A stepwise fuzzy linear programming model with possibility and necessity relations

Linear programming (LP) is the most widely used optimization technique for solving real-life problems because of its simplicity and efficiency. Although conventional LP models require precise data, managers and decision makers dealing with real-world optimization problems often do not have access to exact values. Fuzzy sets have been used in the fuzzy LP (FLP) problems to deal with the imprecise data in the decision variables, objective function and/or the constraints. The imprecisions in the FLP problems could be related to (1) the decision variables; (2) the coefficients of the decision variables in the objective function; (3) the coefficients of the decision variables in the constraints; (4) the right-hand-side of the constraints; or (5) all of these parameters. In this paper, we develop a new stepwise FLP model where fuzzy numbers are considered for the coefficients of the decision variables in the objective function, the coefficients of the decision variables in the constraints and the right-hand-side of the constraints. In the first step, we use the possibility and necessity relations for fuzzy constraints without considering the fuzzy objective function. In the subsequent step, we extend our method to the fuzzy objective function. We use two numerical examples from the FLP literature for comparison purposes and to demonstrate the applicability of the proposed method and the computational efficiency of the procedures and algorithms. © 2013-IOS Press and the authors. All rights reserved.

Fuzzy Multi-Objective Linear Programming and Simulation Approach to the Development of Valid and Realistic Master Production Schedule

Master production schedule (MPS) plays an important role in an integrated production planning system. It converts the strategic planning defined in a production plan into the tactical operation execution. The MPS is also known as a tool for top management to control over manufacture resources and becomes input of the downstream planning levels such as material requirement planning (MRP) and capacity requirement planning (CRP). Hence, inappropriate decision on the MPS development may lead to infeasible execution, which ultimately causes poor delivery performance. One must ensure that the proposed MPS is valid and realistic for implementation before it is released to real manufacturing system. In practice, where production environment is stochastic in nature, the development of MPS is no longer simple task. The varying processing time, random event such as machine failure is just some of the underlying causes of uncertainty that may be hardly addressed at planning stage so that in the end the valid and realistic MPS is tough to be realized. The MPS creation problem becomes even more sophisticated as decision makers try to consider multi-objectives; minimizing inventory, maximizing customer satisfaction, and maximizing resource utilization. This study attempts to propose a methodology for MPS creation which is able to deal with those obstacles. This approach takes into account uncertainty and makes trade off among conflicting multi-objectives at the same time. It incorporates fuzzy multi-objective linear programming (FMOLP) and discrete event simulation (DES) for MPS development.

A fuzzy AHP based integer linear programming model for the multi-criteria transshipment problem

Purpose – The purpose of this research is to develop a holistic approach to maximize the customer service level while minimizing the logistics cost by using an integrated multiple criteria decision making (MCDM) method for the contemporary transshipment problem. Unlike the prevalent optimization techniques, this paper proposes an integrated approach which considers both quantitative and qualitative factors in order to maximize the benefits of service deliverers and customers under uncertain environments. Design/methodology/approach – This paper proposes a fuzzy-based integer linear programming model, based on the existing literature and validated with an example case. The model integrates the developed fuzzy modification of the analytic hierarchy process (FAHP), and solves the multi-criteria transshipment problem. Findings – This paper provides several novel insights about how to transform a company from a cost-based model to a service-dominated model by using an integrated MCDM method. It suggests that the contemporary customer-driven supply chain remains and increases its competitiveness from two aspects: optimizing the cost and providing the best service simultaneously. Research limitations/implications – This research used one illustrative industry case to exemplify the developed method. Considering the generalization of the research findings and the complexity of the transshipment service network, more cases across multiple industries are necessary to further enhance the validity of the research output. Practical implications – The paper includes implications for the evaluation and selection of transshipment service suppliers, the construction of optimal transshipment network as well as managing the network. Originality/value – The major advantages of this generic approach are that both quantitative and qualitative factors under fuzzy environment are considered simultaneously and also the viewpoints of service deliverers and customers are focused. Therefore, it is believed that it is useful and applicable for the transshipment service network design.

