NATIONAL BUREAU OF STANDARDS REPORT

1 0 695

THE NATIONAL BUREAU OF STANDARDS'

LINEAR AND QUADRATIC PROGRAMMING

SUBROUTINES

Technical Report

to

Computer Services Division

Center for Computer Sciences and Technology

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

2053588 February, 1972

NBS REPORT

10 695

THE NATIONAL BUREAU OF STANDARDS’

LINEAR AND QUADRATIC PROGRAMMING

SUBROUTINES

W. G. Hall

R. H. F. Jackson

P. B. Saunders

Applied Mathematics Division

IMPORTANT NOTICE NATIONAL BUREAU OF STA

for use within the Government, f and review. For this reason, the whole or in part, is not authori; Bureau of Standards, Washingtor

the Report has been specifically p

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

; accounting documents intended

ubjected to additional evaluation

listing of this Report, either in Office of the Director, National

the Government agency for which

pies for its own use. <NBS>

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
.
ACKNOWLEDGEMENT

The authors would like to acknowledge the aid provided by

Richard Ku of the Technical Analysis Division; not only for proofing

the manuscript and providing numerous unusual test problems, but also

for playing the role of "typical user" on whom we tried our ideas.

ABSTRACT

This report documents one phase of an effort to provide users,

of the facility operated by the National Bureau of Standards* Computer Services Division, with reliable , well-tested , clearly-described

solution algorithms for selected frequently-arising classes of special mathematical problems. The report presents algorithms for the

simplex and revised simplex methods of linear programming, as well

as their adaptations to quadratic programming. Set up as subroutines,

the present versions of these codes use internal storage only, with

resultant limitations on the size of the problems which can be treated.

Key Words : Algorithms, linear programming, nonlinear programming,

quadratic programming.

CONTENTS

1; Introduction . . . . . . 1

2. The Simplex Subroutine 3

3. The Revised Simplex Subroutine 8

4. Solving Quadratic Programs 9

5. The Parameters 12

6. Output 22

7. Additional Work and Future Plans 30

8. References .............. .32

Appendix A: T.ie "Almost Linear" Kuhn-Tucker Conditions

for Quadratic Programming .33

Appendix B: Listing of RVSMPX ......... 36

Appendix C: Listing of SIMPLX . .50

Appendix D: Timing Considerations . .62
. - .

.• .

•

• •' .

• • .

'

*
- 1 -

1. INTRODUCTION

The "package" documented in this report consists of two subroutines : SIMPLX, which is an implementation of the simplex method

for linear programming, and RVSMPX, which is an implementation of the

revised simplex method for linear programming. In addition, both subroutines can solve some types of quadratic programming problems.

The primary goal during the development of this package was to

provide reliable user-oriented subroutines for solving linear programming problems. Other desirable program attributes, such as efficiency

with respect to core usage and time, and the extension of problem size,

were considered to be of less importance. An abundance of monitoring

prints is available; each is optional and may be printed or suppressed

independently of the other outputs. All output appears on unit 6;

there are no other references to peripheral devices within the subroutines. Since each program is a subroutine, there is a return to the

calling routine regardless of whether the exit is normal (a solution has

been found) or abnormal (the problem is infeasible, unbounded, ill-stated,

or numerically unstable). Hence the user mus

t

, in his calling routine,

test a status parameter upon return from the subroutine to determine the

cause of the return.

The calling statements for the subroutines are as similar as

possible. In all of these, A is the augmented constraint matrix, X

is the output vector, L contains the output switches, etc. In fact,

in the interests of standardization, the two subroutines were designed
- 2 -

to be as similar as possible with respect to variable names, errorhandling, and output messages, without sacrificing any speed or efficiency intrinsic to the respective algorithms.

It is due to this similarity between SIMPLX and RVSMPX, and to an

attempt at avoiding duplication, that the documentation here presented is in a somewhat "factored" form : sections 4, 5, and 6 contain

information that is common to both subroutines and would have been duplicated had it not been "factored out" into its own section.

It is well-known [7] that the revised simplex method is faster and

more efficient computationally than the original simplex method. The

main reason for the inclusion of the simplex method is that since its

computations are actually performed in the memory space occupied by the

A matrix (the A matrix is the simplex tableau) and there is no need to

store separately the inverse of the basis matrix, SIMPLX can solve somewhat larger problems than RVSMPX. This should be advantageous to the

user whose problem is slightly too large for RVSMPX but not large enough

to warrant the use of peripheral storage.

In the interests of completeness we mention here the rule used by

RVSMPX and SIMPLX to break ties for the variable to leave the basis.

The variable that is chosen to leave the basis when there is a tie is the

variable whose pivot element is largest in magnitude. Although this

rule does not guarantee the prevention of cycling, practical experience

thus far has shown it to be effective.

This report is intended as a user's manual for RVSMPX and SIMPLX.

The reader should have some familiarity with both linear and computer programming. Although some parts of the text do require more than a basic

familiarity, the user will normally have no need to understand these sections.
-3-

2. THE SIMPLEX SUBROUTINE (sIMPLX)

SIMPLX finds, using the simplex method for linear programming, the

maximum value of a linear objective function subject to a set of linear

constraints with non-negative variables and non-negative right-hand-sides . The problem is:

maximize

nZ

j=l

c x

J J

subject to: p

n

j=l

n

X < b . , i = 1, 2, » • • f 'C i l

X > b . > i = £ + 1 , t + 2

,

i l

X b . , i = l + g + 1, l +

i l

x . J > 0 , J = 1, 2, • • • j Q

b. > 0 , i = 1, 2, • • • ,m

£+ S + e

where n is the number of "real" (i. e. , original) variables; m , which

is equal to £ + g + e , is the total number of constraints; £ is the

number of contraints with <_ signs (LE constraints) ; g is the number

of constraints with >_ signs (GE constraints) ; e is the number of

constraints with = signs (EQ constraints) . Note the requirement that

all b > 0 . i ~

Within the SIMPLX subroutine, the constraint set is transformed

into a system of equations with an initial basic feasible solution

through the addition Q f slack variables for the LE constraints,

surplus variables for the GE constraints, and artificial variables

for the GE and EQ constraints. The subroutine then proceeds with

the "Two-Phase Method" for solving the problem.
-4-

During Phase I, an artificial objective function is constructed in

such a way that when simplex iterations have driven it to zero, the

basis contains no non-zero artificial variables , and is a basic feasible

solution to the original problem However, if the maximum of the artificial

objective function is not zero, there is no solution to the original problem, and that problem is declared infeasible. On the other hand, if at

the end of Phase I one or more artificial variables remain in the solution

set (basis) at a zero level, then the constraints associated with those

artificial variables are redundant (i.e„, they are not needed). These

variables will remain in the basis at a zero level throughout Phase II,

and will be members of the final solution. Note that this allows an

arbitrary relationship between the number of constraints and the number

of variables in the original problem. Phase I is not required if all the

constraints in the original systems are LE constraints, since the slack

variables alone provide an initial basic feasible solution in this case.

