ZERO-ONE INTEGER PROGRAMMING 763 by RATNAM VENKATA CHITTURI B. S.(Meoh. Engg), Andhra University Waltair, India, 1961; A MASTER'S REPORT submitted in the partial fulfillment of the requirements for the degree MASTER OP SCIENCE Department of Industrial Engineering KANSAS STATE UNIVERSITY Manhattan. Kansas 1967 Approved by: tg&£2^~^ Major Professor TABLE OP CONTENTS Page INTRODUCTION 1 ZERO-ONE INTEGER PROGRAMMING PROBLEM FORMULATION .... 9 CUTTING PLANE METHODS 11 PARALLEL SHIFTS OF THE OBJECTIVE FUNCTION HYPER PLANE . 22 COMBINATORIAL METHODS lj.1 ADDITIVE ALGORITHM OF BALAS (WITH SOME MODIFICATIONS) . 62 SUMMARY 9I4. CONCLUSION 95 ACKNOWLEDGEMENTS 97 REFERENCES 98 APPENDICES 100 APPENDIX I 101 APPENDIX II 109 APPENDIX III 116 APPENDIX IV 127 INTRODUCTION Linear programming was developed in the early 50's. Since then, it has become a powerful mathematical tool in solving a number of military, economic, and industrial problems. Prima- rily, linear programming helps in selecting a best schedule of actions among many conflicting alternatives and may be applied to systems having the following characteristics: 1. Some objective to be optimized, such as maximum profit or minimum cost. 2. There are a number of variables to be considered simultaneously. 3. In addition to many variables, there are a number of constraints on the system which the solution must satisfy. These restrictions are linear and may represent such quantities as production capacity or limited sales. In general, linear programming problems are solved by the simplex method. In some situations, the simplex solution which allows for fractional values of the variables may not provide the solution that is desired. For example, in an airline sched- uling, it makes little sense to speak of an optimal solution in terms of fractional units of planes or people. It is important to note that the integer solution obtained by rounding off the fractional values does not usually constitute an optimal integer solution. This fact led to the development of integer linear programming in 1959, which guarantees an optimal integer solution. In some situations, the variables in the integer solution may take only a zero or one value. This may occur in scheduling and machine sequencing. 3ecause of the special form of the solu- tion and because of computational difficulties with the current integer programming algorithm, much work has been done to find an efficient algorithm for solving the zero-one integer program- ming problem. The purpose of this paper is to review some of the better known zero-one algorithms and discuss their respec- tive advantages and disadvantages. A number of practical prob- lems can be classified as zero-one integer programming problems The following are examples of these type of problems: a. Scheduling of jobs through a production facility. b. The machine loading or sequencing problem. c. Problems where decision A or B must be made, but not both A and 3. These problems are special cases of linear programming problems in which the decisions are of the "either - or" type and where a linear objective function is to be optimized and the system is subject to restraints. To simplify the discussion, the zero-one problem is presented in the manner of the linear pro- gramming (L.P.) problem. As mentioned earlier, the objective of a linear programming problem is to maximize or minimize a linear objective function subject to a set of linear equality or inequality constraints governing the system. Linear Programming The general form of a linear programming problem is stated as: minimize Z - CX subject to AX \ b (1) X >. where C, X and b are vectors and A is an n-dlmensional matrix and Z is a scalar. From elementary algebra, it is known that a system of lin- ear equalities either has a unique solution or has no solution at all. However, a system of linear inequalities may have an infinite number of solutions or a finite number of basic solu- tions. A basic solution to a linear programming problem having m constraints and n variables, where n > m, is the one in which, at most, m variables take values greater than zero and the re- maining variables are set equal to zero. It is important to note that a solution mu3t be non-negative or feasible, that is x ** j * 0» J 1, 2, 3 n. A feasible solution which yields a maximum or minimum value for the objective function is called an optimal solution to the linear programming problem. If X' is an optimal feasible solution, then Z* = CX' , X' >. and AX' -b>0. •"- where Z » the optimal value of the objective function. Generally linear