Frequency planning as a set partitioning problem

NOAK '79 prize-winning paper

Frequency planning as a set partitioning problem H. T H U V E

over the various links of the system. In the design and operation of radio communication systems it is therefore a strong incentive to minimize the total system interference. The international Telecommunication Union (ITU) maintains a register of the use of the radio frequency spectrum, and it is the responsibility of the planners of new radio systems to select frequencies in accordance with ITU regulations and to avoid mutual interference with existing systems already registered with the ITU. This paper presents a method to aid that effort. The method has been specifically developed for frequency planning of satellite communication systems that often has to be introduced in areas where the frequency bands in question are extensively used by terrestrial services like line-ofsight radio links. (Although the paper makes specific reference to satellite communication systems it is clear that the methods presented are equally relevant to troposcatter and line-of-sight systems). In this context the radio communication system to b~ planned is defined by the transmit and receive sites and the links that must be established between these and also the frequency band available for that purpose. A satellite communication system includes

SHAPE Technical Centre, The Hague, The Netherlands

Received April 1980

Some aspects of a new computerized method for automatic generation of frequency plans for radio communication systems are presented. The emphasis is on problem formulation where the frequency planning problem is recast as a set partitioning problem. The objective is minimization of total system interference. A solution algorithm that has been found useful in practical applications is presented. An alternative algorithm is also demonstrated.

1. Introduction

The part of the electromagnetic spectrum that is used for radio communication purposes is becoming increasingly congested. It must be expected that this trend will continue in the foreseeable future especially in view of the growing number of geostationary communication satellites. A consequence of this trend is an increasingly severe interference environment. The quality and performance of a communication system is of course a function of the interference The subject of frequency planning was brought to the author's attention by Dr. P. Bakken who also has been instrumental in the testing of the various versions of the computer implementation. Mr. R. Lorentzen has contributed suggestions and guidance on mathematical programming theory on numerous occasions. Both have been kind enough to review the present paper, The computer code including the several revisions and extensic:~lswas written by Mr. D. van Rooijen. Prize-winning paper f r o m the NOAK'79 congress, selected by the jury consisting of B~rgerRapp (chairman), Erik Engebretsen, Stein Efik Grenland, Svein Arne Jessen.

"- ........

--'v'- ......

TRANSMIIIERS:

~

" ~ " - - - - V

......

~

' ~

RECEIVERS:

ic {1.2 . . . . . ,ol

jE I~.~ ..... .O]

LINKS! : 1,2 . . . . . ,l GROUND TERMINALS g : l , 2 " ' " ~,G

© North-Holland Publishing Company Eur.opean Journal of Operational Research 6 (1981) 29- 37

Fig. 1. Elements of a satellite communication system. 29

30

H. Thuve / Frequeno, plannhzg

the satellite as an additional element through which all links are routed (see Fig. 1). A frequeno' plan is an assignment of a separate carrier frequency interval, within the available channel, to each link in a radio communication system. The purpose of this paper is to present some aspects of a new computerized method for automatic generation of frequency plans. The emphasis is on a problem formulation where the frequency planning problem is recast as a set partitioning problem. A solution algoritlun that has been found useful in practical application is also presented. The aim of the problem formulation and solution method is to determine optimal frequency plans in the sense that the total system interference is minimized. Total system interference is a somewhat ambigious term and will be described further in Section 2 ,along with a technical problem definition. The solution algorithm presented in Section 4 does not guarantee optimality. Its deficiencies are demonstrated in Appendix A and an alternative solution algorithm is demonstrated in Apper, dix B.

2. The frequency planning problem In Section 2.1 we present some further terminology and define the technical restrictions imposed on the frequency plan. In Section 2.2 we define the measure of quality or total system interference according to which we define optimality.

Z 1. Restrictions Each ground terminal in a satellite communication system may in general act as both trammitter and receiver, G is the total number of ground terminals in the system. Each link l is defined by an ordered pair of ground terminals (i, ]). There can be at most G(G - 1) links specified for the system such that L ~ G(G - I), where L is the number of links in the system. The required band vddth Bt is specified for each link t - 1,2, ..., L. Let X1 be a variable representing the transmitter centre frequency for link I. The signal processing in the satellite includes a fixed frequency shift S(ISI is larger than the available bandwidth) such that the receiver centre frequency for link l is X! + S. A J?equency plan is therefore defined by the unique values assigned to each centre frequency Xt for all links l = 1, 2, ..., L.