A few words need to be said about the redundancy indication mentioned above. At the "textbook" end of Phase I (all the indicators , the

"z - c ," non-positive), the NBS simplex and revised simplex subroutines ’ j J

go one step further. If at that point there are any artificial variables

in the basis at an "effectively" zero level (see section 5.9), an attempt

is made to remove them by replacing them with any non-basic, non-artificial

variable whose indicator is zero. Any artificial variable removed in this

way was in the basis as a result of degeneracy, and any artificial

variable that remains in the basis at a zero level after this procedure,

is present as a result of redundancy.
-5-

Pnase II consists of simplex iterations whose task is to maximize

the real (original) objective function. If no maximum exists, the

problem is declared unbounded.

SIMPLX consists of approximately 400 FORTRAN V statements that

compile into slightly less than 2,000 computer words. Data storage

requires approximately mn-K5m+5n words. With a small main program

on the NBS UNIVAC 1108, under the EXEC II operating system, mn may be

as large as 44,000.

The calling statement is as follows

CALL SIMPLX (A,MA,MT,NT,L,X,T0LP,KQP).

For an explanation of the parameters in the calling statement and

a discussion of their meaning on entering and exiting from the subroutine, see Section 5.

For the purpose of illustration, and to give the user an idea of

what the output from a run will look like, the output from an example

problem is included here. The problem is:

subject to :

1.5x.

2

+ 5 *3 +2 XL4

3x^ + 2x

2 + X

3 + 4x

2x

i + X

2 + 5x3

4 X

2 x

i + 6 x

2

- 4x

3 + 8x

x

i

9 X

2

9 X3, X

4 - w

4

4 ~

4’ >

The A matrix was set up as follows
The other

MA=100

MT=5

NT=5

3 2 1 4 6

2 1 5 1 4

2 6 -4 8 0

1 1.5 5 2 0

0 0 0 0 0

input parameters were:

L(l)=2 L(4) through L(14)=1

L(2)=0 T$LP=0

L(3)=0 K QP = 0

The following page contains the output from this run
PHASE 1

UERAIiON

1.

PIVOTS

6.00000

OR

JEETT

ME

FUMCTIOM=

.00000000

X(

4)

P-NTFi>ED

TH^

RASTS»

A(

1)

L^FT

-Q

A -7-

ii

C\J

LT n

ij <r< 3 3 o o o o 3 3 3 O • •3

II:}

U>—i

l/l < X X X X X *****• ******X***X* • c * IT O * Z C * c c * *-> c * K O X < O * cr c * L.' • * i- * >-h x* — II ** z * c x L_ *-> * a i— * Lu <_> * 5 z * T * UJ IL * X ¥ L Li * X > X I- I—I * t- * u * LJ * ~> * cc x o *¥ C _J * o c * O li, * C K * O * c * c. * o • * • .H X*** z ¥ C * 1-1 * III- * < * z or * c u, * i-i i— * i— • * u * z i- * o <r * u x¥ LU * >

t-l c ¥ U- c *¥¥***X**¥¥¥¥¥¥**¥¥¥¥*X*X**XX*X*X¥¥¥¥X*¥¥¥¥¥*¥***¥¥X**XXxXXxxXXXXX¥¥XX*XXXXXXx X (J 3 X X LJ O X X “3 • X x rr cr x x c x X X X X X l/l X x «r x x • x X —I X x i- x X lx! C * X J) > X X <1 — ¥ X X CL X X Q_ X X 3 X X U_ Z> X X O 5 X X t-l X X O Z X X Z >-> X x Lxi x X X * X X X X X x x X X x x x x X X X X X x x x X X X X X X X X X X ccc3oZ

IICM 3oc3o

nr<>

II 3 3oo33•3 u—» n

=1 m X

IT

I—I

l/l ccLuX

Cl

LJa

li. I—z

Ll.

fO

vO m

sO

K)3

tnXMcozX

Llluo

IdX

crc

crco

in

in

il i—o>

HH

0.

rf QC

Id

I-M

Ul inrrx

iT rOn

iOK'

II3X

fO

CM

"\l fmd

_imc

-H l| TC < tO

CM33O

<-¥ X m

ra

x * * x

x

LJ

—— X X X X X X

EMl)

OF

PHASE

2.

03

JECTI

V/E

FUNCTION =

4.3636363

TH^RE

WERE 1I

TFR

ATT

ONS

MIHIMLH

PIVOT

WAS

5.

5000

AT

ITERATION

1.
- 8-

3. THE REVISED SIMPLEX SUBROUTINE (RVSMPX)

The formulation of the problem, restrictions pertaining thereto,

and definitions of the variable* are the same here as that in section 2

for SIMPLX. The method of solution is, of course, different. The

method used is the revised simplex method as presented in chapter 3 of

[3], with one important modification. The modification is the provision

for re-inverting the basis matrix every [m/2]+5 iterations, in order to

reduce round-off error. (The value [m/2]+5 is one that appears in other

codes, see[l], and has been shown in practice to be a reasonably good

choice. However, if the user feels another value is more appropriate, he

may use his value by changing the value of INVC on card number 6 of the

subroutine.) A discussion of the advantages of re-inverting the basis

matrix appears in section 7-8 of [4]. This provision also allows for

the '’restart” or "advanced start" capability whereby the user may choose

to supply the subroutine with an initial basic feasible solution, if known,

in order to speed up the algorithm. For instructions on how to use this

advanced start capability, see section 5.10.

RVSMPX consists of approximately 450 FORTRAN V statements that

compile into slightly more than 2,000 computer words. Data storage is

2 approximately mn + m + 2n + 3m computer words.

The calling statement is

:

CALL RVSMPX (A,MA ,B , MB , MT, NT, L, X, T0LP, INV, KQP)

.

