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NASA TECHNICAL NOTEH13- 33SJ7NASA TN D-7457

ROUND-OFF ERRORS IN CUTTINGPLANE ALGORITHMS BASED ONTHE REVISED SIMPLEX PROCEDURE

by John Edd Moore

George C. Marshall Space Flight Center

Marshall Space Flight Center, Ala. 35812 I

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • OCTOBER 1973

https://ntrs.nasa.gov/search.jsp?R=19730024784 2018-07-16T20:43:58+00:00Z

1. REPORT NO.

NASA TN D-7lt574. TITLE AND SUBTITLE

1 2.' GOVERNMENT ACCESSION NO.

ROUND-OFF ERRORS IN CUTTING PLANE ALGORITHMSBASED ON THE REVISED SIMPLEX PROCEDURE*

7. A U T H O R ( S )

John Edd Moore9. PERFORMING ORGANIZATION NAME AND ADDRESS

George C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama 35812

12. SPONSORING AGENCY NAME AND ADDRESS

National Aeronautics and Space AdministrationWashington, D.C. 20546

3. RECIPIENT'S CATALOG NO.

5. REPORT DATEOctober 1973

6. PERFORMING ORGANIZATION CODE

8. PERFORMING O R G A N I Z A T I O N REPORT K

M117

10. WORK UNIT NO.

1 1. CONTRACT OR GRANT NO.

13. TYPE OF REPOR'i & PERIOD COVERED

Technical Note

14. SPONSORING AGENCY CODE

15. SUPPLEMENTARY NOTESPrepared by Aero-Astrodynamics Laboratory, Science and Engineering* This work is included as an appendix to the author's Ph.D. dissertation which is in the area ofLinear Integer Programming.

16. ABSTRACT

This report statistically analyzes computational round-off errors associated with the cuttingplane approach to solving linear integer programming problems. Cutting plane methods require thatthe inverse of a sequence of matrices be computed. The problem basically reduces to one ofminimizing round-off errors in the sequence of inverses. Two procedures for minimizing thisproblem are presented, and their influence on error accumulation is statistically analyzed. Oneprocedure employs a very smallprocedure is a numerical analysis

tolerance factor to round computed values to zero. The othertechnique for "reinverting" or improving the approximate inverse

of a matrix. The results indicate that round-offemploying a tolerance factor which reflects thecalculation and by applying the

accumulation can be effectively minimized bynumber of significant digits carried for each

reinversion procedure once to each computed inverse. If 18significant digits plus an exponent are carried for each variable during computations, then atolerance value of 0.1 X 10"12is reasonable.

The prerequisite for reading this report is a working knowledge of the simplex method andthe revised simplex algorithm in particular.

17. KEY WORDS

Round-off ErrorsCutting Plane AlgorithmsRevised Simplex AlgorithmInteger Linear ProgrammingMathematical ProgrammingNumerical Analysis

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TABLE OF CONTENTS

Page

INTRODUCTION 1

LINEAR INTERGER PROGRAMMING AND CUTTING PLANEMETHODS 2

ROUND-OFF ERRORS 3

SUMMARY 25

REFERENCES 26.

111

LIST OF ILLUSTRATIONS

Figure Title Page

1. Illustration of cutting methods 10

2. Cutting algorithms based on revised simplex 11

3. Histograms of round-off error associated with number ofproblem variables 17

4. Histograms of round-off error associated with density of theproblem constraint matrix 18

5. Histograms of round-off error associated with relative magnitudeof the constraint coefficients 20

6. Histograms of round-off error associated with the procedurefor improving the inverse of basic matrix 21

7. Histograms of round-off error associated with the tolerancevalue used to round small computed values to zero 22

IV

LIST OF TABLES

Table Title Page

1. Coefficient Ranges for the Test Problems 4

2. Accumulative Round-Off Error (Low Relative Magnitude ofConstraint Coefficients and 40 Percent Dense Constraint Matrix) ... 6

3. Accumulative Round-Off Error (Low Relative Magnitude ofConstraint Coefficients and 75 Percent Dense Constraint Matrix) ... 7

4. Accumulative Round-Off Error (High Relative Magnitude ofConstraint Coefficients and 40 Percent Dense Constraint Matrix) ... 8

5. Accumulative Round-Off Error (High Relative Magnitude ofConstraint Coefficients and 75 Percent Dense Constraint Matrix) ... 9

6. Sample Medians 14

7. Sample Ranges 15

8. Results of Factorial Experiment 16

9. Chi-Square Values for Interactions Among the Factors 23

10. Round-Off Errors Associated with Different Combinations ofTolerance Values and Constraint Coefficient RelativeMagnitudes 23

