Single item lot-sizing with non-decreasing capacities

We consider the single item lot-sizing problem with capacities that are non-decreasing over time. When the cost function is i) non-speculative or Wagner-Whitin (for instance, constant unit production costs and non-negative unit holding costs), and ii) the production set-up costs are non-increasing over time, it is known that the minimum cost lot-sizing problem is polynomially solvable using dynamic programming. When the capacities are non-decreasing, we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot-sizing problem to optimality when the objective function satisfies i) and ii). The formulation is based on mixing set relaxations and reduces to the (known) convex hull of solutions when the capacities are constant over time. We illustrate the use and effectiveness of this improved LP formulation on a new test instances, including instances with and without Wagner-Whitin costs, and with both non-decreasing and arbitrary capacities over time.lot-sizing, mixing set relaxation, compact reformulation, production planning, mixed integer programming

Fuzzy Simultaneous Congruences

We introduce a very natural generalization of the well-known problem of simultaneous congruences. Instead of searching for a positive integer $s$ that is specified by $n$ fixed remainders modulo integer divisors $a_1,\dots,a_n$ we consider remainder intervals $R_1,\dots,R_n$ such that $s$ is feasible if and only if $s$ is congruent to $r_i$ modulo $a_i$ for some remainder $r_i$ in interval $R_i$ for all $i$. This problem is a special case of a 2-stage integer program with only two variables per constraint which is is closely related to directed Diophantine approximation as well as the mixing set problem. We give a hardness result showing that the problem is NP-hard in general. By investigating the case of harmonic divisors, i.e. $a_{i+1}/a_i$ is an integer for all $i<n$, which was heavily studied for the mixing set problem as well, we also answer a recent algorithmic question from the field of real-time systems. We present an algorithm to decide the feasibility of an instance in time $\mathcal{O}(n^2)$ and we show that if it exists even the smallest feasible solution can be computed in strongly polynomial time $\mathcal{O}(n^3)$

Mixing sets linked by bidirected paths

Recently there has been considerable research on simple mixed-integer sets, called mixing sets, and closely related sets arising in uncapacitated and constant capacity lot- sizing. This in turn has led to study of more general sets, called network-dual sets, for which it is possible to derive extended formulations whose projection gives the convex hull of the network-dual set. Unfortunately this formulation cannot be used (in general) to optimize in polynomial time. Furthermore the inequalities definining the convex hull of a network-dual set in the original space of variables are known only for some special cases. Here we study two new cases, in which the continuous variables of the network-dual set are linked by a bi- directed path. In the first case, which is motivated by lot-sizing problems with (lost) sales, we provide a description of the convex hull as the intersection of the convex hulls of 2^n mixing sets, where n is the number of continuous variables of the set. However optimization is polynomial as only n + 1 of the sets are required for any given objective function. In the second case, generalizing single arc flow sets, we describe again the convex hull as an intersection of an exponential number of mixing sets and also give a combinatorial polynomial-time separation algorithm.mixing sets, extended formulations, mixed integer programming, lot-sizing with sales

Mixed n-step MIR inequalities: Facets for the n-mixing set

AbstractGünlük and Pochet [O. Günlük, Y. Pochet, Mixing mixed integer inequalities. Mathematical Programming 90 (2001) 429–457] proposed a procedure to mix mixed integer rounding (MIR) inequalities. The mixed MIR inequalities define the convex hull of the mixing set {(y1,…,ym,v)∈Zm×R+:α1yi+v≥βi,i=1,…,m} and can also be used to generate valid inequalities for general as well as several special mixed integer programs (MIPs). In another direction, Kianfar and Fathi [K. Kianfar, Y. Fathi, Generalized mixed integer rounding inequalities: facets for infinite group polyhedra. Mathematical Programming 120 (2009) 313–346] introduced the n-step MIR inequalities for the mixed integer knapsack set through a generalization of MIR. In this paper, we generalize the mixing procedure to the n-step MIR inequalities and introduce the mixed n-step MIR inequalities. We prove that these inequalities define facets for a generalization of the mixing set with n integer variables in each row (which we refer to as the n-mixing set), i.e. {(y1,…,ym,v)∈(Z×Z+n−1)m×R+:∑j=1nαjyji+v≥βi,i=1,…,m}. The mixed MIR inequalities are simply the special case of n=1. We also show that mixed n-step MIR can generate valid inequalities based on multiple constraints for general MIPs. Moreover, we introduce generalizations of the capacitated lot-sizing and facility location problems, which we refer to as the multi-module problems, and show that mixed n-step MIR can be used to generate valid inequalities for these generalizations. Our computational results on small MIPLIB instances as well as a set of multi-module lot-sizing instances justify the effectiveness of the mixed n-step MIR inequalities

