A tutorial on integer programming mathematical sciences. An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. Aateachinglecturestopics in controlinteger programming 05. Mixed integer programming adds one additional condition that at least one of the variables can only take on integer values. In this paper, we solve a logical puzzle, the battleship puzzle, by integer programming.
Pdf puzzle solving the battleship puzzle as an integer. Irn, where zn is the set of all ndimensional vectors with integer components. A practical, accessible guide to optimization problems with discrete or integer variables integer programming stands out from other textbooks by explaining in clear and simple terms how to construct custommade algorithms or use existing commercial software to obtain optimal or nearoptimal solutions for a variety of realworld problems, such as airline timetables, production line schedules. Integer programming is a combinatorial optimization problem.
Write the linear programming problem corresponding to this situation. Usually, this problem is called the 01 knapsack problem, since it is analogous to a situation in which a. Integer programming is an ideal text for courses in integer mathematical programming whether in operations research, mathematics, engineering, or computer science departments. Integer programming formulation of traveling salesman. But, if some of the basic variables do not have nonnegative integer value, an additional linear constraint called the gomory constraint is generated. Every instance of a combinatorial optimization problem has data, a method for determining which solutions are feasible, and an objective function value for each feasible solution. The lagrangian relaxation method for solving integer programming problems marshall l.
Some food items that are being considered for inclusion in the knap. Special attention is given to the theory behind the algorithms used in stateoftheart solvers. Column generation in integer programming with applications in. Integer programming formulation 1 integer programming introduction. Integer programming formulations for the elementary shortest.
There are 8 courses he is considering to take in the spring term. To cope with this condition we enlarge the ip given above in the following manner. We propose a generic biobjective branchandbound algorithm that uses a problem independent branching rule exploiting available integer solutions and takes advantage of integer objective coe cients. Every instance of a combinatorial optimization problem has data, a method for determining which solutions are feasible, and an. Dec 18, 2009 applied integer programming features a unique emphasis on this point, focusing on problem modeling and solution using commercial software. Integer programming is the class of problems that can be expressed as the optimization of a linear function subject to a set of linear constraints over integer variables. These techniques are outside the scope of our discussion. An introduction to the national football league scheduling. All the decision variables are assumed to be integers. The linear programming model for an integer programming problem is formulated in exactly the same way as the linear programming examples in chapters 2 and. Suppose that jean luc an mba student plans to study 40 hours in a week.
Introduction to integer programming mit opencourseware. In this lecture, show how to solve mixed integer linear programs continue to use ampl format. Pdf integerprogrammingproblemformulation ajay dhamija. We will use the linear programming relaxation to estimate the optimal solutionof an integer programming. A frequent use of integer variables in a model is as a zeroone variable to represent a gonogo decision. If some variables are restricted to be integer and some are not then the problem is a mixed integer programming problem. This makes little sense as a solution, since we can hardly manufacture. Mixed integer linear programming formulation techniques. Many realworld problems could be modeled as linear programs except that some or all of the variables are constrained to be integers. The integrality enforcing capability is perhaps more powerful than the reader at first realizes. It seems hopeful that more efficient integer programming procedures now under development will yield a satisfactory algorithmic solution to the traveling salesman problem, when applied to this model. Basic constructions we begin with a combinatorial optimization problem formulated as the integer program z min cx s.
Integer programming modeling setcovering covereachofmatleast once 1. The feasible region of the linear programming problem is empty. With integrality conditions imposed on the variables it is of importance to limit the number of columns introduced in the integer programming problem. Linear programming problems usually called linear programs, and abbreviated as lps contain a set of decision variables, which are the unknown quantities or decisions that are to be optimized. In biobjective integer optimization the optimal result corresponds to a set of nondominated solutions. Integer programming parameters c, a, b, and decision variables x. It is said to be a mixed integer program when some, but not all, variables are restricted to be. In fact, integer linear programming is an npcomplete problem 63,75.
Pdf a short introduction to integer programming ip. Just for simplicity assume that the feasible set x. A tutorial guide to mixedinteger programming models and. Ones aim in solving logical puzzles is to find the solution by making use of several clues and restrictions. The solution procedures for ips are still under development. Taking an applicationoriented approach, this book addresses the art and science of mathematical modeling related to the mixed integer programming mip framework and discusses the algorithms and associated. The lagrangian relaxation method for solving integer. Note that a negative value for x j would not have a meaning, hence we knowatleastthatx j 0 foralljinj. We wish to select a set of foods that will satisfy a set of daily nutritional requirement. In many settings the term refers to integer linear programming ilp, in which the objective function and the constraints other than the integer constraints are linear integer programming is npcomplete. An introduction to linear programming williams college. More important, perhaps, is the fact that the integer programs that can be solved to provable optimality in reasonable time are much smaller in size. Picking winners in daily fantasy sports using integer.
In any case, the model serves to illustrate how problems of this sort may be succinctly formulated in integer programming terms. Linear integer programming types of integer programming problems pure integer programming problem. Mixed integer linear programming a mixed integer programming mip problem results when some of the variables in your model are realvalued can take on fractional values and some of the variables are integer valued. Such problems are called integer programming problems. The scheme incorporates the two major algorithmic approaches in integer programming, the branch and bound technique and the cutting plane technique. Especially my work on integer programming especially 3, 4, nonlinear duality theory 6, and largescale optimization methods especially 5.