11. Round-Off Errors Associated with Different Combinations ofConstraint Coefficient Relative Magnitudes and Frequencies forApplying the Procedure which Improves the Computed Inverseof the Basic Matrix 24

\12. Round-Off Errors Associated with Different Combinations of

Tolerance Values and Frequencies for Applying the Procedurewhich Improves the Computed Inverse of the Basic Matrix 24

ROUND-OFF ERRORS IN CUTTINGPLANE ALGORITHMS BASED ON

THE REVISED SIMPLEX PROCEDURE

INTRODUCTION

This report statistically analyzes computational round-off errors associated withthe cutting plane approach to solving linear integer programming problems. Cuttingmethods typically employ the revised simplex procedure to augment fractional cuts(secondary constraints) during the search for the optimum integer solution. The revisedsimplex procedure requires that the inverse of the basic matrix be computed at eachiteration. Since the inverse for each successive iteration is computed directly from theprevious inverse, round-off errors tend to propagate. The accuracy of the computedinverse is very important because all the other information associated with an iteration iscomputed from the inverse and the original problem coefficients.

Two procedures for minimizing round-off error accumulation are presented andtheir influence is statistically analyzed. One procedure employs a very small tolerancefactor to round computed values to zero. The justification for the procedure is thatcomputed quantities having absolute values less than the tolerance are most likelyaccumulated errors. The other procedure is a numerical analysis technique for"reinverting" or improving the approximate inverse of a matrix. The value of thetolerance employed and the frequency at which the basic matrix is reinverted are bothshown to have a statistically significant effect on round-off error. The results indicatethat employing a tolerance factor which reflects the number of significant digits carriedfor each calculation and applying the reinversion procedure once to each computedinverse effectively minimizes round-off error accumulation. If 18 significant digits plus anexponent are carried for each computed quantity, then a tolerance factorof 0.1 X 10"12 is reasonable.

Readers of this report are assumed to have a working knowledge of the simplexmethod and the revised simplex algorithm.

LINEAR INTEGER PROGRAMMING ANDCUTTING PLANE METHODS

Cutting methods seek to reshape the continuous solution space by consecutivelyimposing special constraints on it until the required optimum integer solution coincideswith the optimum simplex solution. These methods capitalize on knowledge that thesimplex solution to the continuous problem must occur at an extreme point. If thecontinuous solution is integer, it is the optimum solution. Otherwise, it is an infeasiblepoint that can be eliminated. One or more secondary constraints can be computed fromthe current simplex tableau which, when augmented to the problem, "cut off" a portionof the feasible space which contains the current solution. A secondary constraint, referredto as a cut, does not violate any feasible integer points since it is nothing more than anecessary condition for integrality. The dual simplex method isolates another optimumextreme point as feasibility is recovered. Successive application of cuts produces a feasiblespace with an optimum extreme point satisfying the integrality conditions.

Figure 1 illustrates the application of cutting methods. Only two cuts wererequired in this example to permit the optimum simplex solution to coincide with theoptimum integer solution.

Since most cuts contain fractional coefficients, the addition of each new cut addsto the round-off problem. Therefore, it is desirable to employ a simplex algorithm whichhas a mechanism for purging the tableau of round-off errors after several iterations havetaken place. The revised simplex algorithm has such a mechanism. At any iteration of thisalgorithm, the entire tableau can be constructed from a knowledge of the inverse of thebasic matrix and the original problem coefficients. This means that the tableau can bepurged of round-off errors at any iteration by reinverting the basic matrix.

Figure 2 presents the primary steps required to use the revised simplex algorithmas an integral part of cutting algorithms. Each step is straightforward except for updatingand inverting the basic matrix. The procedure for accomplishing this task depends onwhether a redundant cut was dropped the previous iteration.

Case 1. No redundant cut was deleted the previous iteration. This means that thecurrent problem contains k < m + n cuts, where m and n are the original number of simplexconstraints and variables respectively. Let

aiXi + «2x2 + ... + akxk + S =

be the computed cut in terms of the initial problem variables. The «j are used to

construct a row vector C to be augmented to the bottom of the current basicmatrix BC in forming the new basic matrix Bn . C = (cj , , . . . , cm), where

ci =

«j if j < k and Xi is tne ith basic variable

0 otherwise

4 -

3 -

CUT 1\

\

OPTIMUM INTEGERSOLUTION

INITIAL SIMPLEX^\ SOLUTION . \

\

\

\CUT 2 \

FEASIBLE INTEGER SOLUTIONS

PORTIONS OF INITIAL FEASIBLE SPACE"SLICED OFF BY CUTS

Figure 1. Illustration of cutting methods.

