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* / A s Report No. 302 
 30l> 
 
 7^/^rt 
 
 A CODE FOR ZERO-ONE INTEGER LINEAR 
 PROGRAMMING BY IMPLICIT ENUMERATION 
 (A PROGRAMMING MANUAL FOR ILLIP) 
 
 by 
 Tso-kai Liu 
 
 December 15, 1968 
 
Report No. 302 
 
 A CODE FOR ZERO-ONE INTEGER LINEAR 
 PROGRAMMING BY IMPLICIT ENUMERATION 
 (A PROGRAMMING MANUAL FOR ILLIP) 
 
 by 
 Tso-kai Liu 
 
 December 15, 1968 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 618OI 
 
 This work was submitted in partial fulfillment of the requirements 
 for the degree of Master of Science in Computer Science in the 
 Graduate College of the University of Illinois, December, 1968. 
 
ACKNOWLEDGMENT 
 
 The author is very grateful to his advisor, Professor S. Muroga, 
 for his invaluable guidance and constructive criticism in the 
 preparation of this thesis. The author also wishes to express his 
 appreciation to Mr. C. R. Baugh and Mr. T. Ibaraki for their many 
 helpful suggestions. 
 
 111 
 
TABLE OF CONTENTS 
 
 CHAPTER Page 
 
 1. INTRODUCTION 1 
 
 2. PROGRAM DESCRIPTION 3 
 
 3. SCHEMES FOR AUGMENTING VARIABLE (AGMT-VAR) 13 
 
 k. INPUT 16 
 
 5. OUTPUT 28 
 
 APPENDIX 32 
 
 LIST OF REFERENCES kO 
 
 IV 
 
CHAPTER 1 
 INTRODUCTION 
 
 Recently, the application of integer linear programming (ILP) 
 has "become extensive. Many different problems such as network 
 synthesis problems, traveling-salesman problems, flight -scheduling 
 problems and covering problems can all be formulated into integer 
 linear programming problems. [2] However, probably no algorithm which 
 is efficient on all problems exists at the present time. 
 
 ILLIP (Illinois Integer Programming Code), the computer 
 program which will be presented in this report, is written in FORTRAN 
 IV for solving 0-1 ILP. The algorithm used in ILLIP is basically an 
 extension of the implicit enumeration. [l], [h] Some sophisticated 
 ideas such as maximum sum, minimum sum and pseudo-underlining a 
 variable, have been incorporated to improve the rate of convergence 
 of the algorithm. [6] Furthermore, three different schemes are 
 available for one of the subroutines in ILLIP so that users can choose 
 the one which seems most suitable to their problems. 
 
 The original testing problem which was used to improve the 
 speed of ILLIP is the optimal- NOR network synthesis of switching 
 functions (Boolean functions). [7], [8] The number of inequalities 
 of this problem is larger than the number of variables; the coefficient 
 matrix of inequalities is very sparse and most non-zero coefficients 
 
 0-1 ILP is an abbreviation of an integer linear programming whose 
 variables assume the values or 1 only. 
 
2 
 are -1 or 1; all non-zero coefficients of the objective function are 
 1. Some programming tricks to utilize the features of the above 
 problem are added to increase the speed of ILLIP. Therefore 0-1 ILP 
 problems with sparse coefficient matrices can be efficiently solved 
 by the current version of ILLIP. 
 
 Only non-zero coefficients of inequalities are stored in 
 memory. They are stored both in rowwise and in columnwise. Therefore 
 each coefficient can be fetched efficiently. A flag is assigned to 
 each inequality so that some inequalities can be skipped during the 
 process of checking inequalities depending upon the values of their 
 flags. 
 
 It turns out that ILLIP is very efficient especially for 
 problems with similar features as the above network synthesis problem. 
 Some problems with 193 variables and Ul8 inequalities (problems of 
 synthesizing 6-gate networks for 3-variable functions) were solved 
 within 30 seconds on IBM 36o/75I» Several other types of problems 
 were also solved. The result can be found in [6]. 
 
 The aim of this report is to describe briefly the procedure of 
 the program as well as to provide necessary information for those who 
 want to use it. More detailed description about the algorithm used in 
 the program and some computational results are discussed in [6]. 
 
 A solver of an ILP problem should be aware of the fact that 
 change of inequalities or additional inequalities (e.g. inequalities 
 which preclude solutions obtained from others by permuting variables) 
 often further speeds up computation time. 
 
CHAPTER 2 
 PROGRAM DESCRIPTION 
 
 Any general 0-1 ILP problem can be defined as: 
 
 Minimize c x 
 Subject to b + A x > 0. (2.1) 
 
 c = (c n , c_, ... c ) and x = (x n , x_, ... x ) are n-dimensional vectors, 
 V 1'2 n 1'2 7 n 
 
 b = (b n , b_, ... b ) is an m-dimensional vector, A = [a. .1 is an n by 
 
 1' 2' 7 m ij 
 
 m matrix, where x. = or 1, j=l, 2, . ,n are called variables; c, b., 
 
 a. . are constants and can be any real numbers. The problem is to 
 
 assign or 1 to each variable (each component of x) such that the 
 
 inner product c x is minimum. However, in our program ILL IP, it is 
 
 further assumed that c, b., a. . are integers and all c.'s are greater 
 
 i i ij i 
 
 than or equal to zero. This is still a general form because any 0-1 
 ILP problem can be converted into the above form by proper 
 transformations of coefficients. 
 
 Any or 1 assignment to all variables is called a solution ; 
 any solution such that b + A x > is satisfied is called a feasible 
 solution . The optimal feasible solution is a feasible solution such 
 that c x is minimum. If part of the variables have been assigned to 
 one or zero, it is called a partial solution . The variables which 
 have been set to one or zero in a partial solution are called fixed 
 variables . The rest are called free variables and may be set either 
 to one or zero in the furture. 
 
i* 
 
 The program includes five main subroutines, INTL (iNiTiaL), 
 CHK-IEQ (CHecK InEQualities), AGMT-VAR (AuGMenT-VARiable), BCKTRK 
 
 -.CKTRack), CHNGMM (CHaNGe Maximum sum and Minimum sum). The 
 linkage between them is shown in Fig. 1. The lines connecting 
 CHNGMM with other subroutines mean that CHNGMM may be called by 
 CHK-IEQ, ACMT-VAR or BCKTRK and then returns to the subroutine from 
 
 Ich it was called, 
 (l) INTL 
 
 This subroutine will read in coefficients of objective function 
 and of inequalities, i.e. c, b, A in (2.1). If some variables has 
 been specified to one or zero by the user, it will also set them to 
 one or zero and start with this initial partial solution. If no 
 information about the values of some variables in the optimal feasible 
 solution is known beforehand, the program will start with a partial 
 solution without any fixed variable. 
 
