at subject to Ay - bt 5 0, DTy + @t = I , y 2 0, t 2 0 ,

Introducing slack variables and solving t,he problem by the sjmples met.hod, we find that t is always a basic variable. Let t'he optinial simplex tableau be given by: Nasimiae z = 9 + ,z c,yi

srrbjcct to ylnt g t J

ailyg 5 yf, i E M , r! +- 2 (Zgvf = d , ] € J jt.,

t ~ O , y 1 ~ 0 f o r j E J a n d j r ; : B I , i E M ,

where J is the index set of nonbasitt variables and y,;i. i 6 iK, and t are the basic variables.

yBC = yf, i f 111, t = is, yj = 0, ot,hervise . Note that 6 is posit,ive and if y i / b is integer for evcr.y Hi E XI, then t'he required Rolution to (P) is obtained. If the necessary integrality restrictions are not sat.isfied, let ;jk/d be noninteger for some RI, E N,. We denote the largest intrgcr lcss than y&3 by bk/d] and the smallest integer greater t,hm ;pk/h hy <;r*/d>,. tSinc'e yI ik / t is required to he an intrgcr,

either yIzk/t 5 [7'k/6] or ynk/ t 2 ( y p / t J ) . Lct 11s consider the former. Since t > 0, t>his iriiplirs

The. opfdmnl basic fectsible solution to (Qo) is gi\,t.n hy:

(1)

YHg 5 c ; , k / 4 t

which gives rise to the constraint

It ran easily be verified that Lyk/S] S - ;'k is negative and tlic optimal solution to (Q,,) given by (1) does not satisfy (3). Attg- inenting (2) to (Qo) we obtain one of thc hraiirlies.

Similarly, corresponding to

we obt'ain t'hc constraint

J € J

Introducing (3) to (Qo) we obtain the other branch. This, to- gether wit'h the rules for the selection of branching variables completely describes the branching strategy. Since ot,hcr things remain the same as in the case of DAKIN [5 ] , wc pass on to tlic discussion of further improvements throiigh penalty calcu- lations.

2 (flkg - (;'k/6> d f ) yf 5 ?'k -- <;'k/d) 8 . (:iJ

4.3. Calculation of penolties Suppose that we have obtained tlic continuous opt.iinal nolii t,ion corresponding to a node and the associat'ed objcct'ivc function is given by :

where J + is the index set of nonbasic variablrs antt 2' i n tlic value of the objective function at the node. Suppose that a IICN branch is created by introducing the constraint

P J

Then, the penalty p , for augmenting this constraint is givcln tiy

2 + Y s l Y , i -f where f > 0 -

C+

G'r p = f x L , where

If we have already obhined (t feasible solution t.o (P) satisfying the necessary integrality restrictions with tlhe corresponding value of the objective function as z*, then the above branch can be delected from further consideration if I+ - p 5 z*. This helps us in eliminating many nonpromising solot.ions without much computational effort.

Moreover, since the procedure eonsists of solving linear pro- gramming problems, the node and branch select'ion can also br hased on these penaIt,y calculations as in tlic caw of intrgrr linear programs (see SALKIN [S]).

4.3 Finiteness When a new branch is constructed, owing to t.he corrcspon-

dence between the feasible solutions to (Qj and (P), by t,he addition of a new constraint to (Q.) we implicitly impose either TI 5 6 or xf 2 8 to the corresponding solutions to (P) for some E N , where 8 is an integer. Hence, the finiteness of t,he pro-

cedure is assured by the, boundedness of 8.

4.4 Further discussion Another point which is worth mentioning here is that regard-

ing the redundancy of cert,ain bonnding const,raints. I n DAKIN'S

inethod, if XI; 2s 0 has already been augmented and if xt 5 0' is to ljc augniented siilwxpently, then O1 < 0 and the constraint I',; 0 can be deleted. Similar rules hold good here also. Sup- IJOW that y k - Ut 5 0 has already been augniented and if in addition to t'his we want to augment the constraint yk - OltsO, then 0' < O and the constraint yh - Ot 5 0 can be deleted. This can be done by deleting the row in which the slack variable of the c~onst~raint y k - Ot 5 0 is the basic variable. This helps us in prcvcntii-rg t.lio size of t.he tableau becoming uiiwicldy.