For an explanation of the parameters in this calling statement and a description of their values on entering and exiting from the subroutines, see section 5.

The output messages that may appear are described in section 6.
-9-

4. SOLVING QUADRATIC PROGRAMS.

As was mentioned in the introduction, both RVSMPX and SIMPLX

have the capability to solve quadratic programming problems. This

is accomplished via the parameter KQP, discussed below in section 5.

A quadratic program is of the following form:

maximize : n n n E c .x . + E £ d..xx.

j=l J J i=i j=i 1J 1 J

subject to the same set of constraints that appears in section 2.

There is, however, one important additional restriction—the objective

function above must be concave . If it is not concave, although a

feasible solution to the quadratic program may be found, there is no

guarantee that this solution will be optimal.

The objective function above is concave if

n n E E d x .x

.

i=l j=l 1 J 1 J

for all x , where i=l,2,...,n. This is equivalent to saying that the

i

matrix of the d is negative semi-definite. The quadratic form above

ij

may be shown to be concave by showing that it can be written as the

negative of a sum of linear forms,

n n 2

E E

i=l j=l

d . .x .x . ij i J “ELL (x) ]

. k k

in many applications (e.g., least squares estimation) this is known

a priori. Equivalently, one could attempt to diagonalize the matrix

of the d and check the resulting diagonal elements (the eigenvalues

ij

of the d matrix). If all are non-positive, the matrix is negative

ij

semi-definite. See chapter 8 of [2] for more on this.
- 10-

The method of solution that is used was originally developed

by Wolfe [9], He noted that the Kuhn-Tucker conditions (see Appendix

A and [6]) for the quadratic programming problem are "almost linear ,

and that, under the concavity assumption mentioned above, they are necessary

and sufficient —which means that if one can find a set of , j=l,2.*.n,

that satisfy the Kuhn-Tucker conditions, then those x^ are the optimal x^

for the original problem. Thus, the emphasis was shifted to solving a set of

"almost linear" equations. This was accomplished in [9] by utilizing

Phase I of the simplex method for linear programming with one modification.

That modification is that certain pairs of variables cannot simultaneously

be in the basis; it is implemented by restricting the choice of the

entering basic variable. This approach is used in both RVSMPX and

S IMPLX

.

All that is necessary to solve a quadratic program is to set

KQP to n, where n is as defined in this section, and to set up the

constraint matrix as follows:

r —1 A

1

0 0 0 0 0

A 0 0 0 0 0 b

2

A 0 0 0 0 0

3

-2D a

t

1

a

t

3

-a

t

3

-I n

c

0 0 0 0 0 0

*

* * * * * * *

i

.
- 11 -

where the original constraint matrix has been partitioned into

A , A0 12 and A , consisting of the LE, GE, and EQ constraint coefficients

3

T

respectively* A indicates the transpose of the matrix A, D is

the matrix of the d , and I is an nxn identity matrix. The

i 3 n solution to the quadratic programming problem will be found in X(l)

through X(N) where N is the number of real variables in the original

problem.

For a more rigorous development of the algorithm and an explanation

of some of the information that might be gleaned from the solution,

see appendix A.
- 12 -

5. THE PARAMETERS.

Listed below are the parameters that appear in either one or

bota of the subroutines. After the name of each variable appears

an (R) if the variable is a parameter of RVSMPX; an (S) if the

variable is a parameter of SIMPLX; and an (RS) if the variable is a parameter of both RVSMPX and SIMPLX. All variables conform to

the FORTRAN implicit naming conventions regarding mode.

1. A (RS)

A is the constraint matrix augmented by a right-hand-side

column, an objective function row, and in the case of SIMPLX,

a row to be used (by the subroutine) for the coefficients of the

artificial objective function. The first £ rows of A contain the

LE constraint coefficients, thfe next g rows contain the GE constraint coefficients, and t:e next e rows contain the EQ constraint

coefficients. Row m+1 contains the coefficients of the real objective

function; row m+2 , whose values are not supplied by the user, contains

tne artificial objective function coefficients for SIMPLX, and column

n+1 contains the non-negative right-hand-sides. Hence, A must be

dimensioned at least (mfl)x(n+l) for RVSMPX and (m-f-2 ) x (n+1 ) for SIMPLX.

Pictorially, where ,, *n indicates a value that is not supplied by the

user, A is

:
-13-

a a b

11 12 In 1

a a b

1+1,1 1+1,2 J+l j n l+l

a

• a

0 • a b

-t+g+i

1 -f+g+1 , 2 o • • £+g+l , n 1+g+1

a

• a

o • a b

ml m2 mn m

c c • o c *

1 2 n

* * * *

We note, again, that the last row is not needed for RVSMPX.

Since the A matrix is used by SIMPLX as explicit storage for the

condensed simplex tableau (see pp. 119 of [8]), the original values

in the A matrix will have been completely destroyed by the time

SIMPLX returns to the main program. In most cases there is nothing

in the A matrix of value to the user at this point. For RVSMPX,

however, the A matrix is completely unchanged during the course of

the algorithm.

2. MA (RS)

This variable contains the value of the first dimension of A,

as A is dimensioned in the calling routine; e.g., if A is

dimensioned as A(20,50), then MA should be set to 20 in the

main program before the subroutine is called.

This parameter is unchangetf by either of the subroutines.
-14-

. 3. B (R)

This parameter is a matrix whose dimensions must be at least

(m+2)x(m+2) . Upon exiting from RVSMPX, B will contain the inverse

of the matrix of the current basic columns in ( (B (I , J) , J=1 , M) , 1=1 , M)

;

the negatives of the current values of the dual variables in

(B (M-f-1 , J) , J=1 , M) ; the current values of the basic variables in

(B (I , M-f-2 ) , 1=1 , M) , in an order specified by a portion of the L vector

to be explained later; the negative of the value of the real objective

function in B (M+l , M+-2 ) ; and the negative of the value of the artificial

objective function in B (Mf2 , Mff-2 ) .

4. MB (R)

This contains the value of the first dimension of the B matrix

as it is dimensioned in the calling routine 0 It too is unchanged by the

subroutine

•

5. MT (RS)

MT is the number of rows of information in the A matrix, which

is m-f-2 for SIMPLX and m+1 for RVSMPX. Note that MT is less than or

equal to MA. This variable is not changed by the subroutines.

6. NT (RS)

The value of this variable is the number of columns in A, i.e.,

n+1 . It too is unchanged by SIMPLX or RVSMPX.