SAVE INITIAL FORM OF PROBLEM FOR LATER USE INOBTAINING BASIC VECTORS

DROP INTEGRALITY CONSTRAINTS AND SOLVE BYTHE REVISED SIMPLEX METHOD

IS THE SOLUTION INTEGER?I C.O

- STOP

INO

COMPUTE CUT AND ADD TO CURRENT TABLEAU. OBTAIN CUT IN

TERMS OF ORIGINAL VARIABLES BY SUBSTITUTING FOR SLACK

VARIABLES. ADD THIS FORM OF CUT TO CURRENT STATEMENT OF

PROBLEM IN TERMS OF ORIGINAL VARIABLES

UPDATE CURRENT BASIC MATRIX TO INCLUDE CUT AND REVISE

NO

i i o ill v [_r\oi_

iDETERMINE ENTERING VARIABLE FROM CURRENT TABLEAU BY DUAL

SIMPLEX RULE. USE REVISED SIMPLEX PROCEDURE TO RESTORE

FEASIBILITY TO TABLEAU.

1DOES- OPTIMUM BASIC SOLUTION CONTAIN A SLACK VARIABLE

ASSOCIATE WITH A CUT?

1 YES

REMOVE THIS SLACK' VARIABLE AND ITS ASSOCIATED CONSTRAINT

FROM (i) TABLEAU, (ii) BASIC MATRIX, AND (ill) DEFINITION

OF CURRENT PROBLEM IN TERMS OF INITIAL VARIABLES.

t

Figure 2. Cutting algorithms based on revised simplex.

Also, the column vector (0 , 0 , . . . , 0 , 1) must be added to the right side of BC informing Bn . Then

Bn -

0

C 1

Bn l is obtained from B 1 by the partitioned matrix method [ 1 ].

-C

Case 2. A redundant cut was deleted at the end of the last iteration. Deletion ofa cut means that one row and one column of the current basic matrix BC must be

deleted. However, a new cut is being added to the problem which will add a row and acolumn to BC . Rather than reduce the basic matrix by a row and a column and

immediately add back a row and colymn, it is simpler to replace the old row and columnwith the new'row and column. Note that the column being deleted and the column beingadded are identical. That is, this column corresponds to the slack variable being deletedand the slack variable being added. All the entries in the column are zero except for aone in the row being deleted/added. Thus the problem of finding the new inverse reducesto the following problem.

Given a k X k basic matrix A and its inverse A"1 , construct a new basicmatrix B from A by replacing row r of A with a new row vector C . Compute theinverse of B from a knowledge of A"1 and the new row C . Here C is the row vectordefined in Case 1.

B"1 can be identified by first computing the row vector

p = C A l = («! , a2 , . . . , ak)

Construct a new row vector £ from p as follows:

oij. , -a 2 /QT> • • • ,

where the ith element is -O!j/o:r except for the rth element which is 1/Oj.. Let E be

a k X k identity matrix with its rth row replaced by % . Then,

= A'1 E

This procedure is only a slight variation of the revised method using the product form ofthe inverse. References 2 and 3 both provide a detailed discussion of the product form ofthe inverse.

ROUND-OFF ERRORS

Recall that at any iteration, the revised simplex procedure generates the tableaufrom the inverse of the basic matrix and the definition of the problem in terms of theoriginal variables. This means that round-off can be minimized in the tableau byminimizing round-off accumulation in the inverse. An iterative procedure for improvingthe inverse of a matrix can be used for this purpose.

Let A be the current basic matrix and let Bj be a computed approximation

to A"1 . Then BJ+J , the new approximation, is computed from B- by the matrixequation

Bi+1 = B t(2I - ABj ) ,

where I is the identity matrix. The sequence Bj , BJ+J , Bj+2 , ... converges

to A"1 quadratically if any norm (the Euclidean norm is usually used) of

is less than one. A detailed presentation of this procedure is provided by Reference 4 anda discussion of its use with the revised simplex method is given by Reference 5.

Another technique which has proven to be computationally effective in reducinground-off accumulation is to round all computed values to zero that have an absolutevalue less than a given tolerance. That is, if a is a computed value and I a I < £ where£ > 0 is chosen to be very small, then set a = 0 . The value chosen for £ should reflectthe number of significant digits stored for the computational variables.