An introduction 2 an integer programming model is one where one or more of the decision variables has to take on an integer value in the final solution solving an integer programming problem is much more difficult than solving an lp problem even the fastest computers can take an excessively long time to solve big integer. Perform quick check by relaxing hard part of problem and solve. The diet problem belongs to the family of blending problems. It is also a valuable reference for industrial users of integer programming and researchers who would like to keep up with advances in the field. For the remainder of the course we will look at how to solve problems in. Integer programming an overview sciencedirect topics. An integer programming problem is a mathematical programming problem in which s. First, we present the general form of a linear programming problem. This will be a solution to the mixed integer programming problem if those variables appearing in the basis for the optimal solution that are required to have integer values. Apr 08, 2017 how to solve an integer linear programming problem using branch and bound duration. However, for real problems this approach will take practically in. Application first published as paperback by john wiley, cornuejols and laurence wolsey on integer programming and combinatorial op solving integer programming problems. Glpk integer solver glpk has a very good integer solver.
This book is an elegant and rigorous presentation of integer programming, exposing the subjects mathematical depth and broad applicability. Chapter 9 lagrangian relaxation for integer programming. This problem is called the linear integerprogramming problem. Integer programming formulations mit opencourseware. The technique finds broad use in operations research. The integer programming problem ip is that of deciding whether there exists an integer solution to a given set of m rational inequalities on n variables. For the integer programming problem given before related to capital budgeting suppose now that we have the additional condition that either project 1 or project 2 must be chosen i. Thecase where the integer variables are restricted to be 0 or 1 comes up surprising often. Integer programming formulation examples capital budgeting extension.
The downside of all this power, however, is that problems with as few as. The lagrangian relaxation method for solving integer programming problems 1862 management science 5012s, pp. Column generation in integer programming with applications. We can attempt to solve a mixed integer programming problem in the same way as a pure integer programming problem. Given that z is an objective function for a maximization problem max z min z. We are thus ready to write a general mathematical model for the factory planning problem, whichlooksasfollows.
In the context of linear and mixed integer programming problems, the. The runtime complexity of parametric integer programming is exponential 50. Among unfathomed subproblems, select one most recently created and create 2. The mathematical representation of the mixed integer programming mip problem is maximize or minimize. We now consider some ways to manipulate problems into the desired form. In fact, such problems usually admit many different ilp formulations. An integer programming problem is a mathematical programming problem in which s zn rn where zn is the set of all ndimensional integer vectors i. Constraint inequalities we rst consider the problem of making all constraints of a linear programming problem. They are listed below with the number of hours per week required.
A mixed integer programming problem is a mathematical programming problem in which at least one, but not all, of the components of x. So we next need to see what a linear programming problem consists of. Aateachinglecturestopics in controlinteger programming. Second, we consider basic approaches that have been developed for solving integer and mixed integer programming problems. We use the simplex method to obtain the optimal solution to the related linear programming problem. After examining the current method, we will attack the problem by formulating and implementing our own model using methods learned in operations research, such as integer programming, to create a potentially better, or more optimal, solution. The traveling salesman problem is a combinatorial optimization problem. An analytical method consisting of what amounts to a series of yes 1 and no 0 answers to arrive at a solution. Problems leading to ip 4 enumerative methods for integer programming wiley, new york, anderson and nash linear programming in infinitedimensional spaces. Solving integer programming with branchandbound technique. Finding a particularly suited one is often a decisive step towards the solution of a problem. For an integer programming model p, the linear programming model we get by dropping the require. When the objective function and constraints are all linear in form, then it is a mixedinteger linear program milp.
In many settings the term refers to integer linear programming ilp, in which the objective function and the constraints other than the integer constraints are linear. Basically all modeling languages incorporates integer variables. Integer programming integer variables extends the possibilities of problem solving. Consider an integer programming problem of the form min xn j1 cjxj s. Pdf an integer programming model for the sudoku problem. Geometry of linear ip find, read and cite all the research. A mixedinteger programming mip problem results when some of the variables in your model are realvalued can take on fractional values and some of the variables are integervalued.
A wide range of problems can be modeled as mixed integer linear programming mip. Jan 04, 2018 in this video, first we give a brief introduction about the difference between the linear programming problem and integer linear programming problem. Fisher university of pennsylvania, philadelphia, pennsylvania one of the most computationally useful ideas of the 1970s is the observation that many hard integer program. An integer programming problem in which all variables are required to be integer is called a pure integer programming problem. When some, but not all decision variables are restricted to be integer, this problem called a mixed integer problem and when all decision variables are integers, its a pure integer program. Integer programming formulation 1 integer programming. Important issues in modeling and solving linear problems are infeasibility and. Constraint inequalities we rst consider the problem of making all constraints of a linear programming problem in the form of strict equalities. Integer programming formulations for the elementary shortest path problem leonardotaccari dipartimento di elettronica, informazione e bioingegneria, politecnico di milano, italy abstract given a directed graph g v,a with arbitrary arc costs, the elementary shortest path problem espp consists of. Problem is that integer programs are in general much more di. Ampl, one can easily change a linear programming problem into an integer program. Math3902 operations research ii integer programming p. How to solve an integer linear programming problem.