7. L (RS)

L is a multipurpose vector that provides storage space for a

number of distinct working vectors constructed by the subroutines,

in addition to providing indicators and constants for both entry and

exit

.

It also contains output switches which are used to determine how
-15-

much (if any) output from the subroutines is given . For RVSMPX, L

must be dimensioned at least 14+2m+n+g; for SIMPLX, 14+2(m+n)+g. The

meaning of L(l) through L(14) is the same for both subroutines for both

entry and exit; the differences in the use of the L vector by RVSMPLX

and SIMPLX occur beyond L(14). Although the typical user need not

understand what happens to this latter portion of the L vector, a description is included here for the user who may wish to modify the

subroutines

L(l) : For entry, this must contain the number of LE constraints,

oi L* The exit value of L(l) is the number of non-zero elements in

the original A matrix that are between (-EPS) and EPS and are,

therefore, considered by the subroutines to be equal to zero [see 9

below]. If the exit value of L(l) is positive, the user is providing

the subroutine with information it does not use. If the exit value

of L(l) is large, it is suggested that the user either force the subroutine to use the enumerated small matrix entries by making EPS smaller

(see 9 below), or else not provide these numbers (set them to zero). If

one of these actions is not taken, numerical problems might arise.

L(2) : This should contain the number of GE constraints, or g 5 when

entering the subroutine. The exit value of this variable is the total

number of iterations the subroutine performed before termination. It

includes the iterations for both phases.

L(3) : When entering the subroutines, this is a default print switch.

If L(3) is non-zero, the subroutine sets L(4), L(5) , L(8) , L(12), L(13)

and L(14) to 1, and L(6), L(7), L(9), L(10), and L(ll) to 0. If L(3) is

zero, the user sets L(4) through L(14) to obtain the desired outputs. When

leaving the subroutine, the value of L(3) is very important — it indicates

what caused the termination. A value of 0 indicates an
-16-

optimal basic feasible solution has been obtained; 1 indicates that an

optimal basic feasible solution has been obtained but that numerical

difficulties <$rose during the course of tie algorithm, and that the user

should question t.ie results; 2 indicates that the problem is infeasible;

3 indicates the problem is unbounded; and 4 indicates a system error.

1.(4) : If this parameter is non-zero when the subroutines are

entered, the values of EPS, CEPS, and L(l) are printed. The value of

L (4) remains unchanged by the subroutine.

L(5) : For entry, if L(5) is non-zero, a warning message will be

printed if necessary. This variable, too, is untouched by the subroutine .

L(6) : If L(6) is greater than zero when entering RVSMPX or SIMPLX,

the iteration summary will be printed after every L(6) iterations in

Phase I. Neither of the subroutines change^ the value of L(6)„

L(7) ; If L(7) is greater than zero when entering the subroutines,

t ,e basic variables and their values will be printed after every L(7)

iterations in Phase I. Neither RVSMPX nor SIMPLX changes this variable.

1(8) : If this switch is non-zero when the subroutines are entered,

the final summary for Phase I will be printed. No change occurs in the

value of this switch.

L(9) : This switcn, if non-zero when entry occurs, will cause the

basic variables and their values to be printed at the end of Phase I.

Again, no change occurs in the value of this variable.
-17-

L(10) through L(13) : These variables perform the same task as

L(6) through L(9), except that they refer to Phase II 0 L(14) : When entering the subroutine, if L(14) is non-zero, error

messages will be printed if errors occur. No change in the value of

L(14) occurs.

Values for the remaining portions of the L vector are not supplied

by the user, and, as was mentioned earier, tne exit values of these

variables will probably not be of any importance to the casual user

of these subroutines 0 Nevertheless, in the interests of completeness,

their descriptions are provided here.

RVSMPX and SIMPLX use this storage space differently. The description

of the manner in which RVSMPX uses this space is provided first. To

facilitate the discussion, let

M = m

N = n MG = g

L10FF =14

L20FF = L10FF + n + £ + g + g + e = 14 + n + m+ g

Then, for J = 1 through N + M 4- MG, if L(L10FF + J) equals 0, X(J) is

a real variable; if 1, X(J) is a slack variable; if 2, X(J) is a surplus variable; if 3, X(J) is an artifical variable; if 4, X(J) is an

artificial variable that has been removed from the basis, and, therefore,

from future consideration (see p. 119 of [4]); if 5, X(J) is an artifical

variable that is in the basis at a zero level and cannot be removed. For

"t h

J = 1 through M, if L(L20FF + J) = K, X(K) isthe J basic variable, and

its value appears in B(J,M + 2).
- 18 -

To facilitate the discussion of the manner in which SIMPLX uses

this space, let

L10FF = 14

L20FF = L10FF + n = 14 + n,

L30FF = L20FF +m= 14+n+m

IN0FF = L30FF + g = 14 + n+ m + g,

N0T0FF = IN0FF + m = 14 + n + m+ g + m

L10 = the number of columns in the A matrix after the artificial

variables have been removed by the subroutine,

L20 = the number of rows in the A matrix after redundant constrainhave been implicitly removed by the subroutine.

Then, for J = 1 through L10, if L(L10FF + J) = K, the J

^ column of the

th

current A matrix is the K column of the original A matrix. Also,

for J = 1 through L20, if L(L20FF + J) = K, the row of the current

th

A matrix is the K row of the original A matrix. Since the surplus

and artificial variables for each GE constraint are never needed at the

same time, L(L30FF + J) for J = 1 through MG is set up as follows:

if L(L30FF + J) equals 1, the artificial variable for the GE constraint is still in the basis; if L(L30FF + J) equals 0, the artificial

th

variable for the J GE constraint has been removed from the basis,

and the storage space previously occupied by it is occupied by the surplus

variable for that constraint. For J = 1 through M, if L(IN0FF + J) = K,

"t h. the J basic variable is X(K). Finally, for J = 1 through N, if

L(N0T0FF + J) = K, the J non-basic variable is X(K).

8. X (RS)

This vector is the solution vector. Its dimension should be at

least n + ^+ g-fg+e+1, or equivalently, n + m + g + 1. For

the SIMPLX subroutines, the values in the X vector when entry occurs

have no effect on the subroutine, since SIMPLX initializes X to zero.