 In order to simplify the program, two inequalities will be 
 generated in INTL. One is ZMAX-1 - c x > 0; the other is - ZMIN + 
 c x > where ZMAX is the upper bound of the optimal objective value 
 and ZMIN is its lo^er bound. These two inequalities which are added to 
 the original set of inequalities in (2.1) guarantee that the objective 
 
 That is, the inner product, c x, corresponding to the optimal 
 feasible solution. 
 

 IMTL 
 
 
 
 
 
 
 
 
 
 
 if 
 
 
 
 
 
 CHK-IEQ 
 
 
 
 
 
 
 • 
 
 • 
 
 I \ 
 
 
 
 
 AGMT-VAR 
 
 
 
 o 
 
 
 
 BCKTRK 
 
 J CHNGMM (^ 
 
 - 
 
 
 
 
 
 
 
 
 
 
 Fig. 1. Linkage between main subroutines 
 
value of a feasible solution to be found next is always greater than 
 or equal to ZMIN and smaller than ZMAX. Whenever a better feasible 
 solution is obtained, ZMAX will be updated to the objective value of 
 the new feasible solution. The initial values for ZMAX and ZMIN will 
 be discussed in Chapter h. Therefore, the objective value of a new 
 feasible solution is always smaller than that of the previous one. 
 (2 ) CHK-IEQ 
 
 With a partial solution, CHK-IEQ will check the maximum sum 
 defined below to detect whether some free variables must be one or 
 zero so that every inequality could be satisfied. If it is so, it 
 will set these free variables to their proper values and underline 
 them. The term, " underline" was first proposed by A. M. Geoff rion in 
 [h]. If a fixed variable in a partial solution is underlined, then 
 during backtrack the opposite value of this variable will not be 
 assigned. 
 
 Assume a partial solution with F = {x -* f . , x. -* f . ..., 
 
 J-i J-i Jo ^2 
 
 *•* 
 x. -* f } is given, where x. for j = j , ..., j is assigned to a 
 
 \ K j i k 
 
 binary value f . which is 1 or 0. Let W be the set of the remaining 
 free variables. Now in order to discuss the work done in CHK-IEQ, 
 let us first define the following three vectors. 
 
 A feasible solution now means a solution which satisfies 
 both (2.1) and these two generated inequalities. 
 
 The set of fixed variables. 
 
(a) Partial Sum : y(F) = (y^F), y 2 (F), ..., yjF)) 
 
 y.(F) = b. + E a. . f. i = 1, 2, ...m 
 1 x £F 1J J 
 
 (b) Maximum Sum : u(F) = (u _(F), u (F), ... u (F)) 
 
 u.(F) = b. + £ a. . f . + E a. . 
 
 1 x.eF 1J J x.ew 1J 
 
 a. .X) 
 
 = y.(F) + E a. . i = 1, 2, . . . m. 
 
 x.ew 
 a. >0 
 
 (c) Minimum Sum : 1(F) = U.(F), ^(F) . .., i (F)) 
 
 m 
 
 i.(F) = b. + E a. . f . + E a. . 
 i i x.eF 1J J x.ew 1J 
 
 a. .<0 
 
 = y.(F) + E a. . i = 1, 2, ...m. 
 
 x.ew ^ 
 
 J 
 a. .<0 
 
 ij 
 
 Note that u.(F) is equal, to the left hand side of the i-th 
 inequality in (2.1) corresponding to the solution which is obtained by 
 assigning all free variables of a partial solution with positive 
 coefficients in the i-th inequality to one and the others with negative 
 or zero coefficients to zero. Therefore it is the maximum value that 
 the left hand side of the i-th inequality can achieve over all 
 
solutions with F. Similarly, -0.(F) is its minimum value. Then 
 
 CHK-IEQ will take the following action depending on the value of u(F). 
 
 (a) If u.(F) = for some i, then it will set all free variables with 
 positive coefficients in the i-th inequality to one and 
 underline them; it will also set all free variables with negative 
 coefficients in the i-th inequality to zero and underline them. 
 
 (t>) If u.(F) > for some i, and if there exist a free variable x 
 such that |et. .| > u^F), then CHK-IEQ will set x.. (if a_ > 0) 
 to one and underline it; CHK-IEQ will set x. (if a^ < 0) to 
 
 zero and underline it. 
 (c) If u. (F) < for some i, then there exists no feasible solution 
 with F. The program will transfer to the subroutine BCKTRK and 
 backtrack. 
 
 If u.(F) > for i = 1, 2, ..., m, and if there is no free 
 variable left, then there is only one feasible solution which is 
 uniquely determined by F. The previous feasible solution is 
 replaced by the new one and the program transfers to BCKTRK. 
 Since u(F) and i(F) may change when some free variables are assigned, 
 CHK-IEQ will repeatedly check above (a) through ( d) with all 
 inequalities until no further free variables can be set. Then it 
 will transfer to AGMT-VAR. 
 
 For a problem with many inequalities, the checking process in this 
 subroutine may be very time-consuming. In order to speed up the 
 program, it checks i.(F) for each inequality before the checking of 
 u. (F). If &.{Y) > for some i, then the i-th inequality is always 
 
9 
 satisfied, no matter what values are assigned to the free variables. 
 
 Therefore, the program will skip checking u.(F) for this i. The speed 
 
 of the program has increased considerably by doing this. 
 
 (3) AGMT-VAR 
 
 When CHK-IEQ could neither conclude that there exists no 
 
 feasible solution with the current F nor find that there exists 
 
 only one feasible solution with the current F, the program transfers 
 
 to AGMT-VAR. AGMT-VAR will set one free variable x . to 1 or 
 
 3 
 
 according to one of the schemes in Chapter 3» After assigning a 
 
 free variable, the program will check whether the variable can be 
 
 p seudo -underlined or not and pseudo underline the free variable if 
 
 possible. Then the program will transfer back to CHK-IEQ to check the 
 
 partial solution with a new F which is the previous F with one more 
 
 fixed variable x. added. 
 3 
 
 (k) BCKTRK 
 
 When CHK-IEQ, either concludes that there exists no feasible 
 solution with the current F or find the only feasible solution with 
 the current F, the program transfers to BCKTRK in order to generate 
 a new partial solution and then transfers to CHK-IEQ again. The 
 following scheme to generate a new partial solution is called 
 backtrack. With this backtrack scheme it has been proven in [U] 
 
 * The conditions for pseudo underlining a variable is discussed 
 in [6]. 
 
