Large Linear Programming Problems Research Articles

LINEAR PROGRAMMING MODEL OF PRODUCTION, INVENTORY AND DISTRIBUTION PROBLEM BASED ON RANDOM SAMPLING

A linear programming model using the random sampling technique was used for the optimization of the products in the production line of Nigerian Bottling Company (NBC), Lagos, which has many production facilities and multi-products systems. Products from the company are distributed to a number of depots across the country of which the demand for each product is known. The problem of interest involves determining what product/s should be made, how much of each product should be produced and where production should take place. The objectives of the company are to minimize the total cost as well as maximize the total sales revenue based on the set of decisions including demands, capacity restriction and budget constraints. Linear Programming software solver (LiPs) was used to solve large linear programming problems based on the mathematical model developed. The model improved the profit of the company under study by 5%.

A Hybrid Multi-GPU Implementation of Simplex Algorithm with CPU Collaboration

The simplex algorithm has been successfully used for many years in solving linear programming (LP) problems. Due to the intensive computations required (especially for the solution of large LP problems), parallel approaches have also extensively been studied. The computational power provided by the modern GPUs as well as the rapid development of multicore CPU systems have led OpenMP and CUDA programming models to the top preferences during the last years. However, the desired efficient collaboration between CPU and GPU through the combined use of the above programming models is still considered a hard research problem. In the above context, we demonstrate here an excessively efficient implementation of standard simplex, targeting to the best possible exploitation of the concurrent use of all the computing resources, on a multicore platform with multiple CUDA-enabled GPUs. More concretely, we present a novel hybrid collaboration scheme which is based on the concurrent execution of suitably spread CPU-assigned (via multithreading) and GPU-offloaded computations. The experimental results extracted through the cooperative use of OpenMP and CUDA over a notably powerful modern hybrid platform (consisting of 32 cores and two high-spec GPUs – Titan Rtx and Rtx 2080Ti) highlight that the performance of the presented here hybrid GPU/CPU collaboration scheme is clearly superior to the GPU-only implementation under almost all conditions. The corresponding measurements validate the value of using all resources concurrently, even in the case of a multi-GPU configuration platform. Furthermore, the given implementations are completely comparable (and slightly superior in most cases) to other related attempts in the bibliography, and clearly superior to the native CPU implementation with 32 cores. Keywords—Parallel Computing; Linear Programming; Simplex Method; Multicore Platform; Hardware Acceleration, GPGPU; OpenMP; CUDA

Data driven robust optimization for handling uncertainty in supply chain planning models

While addressing supply chain planning under uncertainty, Robust Optimization (RO) is regarded as an efficient and tractable method. As RO involves calculation of several statistical moments or maximum / minimum values involving the objective functions under realizations of these uncertain parameters, the accuracy of this method significantly depends on the efficient techniques to sample the uncertainty parameter space with limited amount of data. Conventional sampling techniques, e.g. box/budget/ellipsoidal, work by sampling the uncertain parameter space inefficiently, often leading to inaccuracies in such estimations. This paper proposes a methodology to amalgamate machine learning and data analytics with RO, thereby making it data-driven. A novel neuro fuzzy clustering mechanism is implemented to cluster the uncertain space such that the exact regions of uncertainty are optimally identified. Subsequently, local density based boundary point detection and Delaunay triangulation based boundary construction enable intelligent Sobol based sampling to sample the uncertain parameter space more accurately. The proposed technique is utilized to explore the merits of RO towards addressing the uncertainty issues of product demand, machine uptime and production cost associated with a multiproduct, and multisite supply chain planning model. The uncertainty in supply chain model is thoroughly analysed by carefully constructing examples and its case studies leading to large scale mixed integer linear and nonlinear programming problems which were efficiently solved in GAMS® framework. Demonstration of efficacy of the proposed method over the box, budget and ellipsoidal sampling method through comprehensive analysis adds to other highlights of the current work.