For RVSMPX, however, the initial values in the X vector are important

when the advanced start option (see 10 below) is used, and unimportant

otherwise

.
-19-

When termination occurs, X contains the terminal values of the

variables in the original problem, including any slack, surplus, or

artificial variables that were added by the subroutine. The first n elements of X contain the values of x through x . Next in order are

1 n the slack variables for the LE constraints, then the surplus variables for

the GE constraints, the artificial variables for the GE constraints, the

artificial variables for the EQ constraints, and finally, in

X(N 4- ML + 2* MG + ME + 1) , the value of the objective function when

termination occurs.

The scheme by which values are stored in X is as follows: for

each x that is not in the basis, X(I) = -EPS/100; hence, for each

i

I for which X(I) is non-negative, x. is in the basis at its given

level. This scheme was employed so as to permit both easy identification of the basic variables and immediate use of the solution vector

at the same time, with no undue round-off error.

X will contain these values according to the scheme noted above

regardless of the reason for termination. Of course, if termination

is with L(3) greater than 1, all the values in X will be the current

ones and not necessarily optimal or even feasible.

9. T0LP (RS)

This is a parameter which is used in the construction of epsilon (EPS)

and capital epsilon (CEPS), variables that are used as tolerance

parameters. Within the subroutines, [a|<EPS is equivalent to a=0.

The user has the following options regarding the value of EPS:

I T0LP I .Z • I a. .1 If T0LP is less than zero, EPS = 1 .

1

> J 1 1 >

1 m * n
-20-

10

_5

.S.|a. ,|

If T0LP equals 0, EPS = ^

.

m * n

If T0LP is greater than 0, EPS = T0LP.

Tests have shown the default choice (T0LP=O) to be reasonably effective. The word "reasonably" is used here since there are cases when

this choice is inappropriate. It is for this reason that the two other

choices for T0LP and EPS have been provided. Furthermore, to aid the

user in determining when an inappropriate value of EPS has been selected, warning messages have been provided (see section 6) that indicate

that numerical problems have arisen. This may possibly be overcome

by changing the value of EPS.

In any case, CEPS is set to 10*EPS and is used solely to determine when a number is "close to zero".

The value of T0LP is not changed within either of the subroutines.

iO. INY (R)

This is a switch that allows the user to provide RVSMPX with an

initial basic feasible solution. If INV is non-zero, an initial basic

feasible solution will be expected. If INV equals zero, the subroutine

will start from scratch with Phase I.

The initial basis is passed via the X vector as follows: for i=l

through n+^+g+g-fe, if x is a member of the basis, then

X(I) is greater than or equal to zero; if x. is not a member of the

basis, then X(I) is less than zero. The actual values are irrelevent

so long as the signs are correct. RVSMPX will form a basis with these

variables and proceed with Phase I or Phase II, depending on whether

or not the basis contains any artificial variables.
-21-

When RVSMPX terminates, INV may have a value of 0,1, or 2.

In any event, its value is irrelevant.

11. KQP (RS)

This is a switch that allows the user to solve quadratic programming problems. If KQP is zero, the subroutines will consider the problem being solved to be a linear programming problem. In order to solve

a quadratic programming problem, KQP must be set to n, the number of

real variables in the original quadratic programming problem. For a discussion of the quadratic programming algorithm that is used, the

method of setting up the problem, and the interpretation of the results, see section 4.

The value of this switch remains unchanged throughout the subroutine •
-22 -

6. OUTPUT

As indicated in section 5.7, the output messages are controlled

(with one exception) by the values of L(3) through L(14), which are

parameters in the subroutine call. This section dicusses the output messages that may result from the use of RVSMPX or SIMPLX. Since

there are a few messages that are indigenous to one or the other of the

two subroutines, the same notational scheme that was used in section 5

will be used here; i.e., if the message can result from RVSMPX, an (R)

will appear after the name of the message; if the message can result

from SIMPLX, an (S) will appear after the name of the message; and if

the message can result from either one, an (RS) will appear.

1 „ The Negative Right Hand Side Message (RS)

.

AT l c. A £» T ONEe element OF THE right nano side column IS LESS than ZFRO. subroutine terminates.

This message is exceptional in that it is not under control of

any switch in the L vector. Immediately upon entry, the subroutines

examine the problem data. If any entry in column NT of matrix A

(i.e., the right hand sides) is negative, this message is printed,

the value of L(3) becomes 2, and the subroutine returns to the calling

routine. The problem should be reformulated so that all right hand

sides are non-negative.

2 . The Epsilon Print (RS) .

EPSIlON = .-964732-02 CAPITAw EPSILON = . 964732-01 0 N0N-2EP0 ENTRIES ARE EFFECTIVELY E9UAL T3
-23-

This message is printed whenever L(4) is non-zero. The values of

epsilon and capital epsilon are based on T0LP, a parameter in the

calling sequence (see section 5.9). Major ways in which these values

are used are discussed in 5, 9, and 11 below, and in the discussion of

L(l) in section 5.7, The number of non-zero entries in the A matrix

that are "ef f ectively" equal to zero is stored in L(l). For a discussion of this value see section 5.7.

3 . The "infeasible Initial Basis'* Message (R)

..**, ..  t.t ** ********************* m** ********** *******************************

* * EhRjH - V A.X I AjlED SPECIFIED AS COMPRISING «N INITIAL SOLUTION DO NOT FORM A BASIC FEASIBLE SOLUTION. *

* *

* The; ..;.s detected at iteration u or phase 1. at that time the objective function value was -21.000000 *

* A-Jj I-ii FOLi-O U !3 VARIABLES WERE BASIC. *

* *

*. ** ********* *...***«*****»*******«************.******************************************************

dM 3 I C 1 ‘a H i A 15 1

x< 6)- 2.0UGOPU X< (5) = 1 .OQ0O00 x( 10)= 3.000000 X< 12) = 2.000000 X( 14)= 6.000000

5< d)~ . •j.ou'iooo

A ( ->) = ll.OU'iOOO A ( L) = 10.000000

If the advanced start feature is used (INV non-zero), and the

variables specified in the X vector as forming a basis do not form a

feasible basis, this message is printed, (if L(14) is non-zero), L(3)

is set to 2, and RVSMPX returns to the calling routine. The basic

variables that are printed are the variables that had been put into

the basis before the error was detected.

There are two ways in which this error may occur:

(i) The number of non-negative variables passed in the X vector is

not equal to m. However, in some cases when the number is greater than

m, no error will occur. That is, if the m variables that are the

first to enter the basis d_o form a feasible basis, then they are used

as the basis and no error is recorded.
-24-

(ii) The number of non-negative variables passed in the X vector is

equal to m, but these variables do not form a feasible basis. In this

case, at least one of the m basic variables printed will be negative.

