Integrated supply chain planning under uncertainty using an improved stochastic approach

This paper proposes an integrated model and a modified solution method for solving supply chain network design problems under uncertainty. The stochastic supply chain network design model is provided as a two-stage stochastic program where the two stages in the decision-making process correspond to the strategic and tactical decisions. The uncertainties are mostly found in the tactical stage because most tactical parameters are not fully known when the strategic decisions have to be made. The main uncertain parameters are the operational costs, the customer demand and capacity of the facilities. In the improved solution method, the sample average approximation technique is integrated with the accelerated Benders’ decomposition approach to improvement of the mixed integer linear programming solution phase. The surrogate constraints method will be utilized to acceleration of the decomposition algorithm. A computational study on randomly generated data sets is presented to highlight the efficiency of the proposed solution method. The computational results show that the modified sample average approximation method effectively expedites the computational procedure in comparison with the original approach.     

A stochastic programming approach for supply chain network design under uncertainty

This paper proposes a stochastic programming model and solution algorithm for solving supply chain network design problems of a realistic scale. Existing approaches for these problems are either restricted to deterministic environments or can only address a modest number of scenarios for the uncertain problem parameters. Our solution methodology integrates a recently proposed sampling strategy, the sample average approximation (SAA) scheme, with an accelerated Benders decomposition algorithm to quickly compute high quality solutions to large-scale stochastic supply chain design problems with a huge (potentially infinite) number of scenarios. A computational study involving two real supply chain networks are presented to highlight the significance of the stochastic model as well as the efficiency of the proposed solution strategy.     

Tight bounds on indefinite separable singly-constrained quadratic programs in linear-time

This paper focuses on solving two-stage stochastic mixed integer programs (SMIPs) with general mixed integer decision variables in both stages. We develop a decomposition algorithm in which the first-stage approximation is solved by a branch-and-bound algorithm with its nodes inheriting Benders’ cuts that are valid for their ancestor nodes. In addition, we develop two closely related convexification schemes which use multi-term disjunctive cuts to obtain approximations of the second-stage mixed-integer programs. We prove that the proposed methods are finitely convergent. One of the main advantages of our decomposition scheme is that we use a Benders-based branch-and-cut approach in which linear programming approximations are strengthened sequentially. Moreover as in many decomposition schemes, these subproblems can be solved in parallel. We also illustrate these algorithms using several variants of an SMIP example from the literature, as well as a new set of test problems, which we refer to as Stochastic Server Location and Sizing. Finally, we present our computational experience with previously known examples as well as the new collection of SMIP instances. Our experiments reveal that our algorithm is able to produce provably optimal solutions (within an hour of CPU time) even in instances for which a highly reliable commercial MIP solver is unable to provide an optimal solution within an hour of CPU time.     

Proximity Benders: a decomposition heuristic for stochastic programs

In this paper we present a heuristic approach to two-stage mixed-integer linear stochastic programming models with continuous second stage variables. A common solution approach for these models is Benders decomposition, in which a sequence of (possibly infeasible) solutions is generated, until an optimal solution is eventually found and the method terminates. As convergence may require a large amount of computing time for hard instances, the method may be unsatisfactory from a heuristic point of view. Proximity search is a recently-proposed heuristic paradigm in which the problem at hand is modified and iteratively solved with the aim of producing a sequence of improving feasible solutions. As such, proximity search and Benders decomposition naturally complement each other, in particular when the emphasis is on seeking high-quality, but not necessarily optimal, solutions. In this paper, we investigate the use of proximity search as a tactical tool to drive Benders decomposition, and computationally evaluate its performance as a heuristic on instances of different stochastic programming problems.     