10 
 that all possible solutions will be implicitly enumerated. 
 
 In the program, partial solutions are represented by two arrays, 
 X and XS. X indicates what values have been assigned to the fixed 
 variables and XS indicates whether the fixed variables have been 
 underlined or not. Initially, X and XS are empty. Whenever a value 
 is assigned to a free variable, its corresponding variable number 
 with a proper sign is added to X and XS. The variable number in X is 
 negative, if the variable has been assigned to zero; on the other 
 hand, its variable number is positive, if it has been assigned to one. 
 Similarly its variable number is negative in XS, if it has been 
 underlined; its variable number is positive, if it is not underlined. 
 For example, assume a partial solution of a problem with six variables 
 is represented by X = {-2, -3, k) , XS = {-2, 3, -k) . That is, x„ is 
 assigned to zero and underlined, x is assigned to zero and not 
 underlined, Xi is assigned to one and underlined. The rest of the 
 variables are free variables. If x_ is set to one and underlined, 
 the new partial solution becomes 
 
 X = {-2, -3, k, 1}, XS = {-2, 3, -h, -1} (2.2) 
 
 When the program transfers to BCKTRK, the underlined fixed 
 variables at the right of the last not underlined variable are set to 
 free variables, and then BCKTRK sets the last not underlined variable 
 to its opposite value and underlines it. For example, with the 
 
 * For example the variable number of x_ is 5« 
 
 5 
 
11 
 
 partial solution (2.2) BCKTRK will set x and x, to free variables 
 since they are underlined variables to the right of the last not 
 underlined variable x„. Then x is set to one and underlined. The 
 new partial solution will be X = {-2, 3), XS = {-2, -3). 
 
 Sometimes variables are pseudo-underlined. In this case their 
 variable numbers in XS are added by 2000. A pseudo-underlined 
 variable is treated as an underlined variable in BCKTRK if no feasible 
 solutions have been found since the variable was pseudo-underlined. 
 Otherwise a pseudo-underlined variable is treated as a non-underlined 
 variable. 
 
 If all fixed variables are underlined when the program transfers 
 to BCKTRK, all solutions have been implicitly enumerated. The 
 feasible solution in memory is an optimal feasible solution . If no 
 feasible solution has been found, the problem is infeasible. 
 (5) CHNGMM 
 
 In the program, u(F) and £(F) are used very often. In order to 
 
 save time, the program updates their values whenever a free variable 
 
 is assigned to some value instead of calculating them every time they 
 
 are used. When a free variable x. is assigned to 1 or in some 
 
 J 
 
 subroutine, the program will transfer to CHNGMM. If x . is assigned 
 
 J 
 
 to 1, CHNGMM updates u(F) and 2(f) by setting 
 
 Only one of optimal feasible solutions will be obtained. 
 
12 
 u.(F') = u.(F) i.(F') = i.(F) + a. . if a. . > 
 
 u.(F') = u.(F) + a. . i.(F') = i.(F) if a . . < 
 
 l l ij l I N ' ij 
 
 u.(F') = u.(F) i.(F') = i.(F) if a . . = 
 
 If x. is assigned to zero, then CHGMM will set 
 
 J 
 
 u.(F') = u.(F) -a.. i.(F') = i.(F) if a . . > 
 l l io I i ij 
 
 u.(F') = u.(F) i.(F') = i.(F) - a. . if a . . > 
 
 l l l i ij a ij 
 
 u.(F') = u.(F) i.(F») = i.(F) if a. . = 
 
 i l l i ij 
 
 Then the program transfers back to the subroutine from which CHNCMM 
 was called. 
 
CHAPTER 3 
 SCHEMES FOR AUGMENTING VARIABLE (AGMT-VAR) 
 
 (1) Scheme one 
 
 Let i(F) be the minimum sum (as defined on Page 7) corresponding 
 to a set of fixed variables represented by F, and 1(F; x. -*• l) be 
 
 J 
 
 the minimum sum corresponding to a new set of fixed variables 
 
 ■* 
 represented by F* which is obtained by adding x. -* 1 to F. Let n. 
 
 J J 
 
 be the number of inequalities where i.(F) < and £.{F; x. -*• l) > 0. 
 
 X X <J 
 
 AGMT-VAR with this scheme will select the free variable which has the 
 maximum n. over j's and assign the free variable to one. If there are 
 
 J 
 
 two free variables x. and x. such that n. = n. , then the one which 
 has a greater 5 . will be selected where 
 
 5 = Z U.(F; x. -»l) - £.(¥)). 
 J i.(F)<0 x J x 
 
 If S . = 8 . again, then the one with a greater variable number will 
 J l °2 
 
 be selected. 
 
 (2) Scheme two 
 
 This scheme is essentially the same as scheme one except that 
 
 the possibilities of setting a free variable to one or zero are both 
 
 considered. Let n. be the number of inequalities with £.(F) < and 
 
 J i 
 
 * x. is a free variable 
 
 13 
 
1U 
 
 ^.(f; x. -» a ) : .ere a is either one or zero. Let 
 
 / 1 C\ * 
 
 . = Max (n., n. . AGMT-VAR with this scheme will select the free 
 J 3 J 
 
 variable, x. , such that n. = Max (n.). x. will be set to one if 
 
 n . = n . ; x . will be set to zero if n . = n . . If there exist two 
 
 free variables x. and x. such that n. = n. , then the one with 
 3 1 J 2 J l 3 2 
 
 greater 6 . will be selected where 
 
 5. = 2 U.(F; x. -+a.) - i.(F)) 
 J.(F)<0 i J J i 
 
 a . = 1 if n. =n. ; a . = if n . =n. 
 J J J C 
 
 If 5 . = 8 . t then the one with a greater variable number will be 
 J l J 2 
 
 selected. 
 
 (3) Scheme three 
 
 This scheme is a slight modification of Geoffrion's method 
 
 which was based on Balas' method [l]. Let y(F; x. -*• l) be the partial 
 
 J 
 
 sum corresponding to a new set of fixed variables represented by F' 
 which is obtained by adding x. -» 1 to F. Let 
 
 J 
 
 m 
 p . = Z Min (y.(F; x. -* l), 0) 
 3 i=l l J 
 
 If n] = n°, then n = n 1 
 
15 
 
 AGMT-VAR with this scheme will select the free variable which 
 
 has the maximum p . and assign the free variable to one. If two free 
 
 J 
 
 variables have same P ., then the one with greater variable number 
 
 J 
 
 will be selected. 
 
