Zero-one Variables Research Articles

A Fully Distributed Approach to Optimal Energy Scheduling of Users and Generators Considering a Novel Combined Neurodynamic Algorithm in Smart Grid

A fully distributed microgrid system model is presented in this paper. In the user side, two types of load and plug-in electric vehicles are considered to schedule energy for more benefits. The charging and discharging states of the electric vehicles are represented by the zero-one variables with more flexibility. To solve the nonconvex optimization problem of the users, a novel neurodynamic algorithm which combines the neural network algorithm with the differential evolution algorithm is designed and its convergence speed is faster. A distributed algorithm with a new approach to deal with the inequality constraints is used to solve the convex optimization problem of the generators which can protect their privacy. Simulation results and comparative experiments show that the model and algorithms are effective.

Multi-objective branch and bound

Branch and bound is a well-known generic method for computing an optimal solution of a single-objective optimization problem. Based on the idea “divide to conquer”, it consists in an implicit enumeration principle viewed as a tree search. Although the branch and bound was first suggested by Land and Doig (1960), the first complete algorithm introduced as a multi-objective branch and bound that we identified was proposed by Kiziltan and Yucaoglu (1983). Rather few multi-objective branch and bound algorithms have been proposed. This situation is not surprising as the contributions on the extensions of the components of branch and bound for multi-objective optimization are recent. For example, the concept of bound sets, which extends the classic notion of bounds, has been mentioned by Villarreal and Karwan (1981). But it was only developed for the first time in 2001 by Ehrgott and Gandibleux, and fully defined in 2007.This paper describes a state-of-the-art of multi-objective branch and bound, which reviews concepts, components and published algorithms. It mainly focuses on the contributions belonging to the class of optimization problems who has received the most of attention in this context from 1983 until 2015: the linear optimization problems with zero-one variables and mixed 0–1/continuous variables. Only papers aiming to compute a complete set of efficient solutions are discussed.

Improved logarithmic linearizing method for optimization problems with free-sign pure discrete signomial terms

Free-sign pure discrete signomial (FPDS) terms are vital to and are frequently observed in many nonlinear programming problems, such as geometric programming, generalized geometric programming, and mixed-integer non-linear programming problems. In this study, all variables in the FPDS term are discrete variables. Any improvement to techniques for linearizing FPDS term contributes significantly to the solving of nonlinear programming problems; therefore, relative techniques have continually been developed. This study develops an improved exact method to linearize a FPDS term into a set of linear programs with minimal logarithmic numbers of zero-one variables and constraints. This method is tighter than current methods. Various numerical experiments demonstrate that the proposed method is significantly more efficient than current methods, especially when the problem scale is large.

Portfolio Size in Stochastic Portfolio Networks Using Digital Portfolio Theory

The investment portfolio with stochastic returns can be represented as a maximum flow generalized network with stochastic multipliers. Modern portfolio theory (MPT) [1] provides a myopic short horizon solution to this network by adding a parametric variance constraint to the maximize flow objective function. MPT does not allow the number of securities in solution portfolios to be specified. Integer constraints to control portfolio size in MPT results in a nonlinear mixed integer problem and is not practical for large universes. Digital portfolio theory (DPT) [2] finds a non-myopic long-term solution to the nonparametric variance constrained portfolio network. This paper discusses the long horizon nature of DPT and adds zero-one (0-1) variables to control portfolio size. We find optimal size constrained allocations from a universe of US sector indexes. The feasible size of optimal portfolios depends on risk. Large optimal portfolios are infeasible for low risk investors. High risk investors can increase portfolio size and diversification with little effect on return.

Conflict graphs in implicit enumeration

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x1,…,xn are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y1,…,yn defined by y1=1−x1,…, yn=1−xn respectively. For a unified notation the variables x1,…,xn and y1,…, yn together will be denoted by z1,…,z2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables zi and zj both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes zi and zj will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments zi =1 and zj =1 which explains the name conflict graph. In other words the non-directed edge between the nodes zi and zj codes the fact that the inequality zi+zj≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

Optimum loading of machines in a flexible manufacturing system using a mixed-integer linear mathematical programming model and genetic algorithm

Machine loading problem in a flexible manufacturing system (FMS) encompasses various types of flexibility aspects pertaining to part selection and operation assignments. The evolution of flexible manufacturing systems offers great potential for increasing flexibility by ensuring both cost-effectiveness and customized manufacturing at the same time. This paper proposes a linear mathematical programming model with both continuous and zero-one variables for job selection and operation allocation problems in an FMS to maximize profitability and utilization of system. The proposed model assigns operations to different machines considering capacity of machines, batch-sizes, processing time of operations, machine costs, tool requirements, and capacity of tool magazine. A genetic algorithm (GA) is then proposed to solve the formulated problem. Performance of the proposed GA is evaluated based on some benchmark problems adopted from the literature. A statistical test is conducted which implies that the proposed algorithm is robust in finding near-optimal solutions. Comparison of the results with those published in the literature indicates supremacy of the solutions obtained by the proposed algorithm for attempted model.