A constraint generation scheme to probabilistic linear problems with an application to power system expansion planning

In this paper, we first describe a constraint generation scheme for probabilistic mixed integer programming problems. Next, we present a decomposition approach to the peak capacity expansion planning of interconnected hydrothermal generating systems, with bounds on the transmission capacity between the regions. The objective is to minimize investments in generating units and interconnection links, subject to constraints on supply reliability. The problem is formulated as a stochastic integer program. The constraint generation scheme, which is similar to Benders decomposition, is applied in the solution of the peak capacity expansion problem. The master problem in this decomposition scheme is an integer program, solved by implicit enumeration. The operating subproblem corresponds to a stochastic network flow problem, and is solved by a maximum flow algorithm and Monte Carlo simulation. The approach is illustrated through a case study involving the expansion of the system of the Brazilian Southeastern region.     

Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs

This paper considers deterministic global optimization of scenario-based, two-stage stochastic mixed-integer nonlinear programs (MINLPs) in which the participating functions are nonconvex and separable in integer and continuous variables. A novel decomposition method based on generalized Benders decomposition, named nonconvex generalized Benders decomposition (NGBD), is developed to obtain ??-optimal solutions of the stochastic MINLPs of interest in finite time. The dramatic computational advantage of NGBD over state-of-the-art global optimizers is demonstrated through the computational study of several engineering problems, where a problem with almost 150,000 variables is solved by NGBD within 80 minutes of solver time.     

L-shaped decomposition of two-stage stochastic programs with integer recourse

We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.     

On the convergence of cross decomposition

Cross decomposition is a recent method for mixed integer programming problems, exploiting simultaneously both the primal and the dual structure of the problem, thus combining the advantages of Dantzig—Wolfe decomposition and Benders decomposition. Finite convergence of the algorithm equipped with some simple convergence tests has been proved. Stronger convergence tests have been proposed, but not shown to yield finite convergence.In this paper cross decomposition is generalized and applied to linear programming problems, mixed integer programming problems and nonlinear programming problems (with and without linear parts). Using the stronger convergence tests finite exact convergence is shown in the first cases. Unbounded cases are discussed and also included in the convergence tests. The behaviour of the algorithm when parts of the constraint matrix are zero is also discussed. The cross decomposition procedure is generalized (by using generalized Benders decomposition) in order to enable the solution of nonlinear programming problems.     

Decomposition methods for the two-stage stochastic Steiner tree problem

A new algorithmic approach for solving the stochastic Steiner tree problem based on three procedures for computing lower bounds (dual ascent, Lagrangian relaxation, Benders decomposition) is introduced. Our method is derived from a new integer linear programming formulation, which is shown to be strongest among all known formulations. The resulting method, which relies on an interplay of the dual information retrieved from the respective dual procedures, computes upper and lower bounds and combines them with several rules for fixing variables in order to decrease the size of problem instances. The effectiveness of our method is compared in an extensive computational study with the state-of-the-art exact approach, which employs a Benders decomposition based on two-stage branch-and-cut, and a genetic algorithm introduced during the DIMACS implementation challenge on Steiner trees. Our results indicate that the presented method significantly outperforms existing ones, both on benchmark instances from literature, as well as on large-scale telecommunication networks.     

Multicut Benders decomposition algorithm for process supply chain planning under uncertainty

In this paper, we present a multicut version of the Benders decomposition method for solving two-stage stochastic linear programming problems, including stochastic mixed-integer programs with only continuous recourse (two-stage) variables. The main idea is to add one cut per realization of uncertainty to the master problem in each iteration, that is, as many Benders cuts as the number of scenarios added to the master problem in each iteration. Two examples are presented to illustrate the application of the proposed algorithm. One involves production-transportation planning under demand uncertainty, and the other one involves multiperiod planning of global, multiproduct chemical supply chains under demand and freight rate uncertainty. Computational studies show that while both the standard and the multicut versions of the Benders decomposition method can solve large-scale stochastic programming problems with reasonable computational effort, significant savings in CPU time can be achieved by using the proposed multicut algorithm.     

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. The methodology proposed allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour and a half, outperforming other state-of-the-art methods, which exhausted the (144 GB of) available memory in the largest instances.     