CHAPTER k 
 INPUT 
 
 The order of input cards are shown in Fig. 2. Since all symbols 
 in the program are defined as half-word integers, all data except 
 NPRNT and NPNCH~ must be numbers which are in the range from 
 -32768 (-2 -15 ) through 32767 (2 15 -l). All numbers are read in by an 
 FORTRAN I-format. Therefore they must .be right justified to be 
 read in correctly. 
 
 (1) Title Card 
 
 Only one card for the title is allowed. Any suitable name for 
 the problem can be punched in this card from column 2 through column 
 80. 
 
 (2) ZMAX Card 
 
 ZMAX is the upper bound of the optimal objective value and it 
 was described on Page h. It is punched in column 1 through column 5 
 with its least significant digit at column 5» If no information about 
 the upper bound of the optimal objective value is known beforehand, 
 ZMAX must be set to a large number which is greater than the sum of all 
 coefficients in the objective function. 
 
 *. NPRNT and NPNCH which will be explained later are defined as 
 full-word intej 
 For IBM 360/75 
 
 -31 31 
 full-word integers and can range from -2 through 2 -1. 
 
 16 
 
END 
 
 TITLE OF PROBLEM THREE 
 
 El 
 
 TITLE OF PROBLEM TWO 
 
 INITIAL PARTIAL SOLUTION CARD(S) 
 
 INEQUALITIES CARDS 
 
 OBJECTIVE FUNCTION CARDS (S) 
 
 PARAMETER CARD 
 
 ZMAX 
 
 TITLE OF PROBLEM ONE 
 
 17 
 
 Fig. 2 The order of data cards 
 
18 
 (3) Parameter Card 
 
 This card contains nine parameters for the program. Each of 
 them occupies five columns with its least significant digit at the 
 rightmost column assigned to it. 
 
 (a) N 
 
 N, punched in column 1 through column 5, is the number of 
 variables in the problem. Because of the dimensions of arrays 
 defined in the program, the maximum number of variables for a problem 
 is 399* Actually, the program will count the number of variables by 
 itself and use this number throughout the program. However, in order 
 to make sure that users have prepared data in the correct form, they 
 are required to supply N. If the value of N is different from the 
 number counted by the program, an error message will be given in 
 the output. 
 
 (b) M 
 
 M is the number of inequalities of the original problem plus 
 the two created by the program (i.e. ZMIN < c x and -ZMAX +1 < - ex). 
 It is punched in column 6 through column 10. The program will also 
 count this number; however, for the same reason as N, users are 
 required to supply it. Its maximum value is 999* 
 
 (c) ZMIN 
 
 ZMIN, which is the lower bound of the optimal objective value, 
 is punched in column 11 through column 15- If users don't know this 
 lower bound of their problems, they can set ZMIN to zero. 
 
19 
 
 (d) ITER 
 
 ITER, which specifies the limit of the number of iterations, is 
 punched in column l6 through column 20. The number of iterations is 
 defined as the number of times the program has entered in the 
 subroutine CHK-IEQ. If the number of iterations exceeds the value of 
 ITER times fifty, the program will stop. For example, if ITER 
 equals 2, then the maximum number of iterations is 100. Approximately, 
 for a problem of 10 variables and 10 inequalities one iteration takes 
 5 miliseconds. Consequently, the users can estimate the value for 
 ITER depending on how much time they plan to spend on one problem. 
 
 (e) NPRNT 
 
 NPRNT, which specifies how often partial solutions will be 
 printed, is punched in columns 21 through 25. Since partial solutions 
 can give some information about what is going on in the program, 
 users may be interested in knowing it periodically. For example, if a 
 user wants partial solutions to be printed at every 50 iterations, then 
 he can set NPRNT equal 50. On the other hand, if he is not interested 
 in partial solutions, he can set NPRNT equal zero; then he will not 
 get any partial solution printed unless the iteration limit is 
 exceeded. The output format of partial solutions and the information 
 they can give will be discussed in Chapter 5* 
 
 (f) NPNCH 
 
 NPNCH is punched in columns 26 through 30. It specifies haw 
 often partial solutions will be punched on cards. When one wants to 
 
20 
 solve a fairly "big problem, he may not be able to use a computer 
 continuously. Then, if he sets a proper value to NPNCH, the program 
 will punch partial solutions on cards so that he can continue the 
 problem from the stage where the last run stopped to get optimal 
 feasible solution instead of running it again from the very beginning. 
 For example, if NPNCH is set to 50, then the partial solution at 
 every 50 iterations will be punched. Like NPRNT, if NPNCH equals zero, 
 the user will not get any partial solution unless the iteration limit 
 is exceeded. The partial solution at the last iteration will be 
 punched. In addition to each partial solution, a ZMAX card which 
 gives the last value of ZMAX will be punched. An extra card with 
 the corresponding iteration number of the partial solution will also 
 be punched. The order of these cards are shown in Fig. 3» When a 
 user wants to continue his problem, there are two different ways to 
 change his data, according to whether or not he specified a partial 
 solution in the previous run. If the user specified a partial solution, 
 then he only needs to change the ZMAX card and partial solution cards 
 to their last values. If the user did not specify, he also has to 
 change the letter in column 73 of the last inequality card , in 
 addition to placing the last partial solution at the end of data 
 cards and replacing the old ZMAX card by a new one. 
 
 See the description about inequality cards. 
 
21 
 
 PARTIAL SOLUTION AT 100th ITEPATION 
 
 ZMAX 
 
 100 
 
 PARTIAL SOLUTION AT 50th ITERATION 
 
 ZMAX 
 
 50 
 
 Fig. 3 The order of partial solutions punched by the program 
 
 (when NPNCN is specified to 50.) 
 
22 
 (g) SCHEME 
 
 As discussed in Chapter 3 there are three different schemes 
 for assigning free variables. SCHEME, which is punched in column 31 
 through column 35> indicates which scheme of them is going to be used. 
 If SCHEME is set to 1, the first scheme will be used. If SCHEME is 
 set to 2; the second scheme will be used. If SCHEME is set to 3> 
 the third scheme will be used. No other number than 1,2,3, will be 
 accepted by SCHEME and an error message will be printed, 
 (h) COEFLG. 
 