(Note: if a case (i) error occurs, there may be negative basic

variables. There is no cause for alarm; the basis forming procedure

(actually a matrix inversion) allows basic variables to become negative

during its course since, if the variables do form a feasible basis all

the variables will be non-negative at the end of this procedure.)

4 . The "Small Pivot" Message (RS) :

* • WARNING * * SMALL PI VOT . ELEMENT AT ITERATION 10 OF PHASE 1. PIVOT = . 1 23456A-06

This message is printed if the chosen pivot element is between EPS

and CEPS and if L(5) is non-zero. This is considered a warning message;

computations continue, but the user is strongly advised to view the

results critically.

5 . The alternate Pivot Element Message (S) :

* * warning * » IN PHASE 1* THERE HAVE BEEN 10 ATTEMPTS TO find an ALTERNATE PIVOT ELEMENT.

OF THESE* 6 FOUND NO ALTERNATE ELEMENT GREATER THAN CAPITAL EPSILON.

This warning message requires a rather detailed explanation. If the

pivot element that is chosen using the standard simplex criterion is

"close to zero" (between EPS and CEPS) , SIMPLX goes into the alternate

pivot selection routine. This routine scans the A matrix in
-25-

ascending order of the column number, checking the non-basic variables

(including the one chosen by the standard simplex criterion, if necessary)

for one which has a pivot element greater than CEPS. If one is found,

it is used (without scanning the remaining columns) in lieu of the one

found by the standard simplex criterion. If none greater than CEPS is

found, the largest one is kept. (This may be the one originally found

by standard simplex.) If that one is between EPS and CEPS, it is used,

and if L(5) is non-zero, the "small pivot" message (number 4 above) is

printed. If the largest is less than EPS, the subroutine returns to the

column that contained the standard simplex pivot, and sets that pivot

element to zero. That column is then scanned and quotients are again formed

to get a new minimum quotient, thus determining a new pivot candidate.

If this new pivot candidate is acceptable (greater than EPS), Simplex

continues and performs the operation. If it is not acceptable, it too

is set to zero and this latter process is repeated until either an acceptable pivot element is found or until each element in that column is nonpositive and the problem is declared unbounded.

As for the warning message itself, it appears at the end of a phase

and gives the number of times the subroutines left the standard simplex

pivot and went to the alternate pivot selection routine, and the number

of times it returned to the standard simplex pivot from the alternate

routine without having found an acceptable pivot.

6 . The Iteration Summary (RS) :

1 ITERATION 16. PIVOTi 1U.7H80 OBJECTIVE FU\CTIOM= .27186017-04 S< 4) ENT? RED THE PASIF» A< 16) LEFT.
-26-

4" Vi This message is printed after every L( 6 )

l iteration in Phase I

t h if L( 6 ) is positive, and after every L(10) iteration in Phase II if

L(10) is positive,. It is useful in tracing the progress of the algorithm.

7. The Basis Print (RS)„

„ i C.

X ( X( 1)-

o ) - ,&d '061

. 731610

x(x( i)- .367047

7)= .607440

X ( x< a

l* ii

ii .002276

.604561

X( 4) =

X ( Q) = ,0110”

.605376

X ( 5 ) = X( 10)=

• 6°6°60

,1?Q69">

SI 1 ) = b.93iP5o S< 21= .234020 S( 3) = 13.179356 S< 4 ) = .onnoon

A ( L> ) - . OO'.lflflU A ( 7)= .iiounoo. A ( 9) = .000000 At 10)= .oonooo At 15)= .oonooo'

At . h) = .uonniiM

This print occurs after every

th

L(7) iteration in Phase I if L(7) is

positive, after every L(ll) iteration in Phase II if L(ll) is positive,

at the end of Phase I if L(9) is non-zero, and at the end of Phase II if

L(13) is non-zero. It is also printed if the problem is determined to be

infeasible or unbounded and L(14) is non-zero.

This print simply gives the indices and values of the basic variables.

The letter X refers to real variables with indices between 1 and n. The

letter S. refers to slack or surplus variables with indices between 1 and

^ 4- g # The letter A refers to artifical variables, with indices

between 1 and g + e.

8 . The End-of-Phase Summary (RS) .

•^•.***»f t** *****4**<r^***»*«>*************«’  ** ********** ***********************************************

** E'O of phage i. objective -unction = . 27136017-04 tht re were 17 iterations.

Mli'U'VJ'l PIVOT tv AS .20277 AT ITERATION 12. REAL 00JECTIV r FUNCTION = -24.310430

*****.**•**»*. »***«,**#*» *****************«**********<'**•************ + ****************************•*****
-27-

This message is printed at the end of Phase I if L(8) is non-zero,

and at the end of Phase II if L(12) is non-zero, The phrase "REAL

OBJECTIVE FUNCTION =" is printed only at the end of Phase I, since

at that point the objective function is the artificial objective function.

9 . The Infeasible Message (RS).

* I-^n

’ ' 3,JSTRAIVTS ASSOCIATE »TTH THE ARTIFICIAL VARlAPLrr B r * IF L P* ipr t„iC 0NC T c T r,T .0., APr^H. MUSICAL J1PFICJLTIFS HAVE n

* Ij

E£M FNCOUNT^. THE LARGEST E“TRV IN THE najFCTlU r.l^tJoN RnJ "D* Jc?iH “ *

* It °

r °HASE *• AT THAT TIME T,E or,JECTTV E function value -A, -21.0000

* 00

This is printed when the problem is infeasible and L(14) is non-zero. A

problem is said to be infeasible when it is impossible to satisfy one or more constraints. It is detected in the subroutine when all the indicator variables (the z . - c ) are less than EPS and either one or both of

J J the following is true: The value of the artificial objective function

is greater than (g + e)*CEPS, or one or more artificial variables

which cannot be made non-basic have value greater than CEPS.

If the maximum value in the objective function row (the indicators)

is "close to " EPS and/or the objective function is "close to" CEPS

and/or the artificial variables in the basis are "close to" CEPS, then

the infeasibility may result from numerical instability. In this case,

it may be possible to "achieve" feasibility by changing the value of

EPS (see section 5.9). It should be noted, however, that a different

value of EPS could cause the algorithm to follow a different path toward the solution. Thus it is not obvious what the direction and

magnitude of the change should be.