The case for strategic oscillation

We study a “hard” optimization problem for metaheuristic search, where a natural neighborhood (that consists of moves for flipping the values of zero-one variables) confronts two local optima, separated by a maximum possible number of moves in the feasible space. Once a descent method reaches the first local optimum, all sequences of feasible moves to reach the second, which is the global optimum, must ultimately pass through solutions that are progressively worse until reaching the worst solution of all, which is adjacent to the global optimum. We show how certain alternative neighborhoods can locate the global more readily, but disclose that each of these approaches encounters serious difficulties by slightly changing the problem formulation. We also identify other possible approaches that seem at first to be promising but turn out to have deficiencies. Finally, we observe that a strategic oscillation approach for transitioning between feasible and infeasible space overcomes these difficulties, reinforcing recent published observations about the utility of solution trajectories that alternate between feasibility and infeasibility. We also sketch features of such an approach that have implications for future research.

A method of solving fully fuzzified linear fractional programming problems

In this paper, we propose a method of solving the fully fuzzified linear fractional programming problems, where all the parameters and variables are triangular fuzzy numbers. We transform the problem of maximizing a function with triangular fuzzy value into a deterministic multiple objective linear fractional programming problem with quadratic constraints. We apply the extension principle of Zadeh to add fuzzy numbers, an approximate version of the same principle to multiply and divide fuzzy numbers and the Kerre’s method to evaluate a fuzzy constraint. The results obtained by Buckley and Feuring in 2000 applied to fractional programming and disjunctive constraints are taken into consideration here. The method needs to add extra zero-one variables for treating disjunctive constraints. In order to illustrate our method we consider a numerical example.

Rewriting integer variables into zero-one variables: Some guidelines for the integer quadratic multi-knapsack problem

This paper is concerned with the integer quadratic multidimensional knapsack problem (QMKP) where the objective function is separable. Our objective is to determine which expansion technique of the integer variables is the most appropriate to solve (QMKP) to optimality using the upper bound method proposed by Quadri et al. (2007). To the best of our knowledge the upper bound method previously mentioned is the most effective method in the literature concerning (QMKP). This bound is computed by transforming the initial quadratic problem into a 0–1 equivalent piecewise linear formulation and then by establishing the surrogate problem associated. The linearization method consists in using a direct expansion initially suggested by Glover (1975) of the integer variables and in applying a piecewise interpolation to the separable objective function. As the direct expansion results in an increase of the size of the problem, other expansions techniques may be utilized to reduce the number of 0–1 variables so as to make easier the solution to the linearized problem. We will compare theoretically the use in the upper bound process of the direct expansion (I) employed in Quadri et al. (2007) with two other basic expansions, namely: (II) a direct expansion with additional constraints and (III) a binary expansion. We show that expansion (II) provides a bound which value is equal to the one computed by Quadri et al (2007). Conversely, we provide the proof of the non applicability of expansion (III) in the upper bound method. More specifically, we will show that if (III) is used to rewrite the integer variables into 0–1 variables then a linear interpolation can not be applied to transform (QMKP) into an equivalent 0–1 piecewise linear problem.

Fixed trading costs, signal processing and stochastic portfolio networks

In this study, we use zero-one variables to control fixed transaction costs independent of trade size in the portfolio selection problem. The optimal solution to the maximum flow, risk constrained stochastic portfolio network is found using Digital Portfolio Theory (DPT). Digital signals describe return processes and power spectral densities describe variances of long and short horizon returns. We find high fixed trading costs reduce size and affect composition of optimal portfolios for small investors or long holding periods. High risk portfolios are more sensitive to trading costs. Optimal portfolios for active traders or large portfolios are largely unaffected by fixed commission costs.

A new approach based on the surrogating method in the project time compression problems

This paper develops a mathematical model for project time compression problems in CPM/PERT type networks. It is noted this formulation of the problem will be an adequate approximation for solving the time compression problem with any continuous and non-increasing time-cost curve. The kind of this model is Mixed Integer Linear Program (MILP) with zero-one variables, and the Benders' decomposition procedure for analyzing this model has been developed. Then this paper proposes a new approach based on the surrogating method for solving these problems. In addition, the required computer programs have been prepared by the author to execute the algorithm. An illustrative example solved by the new algorithm, and two methods are compared by several numerical examples. Computational experience with these data shows the superiority of the new approach.

Global optimization of separable objective functions on convex polyhedra via piecewise-linear approximation

The problem of minimizing a separable nonlinear objective function under linear constraints is considered in this paper. A systematic approach is proposed to obtain an approximately globally optimal solution via piecewise-linear approximation. By means of the new approach a minimum point of the original problem confined in a region where more than one linear piece is needed for satisfactory approximation can be found by solving only one linear programming problem. Hence, the number of linear programming problems to be solved for finding the approximately globally optimal solution may be much less than that of the regions partitioned. In addition, zero-one variables are not introduced in this approach. These features are desirable for efficient computation. The practicability of the approach is demonstrated by an example.