The computational success of these two techniques and the influence of theproblem characteristics on round-off error have been statistically analyzed. The problemcharacteristics considered were number of variables, density of the constraint matrix, andthe relative magnitude of the constraint coefficients. These three characteristics and thetwo round-off minimization techniques were considered as five factors in anested-factorial experiment. The number of variables is considered to be nested withinconstraint matrix density which is considered to be nested within the relative magnitudeof the constraint coefficients. A factorial experimental design is used since it provides anefficient means of obtaining information about the influence of each factor and aboutpossible interactions among the factors.

The experiment contains 108 distinct combinations of the 5 factors. One iterationof the procedure for improving the inverse of the basic matrix was considered at threelevels — after each simplex base change, after each fifth base change, and after each ninthbase change. Three tolerance levels (0.1 X 10"6 , 0.1 X 10"12 , and 0.1 X 10"18) wereincluded for rounding computed values to zero. Ten variable, twenty variable, and thirtyvariable test problems were considered. All problems had 10 constraints of the type

xl + ai2 X2 vn b i = 1 , 2 , . . . , 1 0

All coefficients for the test problems were randomly chosen from uniform distributions.Table 1 provides the ranges for these distributions. The constraint matrices had either 75or 40 percent densities. Also, the relative magnitude of the a^ was either high (-2000 to

10000) or low (-20 to 80). The 108 experimental conditions come from combining 3factors having 3 levels and 2 factors having 2 levels.

Six observations were taken for each of the 108 experimental conditions. Anobservation was obtained from the application of 30 fractional Gomory cuts to arandomly generated test problem. Each test problem was screened to insure that 30 ormore cuts would be required to solve it. All test problems were pure integer problems.

Since there exists no direct way of measuring accumulated round-off error, anindicator of this error is used. It is obtained as a byproduct of the computations forimproving the inverse of the basic matrix. In this procedure

= A[A-' = AS

represents the residual, while Sj represents the error matrix. Since Rj is available

but Sj is not, the relationship of the norms

TABLE 1 . COEFFICIENT RANGES FOR THE TEST PROBLEMS

I. Problems with 3jj range of -20 to 80

Number ofVariables

101020203030

Density ofConstraint Matrix

40%75%40%75%40%75%

Range forthebj

30 to 11075 to 20080 to 220

150 to 430100 to 330225 to 650

II. Problems with ay range of -2000 to 1 0 000

101020203030

40%75%40%75%40%75%

3 000 to 150004 000 to 25 0006 000 to 30 000

15 000 to 55 0009 000 to 45 000

22 500 to 80 000

A II II

provides the only clue to the magnitude of the actual error. Note that B: = A"1

whenever Rj vanishes. In the absence of any better measure, the average value of the

Euclidean norm llRjII obtained during the application of 30 fractional cuts to each test

problem was selected as a quantitative indicator of the accumulative round-off error. Allcalculations were made in double precision by a UNIVAC 1 1 08 computer which carries18 digits plus an exponent for each double precision variable. As stated at the begining ofthis report the computer code used to generate the data is based on the revised simplexprocedure.

Tables 2 through 5 present the data. The sample medians and ranges are given byTables 6 and 7. The sample means are not presented because many of the samplescontained extreme values which caused the sample means to be poor indicators of centraltendency. The large variations in the ranges clearly indicate that the variances of thedifferent factorial combinations cannot be assumed to be equal. This precludes the use ofthe F-statistic in performing an analysis of variance on the data. However, Wilson [6]developed a Chi-square statistic which is distribution-free and does not require the equalvariance assumption. This statistic can be used to test main factors and interaction effectsin a factorial design.

Wilson's procedure was used to test the following hypotheses:

1. There is no difference in the round-off error accumulation in all 108 factorialcombinations.

2. Round-off accumulation is the same for 10, 20, and 30 variable problems.

3. Round-off error is the same for all three frequencies at which the procedurefor improving the inverse of the basic matrix was applied.

4. All three tolerance values used to round small computed values to zero havethe same effect.

5. A 40 percent dense constraint matrix has the same effect as a 75 percentdense constraint matrix.

6. "High" relative magnitude constraint coefficients have the same effect as the"low" relative magnitude coefficients.

7. There is no significant interaction among the factors.

All the hypotheses were tested using an a (probability of rejecting a truehypothesis) of 0.05. Table 8 gives the Chi-square statistics associated with each hypothesisand the corresponding critical Chi-square value.

Having rejected all the null hypotheses, the following alternative hypotheses mustbe accepted.

