The idea of a fundamental solution to a problem in linear programming is introduced. A method of determining the fundamental solution and of applying this method to the solution of a problem in linear programming is proposed. Numerical examples are cited

Petrunin, S. V.

English

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NASA TECHNICAL MEMORANDUM 
 NASA TM 75556
 
FUNDAMENTAL SOLUTION OF THE PROBLEM OF LINEAR PROGRAMMING AND
 
METHOD OF ITS DETERMINATION
 
S. V. Petrunin
 
Translation oP "Fundamentalnoye resheniye
 
zadachi lineynogo programmirovaniya i
 
metod yiego opredeleniya",\,Acaaemy of' i-e-s
 
_USs ute of race Research. Moscow.
 
'Report PR- 89. 1978,nv.l-15
 
$NASA-TN-75556) FUNDAMENTAL SOLUTION OF-HE N7832752
PROBLEM OF LI1NEAR PROGRAMMING AND METHOD O
 
ITS DETERMINATION (National Aeronautics and
 
Space Administratio): 16 p HC A02/MF A01 
 Unclas
 
CSCL 09B1 G3/61. 31608 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
 
WASHINGTON, D. C. 20546 OCTOBER 1978
 
https://ntrs.nasa.gov/search.jsp?R=19780024809 2020-03-22T02:08:19+00:00Z
S STANDARD TITLE PAGE 
1. 	 ep ort,No. 	 2. Govarnm.nt A c:cession No. 3 Recipient'sCoailog o, 
-NASA TM 75.556J


4. Tiltl. ndfSubtti. 
FUNDAMENTAL SOLUTION OF THE P.ROBLEM 'OF ! °VOR 1978 
LINEAR PROGRAMMING AND.-METHOD OF ITS 6 .fa~ raiainCd 
DETERMINATION


7. AuthorW) 	 8, Performing Organization R.port No, 
.v.Petrunin 	 0. Work Unit No. 
9. Performing Org izolation Name and Address 	 11. ContrFTf'o.Gsrnt No.... 
i- Asw-319

.SCITRAN 
 
Box 5456 	 13. Typao Report .nd Period Covere, 
Santa Barbara, CA 93108		 Translation


12. 	 Sponsoring Ag,.ncy NM.e nd Addess" 
National Aeronautics and Space Administration 14. Sponsoring Agency Code 
W4ashington, D.C. 20546 14. _pon__rln _ 	Agency Code 
13. 	 Supplementary Notes 
Translation of "Fundamentalnoye resheniye zadachi lineynogo


programmirpvaniya i metod yego opredeleniya",' Academy of


197 PP . .....ISciencesTJSSR, tnstitiite-of SpaceResearch, Mosfc-ow, Report PR-3c9, _ 
16. 	 Abstract. 
The ,idea of a fundamental solution to a problem in linear pro­

'We ptq, 6se a-method of determining
gramming is Introduced. 
 
the fundamental sdlution and of applying this method to the 
solutionof a problem in linear programming. Numerical 
examples are cited. . . .. 
17. 	 K.y Words (Selected by Author(s)) 	 18. DOltrlbutic Stafement 
S"Unclassified - Unlimited 
19. Security Clulif. (of this report) 20. Sec. ity Clot tf. (of this page) 21. No. of Pages 22. 
Undlassifed16
Unclassified 
 
I ii ,. "NASA.AIQ 
ORIGUAL AGFIS


OF pOOR QnMJITY 
FUNDAMENTAL SOLUTION OF THE PROBLEM OF LINEAR PROGRAMMING AND


METHOD OF ITS DETERMINATION


S. V.*Petrunin


ABSTRACT. The idea of a fundamental solution


to a problem in linear programming is introduced.


We propose a method of determining the fundamental


solution and of applying this method to the solu­

tion of a problem in linear programming. Numerical


examples are cited.


1. The Notion of a "Fundamental" Solution /3*


We consider a problem in linear programming (LP) in the follow­

ing form. It is required that we minimize the linear form in n


variables xi


under the linear constraints:


•y i" 7-,l'fl-- (2) 
O. (3) 
Number in the margin indicate pagination in the original foreign


text.


1


All methods existing at the present time seek a final collection


of variables {xi I which satisfy (2) and (3) and minimize Z.


It is not practical in any of these to make use of the fact that


some of the xi can be equal to zero for every b.


We will call a collection of variables x a "fundamental" solu­
1


tion if every one of these variables, eVen if only for one combination


of the {bi} will be different from zero.


