A duality criterion for linear programming and its application in an implicit enumeration algorithm for integer programming

(1)

an implicit enumeration algorithm for integer programming

Citation for published version (APA):

Keulemans, W. K. M. (1976). A duality criterion for linear programming and its application in an implicit enumeration algorithm for integer programming. (Memorandum COSOR; Vol. 7601). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1976

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PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 76-01

A duality criterion for linear programming and its application in an implicit enumeration

algorithm for integer programming by

W.K.M. Keulemans

Eindhoven, January 1976 The Netherlands

(3)

gramming has been used at the Eindhoven University of Technology, which was originally developed as a generalization of existing algorithms for linear zero-one programming (Balas [IJ, Glover [7J, Geoffrion [5J, [6J,

cf. Garfinkel-Nemhauser [4J, Ch, IV sections 5, 6 and 7).

It works rather satisfying for problems of "reasonable" size; it arose independantly from other sin:ilar algorithms as found ~n publications by Trotter and Shetty [9J, or Krolak [8J. The algorithm distinguishes

itself by the use of the so-called "duality criterion", a simple expression in the optimal dual multipliers of the linear program corresponding to the original integer program which enables us to find a good estimate of the optimal value of the LP relaxation of partial solutions. This estimate is compared with the value of the best known solution of the integer pro-gramming problem.

The algorithm is of the branch and bound type. In each vertex v

k a number of tests ~s performed that may lead to a restriction of the domain T

k ~ Sk' where Sk ~s the feasible region of the problem in vertex v

k' or to the

fathoming of v

k' Some of the tests are elementary and related to the well-known tests for zero-one programming. The tests based on the duality

criterion replace the usual tests based on the surrogate constraint. The duality criterion results in a new object function with the optimal vertex of the linear program as its origin.

Definition of problems

We will consider the bounded variable ILP:

(1) m~n z

=

c xT

s.t. Ax::; b

(4)

or equivalently T min z = c x s.t. Ax +

Y

=

b

o ::;

x ::; r x integer, y 2: 0

with the corresponding LP-relaxation:

(2) min z

=

cTx

s.t. Ax + Y

=

b

Y 2: 0

Without loss of generality we assume r to be an integer vector. Usually band y also are assumed to be integer although this is not necessary. We solve (2) by using upperbound technique, and denote the optimal solution by

(x,y).

Each variable x. for which

X. =

r., is replaced by

J J J

its complementary variable x! = r. - x., both in (2) and in (I).

J J J

Afterwards these x~ are called x. again. As a consequence of this we

J J

do not postulate that c 2: O. From now on we assume (I) and (2) written in the shape obtained now. Then the upperbounds r. do not playa part

J

anymore in solving (2), so we may write (2) in an equivalent form: m1n z = c xT (or max z = - c x)T

s.t. Ax + Y

=

b

x, Y 2: 0

Let

(x,y)

be an Qptimal solution of (3) and

(u,v)

an optimal solution of

its dual problem:

(4) min w

s.t. A u - vT

= -

c

u, V 2: 0

(5)

The duality criterion

Let (x',y') be feasible for (3), then we have Theorem: AT AT T, T v x + U

Y

< C x - c x Proof: c xT Hence: or: _T b _T _T T_ _T _T

= -

u + u y + v x

=

c x + v x + U Y _T _T T, T_ v x + u y < C x - c x

The resulting inequality is called the "duality criterion". Note that the conditions x, y ~ 0 have been dropped, for all partial solutions of (1) however x ~ 0 is valid, and for all feasible solutions of (1) y ~ 0 1S

valid too. When we substitute cTx problem: (5) m1n z

=

c xT_ + v x_T + UATY s.t. Ax + Y

=

b

o ::;:

x ::;: r x integer, y ~ 0

Thus we have introduced a new object function with the optimal vertex

(x,y)

of (2) at its origin. Now we define the restricted problem of (5):

(6) min z cx+vx+uyT_ _T _T Ax + Y b x integer, y ~ 0

*

Define y. 1 (b - Ax) .1 (i = 1,2, .. ,

(6)

then T~ ~T z

:=

c x + v Q, + m

I

(u.

*

max(O,y~»

i=l ~ ~ ::; m~n ( z)

It is clear now that the duality criterion has to be prefered to the "best" surrogate constraint for two reasons:

1. it is possible to use U and

v

as object function

m

2. because the term

I

(u.

*

max(O,y~» is taken into account, the estimate

i=l ~ ~

z for (6) is a good estimate, therefore it ~s not necessary to solve the linear program more than once.

The implicit enumeration algorithm. In vertex v

k of the search tree the set of feasible points considered will be denoted by:

In v

k the subproblem

(7) c xT ~s considered together with its relaxation

(8) c x,T x ::; r , x integer}.k

k k

Here Q, and r are integer left and right bound vectors for x; Q,0

=

0 and rO

=

r, so

So

= {x/Ax ::; b,

a ::;

x ::; r, x integer}.