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TABLE 6. SAMPLE MEDIANS

Low Relative Magnitude of Constraint Coefficients

40 Percent Dense Constraint MatrixTolerance = 0. X 10"6

Basic Inverse Updated After

First Base Change Fifth Base Change Ninth Base Change

Number of Problem Variables

10

0.63 X 10"13

20

0.20 X 10"12

30

0.13 X 10"7

10

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20

0.76 X 10"' 2

30

0.29 X 10"' °

10

0.11 X 10"' 2

20

0.40 X 10""

30

0.14 X 10""

Tolerance = 0.1 X 10"'2

0.95 X 10"13 0.77 X 10"13 0.26 X 10"12 0.12 X 10"" 0.30 X 10"12 0.26 X 10"" 0.11 X 10"12 0.75 X 10"' ° 0.35 X 10""

Tolerance = 0.1 X 10""

0.61 X 10"13 0.18 X 10"' 3 0.46 X 10"13 0.17X 10"" 0.26 X 10"12 0.40 X 10"12 0.11 X 10"' 2 0.24 X 10"'° 0.16X 10"'°

75 Percent Dense Constraint MatrixTolerance = 0.1 X 10"*

0.16 X 10"12 0.22 X 10"" 0.37 X 10"12 0.16 X 10"' 2 0.89 X 10"9 0.48 X 10"" 0.12X 10"" 0.12 X 10"" 0.18 X 10"'°

tolerance = 0.1 X 10"12

0.25 X 10"12 0.75 X 10"' 2 0.17 X 10"12 0.14 X 10"' 2 0.16 X 10"' ° 0.78 X 10"' ' 0.50 X 10"" 0.22 X 10'" 0.61 X 10"9


0.60 X 10"12 0.65 X 10"'2 0.81 X 10"13 0.98 X 10~13 0.11 X 10"" 0.68 X 10"' 2 0.14 X 10"' 2 0.80 X 10"12 0.45 X 10"' '

High Relative Magnitude of Constraint Coefficients

40 Percent Dense Constraint MatrixTolerance = 0.1 X 10"6

0.22 X 10~« 0.39 X 10^ 0.32 X 10~* 0.76 X 10"'° 0.11 X 10"' 0.17 X 10^ 0.52 X 10"' ° 0.38 X 10"3 0.17 X 10"9

Tolerance = 0.1 X 10"' 2

0.47 X 10"" 0.12 X 10"10 0.67 X 10"" 0.19 X 10"'° 0.73 X 10"'° 0.14 X 10"" 0.92 X 10"' ' 0.1 IX 10"'° 0.40 X 10"'°

Tolerance = 0.1 X 10"' '