Optimistic linear programming gives logarithmic regret for irreducible MDPs

We present an algorithm called Optimistic Linear Programming (OLP) for learning to optimize average reward in an irreducible but otherwise unknown Markov decision process (MDP). OLP uses its experience so far to estimate the MDP. It chooses actions by optimistically maximizing estimated future rewards over a set of next-state transition probabilities that are close to the estimates, a computation that corresponds to solving linear programs. We show that the total expected reward obtained by OLP up to time T is within C(P) log T of the reward obtained by the optimal policy, where C(P) is an explicit, MDP-dependent constant. OLP is closely related to an algorithm proposed by Burnetas and Katehakis with four key differences: OLP is simpler, it does not require knowledge of the supports of transition probabilities, the proof of the regret bound is simpler, but our regret bound is a constant factor larger than the regret of their algorithm. OLP is also similar in flavor to an algorithm recently proposed by Auer and Ortner. But OLP is simpler and its regret bound has a better dependence on the size of the MDP.

Linear programming for large-scale Markov decision problems

We consider the problem of controlling a Markov decision process (MDP) with a large state space, so as to minimize average cost. Since it is intractable to compete with the optimal policy for large scale problems, we pursue the more modest goal of competing with a low-dimensional family of policies. We use the dual linear programming formulation of the MDP average cost problem, in which the variable is a stationary distribution over state-action pairs, and we consider a neighborhood of a low-dimensional subset of the set of stationary distributions (defined in terms of state-action features) as the comparison class. We propose a technique based on stochastic convex optimization and give bounds that show that the performance of our algorithm approaches the best achievable by any policy in the comparison class. Most importantly, this result depends on the size of the comparison class, but not on the size of the state space. Preliminary experiments show the effectiveness of the proposed algorithm in a queuing application.

A linear programming approach for estimating the structure of a sparse linear genetic network from transcript profiling data

Background: A genetic network can be represented as a directed graph in which a node corresponds to a gene and a directed edge specifies the direction of influence of one gene on another. The reconstruction of such networks from transcript profiling data remains an important yet challenging endeavor. A transcript profile specifies the abundances of many genes in a biological sample of interest. Prevailing strategies for learning the structure of a genetic network from high-dimensional transcript profiling data assume sparsity and linearity. Many methods consider relatively small directed graphs, inferring graphs with up to a few hundred nodes. This work examines large undirected graphs representations of genetic networks, graphs with many thousands of nodes where an undirected edge between two nodes does not indicate the direction of influence, and the problem of estimating the structure of such a sparse linear genetic network (SLGN) from transcript profiling data. Results: The structure learning task is cast as a sparse linear regression problem which is then posed as a LASSO (l1-constrained fitting) problem and solved finally by formulating a Linear Program (LP). A bound on the Generalization Error of this approach is given in terms of the Leave-One-Out Error. The accuracy and utility of LP-SLGNs is assessed quantitatively and qualitatively using simulated and real data. The Dialogue for Reverse Engineering Assessments and Methods (DREAM) initiative provides gold standard data sets and evaluation metrics that enable and facilitate the comparison of algorithms for deducing the structure of networks. The structures of LP-SLGNs estimated from the INSILICO1, INSILICO2 and INSILICO3 simulated DREAM2 data sets are comparable to those proposed by the first and/or second ranked teams in the DREAM2 competition. The structures of LP-SLGNs estimated from two published Saccharomyces cerevisae cell cycle transcript profiling data sets capture known regulatory associations. In each S. cerevisiae LP-SLGN, the number of nodes with a particular degree follows an approximate power law suggesting that its degree distributions is similar to that observed in real-world networks. Inspection of these LP-SLGNs suggests biological hypotheses amenable to experimental verification. Conclusion: A statistically robust and computationally efficient LP-based method for estimating the topology of a large sparse undirected graph from high-dimensional data yields representations of genetic networks that are biologically plausible and useful abstractions of the structures of real genetic networks. Analysis of the statistical and topological properties of learned LP-SLGNs may have practical value; for example, genes with high random walk betweenness, a measure of the centrality of a node in a graph, are good candidates for intervention studies and hence integrated computational – experimental investigations designed to infer more realistic and sophisticated probabilistic directed graphical model representations of genetic networks. The LP-based solutions of the sparse linear regression problem described here may provide a method for learning the structure of transcription factor networks from transcript profiling and transcription factor binding motif data.