The separation at v

k is determined by choosing a particular variabele xjO

. h k k

w~t r_jO > Q,jO' and by separating:

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This separation is clearly a partitioning. The path P_k from V

_o

to v

k corresponds to an assignment of integer values to a subset of the variables. The assignment is a partial solution of (I).

k k k

Let V

k be the index set of variables with ~._J _k

=

x._J =_k r. and F_J k be the

index set of the ("free") variables x. with r. > x .• The finiteness

J J J

of T

k for k

=

0,1,2, ••• guarantees the finiteness of the enumeration algorithm.

Fathoming.

A lower (upper) bound for z in v

k is denoted by ~k (zk)' As usual the fathoming cases are:

In the beginning we put

~O

=

cTx,

Zo

=

The case Sk

=

0

is contained in ii) if

n

I

r.

*

max (O,c.).

j=1 J J

we define Sk

=

0

~ ~k

=

00.

such a way that '" k the estimates z(x

jO

Selecting the partitioning variable and branching. For the partitioning (9) the index jo is chosen in

v

J'O

=

max_jEF

(v.).

_J In this way the difference between

'" k k k

and z(x,

O~ 1 + ~.O) is maximal.

J J k+l k+l k

Then we branch to vertex v

k+1 by putting ~jO

=

rjO

=

~jO' Once a vertex

is fathomed the next branching is performed by the classical backtrackting principles.

The tests.

Before separation in v

k some tests will be performed. The problem considered at v k is: (10) m~n z n

L

c.x. j=l J J

(8)

n s.t.

_L

a .. x. + y. = b.

}(i

j=1 1J J 1 1 = 1,2, •••m) y. ;::

°

1 k k x. + s. = r. J J J k L (j 1,2,.••• n) x.

-

x. = = J J J k k _;:: 0, integer x.,s.,x. J J J Substituting x. J k

=

x. + J

£~

J in (9) results in: (11 ) min

z

=

n

L

c.x. + n

L

c.Lk j=1 J J j=l J J n k n k

}(i

s.t.

_L

a .. x. + y. = b.

-

_L

a ..L

=

1 ,2, •••m) j=1 1J J 1 1 j=1 1J J y. ;::

°

1 k k k

_£~

(j 1,2, ..• n) x. + s. = r.

-J J J J k k _;:: 0, integer x., s . J J Now we define k n k

Co

:=

L

c.£. j=1 J J

b~

_:= n k b.

-

_L

a .. £ . 1 1 j=l 1J J

d~

k

_£~

:= r.

-J J J (i 1,2, .••m) (j = 1,2, ... n) whence the problem in v

k becomes:

(12) _{min z}

₌

Co

k + n

L

c.x.k

(9)

n k (lk+(i) :=

L

(max(O,a .. )*d.) j=1 1J J max (LHS .) ,1 XET k n k

L

(min(O,a .. )*d.)

=

j=1 1J J min(LHS.),1 XET k k n k Y k

:=

Co

+ _j=1

L

(min(O,c.)*d.)_J _J

=

T... ~ ... k c X + L v.t. + j= 1 J J . T :-:; m~n c X XES k 1 •2, 1.3 and 2.1, 2. 2) . k change 1n t. J k change in d .• J

A number of tests will be applied for j E F

k (tests 1.1,

by which a left (right)

x~bound t~(r~)

may be changed. A

. k k J k J J k .

carries changes 1n b., d. and cO; a change 1n r. carr1es a

1 J J

Finally, in both cases

(l~(i), (l~(i),

Y

k and

v~

will change too. In

general T

k may shrink after the application of such a test.

The fathoming tests are:

Test 0.1: For i := 1,2, ••• m (lk(i) > bk

(10)

Test 0.2: Y_k > 2 0 ~Vk is fathomed by taking ~k = Yk' k Test 0.3: V

o

> 20 ~ Vk 1S fathomed by taking ~k Test 0.4: for j := 1,2, ••• n k x. $ a. on basis of tests 1 • 1 , 1.2 or 1.3

}~Sk

J J k x. > a. on basis of tests 2. I or 2.2 J J In this case x.k J k

$ a. or x. > a. would lead to fathoming by test 0.1.

J J J Test 1.1: for i := 1,2, ••• m k k k a .. > 0: Ctk(i) + cl.a .. > b. ~ X. 1J J 1J 1 J a .. 1J Test I .2: 0: k k 2 0 - Yk c. > _Y k + d.c. > 20 ~ X. $ J J J J c._J k Test 1.3:

V.

0: k k k 2 0 - V

o

:J. _Va ₊ d.c. _> 2 0 ~ X. $

V.