Optimum Scheduling of Energy Production Using Hybrid Decomposition Methods - A Comparative Study.(Dept.M)

The production of energy exists in different shapes and forms, e.g. thermal, mechanical, electrical or other forms of energy. Each of these forms exists in different types and arts. The production and scheduling of energy types are decision processes which are basically concerned with the adoption or more precisely the allocation of the natural and industrial resources in order to best satisfy marketing and customer requircinents at minimum best economic conditions. There are many reasons that make energy production planning a challenging problem. The variable structure of both the demand and costs, the difficulty to precisely forecast the demand at the very detailed level, the less flexibility to modify the operating conditions, and that there are usually long and uncertain delays in obtaining the industrial resources (new machines, workers training, subcontracting capacities, raw materials item components, etc.) Researchers and seientists have paid a great part of attention for developing scheduling and planning systems to support such decision making processes. Two different approaches for production planning and scheduling are known. The first, called the monolithic approach, formulates the problems as a large scale mixed-integer linear programming problems and is usually solved approximately using Lagrangean relaxation to the mixed integer linear program. The second approach, called the hierarchical approach, partitions the planning and scheduling problem into a hierarchy of smaller subproblems. The upper hierarchy deals with strategic decisions for the planning horizon, while the lower hierarchy deals with the more detailed short-termed scheduling. The computational effort required for the monolithic approach is generally greater than that required for the hierarchical approach. The hierarchical approach may require less detailed demand input data. Also the uncertainty is better treated by the hierarchical approach. In this study, a comparison is presented between two different hybrid procedures for solving in hierarchical energy production planning and scheduling problem. A simplification in one of them is made. In the first procedure, hierarchical subproblems are included in an overall mixed integer linear p rogramming formulation. Accordingly, the problem is partitioned into two hierarchics; families in the lower hierarchy, which, in turn, are aggregated into types in the upper hierarchy. In this procedure, the set of inventory constraints are priced out using Lagrangeau multipliers in the main objective function, which is then partitioned into two subproblems. The first subproblem is an optimal control problem on the family level (lower hierarchy) while the other subproblem is a linear program on the type level (upper hierarchy). The two subproblems are linked together by an inventory aggregation constraint. By this approach, a feedback is included automatically in the solution procedure, but it does not pass directly from the lower level to the upper level since both subproblems are solved in parallel. In the second procedure, instead of using Lagrangean relaxation with respect to a group of constraints in the original model, the model is first partitioned, ill which cuts are priced out by means of a set of Lagrangean multipliers into the subproblem generated on the family level. Accordingly, one subproblem is trivial and the other subproblem is an incapacitated lot-sizing oplinal control problem. The size of this subproblem is exactly the same as that of the equivalent subproblem in the previous procedure. In both procedures, the optimal control subproblem is solved using dynamic programming and the set of Lagrangean multipliers are updated using subgradient optimization algorithm. Results show that both algorithms are quite acceptable and efficient. A comparison between the results of both algorithms shows considerable improvement compared to the result found in the literature. The enhancement of these results may be returned to the fact that step sizes are more suitably chosen and to the simplification done in the solution procedure.

Locating Binding Constraints in LP Problems

In this work, a new method is presented for determining the binding constraints of a general linear maximization problem. The new method uses only objective function values at points which are determined by simple vector operations, so the computational cost is inferior to the corresponding cost of matrix manipulation and/or inversion. This method uses a recently proposed notion for addressing such problems: the average of each constraint. The identification of binding constraints decreases the complexity and the dimension of the problem resulting to a significant decrease of the computational cost comparing to Simplex-like methods. The new method is highly useful when dealing with very large linear programming (LP) problems, where only a relatively small percentage of constraints are binding at the optimal solution, as in many transportation, management and economic problems, since it reduces the size of the problem. The method has been implemented and tested in a large number of LP problems. In LP problems without superfluous constraints, the algorithm was 100% successful in identifying binding constraints, while in a set of large scale LP tested problems that included superfluous constraints, the power of the algorithm considered as statistical tool of binding constraints identification, was up to 90.4%.

Parallelizing the dual revised simplex method

This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems. One approach, called PAMI, extends a relatively unknown pivoting strategy called suboptimization and exploits parallelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible. Computational results show that the performance of PAMI is superior to that of the leading open-source simplex solver, and that SIP complements PAMI in achieving speedup when PAMI results in slowdown. One of the authors has implemented the techniques underlying PAMI within the FICO Xpress simplex solver and this paper presents computational results demonstrating their value. In developing the first parallel revised simplex solver of general utility, this work represents a significant achievement in computational optimization.