 COHFLG, is punched in columns 36 through ko. It tells the 
 program how many columns each coefficient in the objective function 
 card and inequality cards (i.e. c, A, b in definition (2.1)) will 
 occupy. COEFLG can only have two values: 8 or k. Generally, if the 
 maximum magnitude of coefficients in a problem is k digits or less, 
 COEFLG will be assigned to k. However, for some problems, if the 
 number of digits (including minus sign) of some coefficients is more 
 than k but does not exceed 8, COEFLG has to be assigned to 8 even if 
 6 or 7 columns will be sufficient, 
 (i) TYPE 
 
 TYPE, is punched in columns Ul through h-5. It indicates how 
 inequalities will be supplied. In order to describe the input format 
 of inequalities let us rewrite them as below: 
 
23 
 
 Col. Col. 1 Col. 2... Col. n 
 
 Rowl b x + a 11 x 1 + a^ ^ + . +a ln x n >0 
 
 Row 2 b 2 + a 2iXl + a^^ + a 2n X n ^ ° (lul > 
 
 0*0*«****0*00««0*«*«**0«000«««**«0***«0»000*000 
 
 Row m "b + a _ x_ + a x_ +.......+ a x >0 
 
 m mil m2 T? mm n — 
 
 Only non-zero coefficients need be supplied. Users may supply 
 inequalities in two different ways. One is to supply them raw by row 
 i.e. list non-zero coefficients, and their column numbers in each 
 row and supply them in ascending row number order. Another is to 
 list non-zero coefficients and their ro*r numbers in each column 
 and supply them in ascending column number order. For example, if 
 one wants to solve two problems which are different in only a few 
 inequalities, then he can supply inequalities row by row. Each time 
 he switches from one problem to another, only a few cards 
 corresponding to the different inequalities need be changed. Similarly 
 if coefficients of some columns will be changed in future, columnwise 
 input is more desirable. If the rowwise input is desired, TYPE should 
 
 * The column number of a. . in (k.l) is o* 
 
 ** The row number of a. . in (k.l) is i 
 
 10 
 
2k 
 be assigned to zero or any negative number. Otherwise, it should be 
 any positive number. 
 (3) Objective Function Card(s) 
 
 Only the first 72 columns of each card for objective function are 
 used. Columns 73 through 80 will not be read into a computer and these 
 columns can be used as comment for the user's convenience. The first 
 72 columns of each card is divided into a certain number of entries 
 where the number of columns for each entry depends on the value of 
 COEFLG. If COEFLG is set to k, then each entry consists of eight 
 columns, and each card can have nine entries. They are located in 
 columns 1-8, 9-l6, 17-2*+ etc. The value of coefficient is on the first 
 four columns of each entry. Its corresponding column number is on the 
 last four columns of each entry. Only non-zero coefficients have to 
 be specified. They have to be arranged in ascending column number 
 order. The coefficient of smaller column number should be before 
 those of larger column numbers. If the number of non-zero coefficients 
 of an objective function is greater than nine, more than one card must 
 be used. The termination of the objective function is indicated by a 
 negative number in the last four columns of the entry following 
 immediately after the entry of the last non-zero coefficient. If an 
 objective function has exactly nine non-zero entries, one extra card 
 with a negative number in the last four columns of its first entry is 
 needed just for indicating termination. For the case of COEFLG equal 
 8, every thing is the same except that each entry consists of twelve 
 
25 
 columns and that each card can only have six entries. In this case, 
 the value of coefficient is in the first eight columns of each entry 
 and its column number is in the last four columns. Here, readers 
 should refer to the example given in the Appendix. 
 (h) Inequality Cards. 
 
 In the same way as objective function cards, the first 72 columns 
 of each card for inequalities are divided into many entries. The 
 relation between the number of columns for each entry and the value of 
 COEFLG is the same as that for the objective function. Therefore it 
 will not be repeated here. In the following discussion we assume that 
 COEFLG is set to h. There are two ways to supply inequalities 
 depending on the value of TYPE. 
 ( a) If TYPE is non-positive 
 
 This means that the inequalities should be supplied row by row. 
 Since COEFLG is k, there are nine entries on each card and each entry 
 consists of eight columns. The non-zero coefficients of each row are 
 listed according to ascending columns number order with their values 
 in the first four columns of each entry and their column numbers in 
 the last four columns. Like the objective function, a negative 
 number has to be used right after the last non-zero entry in order to 
 show the termination of each inequality. The next inequality has to 
 start in a new card. The program will number the rows as 1st row, 
 2nd row etc. according to the order they are supplied. 
 
26 
 
 (b) If TYPE is positive 
 
 Inequalities are supplied column by column if TYPE is positive. 
 
 The format of cards is exactly the same as for rows. However the 
 
 non-zero coefficients of each column are listed according to their 
 
 row order with their values on the first four columns of each entry 
 
 and their row numbers on the last four columns. The cards for each 
 
 column should be arranged in column order; card(s) for column 
 
 (column for b.) first, card(s) for column 1 (column for a. n ) second 
 l ll 
 
 etc. Furthermore non-zero coefficients of each column nust be arranged 
 in ascending row number order. 
 
 Different from the objective function card(s), column 73 of the 
 inequality cards has to be blank except the last card. For the last 
 card, if no initial partial solution will follow, column 73 should be 
 the letter 'E'. If some partial solution will follow, column 73 should 
 be the letter 'S'. 
 
 Note that all inequalities have to be transformed to the form (k.l) 
 before preparing data. The maximum number of non-zero coefficients for 
 each row is 399 a^d for each column is 999* The total number of 
 non-zero coefficients of all inequalities can not exceed 10000. 
 (5) Initial Partial Solution Card(s) 
 
 The initial partial solution cards use the first 75 columns which 
 are divided into 15 entries. Each entry consists of five columns. As 
 it was described on Page 10 every variable number appears in XS must 
 also appears in X, but it may have different signs in X and XS. 
 
27 
 Therefore the first four columns of each entry is used to indicate 
 the variable number and its sign in XS; the fifth column is either 
 1 or 0. If it is 1, the variable number with the same sign as in XS 
 appears in X. If it is 0, the opposite sign is assumed in X . The 
 termination of partial solution is indicated by a blank entry. All 
 entries after the blank entry must be blank. Otherwise an error will 
 be introduced into computation. If a partial solution has exactly 
 fifteen fixed variables, then a blank card has to be added in order 
 to show the termination. 
 
 Note if no initial partial solution is desired, the initial 
 partial solution card is not needed. However, the last card for 
 inequalities must have an 'E' in its 73 column. Then, the program 
 will start with a solution having no fixed variables. 
 (6) End Card. 
 