J J J J J Test 2.I : for 1 := I ,2, ..• m - . k 0: Ct k(i) k k k

_d~

Ctk(1) - bi a .. <

-

d .a .. > b. ~ x. ;:: 1J _{J 1J} 1 _J _J a .. 1J Test 2.2: 0: k k

d~

Y_k - 2 0 c. < _Y k + d.c. > 20 ~ X. ;:: J J J J J c. J Test 3. for c. ;:: 0 determine succesively:

J k

_(d~

b~

-

Ct~

(i) f jl := m1n 1 ) i

I

a .. <0 J a .. 1J 1J k

min(f~

I

,d~)

f j2 := J J

f~

:=

max(0,[f~2J)

; J

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Conunent:

Tests 1.1 and 2.1 represent the elementary necessary conditions for feasibility in Sk:

a) x.a ..k ::;

b~

- (1k (i) (a .. > 0)

J 1J 1 1J

b)

(d~

- x.)a ..k ::;

b~

- uk(i) (a .. < 0)

J J 1J 1 1J

Tests 1.2, 1.3 and 2.2 are necessary conditions for feasible solutions in Sk with cx ::; z00

k

Test 3 needs some explanation, when f. > 0 then: J

a) a .. < 0

~ b~

- f.a ..

~

(1k+(i)

~

0,

1J 1 J 1J

to guarantee

b~ ~

0 for all successor vertices x. =

1 J

f~

.1 1S suff icient

b) a .. > 0, X. > f. ...k b. - x.a ..k < b. - La ..k k

1J J J 1 J 1J 1 J 1J

thus

b~

-

x.a .. ~ 0 ... b.R,

-

f .a ..k ~ 0 (.Q, > k)

1 J 1J 1 _{J 1J}

Therefore~ when x.

=

_{xli 1}

'f

j , x. >

f~

1S a feasible solution of (1)

1 _J _J

then x.

₌

_{xl i i}

'f

J , x.

f~

1S also a feasible solution of ( 1) •

1 _J _J

The program.

The program structure.

The integer program consists of two ma1n parts a) a dual linear programming algorithm

b) the real integer programming algorithm

Adual linear programming algorithm suffices. If a problem is not dual feasible, it is made dual feasible by introducing complementary variables for these variables which disturb dual feasibility. The dual algorithm delivers:

1. copt (= cx)

2~ u (=

G)

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The dual algorithm uses upperbound-technique. After termination of the algorithm v contains the

v.

belonging to the original variables. If

v.

< 0

J J

for some j then the integer algorithm will also introduce a complementary variable Xl

=

r. - x .•

J J J

The integer linear program itself consists of three important parts: 1. the procedure backstep,

this procedure takes care of backtracking 2. the procedure forwardstep,

in this procedure test 3 is performed, the partitioning variable ~s chosen, and the separation is executed

3. the body,

the body itself consists of two parts: a. the initialization of the ILP

b. the remaining tests.

Administration of the vertices.

The arrays x and wand the integer variable r contain all information about the path P

k from starting point to the actual node. The variable r contains the length of P

k. The array w contains + the number of the variable if its upperbound has been changed by separation, - the number of the variable if its lower (upper) bound has been changed by the additional tests or by backtracking. The x. are numbered from 1 to nt

J

the Y. from n + 1 to n + m. The array x contains + the amount by which

~

the lowerbound has been increased, or - the amount by which the upper-bound has been decreased. The array lowbo contains the partial solutions.

Procedure headings.

The integer programming procedure is called IP, its parameters are:

1 • numc: the number of columnvectors n 2. numr: the number of slacks m

3. sum an upper estimate for the length of P_k

4.

rhs the original right hand side ( J

_*

_m)

(13)

5. cost: the original objectfunction ( I

_*

_n) 6. upb the upperbounds (r. ) of x. (I

_*

_n)

J J

7. mat the m

*

n matrix A

8. sol contains after termination the optimal solution of (I)

9. maxc: contains after termination the optimal value of the solution. 10. copt: the optimal value of the LP relaxation

11 lp the procedure lp

The linear programming procedure is called LP, its parameters are: I. copt, as 10. above

2. 3.

4.

5.

u the optimal dual multipliers

v the optimal primal multipliers

rhs, mat, cost, upb, numc, numr have the same meaning as for IP sol : after termination of LP sol contains the optimal solution of

the lp-relaxation (2).

Implementation of the tests.

In the program c 1S stored in the zeroth row, and v ~s stored in the

m + !-th row of the matrix A. We define the right hand sides in vertex v k: Furthermore we define and _{b m}k+₁ := n k

I

(max(O,c.)*d.)

=

j=1 J J ct~(0) n k :=

I

(min(O,c.)*d.) = j=! J J n k

I

v.d.