programming problems are solved by the simplex method developed by Dantzig /K7 Integer Programming A linear programming problem becomes an integer programming problem when one or more of the variables are required to have Integer solution values. Hence it can be shown that the problem (1) becomes an integer programming problem when the following restriction is added: = x x : x 1 2 J ' °' *' J £ (2) j [ j j ' ' ' • ' ] where J C - N , N ^1, 2, 3, . , n] . Beale Equations To better understand the methods of integer programming, it Is helpful to review Beale Equations. E. M. L. Beale f§7 was the first to represent the linear programming problem by equal- ities in non-negative variables. If a slack vector X2 is intro- duced into problem (1), it can be rewritten as minimize a a X, oo o 1 subject to - A^ + IX2 b (3) x 1( x2 where a « ( a is an Q oi' 1 x n row vector, X]_ = (x«) is an n x 1 column vector « In this paper c Is meant to be for inclusion, where as c. is for strict inclusion. — , X = X is an 2 ^ n+I^ m x 1 slack (column) vector A = l ^ a ij"' is an m x n matrix of coefficients I = an m x m identity matrix b = (b^) is an m x 1 requirement vector. 3y rearranging the vectors in the above problem, it Is put In the following form. Minimize jc = a - a X Q 00 1 subject to Xo = b - A-jX-i >. xlt x2 0. where x is the value of the objective function. Now an Integer programming problem can be stated in Beale Equation form as follows: Find x , x^, X£, . x , x , n n+ ]_ k with x,, n+m j' n minimize x„ = 4v_ + 7~ a (-* 00 ' oj 5 ) 3-1 n x a + a n+l " io JZ ij ( -*3 } 3-1 CW X^ <= z an integer J e J, J N i x. = non-basic variables x = basic variables n+i_ aj» - coefficients with Integer values * 1> 2, 3, . • , m J - 1, 2, 3, .... n. This problem can be solved by the simplex method and If the solution obtained has the Integer property, it Is the optimal integer solution. This Is usually not the case, thus other methods are required to solve this problem. The most common method is a modification to the simplex method, due to Gomory /TO, 117 which requires the addition of additional constraints to the optimal simplex solution and force the solution to be integer valued. This Is one of the cutting plane methods which will be discussed later in detail. Zero-one Integer Programming A special case of integer linear programming is called zero-one integer linear programming which is the author's prime concern In this paper. This special case occurs when one or more of the variables are restricted to the integers or 1. For simplicity, this zero-one integer linear programming Is called the zero-one problem. Now It is evident that problem (2) becomes a zero-one problem when the following restriction is added to it. X " Xj j.e *j : - 0, 1 ; J . (5) J where J e N { N - . 1, . n , 2, 3, . , ] . Due to the special nature of the problem, the solution space is . restricted to the unit cube in hyper-space for those variables which are restricted to be or 1. There are a number of algo- rithms for solving, these zero-one problems and they are classi- fied into the following four groups . They are 1. Cutting plane methods. 2. Parallel shifts of the objective function hyper-plane 3. Boolen Algebra methods, li. Combinatorial methods. Literature Survey G-omory /To,ll7 was successful in solving problem (2) by using the cutting plane methods. He developed two algorithms which guarantee an integer optimal (if one exists) solution in a finite number of steps. These algorithms can be utilized to solve zero-one problems simply by adding the additional restric- tion denoted by (5). The second method involves in obtaining a non-integer optimal solution by simplex method, and then adding restraints which forces the solution to integer that is or 1, and resolving the problem. This additional restraint, at each iteration, is obtained from the objective function. The idea is to force the objective function value to be integer and then find the integer solution (if it exists) which corresponds to it. This method was developed by Elmaghraby /57. The boolen algebra methods are due to R. Portet /Jk±J and R. Camon /I3.7 • These are not discussed in this paper because of their special nature. The principal combinatorial methods which are used for solving zero-one problems, are due to 3alas {TJ and Glover {$,$/. Purpose In this paper, an attempt is made to review some of the above methods which include the cutting plane, parallel shifts of the objective function hyper-plane and combinatorial methods. The formulation of a zero-one problem is discussed in the first section.