 The program has the capability to run many problems consecutively 
 just by concatenating the data decks serially. The END card with 
 'END' in its first three columns is put at the end of the last 
 problem in order to terminate execution of the program. 
 
 * 
 
 Refer to the partial solution card of the example in Appendix 
 
CHAPTER 5 
 OUTPUT 
 
 The output of this program will give not only the answer of 
 the problem but also some algorithm statistics : the total number of 
 iterations, the number of backtracks, the number of underlined 
 variables added, which may give the user some idea of the efficiency 
 of the algorithm. The number of backtracks is the number of times 
 the program went to the subroutine BCKTRK, i.e. the program either 
 finds the only feasible solution with a certain F (defined in page 6) 
 or concludes that no feasible solution with a certain F exists. The 
 number of underlined variables added is the number of times that the 
 program discovers a variable has to be assigned to a certain value 
 (l or 0). Therefore, the bigger these two numbers compared with the 
 total number of iterations, the more efficient the program is. 
 
 First a name for the problem will be printed exactly as it is 
 punched on the title card. The number of variables, the number of 
 inequalities and the scheme used to assign free variables will be 
 printed next. The first two numbers will be those supplied by the 
 user. If they are different from the value calculated by the program, 
 a message asking the user to check his data will be given. Then, the 
 objective function and inequalities of the problem will be printed. 
 The inequalities are transformed to the form -b < A x. The '<' sign 
 is actually printed as 'LE' (less than or equal to). Only variables 
 
 28 
 
29 
 with non-zero coefficients are printed. For a positive coefficient, 
 the program will print an '+' sign before the value. However for a 
 negative coefficient, the program also will print an '+' sign before 
 the '-' sign of the negative value. . If there is an initial partial 
 solution, it will be printed in the form as X and XS described on 
 page 10 . In addition to the initial partial solution, the partial 
 sum (y(F) of Def . (C) on page 7 ) for each inequality will also be 
 given They are listed according to the row order. In order to 
 check each inequality for correctness, the user can supply a trivial 
 feasible solution initially (if there is one). If some y. (F) is 
 found to be negative, the user will find that the i-th inequality is 
 wrong. 
 
 If NPRNT is not zero, then partial solutions at the specified 
 numbers of iterations will be printed. In addition to partial 
 solutions, some other statistics, such as number of backtracks and 
 number of underlined variables added, up through the specified 
 iteration will also be given,, 
 
 From a partial solution printed, the user can evaluate how many 
 solutions have been implicitly enumerated at that stage as well as 
 what values have been assigned to those fixed variables. For 
 example, if a partial solution of a problem of 20 variables is given as 
 
 X = {1, A, 2, 7, -6} 
 
 xs = {l, k, -2, 7, -6}, 
 
 * Refer to the output in the Appendix. 
 
30 
 then from X we knew X_, "<. , x , X_, x, are fixed variables and 
 
 :■: , :-: , :•' are assigned to one and Xi, x,- are assigned to zero; from 
 
 XS we "know x , and x,- are underlined. Since "underline" means that 
 
 the partial solution with the variable assigned to the opposite value 
 
 has been implicitly checked, the partial solutions with 
 
 F = {x. -* 1, Xp -»■ 0, x. -»• 0} (due to -2 in XS) and with F ■ 
 
 {x.. -* 1, Xp->lx, -»0x>-»lx 7 -»l) (due to -6 in XS) in the above 
 
 example have been checked. Since free variables in a partial solution 
 
 can be assigned to 1 or 0, and since there are three fixed variables 
 
 in the first partial solution with F and five fixed variables in 
 
 the second partial solution with F , the total number of solutions 
 
 with F or F i.e. the total number of solutions which have been 
 
 implicitly enumerated, is 2 +2 
 
 If the problem is solved with an optimal feasible solution, 
 
 the optimal feasible solution will be printed. It is represented by 
 
 two lines. The first line contains the variable number and the second 
 
 line contains its corresponding value. For example, if an optimal 
 
 feasible solution is x_ = x„ = x, = x r = 1, x_ = x_ = 0, then it will 
 
 1 3 *+ d 2 5 
 
 be printed as 
 
 VAR. NO 1 2 3 h 5 6 
 VALUE 10 110 1. 
 
 In addition to the optimal feasible solution, the value of the 
 objective function corresponding to the optimal feasible solution 
 (optimal objective value), the total number of iterations, the number 
 
31 
 of iterations when the optimal feasible solution was found, the number 
 
 of backtracks and the number of underlined variables added will also 
 
 be given. If the problem doesn't have any feasible solution, then the 
 
 program will print that the problem is infeasible and give some 
 
 algorithm statistics. 
 
 If the problem is not solved within the maximum number of 
 
 iterations specified, the program will print that the number of 
 
 iterations specified is exceeded. The best feasible solution which 
 
 has been found (if any) and some algorithm statistics will also be 
 
 given. 
 
32 
 APPENDIX 
 EXAMPLE 
 Assume we want to solve the following problem. 
 Minimize 
 
 8^ + k^ + 2x- + x, + 8.^ + Hxq + 2^ + x^ + 8^ + 
 
 kx 12 +2x 13 +X lk 
 
 subject to, 
 
 -5U -2x 1 -3x 2 + & 3 + ^ + 2x 5 + x 6 + l6x^ + 8x g + hx + 2x 1Q 
 + l6x-- + Px^ + Ux^o + 2x ±1+ > 
 
 -5 1 * - 3c-l - 2x2 + l6x 3 + 8x^ + kx^ + 2x 6 + 8x_, + ^Xg + ^x 
 + x 10 + l6x u + 8^ + iflc 4- 2 X;L ^ > 
 
 -15 + 9x, + 8x, + Ux, + 2x c + x< :> o 
 1 3 4 5 o — 
 
 -15 + 9Xp + Sx,^ + l4x Q + 2j£ + x 1Q > 
 
 This problem has 14 variables (N=lU), 6 inequalities (M=6) 
 i.e. the above 4 plus 2 generated by the program, (i.e. ZMIN < c x 
 
 and - ZMAX + 1 < - c x. ) If we don 't know the lower bound and upper 
 bound of the optimal value of the objective function, then we will 
 
 set ZMAX to U6 (1 plus the sum of all coefficients in the objective 
 
 This is a problem solved by Haldi in [5]. 
 