0.74 X 10"12 0.22 X 10"" 0.26 X 10"'° 0.49 X 10"" 0.10 X 10"'° 0.71 X 10"" 0.23 X 10"' ' 0.19 X 10"' ° 0.24 X 10"'°


0.12 X 10"'° 0.28 X 10"" 0.11 X 10"1 0.13 X 10"'° 0.13X 10"9 0.20 X 10"8 0.36 X 10"' ° 0.22 X 10"8 0.39 X 10"9

Tolerance = 0.1 X 10"12

0.36 X 10"" 0.67 X 10"" 0.92 X 10"' ' 0.13 X 10"'° 0.69 X 10"'° 0.74 X 10^ 0.16 X 10"9 0.41 X 10"9 0.11 X 10"1


0.32 X 10"" 0.10 X 10"' ° 0.25 X 10""> 0.36 X 10"" 0.34 X 10"'° 0.20 X 10"'° 0.10X 10"'° 0.28 X lb"10 0.13X 10"9

14

TABLE 7. SAMPLE RANGES

Low Relative Magnitude of Constraint Coefficients

40 Percent Dense Constraint MatrixTolerance = 0.1 X IO"6

Basic Inverse Updated After

First Base Change Fifth Base Change Ninth Base Change

Number of Problem Variables

10

0.74 X 10""

20

0.42 X IO"9

30

0.61 X IO"5

10

0.40 X 10""

20

0.61 X 10"'°

30

0.52 X 10"1

10

0.42 X 10""

20

0.14X IO"3

30

0.23 X 10"'°


0.15 X 10"' 2 0.18 X 10"" 0.30 X 10"12 0.15 X 10"'° 0.23 X IO"9 0.16 X 10"'° 0.19X 10^ 0.33 X 10^ 0.71 X 10"'


0.19 X 10"' 2 0.84 X IO"8 0.87 X 10"" 0.70 X 10"' 0.10 0.17 X 10"' ° 0.27 X 10"' 2 0.18 0.45 X 10"'

75 Percent Dense Constraint MatrixTolerance = 0.1 X IO"6

0.33 X 10"8 0.27 X IO"8 0.83 X IO"8 0.21 X 10"'° 0.54 X IO"7 0.74 X 10"'° 0.17X 10^ 0.14X IO"8 0.27 X IO"5


0.14 X 10"'° 0.31 X 10"' ' 0.26 X 10"'° 0.17 X 10"* 0.10 X IO"9 0.1 IX IO"9 0.21 X IO"9 0.14 X IO"10 0.88 X 10"


0.90 X 10"'° 0.1 3 X 10"" 0.27 X 10"'° 0.22 X 10"" 0.55 X 10"' ' 0.26 X 10"'° 0.87 X 10--'° 0.20-X 10"8 0.68 X 10"'

High Relative Magnitude of Constraint Coefficients

40 Percent Dense Constraint MatrixTolerance = 0.1 X 10"*

0.61 X 10"3 0.22 X 10"' 0.2 IX 10"' O.SOX IO"4 0.78 X 10"' 0.32 X 10"' 0.34 X 10"' 0.11 0.16 X 10"2


0.22 X 10~" 0.29 X 10"'° 0.20 X 10"'° 0.45 X 10"* 0.92 X IO"3 0.32 X IO"5 0.22 X IO"7 0.29 X IO"6 0.34 X IO"9

Tolerance = 0.1 X 10~18

0.25 X IO"9 0.16 X IO"9 0.35 X IO"2 0.75 X IO"9 0.19 X IO"4 0.21 X 10"'° 0.96 X 10"" 0.24 X IO"8 0.56 X Iff*


0.42 X Iff* 0.67 X IO"2 0.14 X 10"2 0.27 X IO"3 0.86 X 10"* 0.13 X 10"' 0.37 X 10"" 0.38 X IO"5 0.69 X Iff


0.14X 10"' ° 0.26 X l<f 0.37 X 10^ 0.20 X 10^ 0.97 X 10"9 0.18X IO"3 0.75 X 10^ 0.31 X 10"6 0.38 X IO"2

Tolerance = 0. X IO"18

0.18X IO"9 0.23 X IO"9 0.62 X 10"'° 0.51 X 10^ 0.37 X 10"* 0.22 X 10"9 0.99 X 10"'° 0.85 X 10"9 0.21 X 10^

15

TABLE 8. RESULTS OF FACTORIAL EXPERIMENT

HypothesisNumber

1234567

Degrees ofFreedom

107222118

Computedx2

249.3315.8215.2619.704.17

124.4769.91

Criticalx2

133.265.995.995.993.843.84

15.51

Decision

RejectRejectRejectRejectRejectRejectReject

1. All five factors have a significant effect on round-off error accumulation.

2. There exists a significative interaction among the factors.

The remainder of this report is devoted to an analysis of the individual factor andinteraction effects. The results of this analysis should indicate which combination of thetwo round-off error minimization techniques produces the best results. The individualfactor effects will be discussed first.

The 648 observations of round-off error, on which the above analysis of variancewas based, can be divided in different ways and used to develop histograms for thedifferent levels of each factor. The observations were first divided into three groupsaccording to the number of problem variables. Figure 3 gives the histograms developedfrom these groups. Note that in all three histograms the median provided a much betterestimate of central tendency than did the mean. This is a common characteristic of datacollected in this experiment.

One would expect round-off error to increase with the number of problemvariables. Figure 3 indicates a significant increase in both central tendency and variationbetween 10 variable and 20 variable problems. There appears to be no noticable changebetween 20 and 30 variable problems. These results can be partially explained by notingthat round-off accumulation is more directly related to the size of the basic matrix thanto the number of variables. This is true because the entire simplex, tableau is generatedfrom the original data and the inverse of the basic matrix. Since all the test problems had10 original constraints, then the average size of the basic matrix for the 20 and 30variable problems was not significantly different.

Next, the error observations were divided according to density of the constraintmatrix and histograms were constructed. Figure 4 presents the results. Larger errorsappear to occur more frequently in the 40 percent density problems. Also, the mean andstandard deviation of the histogram for the 40 percent density problems are larger thanthe mean and standard deviation of the histogram for the 75 percent density problems.

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The two histograms associated with relative magnitude of the constraintcoefficients are presented by Figure 5. They clearly indicate that round-off error increaseswith the relative magnitude of the constraint coefficients.