The multicriteria method for environmentally oriented business decision-making

Stimulated by the expressed managers? need for some completed methods for environmental management in enterprises, we present the method for environmentally oriented business decision-making. It is based on simulations where optimization models of business processes are used as scenarios. The possibilities for an integrated approach to environmental protection are introduced and ? decomposed according to the type of the considered element by using zero-one variables ? included in the optimization models. The method is completed for multicriteria decision-making where in the simulations obtained optimal values are included. In a real-life case where the Analytic Hierarchy Process technique is used to evaluate environmentally oriented business processes, special attention is given to criteria and weights: we consider preferences and survey findings on the environmental impact of business processes in the enterprise, survey findings on environmental management in the processing industry, and ecobalances.

On polynomial complexity of a stochastic algorithm for mixed zero-one programs

A stochastic (randomization) algorithm for solving mixed zero-one programs in a yes–no form is presented. Under a set of conditions, we show that any mixed zero-one program is such that either a feasible solution for the problem is obtained or it is classified as infeasible. The algorithm is such that its random running time has its kth moment, k=1,…,j, for any j, bounded by a polynomial in n, the number of zero-one variables.

COMPUTATIONAL METHODS THROUGH GENETIC ALGORITHMS FOR OBTAINING STACKELBERG SOLUTIONS TO TWO-LEVEL MIXED ZERO-ONE PROGRAMMING PROBLEMS

In this paper, we develop computational methods for obtaining Stackelberg solutions to two-level mixed zero-one programming problems in which the decision maker at the upper level controls zero-one variables and the decision maker at the lower level controls real variables. To illustrate two-level mixed zero-one programming problems, we formulate a facility location and transportation problem as a two-level mixed zero-one programming problem. We develop computational methods through genetic algorithms for obtaining Stackelberg solutions. To demonstrate the feasibility and efficiency of the proposed methods, computational experiments are carried out and comparisons between the methods based on the branch-and-bound techniques and the proposed methods are provided.

A time-cost optimization of the national postal distribution network

The postal distribution network in the Czech Republic consists actually of 69 local transit centres (according to the administrative division of the Czech Republic). Each of them operates the post offices in the given region. Some of the important local transit centres can be selected as sorting centres—transhipment points for postal consignments. The aim of this paper is to present a bicriterial optimization model for the selection of the subset of transhipment points from the set of transit centres and choosing their appropriate equipment, such as sorting machines. Another important output of the model is the assignment of the transit centres to the sorting centres. There are two basic optimization criteria defined in this model: minimization of the sum of investment and operational costs connected with building and operating of the sorting centres and the maximization of the percentage of postal consignments delivered to the appropriate post office in the desired time (next day). The proposed model contains several thousands of zero-one variables and constraints. The model has been verified, and the results accepted. Copyright © 1999 John Wiley & Sons, Ltd.

Multigroup Discriminant Analysis Using Linear Programming

In this paper we introduce a nonparametric linear programming formulation for the general multigroup classification problem. Previous research using linear programming formulations has either been limited to the two-group case, or required complicated constraints and many zero-one variables. We develop general properties of our multigroup formulation and illustrate its use with several small example problems and previously published real data sets. A comparative analysis on the real data sets shows that our formulation may offer an interesting robust alternative to parametric statistical formulations for the multigroup discriminant problem.

A simple Tabu Search method to solve the mixed-integer linear bilevel programming problem

Multilevel programming is characterized as mathematical programming to solve decentralized planning problems. The models partition control over decision variables among ordered levels within a hierarchical planning structure of which the linear bilevel form is a special case of a multilevel programming problem. In a system with such a hierarchical structure, the high-level decision making situations generally require inclusion of zero-one variables representing ‘yes-no’ decisions. We provide a mixed-integer linear bilevel programming formulation in which zero-one decision variables are controlled by a high-level decision maker and real-value decision variables are controlled by a low-level decision maker. An algorithm based on the short term memory component of Tabu Search, called Simple Tabu Search, is developed to solve the problem, and two supplementary procedures are proposed that provide variations of the algorithm. Computational results disclose that our approach is effective in terms of both solution quality and efficiency.

A hybrid dynamic programming/branch-and-bound algorithm for the multiple-choice knapsack problem

Dynamic programming and branch-and-bound methodologies are combined to produce a hybrid algorithm for the multiple-choice knapsack problem. Lagrangian duality is used in a computationally efficient manner to compute tight bounds on every active node in the search tree. The use of Lagrangian duality also enables the use of a reduction procedure to reduce the size of the problem for the enumeration phase. Computational experience with up to 200 multiple-choice sets and 20000 zero-one variables is reported. The computational experience indicates that the resulting algorithm is faster than the best published algorithm and is simpler to code.

The convex hull of a linear congruence relation in zero-one variables

A class of optimization problems over subsets of zero-one vectors of then-dimensional unit cube given by a special linear congruence relation is considered. The general problem is formulated as a zero-one linear program, minimal and complete descriptions of the associated polytopes by linear inequalities are derived and an\(\mathcal{O}(n \log n)\) time algorithm for the optimization problems is given. Since the number of inequalities that completely describe the polytope grows exponentially withn, we also give a separation algorithm that identifies violated inequalities in time\(\mathcal{O}(n^2 )\). A particular variation of the bin packing problem is a special case of our problem and can thus be solved in polynomial time.