On the bottleneck linear programming problem

In this paper, the results on primal methods for Bottleneck Linear Programming (BLP) problem are briefly surveyed, the primal method is presented and the degenerate case related to Bottleneck Transportation Problem (BTP) is explicitly considered. The algorithm is based on the idea of using auxiliary coefficients as is done by Garfinkel and Rao [6]. The modification presented for the BTP rectifies the defect in Hammer's method in the case of degenerate basic feasible solution. Illustrative numerical examples are also given.

Finite field computation technique for exact solution of systems of linear equations and interval linear programming problems

An error-free computational approach is employed for finding the integer solution to a system of linear equations, using finite-field arithmetic. This approach is also extended to find the optimum solution for linear inequalities such as those arising in interval linear programming probloms.

A generalized Linear Programming based approach to optimal divisible load scheduling

In this paper we propose a general Linear Programming (LP) based formulation and solution methodology for obtaining optimal solution to the load distribution problem in divisible load scheduling. We exploit the power of the versatile LP formulation to propose algorithms that yield exact solutions to several very general load distribution problems for which either no solutions or only heuristic solutions were available. We consider both star (single-level tree) networks and linear daisy chain networks, having processors equipped with front-ends, that form the generic models for several important network topologies. We consider arbitrary processing node availability or release times and general models for communication delays and computation time that account for constant overheads such as start up times in communication and computation. The optimality of the LP based algorithms is proved rigorously.

An efficient algorithm for linear programming

A simple but efficient algorithm is presented for linear programming. The algorithm computes the projection matrix exactly once throughout the computation unlike that of Karmarkar’s algorithm where in the projection matrix is computed at each and every iteration. The algorithm is best suitable to be implemented on a parallel architecture. Complexity of the algorithm is being studied.

Positional Consensus of Multi-Agent Systems Using Linear Programming Based Decentralized Control with Rectilinear Decision Domain

In this paper we develop a Linear Programming (LP) based decentralized algorithm for a group of multiple autonomous agents to achieve positional consensus. Each agent is capable of exchanging information about its position and orientation with other agents within their sensing region. The method is computationally feasible and easy to implement. Analytical results are presented. The effectiveness of the approach is illustrated with simulation results.

Solving linear programming problems exactly

A linear programming problem in an inequality form having a bounded solution is solved error-free using an algorithm that sorts the inequalities, removes the redundant ones, and uses the p-adic arithmetic. (C) Elsevier Science Inc., 1997

Machine-tool selection and operation allocation in FMS: solving a fuzzy goal-programming model using a genetic algorithm

The decision-making process for machine-tool selection and operation allocation in a flexible manufacturing system (FMS) usually involves multiple conflicting objectives. Thus, a fuzzy goal-programming model can be effectively applied to this decision problem. The paper addresses application of a fuzzy goal-programming concept to model the problem of machine-tool selection and operation allocation with explicit considerations given to objectives of minimizing the total cost of machining operation, material handling and set-up. The constraints pertaining to the capacity of machines, tool magazine and tool life are included in the model. A genetic algorithm (GA)-based approach is adopted to optimize this fuzzy goal-programming model. An illustrative example is provided and some results of computational experiments are reported.

Register Allocation and Optimal Spill code Scheduling in Software Pipelined Loops using 0-1 Integer Linear Programming Formulation

In achieving higher instruction level parallelism, software pipelining increases the register pressure in the loop. The usefulness of the generated schedule may be restricted to cases where the register pressure is less than the available number of registers. Spill instructions need to be introduced otherwise. But scheduling these spill instructions in the compact schedule is a difficult task. Several heuristics have been proposed to schedule spill code. These heuristics may generate more spill code than necessary, and scheduling them may necessitate increasing the initiation interval. We model the problem of register allocation with spill code generation and scheduling in software pipelined loops as a 0-1 integer linear program. The formulation minimizes the increase in initiation interval (II) by optimally placing spill code and simultaneously minimizes the amount of spill code produced. To the best of our knowledge, this is the first integrated formulation for register allocation, optimal spill code generation and scheduling for software pipelined loops. The proposed formulation performs better than the existing heuristics by preventing an increase in II in 11.11% of the loops and generating 18.48% less spill code on average among the loops extracted from Perfect Club and SPEC benchmarks with a moderate increase in compilation time.