Cross decomposition for mixed integer programming

Many methods for solving mixed integer programming problems are based either on primal or on dual decomposition, which yield, respectively, a Benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via Lagrangean relaxation. These methods exploit either the primal or the dual structure of the problem. We propose a new approach, cross decomposition, which allows exploiting simultaneously both structures. The development of the cross decomposition method captures profound relationships between primal and dual decomposition. It is shown that the more constraints can be included in the Langrangean relaxation (provided the duality gap remains zero), the fewer the Benders cuts one may expect to need. If the linear programming relaxation has no duality gap, only one Benders cut is needed to verify optimality.     

Inexact solution of NLP subproblems in MINLP

In the context of convex mixed integer nonlinear programming (MINLP), we investigate how the outer approximation method and the generalized Benders decomposition method are affected when the respective nonlinear programming (NLP) subproblems are solved inexactly. We show that the cuts in the corresponding master problems can be changed to incorporate the inexact residuals, still rendering equivalence and finiteness in the limit case. Some numerical results will be presented to illustrate the behavior of the methods under NLP subproblem inexactness.     

Stochastic uncapacitated hub location

We study stochastic uncapacitated hub location problems in which uncertainty is associated to demands and transportation costs. We show that the stochastic problems with uncertain demands or dependent transportation costs are equivalent to their associated deterministic expected value problem (EVP), in which random variables are replaced by their expectations. In the case of uncertain independent transportation costs, the corresponding stochastic problem is not equivalent to its EVP and specific solution methods need to be developed. We describe a Monte-Carlo simulation-based algorithm that integrates a sample average approximation scheme with a Benders decomposition algorithm to solve problems having stochastic independent transportation costs. Numerical results on a set of instances with up to 50 nodes are reported.     

Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs

We consider a class of two-stage stochastic integer programs with binary variables in the first stage and general integer variables in the second stage. We develop decomposition algorithms akin to the $L$ -shaped or Benders’ methods by utilizing Gomory cuts to obtain iteratively tighter approximations of the second-stage integer programs. We show that the proposed methodology is flexible in that it allows several modes of implementation, all of which lead to finitely convergent algorithms. We illustrate our algorithms using examples from the literature. We report computational results using the stochastic server location problem instances which suggest that our decomposition-based approach scales better with increases in the number of scenarios than a state-of-the art solver which was used to solve the deterministic equivalent formulation.     

Mean value cross decomposition applied to integer programming problems

The mean value cross decomposition method for linear programming problems is a modification of ordinary cross decomposition, that eliminates the need for using the Benders or Dantzig-Wolfe master problems. As input to the dual subproblem the average of a part of all known dual solutions of the primal subproblem is used, and as input to the primal subproblem the average of a part of all known primal solutions of the dual subproblem. In this paper we study the lower bounds on the optimal objective function value of (linear) pure integer programming problems obtainable by the application of mean value cross decomposition, and find that this approach can be used to get lower bounds ranging from the bound obtained by the LP-relaxation to the bound obtained by the Lagrangean dual. We examplify by applying the technique to the clustering problem and give some preliminary computational results.     

Dual decomposition in stochastic integer programming

We present an algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation. The approach can be applied to multi-stage problems with mixed-integer variables in each time stage. Numerical experience is presented for some two-stage test problems.     

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.     

A binary decision diagram based algorithm for solving a class of binary two-stage stochastic programs

We consider a special class of two-stage stochastic integer programming problems with binary variables appearing in both stages. The class of problems we consider constrains the second-stage variables to belong to the intersection of sets corresponding to first-stage binary variables that equal one. Our approach seeks to uncover strong dual formulations to the second-stage problems by transforming them into dynamic programming (DP) problems parameterized by first-stage variables. We demonstrate how these DPs can be formed by use of binary decision diagrams, which then yield traditional Benders inequalities that can be strengthened based on observations regarding the structure of the underlying DPs. We demonstrate the efficacy of our approach on a set of stochastic traveling salesman problems.