Figure 6 presents the histograms associated with the procedure for improving theinverse of the basic matrix. Both central tendency and variation decrease as the procedureis applied more frequently. However, the range of these data was chosen to narrow toshow the importance of the procedure. The experience of' this writer indicates thatround-off error accumulates very fast when the procedure is not applied at all.

Finally, the observations were divided according to the tolerance factor applied.The histograms are presented by Figure 7. A tolerance value of 0.1 X 10~6 appears tobe too large. It apparently causes valid data to be lost, thereby adding to the round-offproblem. However, a review of Tables 2 through 5 reveals that it is an acceptabletolerance for problems which have low relative magnitude constraint coefficients. That is,only one of the 108 error observations for problems with low relative magnitudeconstraint coefficients was greater than 0.7 X 10"s.

A tolerance of 0.1 X10"18 creates a problem that is not revealed by Figure 7. Itallows very small round-off errors to remain in the tableau. When one of these smallvalues is selected as the pivot element, the problem completely degenerates. Thisphenomenon occurred in over half of the problems that had high relative magnitudeconstraint coefficients and 40 percent dense constraint matrices. A total of 73 randomproblems with high relative magnitude constraint coefficients and 40 percent denseconstraint matrices had to be generated to obtain 18 test problems that would not totallydegenerate before 30 cuts were applied when a tolerance of 0.1 X 10~18 was used. Thephenomenon occurred very infrequently in the other test problems when a toleranceof 0.1 X 10~18 was applied.

Figure 7 indicates that 0.1 X 10"12 is an acceptable tolerance value. It is smallenough to prevent valid data from being" lost, yet it is large enough to prevent thephenomenon discussed above.

The discussion will now turn to an analysis of the interactions among the factors.Wilson's Chi-square statistic can also be used to test first order interactions. Since thenumber of problem variables and the density of the constraint matrix were considered tobe nested factors, only the interaction among relative magnitude of constraintcoefficients, tolerance value, and the procedure for improving the inverse of the basicmatrix will be analyzed. Table 9 gives the Chi-square statistics assuming an a of 0.05.Each null hypothesis assumes that no interaction exists.

Having accepted the alternative hypothesis that all three interactions aresignificant, a detailed inspection of each interaction is in order. An interaction can bestbe inspected by dividing the data into subsamples according to factorial combinations ofthe levels of the factors involved. The interaction effect can then be analyzed bycomparing the statistical characteristics of the subsamples.

Table 10 presents the medians and ranges of the subsamples for the interaction oftolerance value and relative magnitude of the constraint coefficients. These data indicatethat a tolerance value of 0.1 X 10~12 is more desirable than the other two values

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E

22

TABLE 9. CHI-SQUARE VALUES FOR INTERACTIONS AMONG THE FACTORS

InteractionHypothesis3

R X TR X PTX P

Degrees ofFreedom

558

Computedx2

148.07140.0744.67

CriticalX2

11.0711.0715.51

Decision

RejectRejectReject

a. R means relative magnitude of constraint coefficients; T means tolerancevalue; and P means procedure for improving the inverse of the basic matrix.

TABLE 10. ROUND-OFF ERRORS ASSOCIATED WITHDIFFERENT COMBINATIONS OF TOLERANCE VALUES

AND CONSTRAINT COEFFICIENT RELATIVE MAGNITUDES

Relative Magnitude ofConstraint Coefficients

Low

High

Tolerance Values

1.0 X 1(T6

0.93 X lO'1*3

(0.14 X 10"3)

0.57 X 1Q-9

(0.1066)

0.1 X 1(T12

0.47 X 1(T12

(0.0709)

0.17 X 1(T10

(0.0038)

0.1 X 1(T18

0.20 X 1(T12

(0.1838)

0.97 X KT11

(0.0051)

a. Median (range) from sample of 108 observations of round-off error.

because it has the smallest range for high relative magnitude coefficients and the secondsmallest range for low relative magnitude coefficients. Also, its median is acceptable inboth cases. The ranges for the other two tolerances are not consistent between the twocoefficient cases. As noted previously, 0.1 X 10"6 is an acceptable tolerance to apply toproblems that have low relative magnitude constraint coefficients.

The interaction between the frequency at which the basic matrix "reinversion"procedure is applied and the relative magnitude of the constraint coefficients is indicatedby the data in Table 11. These data clearly indicate that decreasing the frequency atwhich the reinversion procedure is applied increases both the median and range of theround-off error.