= j=1 ] J n k

I

a 1 .d. j= 1 m+ ,J J

(14)

Test 1.1, 1.2 and 1.3 are now combined: Test 1: for i:= O,l, ...m, m+ I:

a .. 1.J > 0: ak- (i) + d

~

a. . J 1.J k > b. ~ x. ~ 1. J k - . b i - ak(1.) a .. 1.J

Every time when a better solution has been found bk

k k

a: (i)

0

be adjusted, bm+1 is adjusted too if b. > for 1.

This is done as follows:

Test 2.1 and 2.2 are combined too: Test 2: for i := 0,1, ••. m, (m + I):

a.. < 0: a k- (i) - d

~

a.. > 1.J J 1.J

b~ ~

1. x.J

~ d~

J ak(i)

-b~

a .. 1.J k and b_m+I have to some i (I ~ i ~ m).

a) a better solution has been found, with value za then:

b)

b~

>

a~(i)

then:

bk_{m+1 :=} b_{m+1 -}k (b k +( . ) ) and bk. .' =

i - a k 1.

*

ui 1.

In this way no feasible successor vertices of v

k are omitted and

bk k

(15)

[IJ E. Balas, An additive algorithm for solving linear programs with zero-one variables. Opns. Res. 13 (1965)

517-546.

[2J E. Balas, Discrete programming by the filter method. Opns. Res. 15 (1967) 915-957.

[3J B. Fleischmann, Computational experience with the algorithm of Balas Opns. Res.

II

(1967) 153-155.

[4J R.S. Garfinkel and G.L. Nemhauser. Integer programming. Wiley &Sons,

1972 •

[5J A.M. Geoffrion, Integer programming by implicit enumeration and Balas, method. SIAM Rev.

I

(1967) 178-190.

[6J A.M. Geoffrion, An improved implicit enumeration approach for integer programming. Opns. Res. 17 (1969) 437-454.

[7J F. Glover, A multiphase-dual algorithm for the zero-one integer programming problem. Opus. Res.

II

(1965) 879-919.

[8J P. Krolak, Computational results of an integer programm1ng algorithm, Opns. Res.

1I

(1969) 743-749.

[9J L.E. Trotter and C.M.S. Shetty, An algorithm for the bounded variable integer programming problem. Dept. of Industrial Engineering, Georgia Inst. of Technology, 1971.

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( M'?P~Y

p..,,,.

IIOQn. ')0 V. SOL. COSTIt }. MATfld n

9vGPJ I'JTEG"o T. J. 1(. r:OLIIJ~. ROW'Jp. IIJILL. <;U'l: OVAL F. UTYO~<:. TH"TA. QUOT. DIV~T:

ROaLEA~) '~r~ITT. DII\]='''o:

APOlly Arf):"'J'-4o. n:\/,I"Ie}. :JPC 1 : 'lJU"1C+NU'40 }. ~A"{1 :'lJUMPJ...

"n'IQOS[ 1 :'JIJMC1:

Q()()'_fA\I AOolIY Ln".RI1 :NQ"1r:+I\JIJl.lQ}; <;')I.\:=\lI)"'C+"III'-4q: A(O.O}:=O; NTLL:=-1;

fnR r:=l <:T"='! 1)'IlTTL SIJ'" f)O LOWEP£T):=ToUF: F'1Q I: =1 eTn' ] 'It,JT IL NU"lP 00

FnR .):=1 -::T";:> 1 "~ITTL I\JU"'C nl) AIY.Jl:="1ATfI.JH F0q T:=1 qV;:> 1 ""HTl NU"1(' flO ACO.Il:=C'lST[IJl i>JOTT " (liP. ". 'IoQ'));

Fl)o 1:=\ cTFo i '!"lTTL "11)'10 f)r) A(I.()]:=PHSCIJl F')O I:='1,,,..r+1 C:T"P 1 tJ"JTIL <;'J'" no JP(IJ:='iilc:;n; V00 J:=1 cTv:> 1 'I~JT1L '!J"'1C f)O 'IO'JRA<;CJ1:=J: Fn;;> 1:=1 <:T':> , ""JTTl "J""'o f/r) qA<;[TJ:=I+~JUMc:

Fno T:=1 <:TVo 1 'INTTL IIJ'J"C no IF Acndl < 0 TY""J QFGPJ F:",'loCIP /.1);,/"PCf1:=FIILC;v;

FnR J:",n sT"o 1 U"JTTL \lUMO nO

qE~TY IIrJ.Ol:="-F"ACJ.Il; ACJ.11:= .... (-]) ElIJn I': 'II) ;

DIIII"I':O: ",,,,tC;":

[) ...) ~FGT'J ,q.I'<~S:=n: pmJ"Jo:=IIJTLL:

.1)0 1:=1 ST.D 1 1J~ITYl 'lU",q no IF IlC,J.f)} < MIIoJPHC; THf!ll ::JEST"! 'AI~ioHs:=",r,'.()]: O()"'Jq:=J ':"J[1:

IF on,.'''o \l.O "IL'_ THF\l ::JI':GT"

nn q"SI\l THFT8:=.SO; rO/.NO:=NILlI PIVOT:=O; ~INIT:=FALSf;

FOo .1:=1 ST.P 1 U!IITIL \lU~C on qEGIN IF F:=ArPnwNP.jJ <

°

THFN

IF o"nT:=-AIO,JI/ACP::tWNR.JI < THETa OR (")l/OT=TYFTA AND F < PIVI)T) THE"!