If this message is printed, it is always followed by a print of the
-28-

variables which are basic at the time the error was detected.

Whether or not this message is printed, L(3) is set to 2 and the

subroutine returns to the calling routine.

10. The M Re-inversion Infeasibility " Message „ ••••‘-m1 *-********************************* ***************************** *****************

ERRuR - T He. PRO.iLEM IS INFEASIBLE* I NFEASIBILITY INDICATED DURING RE-INVERSION OF THE BASTS MATRIX,

THE ERROR .*nS jETECTED AT ITERATION 10 OF PHASE l. AT THAT TIME THE OBJECTIVE FUNCTION VALUE WAS -21,000000

And TH w FOLLOWi IG VARIABLE:* »JERE BASIC. ****.***************************•******+****************************************************************.*

This message is printed, when L(14) is non-zero, if one or more

basic variables became negative as a result of re-inverting the basis

matrix. This error, in every case, is a result of numerical difficulties

and is an indication that the problem, as it is posed, is just not

numerically tractable. To overcome this problem, on might try changing

the value of EPS (see section 5.9) or even INVC (see section 3); however, it is suggested that the user modify his problem to make it more

numerically stable. Techniques such as normalization, scaling values,

removing redundancies, etc. can be used to accomplish this.

11 . The "Unbounded " Message (RS).

****** ************** ************************************************************************** ********** ****************

* E3R0R - THE PROBLEM is UNBOUNDED. THE VARIABLE X( 13) CAN ASSUME AN ARBITRARILY LARGE VALUE* THEREBY YIELDING

* A si ARBITRARILY LARGE VALUE OF THE Q3UECTIVE FUNCTION.

* The ERROR aAS DETECTED at ITERATION 4 of PHASE 2. AT THAT TIMr the OBJECTIVE -UNCTION value WAS 43138. 5*9

* And THE FOLLOWING VARIABLES a'ERE 3 ASIC

.

***«*-* ***.*************.********+*****************************************************

*

*

*

**

• * *************** *******************

This error message is printed, when L(14) is non^zero, if the

problem is found to be unbounded. A problem is unbounded when at least

one variable, together with the objective function, can assume an

arbitrarily large value without violating any of the constraints.

Unboundedness is detected in the subroutines when all the entries in the

column associated with the variable that is about to enter the basis are
less than EPS. It is this variable that is identified as "causing”

the unboundedness

In some problems (those that are numerically unstable), this situation

might be alleviated by changing the value of EPS (see section 5.9). However, as was mentioned in 9 above, the direction and magnitude of the

required change are not obvious.

If this message is printed, it is always followed by a print of the

variables that were in the basis at the time the error was detected.

Whether or not this message is printed, L(3) is set to 3 and the

subroutine returns to the calling routine.

12 . The "Computational Inconsistency” Message (RS )

„

************************* *************** > + •***¥********* + ******************+  + + +

tl.Ju - COMPUTATIONAL INCONSISTENCY INDICATED AT THE END 0E PHASE 1. TH^ ALGORITHM WILL CONTINUE- WITH PHAS‘D », UJT ThL UslH I S ADVISLU TO CKlTICIZE THE RESULTS. ** ****.**.****** *****************.***************. *********************************. .*******.*********. .**

If L(14) is non-zero, this message is printed at the end of Phase I

if one or more artificial variables which cannot be removed from the

basis is between EPS and CEPS, or if the artificial objective function

is between EPS and CEPS. This situation is an indication of possible

numerical difficulties. The user is advised at least to check the

solution for feasibility
-30-

7. ADDITIONAL WORK AND FUTURE PLANS

The subroutines described in this report are part of a longer-term

effort, in which the goal of the Applied Mathematics Division’s Operations

Research Section is to provide reliable user-oriented solution algorithms

for those mathematical optimization problems which are of particular

importance for operations-research applications (they often arise in

other contexts as well). In addition to SIMPLX and RVSMPX, we also have

operational on the NBS computer:

(a) a 0-1 integer programming routine acquired from the RAND

Corporation

;

(b) a subroutine to solve transportation problems, obtained from

the Carnegie-Mellon University;

(c) a prototype code implementing the Dantzig-Wolf decomposition

principle for linear programming;

(d) a quadratic programming algorithm, limited to ’’separable"

objective functions ( no cross-products of variables), that solves larger

problems than SIMPLX or RVSMPX,

(e) a univariate dynamic programming algorithm, developed at

Johns Hopkins University and supported by the National Bureau of Standards

under Contract CST-1279,

(f) a multivariate dynamic programming algorithm, obtained from

Johns Hopkins University , and

(g) several algorithms for finding shortest paths between pairs

of nodes in networks
-31-

Except perhaps for (e) and (g) , these items have not been tested

or documented to the standards set for SIMPLX and RVSMPXo Future

work, in addition to such "completion” efforts for selected items from

the above list, may include:

(A) testing of the new UNIVAC mathematical programming package

(Functional Mathematical Programming System),

(B) as appropriate, extension of preceding items (quite likely

beginning with RVSMPX) to handle much larger problems through use of

peripheral storage,

(C) provision of capability for parametric and sensitivity

analyses

(D) extension of (d), above, to more general separable nonlinear

problems, and

(E) provision of a mixed-integer programming capability.
- 32 -

8. REFERENCES

1. Clasen, R. "Using Linear Programming as a Simplex Subroutine,"

Rand Report P -3267, Nov . 1965.

2. Finkbeiner, D.T., Introduction to Matrices and Linear Transformations , San Francisco, W.H. Freeman, I960.

3. Garvin, W.W., Introduction to Linear Programming , New York, McGraw-Hill,

1960.

4. Hadley, G., Linear Programming, Reading, Mass., Addison-Wes ley , 1962.

5. Hadley, G., Nonlinear and Dynamic Programming , Reading, Mass., AddisonWesley, 1964.

6. Kuhn, H.W. and A.W. Tucker, "Nonlinear Programming", in Proceedings

of the Second Berkeley Symposium on Mathematical Statistics and Probability , Jo Neyman, ed . , Berkeley, Cal., University of Cal. Press,

1951, pp. 481-492.

7. Wagner, H.M., "A Comparison of the Original and Revised Simplex Methods,

Operations Research , 5(3), 1957.

8. Wagner, H.M., Principles of Operations Research , Englewood Cliffs, N.J.,

Prentice Hall, 1969.