Before proceeding to the description of the method for finding a


fundamental solution, we observe that with every problem in linear


programming, we can compare another LP problem, called the dual [il. /


In particular, for the problem (1) - (3), the dual will be the


problem in which it is required to find the maximum of the linear


form in the variables yI, ..ym


m(4)


if F' 
Condition (5) with the addition of new variablesijJ .(: can be


written in the form:


4Ci2 -- C' 4 . 
The optimal values of the variables wi and xi, in accordance with the


theorem on weak complementary slackness [23, possesses the property


that:


2 
ORIGINAL PAGE ISOF POOR QUALITY 
Moreover, if for the direct and dual problems there exist


feasible solutions, then according to the theorem on strong comple­

mentary slackness [2], we can find at least one pair of variables x2


and w2 such that, if w2 = 0, then x2 > 0.
 

These properties are used for finding the fundamental solution.


2. 	 The Method of Determining the Fundamental


Solution


We turn directly to a description of the method. We write the


constraints (2) in the form of a canonical system which, according


to [1], can be described thus: /5


"A canonical system with an ordered subset of variables, called


basis variables, is a system where, for each j, the jth basis variable


has coefficient one in the jth equation and zero coefficients in the


other equations.,"


In the canonical form, an LP proglem can be formulated thus:


To find the minimum of


C4X 	 (7) 
=


under the constraints


(8)


We repfesenft all the in-depfdent variables -xi-in the form: 
-. Z+ 	 (9) 
where and y - new variables,


ni - some parameters.


3 
The basis variables x. and the objective function Z can be
a 
written:


+ K 
where


Combining equations (9) and (10):


or because . X -0 /6 
.4. 
• _ A /-/2) (1 
#t .7 t - (13) 
We assume that in (12) and (13), and have constant values


corresponding to the minimum of Z. Then, comparing (12) and (13)


with (5), we see that, due to the arbitrariness of y, expressions


(12), (13), and (11) represent the dual problem for the following


primal problem, in which is sought:


Ile-­
(the objective function is written with precision up to a constant)


in the presence of the one condition:


14 
ORIGINALF -POOR QUALITYPAGE IS 

OK J=Zc 14


The values of the variables in the primal, as well as the dual,


problem must be non-negative. Keeping this in mind, we construct


equation (14) in such a manner that all the ratios


IC L2-- and

except one, will be nega tive. __Then the-solution, if it exists, will


be determined only by the term ym, for which the coefficient:


or


will be positive. For such an equation, YA > 0 , while the remain- /7 
ing 0,YjxQ.In the dual problem, however, mx = 0, on the
 con­

trary, while the remaining Z O0 . In this way, if we succeed 
in finding a collection of coefficients ni, such that all Z,- al


i'I- (or and ' ), except the 
one for ym, will not be larger than zero; then we can set x= 0.


Here there can be two cases: either - . or P14> k - In 
the first case, it suffices to set 0 and continue the analysis 
not changing the remaining coefficients. Itf Pm> I, then it follows 
that we can transform the initial system of equations by setting 
5
 
xm = 0 and making it one of the independent variables in the basis. 
By this procedure, the number of independent random variable de­
creases by one. For the new system we again construct the coeffi­
cients:


and i12 l-cic 
and we undertake the same work.


3. A Numerical Example


To find the minimum 2'X -- under the conditions: 
+ 
we set:/8


Z., 
Then the auxiliary equation has the form:


It is necessary to require that one of the coefficients


be greater than zero and the rest - less than zero. 
We assume that:


6 
ORIGINAL PAGE IS 
/KA > OF.POOR QUALIY 
We firs-t explain that the independent variables do not vanish. For


this, we assume that:


We consider the system of inequalities:


-.- >% .-h," .1Lj 09 - , 
* A 0 0 or V1O,3 
From the first two inequalities, it follows that- /Vt 4 0,. 
Adding the first and third inequalities, we have:


Because V this inequality will hold if > In order-
to evaluate the sign of nl, we write:


3A 
/VIV.: ,*. .3t 
From these inequalities, it follows that, if we assume \v .\? A , 
then we can consider that Vt C. From this, i 
- v'. +Vv4 5 -0 
Therefore,


y;- Q and C). 
7 
Returning again to the system of inequalities, setting n3 = 0: 
_- pv ,4


These inequalities are inconsistent. We assume that V


Then,


Yy V. ,4 0~.,..7..._ 
 -. . ...
 