-5. Coil c I 11 (ti iig r e niar k s

,\ccordiiig to Tonir,~ N 1101, DAKIN'S niethod with peiialty cal- culations due to HEALE and SMALL [3] is the most effective method for integer linear programs. Hence this niethod for integcr linear fractional programs seems to be quite promising. We conclude this paper with il. small nunierical example to illustratc the nictliod.

l'roblc1n 1 :

s i i h j c c t to

Sol II t,i 0 1 1 : Thc eciiiivalts1lt p r o b l e ~ i ~ is: ~' l 'O1J~l ' l l l 2: Musiniiac 2 := 3y1 - 2.//2 i y3 s i I 11 j ect to

Oy, ~- 4!/, - 1 S?:! i y,, ~~~ 151 = 0 , 2!jl -1 y2 - 2y3 1 y, - 7t 1.y 0 ,

2y1 3y, $- 5y, -; t :== 1 . i 2y2 - i- g3 -1- y6 -- !)t = 0 ,

yj ;< 0, j = 1 0, yJ t , y2/t integers . We solrc the problem obtained from Problem 2 by omilt'ing

t,lic integer rcstrict,ions on yl/t and y2/t. The optimal simples tableau can bc presented by ~ncans of Problem 3 which is as follo\vs : I'roblen1 3 : JLixniizc J - 5/4 -- ti5/12y3 - 71/12y3 - 1/12y4 salJjcY;t to y1 . . 1 ~ 41 /3tiyd . i y j ' 1 34/%y2 - :3.5/'3y3 4/Y& - 1/3 , gti -1 5q9y2 l/Yyy -- 5/Yy4 = ?/3.

yj 2 0, j = l , 2 , 3 , 4 , 5 , 6 , t 2 0 .

H3/36y3 - 1 I/3tiy,3 - 5/13 ,

t -1. 13/lSy2 i - 7/lt)?/, - l/l%y4 = 1/6,

Thc solut~ion is y1 = 5/12, y5 = 1/3, y6 = 2/3, t = l/ti, y j = 0 , ot,licrwise, wit,li the value of z eqiial t,o 5j4. Since yl / t =: R/2, taliis is not thc one corresyonding to tlic requircct solutioii to 1'lUlj1(3131 1.

Aiignieiitiiig yI .. Bt 2 0 to l'roblctn 3, i.c., itiigiiiciitiiig

I1/3(iy, - - 55/3Ciy, 5/3Cil/, 1 - 81 = - - l / l B

\vhcrc 2; 0 is the slack variable, wc obtain t,lio solutioii y, =: 415, y4 = 3/5, y6 = 315, y6 = 1, t = l/S, g j = 0, othcr- wiw. Here y1/t = 2 a i d therefore an integer feasible solution t o Problem 1 is obtained. The solut,ion is rl = 2, r2 = 0, x3 = 0 wi th the value of z equal to G/5.

The other branch is obtained by adjoining the constraint y1 -- 3t 2 0 to Problem 3. i.e., wc adjoin

-37/3Gy, - 1 - 41/3tiy, -+ 7/3Gv4 + 62 -1/12

where sp 2 0 is the slack variable. penalty p for adjoining this uonstraint is:

1' := (1/12) X (65/12) x (36/37) 115/148.

Sirice 5/4 - 65/148 < G/5, this branch is deleted from further consideration.

Hence t,he vptinial solut,ion t,o I'roblem 1 is :rl = 2 , :r2 .z 0, %' a -. - 0 ,,' \it'll the objective function valoc cqtral to S/5.

40 y,-IXhI, lid. 59, 11. 10

It e f c r e n c e s 1 A I ~ I ~ A I Y A I , , S. ('., On i~~bcgcr hulutiuns to linear t'rtlrtional funcliun;ils by

;L 1ir;incli iind bound tcclluiquo, Aeta Clieneia Jndica 2, $5 .- 7s (l!l76). 2 AtiIlA%vAL, S. C . , Ail nlt.crrlut,ivc iiiethod on integer wlutioiia to lincar fiac-

tioiiall'iiiict.iunaia11~ a branchand tround tecllilique, ZA31dI57, 52-53 ( l v i i ) . 3 BEALE, E. M. I,.: YYALL, 1%. E., Mixed integer programming by B bra11c11

and bound technique, l'roc. IFIP. Congr. 2 (ed. W. A. PALINICM), 45u-451 (1Dti5).