9. Wolfe,P., "The Simplex Method for Quadratic Programming," Econometrica ,

27, 1959, pp. 382-298
-33-

APPENDIX A: THE ’’ALMOST LINEAR" KUHN-TUCKER

CONDITIONS FOR QUADRATIC PROGRAMMING

One form (see section 6-3 of [ 4 ]) of the Kuhn-Tucker necessary

conditions for the quadratic programming problem given in section 4

to have a solution at x (j = 1 through n) is

j

n

lo c

j

+ 2

kil

d jk

X

k ~±=i \

a

ij = °’ if X

j

> °» for J = 1 through n

n m

c

j

+ 2

kild

jk

X

k "ill

>

ia

i i -

°* if X

1 = 0, for j = 1 through n

2 . n

I! a . .x . j=l

^ J = V if 0, i = 1

n

.IL a x

J =1 ij j

if h

= 0, i = 1

nHa x j=l ij j •H

•O II if A. <

1

0, “0 11 •H

n

-.a. x. > b., if X. = 0, ± = l

J=1 iJ J “ 1 1 Ha x = b , j=l ij j i

i = l

through JL

through l,

+ 1 through £ + g

+ 1 through £ + g

+ g + 1 through m

3. X > 0, i = 1 through £

i

X. <0, i = £ + 1 through l + g

1 —

unrestricted, i = X + g + 1 through m

x > 0, j = 1 through n

j ~

As was noted earlier, these conditions are also sufficient when the

objective function is concave. Therefore, the problem reduces to one of

finding x (j = 1 through n) that satisfy conditions 1-3 above for some

j

set of X. (i = 1 through m)
-34-

These condtions may be written as:

n m

1* . c . + 2 E d x - S X- a . • + Y. = °y 3 = 1 through n

J k=l ik k i-i^i lj l

x .y = 0, j = 1 through n

J J

nX a. .x . + s . = b , i = 1 through t

J=1 iJ J i i nX a. x . - s . = b , i = JL + 1 through ^ + g J=1 iJ J l i

n

X n a. x . = b., i = X + g + 1 through m

J- 1 iJ J i

X. s. =0

i i

t i = 1 through JL + g

ot \ > 0, i = 1 through Jf

* i ~

\ < 0, i = l 4- 1 through £ + g

i ~

X

i unrestricted, i = JL + g + 1 through m

x > 0, j = 1 through n

j ~

y > 0, j = 1 through n

s > 0, i = 1 through t + g

i —

With the exception of the requirements x^.y^. = 0 and . = 0, these conditions

form a set of simultaneous linear equations. Futhermore, the variables

either are, or can be made to be, non-negative. In order to make all the

variables non-negative, substitute X = y . , i = 1 through JL; = -u ,

i = JL + 1 through JL + g; = u -v , i = X + g + 1 through m, into (l f )— (3 f )

to get:
-35-

n l

c + 2 E d x - S u . a . . j k=l jk k i=l i iJ

+ "4* u a i=m i IJ

m m

- 2 ua + 2 v . a . . + y . = 0, j=l through n

i = xf+g+l i ij i=^+g+l 1 iJ J

2". £ a x + s = b , i = 1 through £

j=l ij j i i

n2 a x - s = b , i = £ + 1 through £ + g j=l ij j i i

3 .

n

.2 a x - = b,i = £+ g+ l

J=1 ij J i

u

± > 0, i = 1 through m

x. > 0, y.> 0, j = 1 through n

J “ Js. >0, i = 1 through t + g

through m

v^>0, i = £-fg+l through m

x .y . = 0, j = 1 through n

J J u s. = 0, i = 1 through £ + g

l i

These conditions are a set of simultaneous "almost linear" equations

in non-negative variables. The solution to these equations can be found

with Phase I of the simplex method with a modification to account for the

nonlinear equations above. That modification is that x and y are not

j 3

allowed to be simultaneously in the basis; nor are s. and u .

This modification has been made to both SIMPLX and RVSMPX and is

available with the use of the parameter KQP (see section 5.11). Note

that in the discussion of this option in section 4, the do not appear.

This is so because RVSMPX and SIMPLX store slack and surplus variables

implicitly. Therefore it is necessary only to put the original constraint

coefficients in this part of the matrix
- 36 -

APPENDIX B

:

LISTING OF RVSMPX
37

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APPENDIX C:

LISTING OF SIMPLX
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-62-

APPENDIX D: TIMING CONSIDERATIONS

In order to provide the reader with some idea of the time involved in solving linear programming problems with RVSMPX and SIMPLX,

we include the following table of problems that were run on the National

Bureau of Standards* UNIVAC 1108. Each of the problems was randomly

generated in such a way as to be bounded and feasible; and for each

problem, all constraints were "greater than" constraints, so that a full

Phase I was required

.

Note that, as n increases from 20 to 120 while m is held fixed at

20, RVSMPX becomes faster than SIMPLX. The same is true for m held fixed

at 50 and n increasing from 75 to 200. This illustrates the concept

alluded to in section l^and discussed at length in [7]. Basically, the

concept is that for n > 3m, the revised simplex method unequivocally

appears to be better computationally. Futhermore, in [7], Wagner argues

that even for smaller n the revised simplex method is more desirable

than the standard simplex method
TABLE

I:

RESULTS

OF

SAMPLE

RUNS - 63 -

cn2 p o o 00 o X X 05 X X X

0-H X rH in oo <M 05 o n CM in o

a-p rH rH rH rH X CO in rH r—

X d o o o o O o o o rH CM CM 0 p • • • o • o • o o O • S 0

•H-P Ehm

Cfi 'h d

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d-P

^hh

'D0 X X o X X X X X X o X

Cfl 0 X X X 05 o X X X X

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0-H 1> X rH X o o X X rH a-p X X o 05 o X X rH

a3 rH rH X X X X X X X t> 05 0 p o o o o o o o o rH rH rH S 0 0 O • o •H-P HM

x>a

CQ 'H d o o

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rH X X

T30 X X

Cfi 0 X l> X

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rH EH • • • X o X X

05 rH 05 o 05 LO X • • • • rH rH CM X O X X X

rH o X X X X o o • • • X X X X o o o o o o o o X o o

dl X X X X X X X X

rH l> X

rH oX S 

o o in o o o o o o o o

rH rH rH X X X X X X X X

Note:

the

times

listed

above

are

in

seconds
. ?