The frequency interval assigned to a link I is the interval [Xt - ½Bt, Xt + ½Bt]. A feasible frequency plan is an assignment of values to all Xi such that the following restrictions are satisfied: (a) Channel availability. All frequency intervals assigned to the links in the system must be within the available communication channel as specified by the satellite or other considerations i.e.

F L +½B l ~ X l ~ F U - I B 1

(1)

where F L, F U are the lower and upper limits of the channel. One or more intervals within the channel may be reserved for other purposes and therefore not available to the system links. The effect is to break the channel into two or more separate frequency regions. The method described here also handles that situation, but the necessary notation is omitted for clarity. The reserved intervals may be regarded as assigned to notational links with the appropriate bandwidths and pre-determined centre frequencies such that (1) represents no loss of generality. (b) Transmitter and receiver availability. The frequency interval assigned to a link must be within a region available at both transmitter and receiver (taking into account any frequency shift S) as determined by technical or other considerations, i.e. TLi + ½Bl < X 1 < Tui -" I nl ,

(2)

RLj + ½Bl <~X l + S < ~ R u j - I B l ,

(3)

where TLi, Tui, R Lj, R u/are the lower and upper limits imposed by the transmitter and receiver respectively. Each I corresponds to the ordered pair (i, j}. Intervals reserved for other purposes at the transmitter and receiver are also accounted for. (c) F'requency separation. A frequency interval assigned to one link may not overlap any interval assigned[ to any other link, i.e. Xt -

½BI >>" X . ,

+ -IB z .,

or

Xt + ½BI <~Xm -½Bin

)

l = 1 2, ,

)

m= 1 , 2 , . . . , L ,

"",

L, (4)

l :/: m

Thus, frequency planning is a combinaltional problem over the continuous frequency range !FL, Fu] with or without reserved intervals embedded. The relationship

~ B t <<.Fu - FL is an obvious pre-requisite for a feasible frequency plan.

H. Thuve /Frequcncy planning

31

I given the frequency assignment Xt, i.e.

P IX I )

L

I I I

Z =~ I

I ,

I ........... ) . . . . . . . . . . . .

FL

I I

F:tJ

Fig. 2. Link interference function.

Z 2. Platming ob/ective The radio interference caused by a stationary transmitter to other registered services is a function of the transmitter frequency. Let ti(J ) be the transmitter interference at ground station i as a function of frequency f over the channel [FL. Fu ]. Similarly, let ri(y) be the receiver interference at ground station ] as a function of frequency f over the i,aterval [F L + S , F U +S]. The unit of measure for both functions may be the calculated mean time of interference per year, the number of other radio sources with which the terminals interfere or any other measure suitable for the application at hand. The functions .will in general by spikey and highly non-linear. Let the term link interference represent a combined measure of the transmitter and receiver interference at the link terminals. The link interference is a function Pi(Xl) of the centre frequency and can be shown to be determined from the terminal interference functions and the specified bandwidth, i.e. Pt(Xa) = P[ q(Xt), ri(Xt + S), Bt] for Bt and S specified, where i and ] are the terminals of link I. The manner in which the transmitter and receiver interference and/or any other arguments are combined to form the link interference function is immaterial to the method presented I. This function will again, in general, be spikey and highly non-linear (see, e.g., Fig. 2); similar to the transmitter and receiver interference functions. We define the total system interference Z as a weighted sum of the link interference for each link

(5)

)t/lel(Xl),

i=1

where Wt is the weight or relative priority of link 1. The assignment of centre frequencies Xt to each link I -- 1,2, ...,L subject to (1), (2), (3) and (4) such that (5) is minimized, is defined as the optflnal frequency plan. Finding the optimal frequency plan as defined is referred to as problem P.

3. A discrete problem formulation In view of the disjunction (4) and the highly nonlinear form of the link interference functions it is desirable to transform the problem into a more manageable form. We do this by replacing the continuous frequency range by a discrete range of K equally small frequency un.:,t intervals of bandwidth A, 2 where K is the largest integer such that

A . K <~Fu - F t • Similarly we replace the required link carrier intervals by a bandwidth index Nt, where Nt is the smallest integer such that

A. Nt >~Bt • The weighted link interference functions over the range [FL, Fu ] is replaced by step functions of the frequency index, i.e. WlPt(Xl) is replaced by Pz(k) = Pkzfork = I , 2 , . . . , K - N t + 1. Each variable Xt representing the link carrier centre frequency is replaced by a set of binary variables

Xta -

1, iff the interval represented by indices {k, k + 1, ..., k + Nt .... 1 } is assigned to link 1, 0 otherwise.