QFr.I\l THF'TA:=OUOT; PIVOT:=.; COlNR:=J E!IID

TV rJLNR NEO "JILL

THF~I qFGIN TYfTIl:=IICROwNR.01 / A{ROWMP.COLNPH TF THfTA > UPClIION9AS[COLNRll THf\!

q"GYN ~I'IIT:=TPIJf; .;=UP("JONRAC;CCOLNOlJI

FOR J:=0 sn:: D 1 'JNTTL NU~o f)fl qE~IN ACJ.Ol:=*-F*A[J.COLNRl 1 ACJ.Cf)l!llRl:=*~(-1) F'In: lOWI':R['lJOt,JRASCCOLNRll:= ~ LOWERCIIJONRASCCOLIIJR}1; PI!') t'""!n VI<;. PAUDFR:=TR~JE V"')

U~ITTL PAuPFR 00 ., ~I!IITT: I • ., DAIJPFR THEN

R.r.l"J FOO .1:=0 STED 1 UNTIL "JUMP

no ACJ.Ol:=*-THETA~A(J.cnlNRl1 ACRO~!IIQ.01;=THETA; ~:=BAC;CQOW"JR1I

RAC;[R0\J/lIIR1:="JONijAC;CCOlNRl: NONRA<:;tCOlIIJR):=KI FOR 1:=1 STFP } UNTIL COLIIIR-I.

COL'IIP+l STEP 1 UNTIL' NUMr 1)0 A(RnwNR.Jl:=*/P1VOH

:m lP 40 LD ·50 LP ~{}. LPe 10 Lp ilO . LD Q() LP 100 lP 110 LP 120 LP 130 LP 140 LP 150 LP 160 lP 110 LP IRO lP 190 LP ?OO lP "10 LP 2?0 lP

no

LP ;:040 LP ;:>':;0 lP ;:060 LP

no

LP ?RO lP 290 lP 300 lP 310 LP 120 LP 330 lP 340 LP 350 LP 3~0 LP 370 LP 3,,0 LP 390 lP 400 LP 410 LP 420 LP 410 LP 440 LP 450 LP 4f>0 LP 470 LP 4g0 LP 4QO LP 500 LP '510 LP 5?O Lo 530 LP '540 LP 5'50 LP 560-LP Slit·· LP 5130 ~ I

(17)

LP ~JO tP 6;:>0 LP 1)30 lP MO LP 60;0 lP AMl LP 670 LP I,~O LP 6<:)0 lP 700 lP 710 lP 720 . LP 730 lP 740 LP 7'50 lP 7M lP 770 lP 1 .. 0 LP 7<:)0 LC ~no LP '110 lP fPO lP A30 LPA40 LP ACiO LP ..1)0 LP '170 LP 8'10 LD A90 LP 900 UP[IHLP CliO LP 9;>0 LP 910 LP 940 ElSI: R~GIN F:=A(I.COLNR),

rr:-

F IIJEQ-O-THEIIJ

FOR J:=l STEP 1 U'lTIL COLNR-1. COlNR.l STEP 1 \..INTI!. NU"1C 00 ~Fr,IN A[I.Jl:=*-F*A(ROWNR.JJ:

- TF ARSIAIT.J-Jl < ,;,l-f, THI:N AfT.Jl:=o E"ID

F!\Jn:

FOR T:=O STF.P 1 UNTIL NJMR DO A[!.COLNR1:=*/I-PTVOTl;

AfROWNR.COLNR}:=1/DfVOT; .

F[)R 1:=1 STEP 1 UIljTIL NUMR r'ln IF AfI.O) > UPfRASfIl)

THEN REGIN ArT.OH=lJP[BAS(IJI-Af!.On

LOWERr8AS(IIJ:= ~ LOWEP[8ASfI)J:

!"OR JI=l C;Tf!:P 1 <JNTIL fIllj"'lC no AfT.Jl:=**I-J) r:::NO

E'JD ENO

UNTIL POWMP

=

NIll OR PAUPER; COPT:=-Afo.Ol;

FOR 1:=1 c;TEP 1 "NTIL NUMC on

REGIN

K:=NON~AS('J:--IF K' ,lOq N!)MC

THF"N II(K):: IF LOUlER(K} THEN A(0.T1 Fl<:;F--Ho.T) ELSf II(K-IIlUMC}:=AfO,II

pm:

Ff)R 1:=1 c;Tf P 1 \INTIL NI.IMP no

RIOGIN

K'::RASfTl:-IF K LFQ ~1!J"C THE"! V[O:=O ELSE 1j{K'-NUMC1:=0

E\lO:

FOR 1:=1 STFD 1 UNTTL illUMe no SOL(IJ:=!F LO~ER[I) THEN 0

FOR I:=l <;TF:l 1 JlNTIL N\J"lP r)f)IF K:=RASn) tI':0 I\HJ~C T~IO'"

(18)

( RFr,I'J I\lTFGFR T, .J. 1<'. L, p. p. c;. FREE. LOw. UP. OEL TA. x.... ITJ;:Q. NR:

PEAL 7r. 7n. NFr.r. MAXO. F, FA: ROOLEAN FTN. RAr~. FEAC;IQLE;

ARpllY x. -J[l:C:;U"'I. T"Inl:)(. OANK. LO",RO, I/PI'lO. V{1:IIJI/I.ICI. A!O:""I"10_{+1. ,,:N·IMCI. "JEG. POS.} R. I)(O:NU"IR_+l11 ROOLFAN AOPAY LOwERfl:NU"lCl:

RO'lL'CII'J poOrFnlJoF 8AC.-:c;TEP:

qEGIN INTEG-o T. K, L. P. 0: pEAL F; 8()OLEAN pEAr)V: qFAr'\Y:=FAI c:;-:

no qFGT~ TF ~:=wrol > n

THEN T. (~[01-"EG(OI > 0 A"O 11[\10.1(1 < Q(\I0)1

T'-IF"! 0>'1;1'1 O>'IIr)Y:=TPlj>': o:=-X[OI-I: x",:=oA'jl((I(I-1; Fno L:=O c:;T'C o I UNTIL NO no R[Ll:=*-A[L.K]: lO",qOII<):=*+I:

T' 0 > 1

THF'J IF ~[o-I)= -I( 11"10 X[0-11 > n

THFN X[o:=o-11:=~+1

FlC;F qFr,I", 101(01:=-1<'1 X(P):=1 [NO >'LSF qEGTN ~f01:=-I(: )([Pl:=1 ENO:

TF P > 0 TH>"J

REGTN w(O:=*+11:=I(: X(OI:=-P: TTEo:=*+1 ENn

...·m

"", c:;F pEGTN '''[ol:=-K; UPI30[0(]:=*+X(Rl E"ln FLc:;"" PFr,T" P::)«(OI: I(::-K': 0:=0-1; O:=AR<;(O):

p" ~ I f ' ) N,,"Ie T'-IFN R>'r,TN

Fno 1:=0 STEo 1 UNTIL NR nn TF F:=a(I.~1 ~~0 0 THEN OFGI~ TF F < 0 TH>'N N~G£Il:=*+0~F FLc:;F POc:;[T1:=*+0-FI

IF P > 0 THEN 8(I1:=*+0*F

F"\/n:

IF D > 0 THEN LO~~O[~J:=*-o ELSE UP~O[~1:=*-P

F"\lf)

E' c:;r REGTN I<':=K-NU"'C: Rr<:I:=*+n; 8('IoJ:=*+Q*I)[o(1 E~lO

"'''f") FNO U~TIL RFAny nR 0

=

0: i'<lIC'><:STEP:"'P=O F'J() "lllC<<;TFo: Dp~rEnj~E F0P~APD~TFo;

qFr,TN JNTEGFo I. J. 1(. l. FR~>'. c:;: oEAL F: ~OOLFAN OFPEIIT;

~~nLEA"l A~RIIY 8IN~(0:NP11

1'<:0:=*+11

n0 REGIN pFPEAT:",FIIL~F;

'OR J:~o C;TFP 1 UNTIL NR nn gINOrJl:=R{JJ < pnC:;[Jl; FOR c:;:=1 STFo 1 UNTTL XM no

RFGrN T:=INDFV{SJ:

TF FREE:=(UDRn[Il-lnW~n[Tl) > 0 THEN IF A(o.I1 GEQ 0 THEN

qE~T"1 FA:=FpEE:

FOR L:=O STEo J U"ITIL No 00 TF F:=II(ldJ < 0 THE'" TF 81'-'{HLJ THE'II

TF R[Ll- FREE*F > POS£Ll

THE'" REAIN FI=FAFF-CI'l[ll-POSfLJ)/F; IF F < FA THEN FA:=F IP 60 YP 70 IP 1\0 To c/o IP }l'10 IP 110 10 PO IP 110 1° 140 10 1SO IP 1"-0 IP 170 IP 1 ".'0 1° 1~O IP ;:>00 IP 210 TP 220 IP 210 Ie 240 IP ::>50 1° 2'>0 IP no IP 2Rn Ie ?Qfl 10 300 To 310 10 320 1° 310 IP 340 0' YP 350 IP 31'>0 Ie 170 10 3OlO 10 390 IP 400 10 410 IP 420 1° 410 IP 440 To 4<;0 To 4'>0 Ie 470 10 4'10 10 490 10 ₅₀₀ TO '>10 IP ')?O TO <;10 IP '540 IP 5'>0 1° 560 ·10 '570 Ip 5~0 , .'~'~;'t:. ii'

