Integer Programming Code Research Articles

Flexible Mixed-Integer Programming Formulations for Production Scheduling Problems

Abstract We present an approach to modeling practical dynamic production planning problems with product sequencing considerations that yields a significant reduction in the number of required binary variables over traditional approaches and improves the structure of special ordered sets which can be exploited in state-of-the-art mixed integer programming codes. In addition, the variable definition enables the modeler to develop a wide variety of objective functions, thus facilitating model development and interpretation.

An easy solution for a special class of fixed charge problems

The fixed charge problem is a mixed integer mathematical programming problem which has proved difficult to solve in the past. In this paper we look at a special case of that problem and show that this case can be solved by formulating it as a set-covering problem. We then use a branch-and-bound integer programming code to solve test fixed charge problems using the setcovering formulation. Even without a special purpose set-covering algorithm, the results from this solution procedure are dramatically better than those obtained using other solution procedures.

New sufficient optimality conditions for integer programming and their application

The purpose of this report is to present a new class of sufficient optimality conditions for pure and mixed integer programming problems. Some of the sets of sufficient conditions presented can be thought of as generalizations of optimality conditions based on primal-dual complementarity in linear programming. These sufficient conditions are particularly useful for the construction of difficult integer programming problems with known optimal solutions. These problems may then be used to test and/or “benchmark” integer programming codes.

An Application of Facilities Location Theory to the Design of Forest Harvesting Areas

Abstract The problem of designing forest harvesting units and assigning logging equipment to those units can be formulated as a facilities location problem which exhibits a unique structure that has not previously appeared in the literature. This structure is referred to here as a “cascading fixed charge” structure; for small problems, exact solutions can be obtained by means of 0–1 integer programming systems. Problems of practical size for use in forest planning, however, would involve thousands of 0–1 variables and constraints. Such problems are intractable for presently available integer programming codes. This paper presents an approximation algorithm which has been developed to solve facilities location problems exhibiting the special cascading fixed charge structure. Experience with an application of the algorithm to an actual forest planning area is cited.

Integer programming models and codes in the urban environment

It is well known that many real world problems, in particular, many urban problems, can approximately be modelled as linear programs. The representations often become more precise when, in addition, some or all of the variables are integer constrained (e.g. a solution which indicates that 0.67 schools should be built is of little use to the practitioner). Hence, integer programming is of substantial importance in urban science. This article describes several general integer programming models for which efficient computer codes are available. In each case, applications in an urban environment are discussed. This list is not intended to be exhaustive, but rather to acquaint the urbanologist with the models, their possible uses, and the available computer packages.

Design of Optimal Switching Networks by Integer Programming

The design of optimal logic networks is formulated as integer programming (IP) problems. This formulation has the following advantages over other methods of logic design. 1) General feed-forward networks can be dealt with rather than two-level or three-level networks usually treated in conventional switching theory. 2) Network restrictions such as fan-in and fan-out restrictions are easily incorporated. 3) Various gate types such as NOR, NAND, AND-OR combination, NOR-AND combination, and those gates with NOR-OR dual outputs can be treated. 4) Various objectives such as the number of gates and the number of connections are minimized. 5) Incompletely specified functions can be handled without additional difficulty. 6) The formulation can be extended to multiple-output networks. To solve the resulting IP problems, the implicit enumeration method of integer programming is found to be suitable. An IP code ILLIP (Illinois Integer Programming Code) is implemented based on the implicit enumeration by incorporating some new gimmicks such as pseudounderlining. Then the ILLIP is used to solve the IP problems for logical design by making use of the inherent structure of our problems. Various optimal networks are derived by a computer as follows: optimal NOR networks and optimal NOR-AND networks for all functions of up through three variables, one-bit adders with various gate types, and others. These results indicate the computational feasibility of the integer programming approach.

An implicit enumeration program for zero-one integer programming

This paper describes some techniques to improve the speed of the implicit enumeration method for solving zero-one integer programming problems. Among these techniques, the most powerful is the one of using a column vector which works as a tag for each inequality, indicating whether or not the inequality should be checked for the current partial solution. A new condition for underlining a variable and the concept of pseudo-underlining are also proposed. These techniques were implemented in the computer programil lip (ILLinois Integer Programming code). The computational results for different types of problems are discussed.

Comment on Briskin's Note

Recently, in a note by Lawrence E. Briskin [Briskin, Lawrence E. 1970. A note on Trauth and Woolsey's integer programming article. Management Sci. 16(11, July) 651.], it was pointed out that LIP1, an integer programming code, gave a wrong answer. I have now accumulated some years of bitter experience in attempting to solve several hundred integer programming problems with an ever increasing set of algorithms. I am therefore prepared to state that a wrong answer should hardly surprise anyone who has done much computational work in integer programming.

Warehouse Location Under Continuous Economies of Scale

Warehouse Location is a nonconvex programming problem involving the geographic placing and sizing of intermediate facilities in distribution studies. The nonconvexities are caused by economies of scale associated with the cost of building and operating the facilities. A heuristic program has been developed for solving warehouse location problems when these economies are representable by continuous concave functions. The paper discusses the heuristics used and computational experience with the program on “practical” problems. On the basis of two numerical examples for which an optimal solution was obtained through a special purpose experimental mixed integer programming code, it is conjectured (1) that near optimal solutions can be achieved using the heuristic program and (2) that optimal sizing and locating of facilities are very sensitive to the shapes of the warehousing cost functions.

A Computer Code for Integer Solutions to Linear Programs

This paper reports on a new computer code (called “LIP1”) for solving integer programming problems, which will be distributed for general use through SHARE. LIP1 was written as an experimental program for the purpose of reexamining and further testing the efficiency of Gomory's original cutting-plane technique. Many trial problems have now been solved with this code. Of particular interest, however, are 25 test problems that have been solved with other integer programming codes (all these other codes incorporate the method of Gomory's more recent all-integer algorithm). The basis for comparing the performance of the various codes is the number of iterations required to solve each problem. For these 25 test problems LIP1 appears generally as efficient as the other codes tested. Moreover, on the more difficult problems LIP1 generally performed better than the other codes. We conclude that Gomory's original cutting-plane method merits further investigation and development as an efficient computational technique.

Approximate Solutions to the Three-Machine Scheduling Problem

This paper summarizes experimental results from applying several computational methods for solving the classic three-machine scheduling model. The basic mathematical problem is to select an optimal permutation of n items, where the objective function employed is the total amount of processing time elapsing for the completion of all n items on three machines. The methods tested are integer linear programming, ordinary linear programming with answers rounded to integers, a heuristic algorithm, and random sampling. Throughout the experimentation n = 6. A dual integer programming code is tested on six trial problems. Consideration is given to studying the phenomenon of input-form sensitivity. The results are not encouraging and confirm previous experimental evidence as to the strong effect of varying the input form. However, the addition of a powerful bound on the objective function has a significant beneficial effect on the convergence of the dual algorithm. A sampling study of the statistical characteristics of this model is made with 100 sets of data randomly generated. Frequency distributions of minimum and mean total processing time are tabulated and analyzed. On this same set of problems, a linear programming approach is examined for effectiveness in producing nearly optimal schedules. Such an approach tends to give answers that are better than would be obtained by a single permutation drawn at random, but the LP rounded solutions average about an 11 per cent increase over the optimal processing time. A heuristic algorithm based on Johnson's method is tested on 20 of the 100 cases and gives excellent results. Finally attention is turned to the method of randomly sampling the permutations, possibly applying a simple heuristic gradient method to improve each sampled schedule. A feature of this method is that elementary probability theory can be applied to yield corresponding statements about the accuracy guaranteed by the approach.