Table 12 shows the interaction between tolerance value and the frequency atwhich the basic matrix reinversion procedure is applied. Once again, a toleranceof 0.1 X 10"12 is shown to be superior to the other two values. Also, applying thereinversion procedure after each base change holds the range of the round-off error wellwithin acceptable bounds. However, each application of this procedure increases theamount of computer time required to solve a problem. Some compromise between

23

TABLE 11. ROUND OFF ERRORS ASSOCIATED WITH DIFFERENTCOMBINATIONS OF CONSTRAINT COEFFICIENT RELATIVE

MAGNITUDES AND FREQUENCIES FOR APPLYING THEPROCEDURE WHICH IMPROVES THE COMPUTED INVERSE

OF THE BASIC MATRIX

Relative Magnitude ofConstraint Coefficients

Low

High

Reinversion Procedure Applied After

First BaseChange

0.15 X lO'12*(0.61 X 10r5)

0.99 X 1CT11

(0.0217)

Fifth BaseChange

0.65 X 10~12

(0.1035)

0.31 X 10"10

(0.0779)

Ninth BaseChange

0.13 X 1Q-11

(0.1838)

0.47 X 1(T10

(0.1066)


TABLE 12. ROUND-OFF ERRORS ASSOCIATED WITH DIFFERENTCOMBINATIONS OF TOLERANCE VALUES AND FREQUENCIES

FOR APPLYING THE PROCEDURE WHICH IMPROVES THECOMPUTED INVERSE OF THE BASIC MATRIX

Tolerance Values

0.1 X 1Q-6

0.1 X 1CT12

0.1 X 1(T18

Reinversion Procedure Applied After

First BaseChange

0.21 X lO'10*(0.0217)

0.14 X 1CT11

(0.22 X 1(T8)

0.84 X 1CT12

(0.0035)

Fifth BaseChange

0.15 X IF10

(0.0779)

0.12X 10T10

(0.92 X 1(T3)

0.36 X IF11

(0.1035)

Ninth BaseChange

0.26 X la10

(0.1066)

0.18 X 10"10

(0.0709)

0.37 X lO"11

(0.1838)


computer time and allowable range for the round-off error may be desirable. Table 12indicates that the reinversion procedure should be applied at least after each fifth basechange, because the round-off error in the table is only the amount which accumulatedduring the application of 30 cuts to each test problem. Several hundred cuts may berequired to solve difficult problems if they can be solved by cutting methods at all.

24

SUMMARY

The first part of this report shows how the revised simplex procedure can be usedin cutting algorithms. The second part defines and statistically analyzes two proceduresfor minimizing round-off error accumulation in a cutting plane code. The following fivefactors are shown to have a statistically significant effect on round-off error:

1. The number of problem variables.

2. The density (percent of non-zero elements) of the problem constraint matrix.

3. Relative magnitude of the coefficients in the problem constraint matrix.

4. The tolerance value used to round small computed values to zero.

5. The frequency at which a matrix procedure is applied to remove round-offfrom the computed inverse of the basic matrix.

Round-off error accumulation was shown to increase with ( l ) an increase in the numberof problem variables, (2) a decrease in the density of the constraint matrix, (3) anincrease in the relative magnitude of the constraint coefficients, (4) a tolerance valueof 0.1 X 10~5 or greater, and (5) a decrease in the frequency at which the basic matrixreinversion procedure is applied. Round-off error can be minimized by applying thematrix procedure to improve the computed inverse after each base change and selectingan appropriate tolerance which is related to the number of significant digits carried foreach variable. When all calculations are done in double precision (16 to 18 digits plus anexponent) a tolerance value of 0.1 X 10~12 is an acceptable value. The selection of asignificantly larger tolerance tends to add to the round-off problem. Selection of asignificantly smaller tolerance reduces the effectiveness of the tolerance by allowingdivision by very small insignificant values.

25

REFERENCES

1. Taha, Hamdy A.: Operations Research An Introduction. The Macmillan Company,New York, 1971, pp. 675-676.

2. Gass, Saul I.: Linear Programming. McGraw-Hill Book Company, New York,1964, pp. 110-113.

3. Llewellyn, Robert W.: Linear Programming. Holt, Rinehart and Winston, NewYork, 1964, pp. 172-178.

4. Householder, Alston S.: The Theory of Matrices in Numerical Analysis. BlaisdellPublishing Company, New York, 1965.

5. Muller-Merbach, H.: On Round-off Errors In Linear Programming. Springer-Verlag,1970.

6. Wilson, Kellogg V.: A Distribution-Free Test of Analysis of Variance Hypotheses.Psychological Bulletin, vol. 53, no. 1, 1956.

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