(19)

~~ IF F::A!L.Il "JEQ 0 T~f"J IF F < 0 T~~'IJ "JFG(L1:=0-FoJP ELSF POS(Ll:=o-F ouo;

"O~,,!T 1 ::0_\10; \II (~;:<>.,1 : : - l : X! D) : :-lIP

• 'Jr")

~\Jr")

U"ITTL .., o~OFAT:

0'1 ~EGT'IJ T:=I'IJ'1FW!WM_l;

rF FO~F":=(IIPQ,",( Tl-L")W~')(Tll > (1 T"E'!

qEGT"J ,.I!o:=<>.11:=T: X(oJ:=-.pEFI

F~o ~:=" <:TFO I U"JTIL 'lJP )0 ~F F:=I\[~.Il "JfQ 0 TWEN

TF" F < 0 TH~N "JFG{<):::o-FoFEoF Fl<:E p')c;r~l:=o-FqI='F<>Ft pm;

x"!:=X"4-1

f\JD

'J'ITrL FO;:T > 1\ I1P w'-' = n:

Fr)p I :=1 <:T~O 1 'INT rL NU'lo nt') IF q[ I J > pn<:f TJ THEN I<FGI'J wro:::".IJ:=-('JllL4C.IH X!oJ:=on<:P1-l=llfl:

~(TJ:=00S[1l: q[~oJ:=".X[oJ"J[IJ

E',)')

F'lJn FnA~AOn<;TFO;

l.D(COoT. l I . V. '=''-1<;. MAT. ('O<;T. UD"'. Nt/ ....e. NU",ch <:Ot I;

F"JR r:=l eTfD _{1 'JNTTL I\JU"C n') TF A=3<:(VrT11} _< _{Ol-S THF\' V(IJ:=O;} Q:="IEGr:=7r;=7D:=ITFo:=n: x"':="Ju~r; I\JA:=NUM_R.l;

FaA T:=1 <:TFo 1 IJ/ljTIL I\JII"L ,'"1(\ LO .. 8:)(1l:::I); Fno J:=n <:T'o 1 'I"JTTL ',jR !")o \I~r;fJJ:=oO<:(JJ:=I);

FOR 1:=1 <;TF::> 1 ,)'.)TTL iIllJ~o f'1)

FOQ J:=l <:TFO 1 "'ITTl "JII"IC r")0 !\,{I.JJ:="'ATfT.J):

\OlDyTE ('JO'l". ". 'J::>RI:

F"OR 1:=1 <;Tl='o 1 !,"'TlL 'Hl'l::> f'\') i'l{!J:"PH<::fIl:

FOR I:=! <:TF::> 1 "NTIL "Jt!"1C i)0 ~IOoIJ:=ro<;rrTP F:,,! ;

F()ri 1:=J ST!,"D 1 J'ITIL "JU"1(' I)('l IF" alOol 1 > 0 TI-lE'J F:=F+A! n. YJ"u::>qnr I 1;

810J:=f: O{'IJ~):=F-cnoT: "AxC:=F+1:

FOP 1:=1 <;T'::> 1 IfNTYL "'INC 1)1) Af'lo.1J:=vf1J:

FOR I :=J <;TFP 1 UNTIL

""J'"

I)')

IF .{"JP.ll < 0 no (a{~p.IJ

=

0 A"JO A[n.IJ < nl THP! BE!;I" LOIolF:prIJl=FALSF: ,,0:=~P'W(Il:

FOO .1:='1 <;T~P ! 'lIlJTYL

,,0

r")n

qfr;1~ R{JJ:="-UP"A{J.rJ; AfJ.IJ:=""(-11 E"O:

EW)

ElSF lnwEPfTl:=TOU~;

FOP I :=l <;TFP 1 ij"JT IL "H)"!'" no 8EG PJ 1)0:=11040 [ 11 ;

FOP L:=O STfP 1 U~TYL

"'0

no IF F::AIL.I} "'FO 0 THE'" IF F < 0 THFN 'JFGILJ:=<>.Uo"F ELSE °I)SfLJ:="+UP<>f; E"ll);

FOR 1:=1 <;TEP I JNTYL "JU'IC no RA'oJK[IJ:=J; FOR 1:=1 ql='o 1 '1NTIL "lLJIoIC on