Simplex Parallelization in a Fully Hybrid Hardware Platform

The simplex method has been successfully used in solving linear programming (LP) problems for many years. Parallel approaches have also extensively been studied due to the intensive computations required, especially for the solution of large LP problems. Furthermore, the rapid proliferation of multicore CPU architectures as well as the computational power provided by the massive parallelism of modern GPUs have turned CPU / GPU collaboration models increasingly into focus over the last years for better performance. In this paper, a highly scalable implementation framework of the standard full tableau simplex method is first presented, over a hybrid parallel platform which consists of multiple multicore nodes interconnected via a high-speed communication network. The proposed approach is based on the combined use of MPI and OpenMP, adopting a suitable column-based distribution scheme for the simplex tableau. The parallelization framework is then extended in such a way that it can exploit concurrently the full power of the provided resources on a multicore single-node environment with a CUDA-enabled GPU (i.e. using the CPU cores and the GPU concurrently), based on a suitable hybrid multithreading/GPU offloading scheme with OpenMP and CUDA. The corresponding experimental results show that the hybrid MPI+OpenMP based parallelization scheme leads to particularly high speed-up and efficiency values, considerably better than in other competitive approaches, and scaling well even for very large / huge linear problems. Furthermore, the performance of the hybrid multithreading/GPU offloading scheme is clearly superior to both the OpenMP-only and the GPU-only based implementations in almost all cases, which validates the worth of using both resources concurrently. The most important, when it is used in combination with MPI in a multi-node (fully hybrid) environment, it leads to substantial improvements in the speedup achieved for large and very large LP problems.

A depth-first search algorithm to compute elementary flux modes by linear programming.

BackgroundThe decomposition of complex metabolic networks into elementary flux modes (EFMs) provides a useful framework for exploring reaction interactions systematically. Generating a complete set of EFMs for large-scale models, however, is near impossible. Even for moderately-sized models (<400 reactions), existing approaches based on the Double Description method must iterate through a large number of combinatorial candidates, thus imposing an immense processor and memory demand.ResultsBased on an alternative elementarity test, we developed a depth-first search algorithm using linear programming (LP) to enumerate EFMs in an exhaustive fashion. Constraints can be introduced to directly generate a subset of EFMs satisfying the set of constraints. The depth-first search algorithm has a constant memory overhead. Using flux constraints, a large LP problem can be massively divided and parallelized into independent sub-jobs for deployment into computing clusters. Since the sub-jobs do not overlap, the approach scales to utilize all available computing nodes with minimal coordination overhead or memory limitations.ConclusionsThe speed of the algorithm was comparable to efmtool, a mainstream Double Description method, when enumerating all EFMs; the attrition power gained from performing flux feasibility tests offsets the increased computational demand of running an LP solver. Unlike the Double Description method, the algorithm enables accelerated enumeration of all EFMs satisfying a set of constraints.

USING COLUMN GENERATION TECHNIQUE TO ESTIMATE PROBABILITY STATISTICS IN TRANSITION MATRIX OF LARGE SCALE MARKOV CHAIN WITH LEAST ABSOLUTE DEVIATION CRITERIA

The Least Absolute Deviation (LAD) method is the one of many methods used to estimate transition probability matrix of Markov Chain. It can be formu lated as a Linear Programming Problem (LP) and solved using its regular state of the art software. However, when the Markov Chain has a large number of states and historical state probabilities data, the corresponding LP size can be very large reaching c omputer hardware/software limitation. The aim of this study is to apply the Column Generation (CG) technique t o solve this large scale LP and to evaluate its extension b eyond direct hardware/software capabilities. In thi s study, the sample state probabilities data were simulated stat istically and two methods were used to solve the pr oblem. The first method was using ‘linprog’ function in MATLAB to solve the related LP that all decision vari ables were considered simultaneously while the others was the CG method expected to require a much less percentage of all variables. As result effectivenes s, both methods solved all test problems resulting equal LADs each. The CG method required more average time. Nevertheless, less than 30% of decision variables were considered in the CG method. The lesser percentages were found as the problem size grew. Moreover, larger size problems beyond direct use of software were solved using the proposed approach.

