A. Garín

An Approach for Strategic Supply Chain Planning under Uncertainty based on Stochastic 0-1 Programming

We present a two-stage stochastic 0-1 modeling and a related algorithmic approach for Supply Chain Management under uncertainty, whose goal consists of determining the production topology, plant sizing, product selection, product allocation among plants and vendor selection for raw materials. The objective is the maximization of the expected benefit given by the product net profit over the time horizon minus the investment depreciation and operations costs. The main uncertain parameters are the product net price and demand, the raw material supply cost and the production cost. The first stage is included by the strategic decisions. The second stage is included by the tactical decisions. A tight 0-1 model for the deterministic version is presented. A splitting variable mathematical representation via scenario is presented for the stochastic version of the model. A two-stage version of a Branch and Fix Coordination (BFC) algorithmic approach is proposed for stochastic 0-1 program solving, and some computational experience is reported for cases with dozens of thousands of constraints and continuous variables and hundreds of 0-1 variables.

A two-stage stochastic integer programming approach as a mixture of Branch-and-Fix Coordination and Benders Decomposition schemes

We present an algorithmic approach for solving two-stage stochastic mixed 0–1 problems. The first stage constraints of the Deterministic Equivalent Model have 0–1 variables and continuous variables. The approach uses the Twin Node Family (TNF) concept within the so-called Branch-and-Fix Coordination algorithmic framework to satisfy the nonanticipativity constraints, jointly with a Benders Decomposition scheme to solve a given LP model at each TNF integer set. As a pilot case, the structuring of a portfolio of Mortgage-Backed Securities under uncertainty in the interest rate path on a given time horizon is used. Some computational experience is reported.

O(n) Procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates

SummaryIn this paper, we describe computationally efficient procedures for identifying all maximal cliques and non-dominated selected subsets of extensions of minimal covers and alternates that are implied by single 0–1 knapsack constraints. The induced inequalities are satisfied by and 0–1 feasible solution to the knapsack constraint, but are tipically violated by fractional solutions. In addition, the procedures described here are used in conjunction with other constraints to further tighten LP relaxations of 0–1 programs. The complexity of the procedures isO(n).

On multistage Stochastic Integer Programming for incorporating logical constraints in asset and liability management under uncertainty

We present a model for optimizing a mean-risk function of the terminal wealth for a fixed income asset portfolio restructuring with uncertainty in the interest rate path and the liabilities along a given time horizon. Some logical constraints are considered to be satisfied by the assets portfolio. Uncertainty is represented by a scenario tree and is dealt with by a multistage stochastic mixed 0-1 model with complete recourse. The problem is modelled as a splitting variable representation of the Deterministic Equivalent Model for the stochastic model, where the 0-1 variables and the continuous variables appear at any stage. A Branch-and-Fix Coordination approach for the multistage 0–1 program solving is proposed. Some computational experience is reported.

An O(nlog n) procedure for identifying facets of the knapsack polytope

An O(nlogn) procedure is presented for obtaining facets of the knapsack polytope by lifting the inequalities induced by the extensions of strong minimal covers. The procedure does not require any sequential lifting of the inequalities. In contrast, it utilizes the information from the maximal cliques implied by the knapsack constraint for determining the combination of the lifting coefficients to generate each facet.

On using clique overlapping for detecting knapsack constraint redundancy and infeasibility in 0–1 mixed integer programs

SummaryIn this note we present fast procedures for detecting knapsack constraint redundancy and infeasibility in 0–1 mixed integer programs by using information from probing analysis and overlapping clique identification. The new procedures improve current preprocessing techniques for size reduction of integer programs.

O(n log n) procedures for tightening cover inequalities

Abstract We present two procedures for tightening cover induced inequalities in 0-1 programs by using knapsack constraints plus some other information from the program. The tightening is obtained by solving successive knapsack problems with all 0-1 objective function coefficients, and using dominance criteria to avoid the explicit solving of some knapsack problems. The new constraints are 0-1 equivalent to and LP tighter than the original ones. Both procedures have O (n log n) complexity, where n is the number of variables with nonzero coefficients in the knapsack constraint, however one of them can strongly reduce the computational effort. Some computational experience is reported.

On using an automatic scheme for obtaining the convex hull defining inequalities of a Weismantel 0–1 knapsack constraint

In this short note we obtain the full set of inequalities that define the convex hull of a 0–1 knapsack constraint presented in Weismantel (1997). For that purpose we use our O(n) procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates, as well as our schemes for coefficient increase based tightening cover induced inequalities and coefficient reduction based tightening general 0–1 knapsack constraints.

On the Dietrich-Escudero approach for solving the 0–1 knapsack problem with a 0–1 objective function

Abstract In this brief note we demonstrate that the Dietrich-Escudero procedure for solving the 0–1 knapsack problem with all 0–1 objective function coefficients is precisely an application of the Dantzig algorithm for solving the 0–1 LP knapsack problem plus an appropriate rounding of the solution.

Some properties of cliques in 0–1 mixed integer programs

SummaryIn this note we present new properties of cliques induced constraints straintsX(Cr+)-X(Cr-) ≤ 1 - |Cr-| for λ εS, whereS is the set of cliques that are implied by 0–1 mixed integer programs. These properties allow to further fixing of 0–1 variables, to detect instances infeasibility and to imply new cliques.