FOR J:=I.l <;TEP 1 U"'TIL "llJ"1C no

IF AINP.Tl < Af",p.JJ THF"l PANK(JJI=<>.J ElSF

IF 4["lP.Il > A[!lJo.JJ THEN AA!lJK(I1:=~.1 ELSE

TP €-40 IP 6'50 10 _I',"tl IP 670 10 1'.80 10 690 IP 100 IP 11 0 10 _1::>0 To 730 IP 740 10 _7::;0 10 7&0 TO 710 10 7"0 10 790 tP AOO 10 RIO IP A::>O 10 R30 10 840 10 830 TP q,-O IP IHO IP RRO IP 8ClO TP Q(l(1 IP 910 TO 9?O IP 930 IP 940 IP 950 TP QI',O IP 970 10 9'10 10 990 1°1000 IPJ010 1°1020 10₁₀₃₀ IP_I040 IP_I00;0 TOI0<'0 .10₁₀₇₀ IP10RO 1°1090 1°1 lIHI 101110 IP1120 101110 1°1140 IPl150 10_11"'0 101110 J'" •

(20)

IF FPCOF:=rIlPRfll!J-LOIolROlt1) > 0 THE"I

P.r.!"! LOw:=n: ,jP:=F"RFF;

1'"")0 u=n <;TFP I l)'IlTTL IIlQ no IF F:=Jl[f..Tl "Jf.t') I) THfN RFr,I'" FA:=(R[L1-NERILll/F;

TCO -;' < n THEN

TF FA:=(FA+FOFJ:1 > LOW THI'""I (J1W:=-ENTIEO(-FA):

TF F > n T4EI\! IF FI < UP THEN UP:=FNTIERIFA)

""\11):

"""lTA:=LIlW+\JO:=-( FPFF-JP) ; IF !)FLT A > Ft~E""

T\.lF~! RACI<:=TOUF FLc;f TF O~LTA > n TYF'J ::lFGJ\I P:=I;

TF LOW> n THI'"'1

QFGP! IoI[R:=*.11:=-I; l(fR):=tOW; lOIolROlT1:=*~LO\ol;

~~J"):

TF uO > 0 T41:""I

"'F(iHI W[R:=o+IJ:=-I; Y.[RJ:=-UP; UP8"lIIJ:=o-!IP END;

1'"0:;> l:=O <;r,p 1 '-'IIlITL "0

00 IF F:=A[l.I1 NF.O 0 THE"I

Q'GI~ TF LOW> n T~FN RIL1:=*-lO~*F~

IF F < 0

T4F/IJ 'IIfGIlJ:=*-OELTA*F fLS":

POC;IL]:=*-OELTA*S-"'''In

F~l'"

'J\lTIL 0 = 1"1 !/P HACI<; IF ~ PArI( THF"

qEGI!\I FOP T:=l sTEo 1 UIIlTIL NUMQ DO IF BIll> POSlIl THEN

RFr:.T"! WIQ:=O+11:=-(~JMC+T); xIR1:=pn~II1-R£11;

QrTJ:=oO<;ITJ: R[NRJ:= .... X[RJ .. IJfIJ

F","': T : ",-1 ;

nil F"fAc:.1R'.F:=8rt:=T"11 GE~ 0 UNTTL I=NR 00 -~ FEAq~3LE; TF" FICASI"'U::

T~F'" pFr.TN ~~1~:=*-Rlnl-l; 8lNR1:=l>-RI01-n.99; RI011=-I~ pnQ 1:=-1 c:.TEP 1 '1"1'IL 'IltJMC 00 IF lO..JI'"P[I1

THEN S"LIT):=LOWROlIl ELSE C;Ol{T1:=UPR £Tl-L"WR n IIl'

TF NEG[O! < 0 THFN F~OWARD<;TEP ELSE BAC~:=TPUE

F-,I')

Flc:.,," FnQ~APnsTEP:

IF XM

=

n TyEN RAC~:=TQUE

E"!O:

IF RACV THEN REGlN RAC~:=FALC;E; FIN:=BAC~STEP [NO:

E~ID: F-m 10; IP1?40 IP1250 lPPM 1°1270 IPli?RO 1P1290 [Pl300 IP₁₃₁₀ t01320 IPJ310 IP1140 IP1350 TOI3f-O IP}370 IP13RO IP)390 IP 1400-1P1410 tP₁₄₂₀ 10₁₄₃₀ IP1440 101450 1?14f1n 1P1470 I Pl4J<l} -1P₁₄₉₀ 101,,>no 1P₁₅₁₀ 1°1570 101530 101~40 IP1550 tPlS~O IP1510 1P-l 5Rt) 101590 IP11',00 101610 IP1620 101630 yt"H;40· IP1650 ID_161;0 10)670 lDl"''In 101690 1P1700