MIP modelling of changeovers in production planning and scheduling problems

The goal here is to survey some recent and not so recent work that can be used to improve problem formulations either by a priori reformulation, or by the addition of valid inequalities. The main topic examined is the handling of changeovers, both sequence-independent and -dependent, in production planning and machine sequencing, with in the background the question of how to model time. We first present results for lot-sizing problems, in particular the interval submodular inequalities of Constantino that provide insight into the structure of single item problems with capacities and start-ups, and a unit flow formulation of Karmarkar and Schrage that is effective in modelling changeovers. Then we present various extensions and an application to machine sequencing with the unit flow formulation. We terminate with brief sections on the use of dynamic programming and of time-indexed formulations, which provide two alternative approaches for the treatment of time.     

A Mixed Integer Programming Formulation for the Total Flow Time Single Machine Robust Scheduling Problem with Interval Data

We consider a version of the total flow time single machine scheduling problem where uncertainty about processing times is taken into account. Namely an interval of equally possible processing times is considered for each job, and optimization is carried out according to a robustness criterion. We propose the first mixed integer linear programming formulation for the resulting optimization problem and we explain how some known preprocessing rules can be translated into valid inequalities for this formulation. Computational results are finally presented. Work funded by the Swiss National Science Foundation through project 200020-109854/1.     

Valid Inequalities for the Lasdon-Terjung Production Model

We consider a very simple integer program involving production of a single item and start-up costs for the standard machines first studied by Lasdon and Terjung. Solving directly as an integer program leads to prohibitively large branch and bound trees. Here, we show how using a simple family of valid inequalities and a heuristic procedure to choose one of these inequalities as a cut, it is possible to reduce substantially the size of the tree, and in several cases to eliminate the need for branch and bound. The valid inequalities used are all simple Gomory cuts. However, they are derived from the initial problem formulation rather than from the optimal linear programming tableau.     

Lot-sizing on a tree

For the problem of lot-sizing on a tree with constant capacities, or stochastic lot-sizing with a scenario tree, we present various reformulations based on mixing sets. We also show how earlier results for uncapacitated problems involving (Q,S_Q) inequalities can be simplified and extended. Finally some limited computational results are presented.     

Valid inequalities for separable concave constraints with indicator variables

We study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints. These models reduce to a mixed-integer linear program (MILP) when the concave constraints are ignored, or to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms designed to solve these problems to global optimality, cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the MILP only. We propose a technique to obtain valid inequalities that are based on both the MILP constraints and the concave constraints. We begin by characterizing the convex hull of a four-dimensional set consisting of a single binary indicator variable, a single concave constraint, and two linear inequalities. Using this analysis, we demonstrate how valid inequalities for the single node flow set and for the lot-sizing polyhedron can be “tilted” to give valid inequalities that also account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot-sizing problems with concave costs. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed-integer nonlinear programs.     

0-1 reformulations of the multicommodity capacitated network design problem

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.     

Two-edge connected subgraphs with bounded rings: Polyhedral results and Branch-and-Cut

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid for the problem and show that these inequalities together with the so-called cut inequalities yield an integer programming formulation of the problem in the space of the natural design variables. We then study the polytope associated with that problem and describe further classes of valid inequalities. We give necessary and sufficient conditions for these inequalities to be facet defining. We study the separation problem associated with these inequalities. In particular, we show that the cycle inequalities can be separated in polynomial time when K≤4. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results.     

Machine scheduling with resource dependent processing times

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.     

Two-stage minimax regret robust uncapacitated lot-sizing problems with demand uncertainty

In this paper, we consider the two-stage minimax robust uncapacitated lot-sizing problem with interval uncertain demands. A mixed integer programming formulation is proposed. Even though the robust uncapacitated lot-sizing problem with discrete scenarios is an NP-hard problem, we show that it is polynomial solvable under the interval uncertain demand set.     

On the relative strength of split,triangle and quadrilateral cuts

Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.     

Two row mixed-integer cuts via lifting

Recently Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible non-negative integer variables (giving the two row mixed-integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In the other cases (maximal lattice-free triangles with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the latter family of triangles, we present sufficient conditions to yield an extreme inequality for the two row mixed-integer infinite-group problem.     

Two machine preemptive scheduling problem with release dates,equal processing times and precedence constraints

We consider a scheduling problem with two identical parallel machines and n jobs. For each job we are given its release date when job becomes available for processing. All jobs have equal processing times. Preemptions are allowed. There are precedence constraints between jobs which are given by a (di)graph consisting of a set of outtrees and a number of isolated vertices. The objective is to find a schedule minimizing mean flow time. We suggest an O(n²) algorithm to solve this problem.The suggested algorithm also can be used to solve the related two-machine open shop problem with integer release dates, unit processing times and analogous precedence constraints.     

Capacitated lot-sizing with extensions: a review

The capacitated lot-sizing problem (CLSP) is a standard formulation for big bucket lot-sizing problems with a discrete period segmentation and deterministic demands. We present a literature review on problems that incorporate one of the following extensions in the CLSP: back-orders, setup carry-over, sequencing, and parallel machines. We illustrate model formulations for each of the extensions and also mention the inclusion of setup times, multi-level product structures and overtime in a study. For practitioners, this overview allows to check the availability of successful solution procedures for a specific problem. For scientists, it identifies areas that are open for future research.      

A polyhedral study of the single-item lot-sizing problem with continuous start-up costs

In this work we consider the single-item single-machine lot-sizing problem with continuous start-up costs. A continuous start-up cost is generated in a period whenever there is a nonzero production in the period and the production capacity in the previous period is not saturated. This concept of start-up does not correspond to the standard (discrete) start-up considered in previous models, thus motivating a polyhedral study of this problem. We introduce a natural integer programming formulation for this problem, we study some general properties and facet-inducing inequalities of the associated polytope, and we state relationships with known lotsizing polytopes.     

Ramsey theory and integrality gap for the independent set problem

We discuss the effectiveness of integer programming for solving large instances of the independent set problem. Typical LP formulations, even strengthened by clique inequalities, yield poor bounds for this problem. We show that a strong bound can be obtained by the use of the so-called rank inequalities, which generalize the clique inequalities. For some problems the clique inequalities imply the rank inequalities, and then a strong bound is guaranteed already by the simpler formulation.