4 UIIARNBS, A,: C o O r I R : TV. W., Progrn~llnling with lincnr f'raetional i'iinct iu- nali;, Naval ltcs. Lugid. Quart 8 . 181 -186 (IUBP).

5 I ~ A K I N . X., 4 trw srirrclt alguritliiii frrr utised iuteger ~ I ~ ~ I ~ ~ X I I I I I I ~ U ~ , COIIL- yutcr . I . S. 250-255 (1DO5).

(i (INANUT, JJ.; UIIAKUT, V., On integcr a i d niiscd iiitcger friiuliuiiiil prugruiuni . ing prul~letns, Ann. Discrete Math. 1, Studips in Iiitcger I'rugrt~n~iiii~~g, (edh. 1'. 1,. ~ I A Y I I E I ~ rt d.), North Hulbnd Publisliiny Uompnny 221 -%<l (1!)77).

7 LAM, A. It.; 1)01U, A. G., An antonlatic lnetliutl fur s o i ~ i n g discrete pru- graniining problenw, X~unonietr i~~i %, 497 -520 (1960).

l1u- spits 1975

S SALKIN, H. M., Integer Programming, Addison WesIey, l<cutliiig,

0 TUYLIN, J., An improved branch and bound nict,llod for integer progralnnliiig, Operat,iona lle?;. ID, 1070-1075 (1Y71).

10 T O M L I X , J., Survey of computational methods fur solving largc sc~ilc syatcn~s. Ucgurtment of Opcrtitiuilv lleacarch, Tcclmicui Hcport 72 -25, Ytunfurd Uliiverdby 1972.

Eingereiclit: 10. 4. lY7H

Aiiachryten: Suiwsii C ~ i a x u i ~ a a m 1 $I. C I I . W ~ I ~ A ~ I O I I . \ u, Deportnient of Mathematics. Indian lristitute of Techno- logy, Hauz Khas, New Delhi, 11002Y, IKDld

Appruxiuiatiurieu fiir die vollstiiudigcu clliptiseheii Nuritial- iutegralo crstcr, zweiter und dritter Uattung

1. E i n I ( ; i t u ~ i g

Der Ausdruck t"[n + BK(k)]

8 [ E ( k ) - (1 - k2) K ( k ) ] ' ~ -. _____ 4 k ) =

K(k) cin vollst~andiges (I,tl:C~::Ni)HRsCh(~s) clliptisclics KOPIIILI- integral crster Oattanng niit den1 Rlodul k , E(k) eiii vollstPiidiges (LEGENDltEsches) ellipt~isches Normalintegral zwciter Gatttcny niit dcni Modul k , unterscheidet, sich - fiir riiclit zu nakie an 1 liegende k-Werte - nur wenig von 1 [I] . Dies erniiiglicllt, (lie Herleil-ung der folgendrn Appi-oxiniat ion K ( k ) fiir K(b):

k(k) - Xl(1 - X : " ) - l I Y -l&j] ; ; x ( k ' - l / * -0$) ~

I

K ( 0 ) = A(!)) - zr2 , (2) /C' == v1 - e2 tkI' %It k koIllyl~lllcllt&l? h d U 1 .

k ( k ) crgibt die folgonde Approxiination k(k) ftir &(k) :

Q ein Paranicter des Tntcgrals dritter Gattung. &Tit fi(p, a) rrliiilt nian ~ ~ ~ j ~ r o x i i i i ~ t t ~ o i i c i i fiir ~ ( 1 ; ) iiii(1 ~ { ( k ) . Mit Approxiinatiotien fiir K ( k ) rind X ( X ) kdnnw u. t ~ . *\pproo"i-

inationen fur die Tntcgrale U

L , , ( ~ ~ ) = J P K ( L ) a k 0