We may now re-state the frequency planning problem as a standard zero-one integer programming problena:

L g--Xl+t Minimize

Z' = ~ 1=1

1 In the cases reported in Section 5 each link interference fanction is a weighted sum of the transmitter and receiver interference measured in number of radio sources with which the terminals interfere.

~

PktA'kt ,

(6)

k=l

2 We do not discuss the choice of A here, but it should be noted that for any practical application the choice must be based on file values BI and the form of the link interference functions. The unit interval must be small enough to give a sufficiently fine resolution.

32

H. Thuve / Frequency planning

subject to K-Nl+ 1

Xkt =1

forl = 1 , 2 .... ,L

(7)

k=l

and L (~ I= 1

k+tVz-I ~ X~)+Sk=I

fork=l,2,...,K,

(8)

~=k

where Xkt = 0 or 1, for l = 1,2, ..., L and k = 1,2, ..., K - N t + 1 and Sk, for k -- 1,2, ..., K, are slack variables. Variable Xkt is not defined for frequency indices k that represent regions of the frequency band excluded by (1), (2), (3) or intervals in the communication channel reserved for other purposes. This problem formulation is referred to as problem P'. The same problem without the binary restriction on Xkt • (s referred to as problem P". Problem P' belongs to the general class of set partitioning problems. Problem P" is a linear programming (LP) problem. We notice that the set of frequency indices (1,2, ..., K) of P' can be made to represent the continuous frequency range [FL, Fu ] of P as closely as desired by making A sufficiently small.

4. A solution algorithm The size of problem P' is determined by the number of l!inks L in the system and the resolution K of the discrete frequency range. Assuming availability of the full frequency range [FL, Fu ] to all links there are NR = L + K equations and NC = (K + 1)L ~ Nt binary variables (excluding the slack variables). By elemeiitary row operations on equations (8) the number of non-zero elements in the problem matrix can be redt~ced to NZ = 2K - L + 4 [(K + 1)L - 2; Nt] which is proportional to 1[A. The number of feasible solutions (distinct frequency plans) is NF = (L + K - Y~Nt)!/(K - Y~Nt)! which reduces to NF = L! for a problem with no slack (Z Nt = K). For any typical application it appears that the problem cannot be solved within practical resource limits by standard zero-one programming algorithms. As an example, for a typical case with L = 102, K = 425 and X N t = 336 there are NR = 527 equations, NC = 43116 binary variables, NZ = 1732 I2 non-zero elements and NF ~ 1.120590868.1021a distinct frequency plans (102! = 0.9614465124. 10162). Note that the -

number of feasible solutions for the same case in the P formulation is infinite. The number of feasible solutions NF in the P' formulation, although not infinite, is very large. (For comparison, there are about 10 79 atoms in the known universe.) We note that the LP problem P" is similar to the classical assignment problem, but the similarity does unfortunately not extend to guarantee binary LP solution values. However, all feasible solutions to P' are basic feasible solutions to P". The followingsimple algorithm based on a sequence of LP problems is therefore suggested: Step 1. Solve problem P", obtaining the solution values

Step 2.

Step 3.

Step 4.

Step 5.

for Xkt and Z". If all Xkz binary - terminate an optimum solution to P' has been found. The set of solution values will in general consist of both binary and fractional values. For each binary solution value Xkt = 1 assign frequency interval {k, k + 1,..., k + Art - 1} to link 1. Among the remaining links identify the one with the largest fractional solution value X~t, i.e. 0 < Xkt < 1. If there are ties, select any one (This may, at least in theory, cause infeasibility. If necessary use some other selection rule). Assign the corresponding frequency interval {k, k + 1, ..., k + N ! - 1} to the link. Form the next LP problem by deleting all variables and equations for the links I and frequency intervals k assigned in Steps 2 and 3. Solve the new LP problem and repeat Steps 2, 3 and 4 until all links have been assigned some frequency interval.