An interactive algorithm for Large Scale Bi level Linear Multiple Objective programming problems through TOPSIS approach

In this paper, we extend TOPSIS (Technique for Order Preference by Similarity Ideal Solution) for solving Large Scale Bi-level Linear Multiple Objective Programming (LS-BL-LMOP) problems. In order to obtain a compromise ( satisfactory) solution to the LS-BL-LMOP problems using the proposed TOPSIS approach, a modified formulas for the distance function from the positive ideal solution (PIS ) and the distance function from the negative ideal solution (NIS) are proposed and modeled to include all the objective functions of both the first and the second levels. An interactive decision making algorithm for generating a compromise ( satisfactory) solution through TOPSIS approach is provided where the first level decision maker (FLDM) is asked to specify the relative importance of the objectives. Finally, a numerical example is given to clarify the main results developed in the paper.

Optimal Involvement in Futures Markets of a Power Producer

This paper addresses the optimal involvement in a futures electricity market of a power producer to hedge against the risk of pool price volatility. The considered trading horizon spans one whole year. Recognizing the highly uncertain nature of future pool prices, a stochastic programming framework with recourse is used to model this decision-making problem. The resulting problem is a large scale mixed-integer linear programming problem. Scenario reduction techniques are used to make this problem tractable. Risk is properly modeled using the CVaR methodology. Results from a realistic case study are provided and analyzed. Some conclusions are finally drawn.

Heurísticas para o problema de corte de estoque unidimensional inteiro

O problema de corte de estoque unidimensional consiste em cortar objetos disponíveis para a produção de itens de modo a atender uma demanda especificada, em que apenas uma dimensão é relevante para o corte (barras, bobinas, etc.). O problema pode ser formulado como um problema de programação linear inteira de grande porte, cuja solução ótima, via de regra, não pode ser obtida na prática, quando tipicamente dezenas de itens devem ser produzidas. Neste artigo tratamos o problema de determinar soluções inteiras para o problema de corte de estoque unidimensional, dando atenção especial a problemas com baixa demanda. Revisamos métodos heurísticos bem conhecidos e algumas variações. Esses métodos são comparados em relação à qualidade de suas soluções, número de padrões de corte e tempo computacional. Nossa análise está baseada na resolução de exemplares gerados aleatoriamente.

Opportunistic supply chain formation from qualified partners for a new market demand

This paper addresses the problem of short-term supply chain design using the idle capacities of qualified partners in order to seize a new market opportunity. The new market opportunity is characterized by a deterministic forecast over a planning horizon. The production–distribution process is assumed to be organized in stages or echelons, and each echelon may have several qualified partners willing to participate. Partners within the echelon may differ in idle production capacity, operational cost, storage cost, etc, and we assume that idle capacity may be different from one period to another period. The objective is to design a supply chain by selecting one partner from each echelon to meet the forecasted demand without backlog and best possible production and logistics costs over the given planning horizon. The overall problem is formulated as a large mixed integer linear programming problem. We develop a decomposition-based solution approach that is capable of overcoming the complexity and dimensionality associated with the problem. Numerical results are presented to support the effectiveness of this approach.

Integer Linear Programming Models for Global Routing

Modern integrated circuit design involves the layout of circuits consisting of millions of switching elements or transistors. Due to the sheer complexity of the problem, optimizing the connectivity between transistors is very difficult. The circuit interconnection is the single most important factor in performance criteria such as signal delay, power dissipation, circuit size, and cost. These factors dictate that interconnections, i.e., wires, be made as short as possible. The wire-minimization problem is generally formulated as a sequence of discrete optimization subproblems that are known to be NP-hard. Hence, they can only be solved approximately using meta-heuristics. These methods are computationally expensive and the quality of the solution depends to a great extent on an appropriate choice of starting configuration and modeling techniques. In this paper, new modeling techniques are used to solve the routing problem formulated as an integer programming problem. The main contribution of this paper is a proposed global routing heuristic that combines the wire length, channel congestion, and number of pins in routes to find the best wiring layout of a circuit. By adding information such as channel congestion and the number of pins in each route as well as the wire length, the quality of the solution is improved. In addition, the solutions of the large relaxed linear programming problems are skewed towards a zero-one solution, resulting in faster convergence. The developed LP models in this paper are useful when solving the global routing problem for two reasons; first, the new interior-point algorithms to solve the LP problem are polynomial in time. Second, “near optimal wiring” is obtained in polynomial time without performing randomized rounding.