: ­
'yZ00


Z. 
for 4e 44 , it follows from the first inequality tha4 .4-.I . Al... 
 
and from the second that , 1W.j. Therefore, choosing n1

such that:

~i'< >1w~vV I 'vzQ
 
we will have:

7>0-,' -- C and 

We return to the original system of equations and assume


From this, %s "F


We apply the proposed method to the system: 
 /10 
8 
* 	 =ORIGINAffj4 PAEi 
OFPPOOR QUA~rpy 
The 	 auxiliary equation has the form:
 

VI V-
-. 
which does not permit us to find the new variable not occurring in


the fundamental solution, Consequently, x!,. x2 , x4 comprise a funda­

,mental solution for the matrix of conditions:


a 	 coefficients C', 4,.....a. of the objective function. 
Insofar asx 1 = 4 (identically) in the fundamental solution, the 
particular solution 	 to the given problem, if one exists, will be


either xl, x or 
4. 	 Using the Method for the Complete Solution


to an LP Problem. A Numerical Example


The example of the previous paragraph demonstrated sufficiently


graphi.cally the use of the method to obtain a .fundamental solution.


However, in many practical cases, the proposed method can be applied


also to obtain the particular solution.


In fact, it was shown above that the fundamental solution to /i


the problem depends 	 only on the matrix of coefficients of the condi­

tions aji and the coefficients ci of the objective function, but it


does not depend on the b.. Consequently, applying the method only


the 	 the primal problem, we cannot find the particular solution,


which depends also 	 on the values of the b.. We return to the dual


3


9 
problem and we find its fundamental solution. It naturally will:


depend on the matrix of coefficients of the conditions and (whibh


for us is especially important) on the coefficients of the objective


function of the dual problem. And these coefficients are functions


of the values b. of the primal problem [1]. Therefore, using the
3


method on the primal and dual problems, we can obtain in a series of


situations the particular solution to th-eroblem. We emphasize that


this method eliminates the possibility of cycling, which sometimes


occurs in the iterative loop of the simplex method. As proof, we


mention the solution to a problem having cycling in its solution by


the simplex method (the so-called-example of Beale El]).


Example . - .. ... 
*+­
-We introduce the new variables:
 

.0ci asas S-
G G 
Then, - - /12 
- + 
10 
OF POOR QJuWORIGINAL pAGE IS 
We set:


We assume:

/S 4(= 
For 
Therefore, 
x3 = 1, and 
2<1/ 
;'.>/ 
Vt < 0 these inequalities are valid. 
. From the original system, 
We assume n 3 = 0, and continue to analyze the system of inequalities: 
vi, 5.- Zfr..L.5 ,- <0O> 
/13
 
11


The system is valid for PL".- 4 0 1 > 04 Conse­

questly, N/ > 2e assume


4/Q


This sytem is valid neither for , .0 Pt- 0 nor for 

I$>QI < . Hence, one of-the-basis variab-es equals zero. We 
assume that > , i.e., -k+t 3 kL O for kI <;P<'0 ,
 
/2


- P- O~ !These inequalities are valid for


-
-,3 oo- ky <o tL% 
Consequently V > O. 2 =-- , and the original system 
0+/:6, 
From this,


P- V 
We denote again: 
 
Then,


" . 7, 
 !b,',


2


/14 
129 
ORIGINAL PAGE IS

OF POOR QUALITY

and the auxiliary equation:


This equation does not permit us to define new variables equal


to zero for arbitrary b. Consequently x1 , x2, x5 are the fundamental


solution to the problem. To determine a particular solution, we con­

struct the dual problem (the objective function is written with


accuracy up to a constant):


or


We assume:


- 06 
The auxiliary equation:


..----- -F --.--- 5-- ...... o _ 3 
Let:


S ..---- These inequalities are valid if4~4 
° (.
, 70 - . .-. 
and v1 4 _ 
0 
Because, > 0 , then and LO 0 Consequently, 
13 
PAGE is
~,ORI1GPNALL- ~%-

;r: ~OF 	 POOR QUAIO 

and the 	 auxiliary equation:


00


Here, -> 0 and Wjf 0. Consequently, only Lo> 0, and, 
returning to the primal problem, we have Z --0 Then the parti­

cular solution will be:


Z K,, W.-jO, 	= 	 1{z 2!<#i, T6 =a"gQ.­
or, in the original variables:


and


Zo. 
The author expresses his gratitude to B. Ts. Bakhshiyanu for his


valuable advice and aid in this work.


- References 
1. 	 Dantzig, G. B. Linear Programming and Extensions-.


Princeton University Press, 1966.


2. 	 Hu, T. C. Integer Programming and Network Flows. Addison-

Wesley, 1974.


1~4




Fundamental solution of the problem of linear programming and method of its determination

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