The algorithm terminates after at most L iterations (number of LP problems solved). However, it does not guarantee optimal solutions to problem P'. In fact, examples can be constructed, where the algorithm" will not even reach a good solution. This is illustrated in Appendix A. Our reason for presenting the algorithm is that it has been found to work very well in practical applications. Let Z' and Z" be the optimal objective function values for problems P' and P". Let Z ' be the objective function value corresponding to the assignment determined by the algorithm. Then

and E = Z" - Z" is an error bound on the P' solution. In the event that E = 0 we know that the algorithm

H. Thuve / Frequency planning has reached an optimal solution to problem P'. For E > 0 further investigation is necessary to determine whether the solution is optimal. For this purpose an alternative algorithm is illustrated in Appendix B.

33

there is no apparant relationship between the problem size and the number of iterations. The number of iterations, assignments per iterations and the quality of the solution (if any) reached by the algorithm are of course determined by the interference function values. However, the relationships are not known, nor are the conditions which the interference functions must satisfy for the algorithm to perform successfully.

S. Computational experience

The algorithm has been implemented in a computer program that accepts the input data in the problem P form (plus the resolution constant A). The program transforms the data into P' format and returns the frequency plan in P form. The implementation is based on the APEX I11 mathematical programming system [ 1] which solves the successive LP problems. Solution statistics for some typical applications are listed in Table 1. Each application is described by the number of communication links L, the number of discrete frequency segments K and the total number of frequency ~;egments 2; Aft required by the links. The statistics presented are the number of assignments per iteration, the P" solution time (iteration 1) and the total solution time in CP seconds on the CDC Cyber 173 Computer at the SHAPE Technical Centre. Cases 1 and 2 are identical except for the interference functions. The objective function value (total system interference) in all applications to date has been such that E = 0, i.e. the algorithm has terminated in optimal solutions. Case 5 is the only application for which a manually constructed frequency plan exists. Application of the algorithm reduced the objective function value to about 25% of the value for the manually constructed plan. Note that the number of iterations required in each case is small compared to the theoretical maximum. Also

6. Comments

The number of problem variables increases proportionally to the number of links and the resolution. In applications where the number of variables becomes excessive, it may be advisable to reduce the problem size by imposing an upper bound on the link interference, thereby eliminating the variables for which Pkt exceeds the bound. This may also be appropriate from other considerations of the communication system performance. The advantage of the present algorithm is that the successive LP problems become progressively smaller such that a solution is reached quickly. The disadvantage is that the solution is not necessarily optimal. As an alternative we tested an algorithm based on a technique similar t~ that of Dantzig cuts [2]. The algorithm reached optimal solution in the test cases. [towever, convergence cannot be proved and the number and size of the successive LP problems lead to prohibitive solution times. The theoretical interest in the present method relates mainly to the form of the transformed frequency planning problem (6), (7), and (8). The geometry of the solution space is especially simple and it is expected that further investization will lead to better

Table 1 Solution statistics PROBLEM DIIWENSION SOLUTION TIME CASE

L

K

~N[

P"

TOTAL

ASSIGNMENTS PER ITERATION

l

2 i

3

4

5

6

i

1

14

60

54

12

73

3

1

1

7

2

2

14

60

54

II

11

3

3

2

1

2

3

40

425

144

i46

396

2?

8

3

2

4

65

425

219

281

665

33

22

10

P

102

425

336

II0?

2081

50

9

13

11

3

1

1o ,

,,,~

2

H. Thuve / Frequency planning

34

algorithms. An effort is currently under way to augment the computer program by an alternative allinteger algorithm with the option of using the solution from the present algorithm as a starting basis. The alternative algorithm is an adaptation of the Rudimentary Primal Algorithm [2] and it is illustrated in Appendix B.

PkA

PkB LINK A

Minimize

Z ' = [Pkt] [Xk.J]T

for

[Pkz] = [1, 1,5, 7 , 8 , 2 , 2 , 9 , 1,6, 1, 1,8, 1, 1], [Xkl ]

=

[XIA , X2A, X3A, X4A, XSA, X6A ,XIB, X2B, X3B, X4B, XSB, X I C , X2c, X 3 c , X4c] ,

LINK C

6

! I ! i

i

i

23456

Some points relevant to the set partitioning form of the frequency planning problem and the solution algorithm of Section 4 are illustrated by a small example. The example has been constructed specifically to demonstrate the shortfalls of the algorithm. Let l = A, B, C be the three links of a communication system 3 and let all other parameters be such that NA = 1, NB =:'2,N c = 3 and g = 6 (k = 1,2, 3, 4, 5, 6) and link interference (objective function coefficients) as illustrated in Fig. A1. The problem in P' format is then:

LINK B 8

i

Appendix A. The transformed frequency planning problem illustrated

Pkc

i

i

2 !L. K

i

!

i

12345

m

i k

-..,--,-,-~ k 123/,

i

Fig. A1. Link interference.

d

I=A

I=B

!

k:

I=C

! 1

2

3

t,

I

Fig. A2. Feasible frequency plan with

l=C

5

1

2

6

k I

X1A =XT.B =X4C = I. I=C

! k"

I

3

! 4

5

6

Fig. A3. Optimal P " solution.

that solution is Z ' = 1 + 9 + 1 = 11. The optimal solution to the LP problem P" (Step 1 of the algorithm) is illustrated in Fig. A3. The corresponding objective function value is Z" = 3 with Xkt = 0 except X~.4 = X2A = X3B = Xsa = X~c = X4c = ½ •

subject to -I I i i i il li,i

i i II

I

. . . . .

'l

I

"

1

I II I I I

__1

•[ X k t ] T:

i 1

i i

II

i

;

I

I

i I

I I il

1

i

1

II

I i

1

I

il

1

II

i i il

i

, i 1

I

i

i

1

i

I

and Xkt = 0 or 1. There are 3 ! = 6 feasible solutions to P' one of which is illustrated in Fig. A2 (the vertical axis is used to indicate the value of the corresponding Xkt variable). The objective function value corresponding to 3 The example may also be seen as a reduced problem remaining after some iterations of the algorithm.

We notice that the optimal P" solution is fractional, i.e. solution values Xkt ~ 0 are fractional for some or all links. The solution algorithm proceeds from the LP solutb,n of Fig. A3. No links will be assigned frequency indices in Step 2 since all solution values Xkt ~ 0 are fractional. The variables X i a , X2A, XaB, XsB, X l c and X4c are candidates for selection in Step 3. Notice that the choice of X2A or Xaa will lead to infeasibility. This deficiency may be corrected by modifying Step 3 (for example, selection of any Xkt with k = 1 or k = K - Nt + 1 will prevent infeasibility). However, the practical value of any such modification is not dear. With some slack (K t> 7) selection of any of the candidates leads to feasible solutions. In practical applications the slack, [100(K - Y, Nt)/K], is typically about 25%. Given that a feasible selection is made the question of optimality remains. From the example it is quite clear that Steps 2 and 3 may lead to non-optimal

H. Thuve / Frequency plann#tg Table A 1 Possible solutions reached by the algorithm, compared to the optimal solution STEP 3 SELECTION Xl A X2A X3B XSB Xlc X/.C

SOLUTION

TOTAL SYSTEM INTERFERENCE (Z')

XIA=X5B=X2c=I INFEASIBLE INFEASIBLE

10

X&A=XsB=XIc=I X6A=x&B =Xlc =1 X3A= XlB =X/.C = 1

OPTIMAL P' SOLU:ION

I

I

I=C I

2 3 /. Fig. A4. Optimal frequency plan. 1

8

X6A=X 1B =X3c=!

I=B

k:

9 9

|=A I

5

6

I

solutions. In fact, selection of any of the ' ~maining candidates XIA, XsB, XIc, X4c will lead to some non-optimal solution. This is illustrated in Table A 1. The table shows that the best solution that could be reached by the algorithm is XaA = XIB = X4c = 1 with Z.'= 8 while the optimal solution is X6A = XIB = X3C = 1 with Z' = 5 (see also Fig. A4).

Appendix B. An alternative algorithm An alternative algorithm may be obtained by adapting the Rudimentary Primal Algorithm [2] which again is an extension of the simplex algorithm. The theory and steps of the Rudimentary Primal Algorithm are recorded elsewhere, e.g. Salkin [2] and need not be repeated here. It is sut'ficient to note that the algorithm proceeds from a primal feasible all integer simplex tableau and maintains primal feasible all integer tableaux until an optimum integer solution is reached. It does this by augmenting the tableaux with Gomory cuts. It is clear that the special structure of the transformed frequency planning problem P' permits certain simplifications. Recall that all matrix elements