Approximating the Minimum Number of Maximum Power Users in Ad hoc Networks

Topology control is the problem of assigning transmission power values to the nodes of an ad hoc network so that the induced graph satisfies some specified property. The most fundamental such property is that the network/graph be connected. For connectivity, prior work on topology control gave a polynomial time algorithm for minimizing the maximum power assigned to any node (such that the induced graph is connected). In this paper we study the problem of minimizing the number of maximum power nodes. After establishing that this minimization problem is NP-complete, we focus on approximation algorithms for graphs with symmetric power thresholds. We first show that the problem is reducible in an approximation preserving manner to the problem of assigning power values so that the sum of the powers is minimized. Using known results for that problem, this provides a family of approximation algorithms for the problem of minimizing the number of maximum power nodes with approximation ratios of 5/3 + e for every e > 0. Unfortunately, these algorithms, based on solving large linear programming problems, are not practical. The main results of this paper are practical algorithms with approximation ratios of 7/4 and 5/3 (exactly). In addition, we present experimental results, both on randomly generated networks, and on two networks derived from proximity data associated with the TRANSIMS project of Los Alamos National Labs. Finally, based on the reduction to the problem of minimizing the total power, we describe some additional results for minimizing the number of maximum power users, both for graph properties other than connectivity and for graphs with asymmetric power thresholds.

Hyper-Sparsity in the Revised Simplex Method and How to Exploit it

The revised simplex method is often the method of choice when solving large scale sparse linear programming problems, particularly when a family of closely-related problems is to be solved. Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product. For a significant number of practical problems the result of one or more of these operations is usually sparse, a property we call hyper-sparsity. Analysis of the commonly-used techniques for implementing each step of the revised simplex method shows them to be inefficient when hyper-sparsity is present. Techniques to exploit hyper-sparsity are developed and their performance is compared with the standard techniques. For the subset of our test problems that exhibits hyper-sparsity, the average speedup in solution time is 5.2 when these techniques are used. For this problem set our implementation of the revised simplex method which exploits hyper-sparsity is shown to be competitive with the leading commercial solver and significantly faster than the leading public-domain solver.

A MILP decomposition approach to large scale optimization in structural design of energy supply systems

An optimal structural design method is proposed to determine the structures of energy supply systems in consideration of their multi-period operation. The optimization problem is formulated as a large scale mixed integer linear programming (MILP) problem with binary variables for selection and on/off status of operation of equipment and continuous variables for capacities and load allocation of equipment. The dependence of the performance characteristics of equipment on their capacities and part loads, as well as that of capital costs of equipment on their capacities, are incorporated into the optimization model. A decomposition method of solving MILP problems with the block angular structure is applied to derive a suboptimal solution close to the optimal one in a short computation time. The meaning and validity of the decomposition method are clarified through an analytical study on an energy supply system with a simple structure. The validity and effectiveness of the decomposition method are also clarified in terms of solution optimality and computation time through a numerical study on another energy supply system with a complex structure.

D.C. programming approach for multicommodity network optimization problems with step increasing cost functions

We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows xa such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(xa)-LB)/{f(xa)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.

Portfolio Optimization under Lower Partial Risk Measures

Portfolio management using lower partial risk (downside risk) measures is attracting more attention of practitioners in recent years. The purpose of this paper is to review important characteristics of these riskmeasures and conduct simulation using four alternative measures, lower semi-variance, lower semi-absolute deviation, first order below targetrisk and conditional value-at-risk.We will show that these risk measures are useful to control downside risk whenthe distribution of assets is non-symmetric. Further, we will propose a computational scheme to resolve the difficultyassociated with solving a large dense linear programming problems resulting from these models. We will demonstrate that this method can in fact solve problems consisting of104 assets and 105 scenarios within a practical amount of CPU time.

Economic Viability Study of a Polymer Electrolyte Fuel Cell Cogeneration System for Household Application by an Optimization Approach.

Economic viability on using a polymer electrolyte fuel cell (PEFC) as a cogeneration unit for household application is investigated. The PEFC cogeneration unit is installed with a thermal storage tank. An optimization approach is adopted to rationally evaluate the viability. Design variables such as equipment capacities are determined together with the system's operational strategies so as to minimize the annual total cost subject to satisfy all the possible energy demands. This optimization problem is formulated as a large scale mixed-integer linear programming problem by using appropriate approximation. Numerical studies are carried out with some parameters such as energy efficiency and capital unit cost of the PEFC cogeneration unit, unit cost of natural gas energy charge and load factor of household energy demands. Through the studies, it is found that the relationships between the ecomomic viability of PEFC cogeneration system and the aforementioned parameters are clarified.