33 
 
 function) and ZMIN to zero. We don't want the total number of 
 iterations to exceed 100 (ITER = 2). We want a partial solution 
 printed and punched at every 20 iterations (NPRNT = 20). Each 
 non-zero coefficient in the objective function and in the inequalities 
 will occupy h columns (COEFLG = k) . The first scheme of assigning 
 free variables will be used (SCHEME = l). We will supply the 
 inequalities row by row (set TYPE to zero or any negative value). 
 We want to initialize a partial solution X = {-5, 2, -8}, 
 XS = {5, 2, -8}, i.e. x =1, x = 0, x fi = and underline x R . Then 
 
 ^2 -' "5 
 the input data for this problem well be: 
 
 8* 
 
 1. Title card 
 Columns 
 1-8 
 
 bEXAMPLE 
 
 Columns 
 
 Columns 
 
 9 ~ 16 bPROBLEM !T ~ 80 blank 
 
 2 c ZMAX card 
 Columns 
 
 1~5 bbbU6 
 3. Parameter card 
 Columns 
 
 1 ~ 5 bbblU 
 16 ~ 20 bbbb2 
 31 ~ 35 bbbbl 
 
 Columns 
 
 6 ~ 80 blank 
 
 Columns 
 
 6 ~ 10 bbbb6 
 
 21 ~ 25 bbb20 
 
 36 ~ ifO bbbbU 
 
 Columns 
 
 11-15 bbbbO 
 
 26 - 30 bbb20 
 
 1+1 ~ U5 bbbbO 
 
 * "b" in column one represents blank 
 
3k 
 
 L. Objective function cards 
 First Card 
 
 Columns Columns 
 
 1 - k bbb8 5 ~ 8 bbb3 
 
 13 «- 16 bbbU 17 ~ 20 bbb2 
 
 25 - 28 bbbl 29 - 32 bbb6 
 
 37 - 1+0 bbb7 hi - 1+1+ bbbU 
 
 1+9 - 52 bbb2 53~56 bbb9 
 
 6l - 6U bblO 65~68 bbb8 
 73 ~ 80 blank 
 Second card 
 
 Columns Columns 
 
 1 ~ h bbbU 5 ~ 8 bb!2 
 
 13 ~ 16 bbl3 17 ~ 20 bbbl 
 
 25~28 bbbb 29 ~ 32 b-99 
 5. Inequality cards 
 First inequality 
 First card 
 
 Columns Columns 
 
 1 ~ 14 b-5*+ 5~8 bbbO 
 
 13 - 16 bbbl 17 ~ 20 bb-3 
 
 25-28 bbb8 29 - 32 bbb3 
 
 36 ~ 1+0 bbbU 141 ~ 1+1+ bbb2 
 
 1+9 - 52 bbbl 53-56 bbb6 
 
 Columns 
 
 9~12 bbbU 
 
 21 - 2k bbb5 
 
 33 ~ 36 bbb8 
 
 1+5 ~ kQ bbb8 
 
 57 ~ 60 bbbl 
 
 69 - 72 bbll 
 
 Columns 
 
 9-12 bbb2 
 
 21 ~ 2k bblU 
 
 33 ~ 80 blank 
 
 Columns 
 
 9-12 bb-2 
 
 21 - 2k bbb2 
 
 33 - 36 bbbU 
 
 1+5 - 1+8 bbb5 
 
 57 - 60 bbl6 
 
35 
 
 6l ~ 6k bbb7 
 73 ~ 80 blank 
 Second card 
 Columns 
 
 1 ~ k bbb^ 
 13 ~ 16 bblO 
 25 ~ 28 bbb8 
 37 ~ ^0 bbl3 
 k9 ~ 52 bbbb 
 Second inequality 
 First card 
 Columns 
 1 ~ k D -5^ 
 13 ~ 16 bbbl 
 25 ~ 28 bbl6 
 37 ~ UO bbbU 
 k9 - 52 bbb2 
 6l ~ 6U bbb7 
 73 ~ 80 blank 
 Second card 
 Columns 
 
 1 ~ k bbb2 
 13 ~ 16 bblO 
 25 ~ 28 bbb8 
 
 65 ~ 68 bbb8 
 
 69 ~ 72 bbb8 
 
 Columns 
 
 5~8 bbb9 
 
 17 ~ 20 bbl6 
 
 29 ~ 32 bbl2 
 
 kl ~ kk bbb2 
 
 53 ~ 56 b-99 
 
 Columns 
 
 5 ~ 8 bbbO 
 
 17 ~ 20 bb-2 
 
 29-32 bbb3 
 
 kl ~ kk bbbU 
 
 5U ~ 56 bbb6 
 
 65 ~ 68 bbbU 
 
 Columns 
 
 5-8 bbb9 
 
 17 ~ 20 bbl6 
 
 29 - 32 bbl2 
 
 Columns 
 
 9~12 bbb2 
 
 21 ~ 2k bbll 
 
 33 ~ 36 bbbl+ 
 
 k5 ~ kQ bhlk 
 
 57 ~ 80 blank 
 
 Columns 
 
 9-12 bb-3 
 
 21 ~ 2k bbb2 
 
 33 - 36 bbb8 
 
 U5 ~ ^8 bbb5 
 
 57 ~ 60 bbb8 
 
 69 - 72 bbb8 
 
 Columns 
 
 9 ~ 12 bbbl 
 
 21 ~ 2k bbll 
 
 33 ~ 36 bbbl+ 
 
36 
 
 37 - UO bbl3 
 
 1+9 - 52 bbbb 
 Third inequality 
 Columns 
 
 1 ~ k b-15 
 
 13 ~ 16 bbbl 
 
 25 ~ 28 bbb*+ 
 
 37 ~ 1+0 bbb5 
 
 1+9 - 52 bbbb 
 Forth inequality 
 Columns 
 
 1 ~ k b-15 
 
 13 ~ 16 bbb2 
 
 25~28 bbbl+ 
 
 37 ~ 1+0 bbb9 
 
 1+9 - 52 bbbb 
 73 S 
 
 Ul ~ 1+1+ bbb2 
 
 53 ~ 56 b-99 
 
 Columns 
 
 5-8 bbbO 
 
 17 ~ 70 bbb8 
 
 29-32 bbbl+ 
 
 1+1 ~ 1+1+ bbbl 
 
 53 ~ 56 b-99 
 
 1+5 ~ 1+8 bbll+ 
 
 57 ~ 80 blank 
 
 Columns 
 
 9-12 bbb9 
 
 21 - 2k bbb3 
 
 33 ~ 36 bbb2 
 
 1+5-1+8 bbb6 
 
 57 ~ 80 blank 
 
 Columns 
 
 9-12 bbb9 
 
 21 - 2k bbb7 
 
 33 ~ 36 bbb2 
 
 1+5 - 1+8 bblO 
 
 57 - 72 blank 
 
 Columns 
 
 5-8 bbbO 
 17-20 bbb8 
 29-32 bbb8 
 1+1 - 1+1+ bbbl 
 53 ~ 56 b-99 
 73 ~ 80 blank 
 
 The characters 'S' in column 73 of this card tells the program 
 that initial partial solution will follow. If no initial partial 
 solution will be supplied, the character 'E' should replace 'S' 
 6. Initial partial solution card 
 
 Columns Columns Columns 
 
 1-5 bbb50 6-10 bbb21 11 - 15 bb-8l 
 
 16 ~ 80 blank 
 
37 
 7. End card 
 
 Columns Columns 
 
 1 ~ 3 END k ~ 80 blank 
 
 If more than one problem is to be solved, this card should be 
 at the end of the last problem. 
 