35

are 1 or 0 and tliat all integer solutions (frequency plans) correspond to basic feasible solutions of problem P". If, therefore, there is a path in the P" solution space of non-increasing objective function values from any integer feasible solution to an optimum integer feasible solution through integer feasible solutions only, then no Gomory cuts will be required. (A final cut will always be required to prove optimality in cases where Z" < Z'). Unfortunately there are cases where no such path cm~ be found and one or more cuts must be generated to bypass fractional basic feasible solutions. The steps of the alternative algorittfm are illustrated using the example described in Appendix A. Step 1. Determine a feasible frequency plata, identil)' the corresponding basic variables and construct the initial simplex tableau as illustrated below. This is generally trivial. The initial basis may, for example, be the frequency plan determined by the algoritbm of Section 4 augmented by the slack variables. (If b is the vector of the non-zero variables of a feasible frequency plan augmented by the slack variables and B is the matrix of the corresponding columns, then B -1 has the property that it is identical to B except for a change of sign on the off-diagonal elements). Fig. Bl is the initial tableau for the example problem determined lrom the initial basis of Fig. A2 augmented by the slack variables. Step 2. Order the tableau columns with negative objective elements in a sequence C1, C2; ..., CN such that PI <~P2 <'<"'" <~PN, where Pn is the objective element of Qz in the current tableau (n = 1,2 .... , N). If no such column - terminate - optimum solution has been found. For column Cl determine the smallest pivot element PI which preserves feasibility. If P l > 1 proceed to column C2 etc., until a pivot element Pn = 1 for n ~
H. Thuve/ Frequencyplanning

36

I ,¢,,,4

I

I

I

I

I

I

I

I

I

I

I

I

I

¢',1

I

('sl

,..~

,..~

,-.~ ¢',,I ¢'sl

u

o ~J

I

I

I

I

I

I

s~4

I ¢,q

I

re1

I

I

I

I

I

I

I

I

I

I

I

I

~0

I

I

("4

I

I

I

G

I

I

G

I

I

I

I

I ¢~

r,,

I

¢,q

,..~ ,.~

I

I

I

I

¢q

t'q

r',l

t~

.~

H. Thuve / Frequency planning

37

Table B 1 Iteration statistics for the example problem Iteration

Entering

Leaving

basic variable

basic variable

1

X3B

s4

2 3

XS~ XIC

sq s1

4 5

X6A XSA

s6 s3

6

X4B

XIAI

7 8

X2A S1

X2B s2

9 1o

x3c XIA

XSA

11

s4

s1

12 13

XIB

X4B

X4C

g

14 15

s3 No negative Pn

s4

Frequency plan at start of iteration

X I A = X2B = X4C = 1, Z = 1 1

X4c

X6A = X4B = XI C = 1, Z = 9

/

basic ~,ariable, Pn = P4c = - 4 . However, P4c we ~i'oceed to Step 4.

X6A = X1B = X 3 c = 1 , Z = 5 (Optimal frequency plan)

=

2 and

Step 4. From column C1 and the dement pl determined in Step 2 identify the corresponding row (source row). Let tj and r be the tableau elements and right-hand side of the s~urce row (t/= P l for j = CI ). From the source row generate the Gomory : cut:

where [a] means the largest integer ~ a and g is the ~ack variable. (See [2] for the necessary proofs and theoretical development). Augment the tableau with Cl as the pivot colunm (entering basic variable) and the cut as the pivot row (g as the leaving basic variable). For the frequency planning problem it is clear that the cut will not improve the objective function value since r = 0 or 1 and hence [r/pt ] = O. For the example problem the cut derived from the tableau in Fig. B2 using row A as the source row is: g-

X 3 A - X 5 A - X 2 B + X 4 C - s I - s2 - 253 -- Ss -- $6 = 0

•

Computational experience with the alternative algorithm is limited, but iteration statistics for the example problem indicate that convergence may be quite slow. The alternative algorithm is therefore not intended to replace the algorithm of Section 4 but rather to supplement it in the case where the error bound is non-zero (E > 0). It should be noted that the alternative algorithm is an adaptation of the Rudimentary Primal Algorithm for which convergence cannot be proved. However, convergence can be proved for the Simplified Primal Algorithm [2] which consists of the Rudimentary Algorithm modified by a number of extensions. To simplify the implementation we have elected to exclude the extensions until computational experience indicates that they are required.

References [1] APEX 11I Reference Manual, 7607000, version 1.1, revision F. 9 January 1977, Control Data Corporation. [2] H.M. $alkin,'lnteger Programming (Addison-Wesley, Reading, MA, 1975).