Facets for Continuous Multi-Mixing Set and Its Generalizations: Strong Cuts for Multi-Module Capacitated Lot-Sizing Problem

The research objective of this dissertation is to develop new facet-defining valid inequalities for several new multi-parameter multi-constraint mixed integer sets. These valid inequalities result in cutting planes that significantly improve the efficiency of algorithms for solving mixed integer programming (MIP) problems involving multimodule capacity constraints. These MIPs arise in many classical and modern applications ranging from production planning to cloud computing. The research in this dissertation generalizes cut-generating methods such as mixed integer rounding (MIR), mixed MIR, continuous mixing, n-step MIR, mixed n-step MIR, migling, and n-step mingling, along with various well-known families of cuts for problems such as multi-module capacitated lot-sizing (MMLS), multi-module capacitated facility location (MMFL), and multi-module capacitated network design (MMND) problems. More specifically, in the first step, we introduce a new generalization of the continuous mixing set, referred to as the continuous multi-mixing set, where the coefficients satisfy certain conditions. For each n’ ϵ {1; : : : ; n}, we develop a class of valid inequalities for this set, referred to as the n0-step cycle inequalities, and present their facet-defining properties. We also present a compact extended formulation for this set and an exact separation algorithm to separate over the set of all n’-step cycle inequalities for a given n’ ϵ {1; : : : ; n}. In the next step, we extend the results of the first step to the case where conditions on the coefficients of the continuous multi-mixing set are relaxed. This leads to an extended formulation and a generalization of the n-step cycle inequalities, n ϵ N, for the continuous multi-mixing set with general coefficients. We also show that these inequalities are facet-defining in many cases. In the third step, we further generalize the continuous multi-mixing set (where no conditions are imposed on the coefficients) by incorporating upper bounds on the integer variables. We introduce a compact extended formulation and new families of multi-row cuts for this set, referred to as the mingled n-step cycle inequalities (n ϵ N), through a generalization of the n-step mingling. We also provide an exact separation algorithm to separate over a set of all these inequalities. Furthermore, we present the conditions under which a subset of the mingled n-step cycle inequalities are facet-defining for this set. Finally, in the fourth step, we utilize the results of first step to introduce new families of valid inequalities for MMLS, MMFL, and MMND problems. Our computational results show that the developed cuts are very effective in solving the MMLS instances with two capacity modules, resulting in considerable reduction in the integrality gap, the number of nodes, and total solution time

Mixed n-Step MIR Inequalities, n-Step Conic MIR Inequalities and a Polyhedral Study of Single Row Facility Layout Problem

In this dissertation, we introduce new families of valid inequalities for general linear mixed integer programs (MIPs) and second-order conic MIPs (SOCMIPs) and establish several theoretical properties and computational effectiveness of these inequalities. First we introduce the mixed n-step mixed integer rounding (MIR) inequalities for a generalization of the mixing set which we refer to as the n-mixing set. The n-mixing set is a multi-constraint mixed integer set in which each constraint has n integer variables and a single continuous variable. We then show that mixed n-step MIR can generate multi-row valid inequalities for general MIPs and special structure MIPs, namely, multi- module capacitated lot-sizing and facility location problems. We also present the results of our computational experiments with the mixed n-step MIR inequalities on small MIPLIB instances and randomly generated multi-module lot-sizing instances which show that these inequalities are quite effective. Next, we introduce the n-step conic MIR inequalities for the so-called polyhedral second-order conic (PSOC) mixed integer sets. PSOC sets arise in the polyhedral reformulation of SOCMIPs. We first introduce the n-step conic MIR inequality for a PSOC set with n integer variables and prove that all the 1-step to n-step conic MIR inequalities are facet-defining for the convex hull of this set. We also provide necessary and sufficient conditions for the PSOC form of this inequality to be valid. Then, we use the aforementioned n-step conic MIR facet to derive the n-step conic MIR inequality for a general PSOC set and provide conditions for it to be facet-defining. We further show that the n-step conic MIR inequality for a general PSOC set strictly dominates the n-step MIR inequalities written for the two linear constraints that define the PSOC set. We also prove that the n-step MIR inequality for a linear mixed integer constraint is a special case of the n-step conic MIR inequality. Finally, we conduct a polyhedral study of the triplet formulation for the single row facility layout problem (SRFLP). For any number of departments n, we prove that the dimension of the triplet polytope (convex hull of solutions to the triplet formulation) is n(n - 1)(n - 2)/3. We then prove that several valid inequalities presented in Amaral (2009) for this polytope are facet-defining. These results provide theoretical support for the fact that the linear program solved over these valid inequalities gives the optimal solution for all instances studied by Amaral (2009)