 The output for this problem is reproduced below. 
 
38 
 
 -r >r -r ■* rr «■ 
 
 — rs, rs, r\, 
 
 j- r j- *•» ■£ C *- —• 
 
 -. J- J ^ — — CC X 
 
 ,r ^. ,* rx. *r ? ^ «r 4 ^ 
 
 ^- — ,*■ — j T 
 
 - . - - - 
 
 <"s, O *N <-> f" *■■ -ffNtJ-rg 
 
 U f*J *N( 
 
 — cr — -r — ( 
 
 — — mm 
 
 U — — — (J* -9 X tT O ^ 
 
 A, * > 4 | 
 
 - > — O — — 
 
 u. — 
 
 I « W 
 
 « w — 
 
 ^ 
 
 
 
 at 
 
 -J u 
 
 _> 
 
 
 
 *- 
 
 a c 
 
 
 
 
 K 
 
 OJ ■* 
 
 LJ 
 
 
 
 u 
 
 u 
 
 W 
 
 — 
 
 — 
 
 < 
 
 
 
 
 
 
 
 < 
 
 
 
 a. 
 
 u. tu 
 
 — • 
 
 u> 
 
 <r 
 
 O 
 
 U * 
 
 
 
 l 
 
 
 
 a 
 
 
 
 • 
 
 • at 
 
 ■« 
 
 
 V> 
 
 c 
 
 LJ ^ 
 
 — j ■ IT 
 
 <\» <NJ Q. O 
 
 — fst **> + \T 4 
 
39 
 
 >* -■ 
 
 m <-* 
 
 f\l -4 
 
 r\, 
 
 00 
 
 (J- 
 
 o — 
 
 cr 
 
 in 
 
 in 
 -4- 
 
 
 00 
 
 < 
 or 
 
 i— 
 
 X- 
 
 < 
 
 
 CO o 
 
 a 
 
 UJ 
 
 ^ CJ 
 
 •— LXJ 
 
 _ l_J 
 a u> 
 
 l_) 
 
 *i oo 
 
 U. cu 
 
 a <»■ 
 
 • a 
 c < 
 z > 
 
 v0 ll 
 
 LP o 
 
 |_) >T «-> 
 
 — > 
 
 
 
 -J 
 
 PI 
 
 t— > 
 
 U_J 
 
 
 
 <•) 
 
 
 
 UJ 
 
 IM 
 
 *— • 
 
 -J 
 
 
 
 oc 
 
 
 
 1—1 
 
 
 
 l/J 
 
 i—t 
 
 — < 
 
 <l 
 
 
 
 UJ 
 
 
 
 u. 
 
 • 
 
 
 -J 
 
 o 
 
 
 < 
 
 z 
 
 
 i. 
 
 
 UJ 
 
 »— 
 
 • 
 
 3 
 
 H- 
 
 a 
 
 -J 
 
 a 
 
 < 
 
 < 
 
 o 
 
 > 
 
 > 
 
 
 UJ 
 
 > 
 
 au 
 u 
 
 00 
 
 z 
 
 u 
 
 <l 
 ex 
 
 u- 
 a 
 
 o 
 
 x. 
 *- z 
 
 a. 3 
 
 o a 
 
 UL 
 
 z 
 
 ULI 00 
 
 JL < 
 
 J* 3 
 
 OO Z 
 
 z u 
 
 •- ►- 
 
 t- J 
 
 < —I 
 
 a u 
 
 UJ l/J 
 
 U. CD 
 
 U *- 
 00 
 
 • < 
 
 O tu 
 Z Li- 
 
 on 
 *: 
 
 < 
 a 
 
 X. 
 
 < 
 UJ 
 
 a 
 
 u 
 
 UJ 
 
 z c 
 
 — a, 
 
 -J u 
 cc o 
 
 UJ <L 
 
 U 
 
 £. oo 
 
 3 UJ 
 
 —I 
 u_ cu 
 
 a «i 
 
 H— 
 
 • oc 
 
 o < 
 
 z > 
 
LIST OF REFERENCES 
 
 Balas, Egon, "An Additive Algorithm for Solving Linear Programs 
 with Zero-One Variables, " Operations Research vol. 13, no. k, 
 pp. 517-5^9, July - August 1965. 
 
 (2) Balinski, M. L. , "Integer Programming: Methods, Uses, 
 Computation, " Management Science, vol. 12, no. 3, 1965* 
 
 Baugh, C. R. , Ibaraki, T. , Liu, T. K. and Muroga, S., "Optimal 
 Network Design Using NOR and NOR-AND Gates by Integer 
 Programming, " to be published. 
 
 ( k) Geoff erion, Arthur M. , "Integer Programming by Implicit 
 
 Enumeration and Bala's Method", SIAM Review , vol. 9, no. 2, 
 pp. 178-190, April I967. 
 
 5) Haldi, John, "25 Integer Programming Test Problems," Working 
 Paper Ho. 1+3, Graduate School of Business, Stanford University 
 December I96U. 
 
 (6) Ibaraki, T. , Baugh, C. R. , Liu, T. K. and Muroga, S., "An 
 
 Implicit Enumeration Program for Zero-One Integer Programming, " 
 to be published. 
 
 7) Muroga, S. and Ibaraki, T. , "Logical Design of an Optimal 
 
 Network by Integer Linear Programming - Part I, " Report No. 261+ 
 Department of Computer Science, University of Illinois, 
 July, 1968. 
 
 (8) Muroga, S. and Ibaraki, T., "Logical Design of an Optimal Network 
 by Integer Linear Programming - Part II, " Report No. 289 
 Department of Computer Science, University of Illinois, 
 December, 1968. 
 
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