Zero-one Linear Programming Research Articles

Numerical experiments with LP formulations of the maximum clique problem

The maximum clique problems calls for determining the size of the largest clique in a given graph. This graph problem affords a number of zero-one linear programming formulations. In this case study we deal with some of these formulations. We consider ways for tightening the formulations. We carry out numerical experiments to see the improvements the tightened formulations provide.

Development of an exact method for zero-one linear programming model

The paper presents a new method for solving the 0–1 linear programming problems (LPs). The general 0–1 LPs are believed to be NP-hard and a consistent, efficient general-purpose algorithm for these models has not been found so far. Cutting planes and branch and bound approaches were the earliest exact methods for the 0–1 LP. Unfortunately, these methods on their own failed to solve the 0–1 LP model consistently and efficiently. The hybrids that are a combination of heuristics, cuts, branch and bound and pricing have been used successfully for some 0–1 models. The main challenge with these hybrids is that these hybrids cannot completely eliminate the threat of combinatorial explosion for very large practical 0–1 LPs. In this paper, a technique to reduce the complexity of 0–1 LPs is proposed. The given problem is used to generate a simpler version of the problem, which is then solved in stages in such a way that the solution obtained is tested for feasibility and improved at every stage until an optimal solution is found. The new problem generated has a coefficient matrix of 0 s and 1 s only. From this study, it can be concluded that for every 0–1 LP with a feasible optimal solution, there exists another 0–1 LP (called a double in this paper) with exactly the same optimal solution but different constraints. The constraints of the double are made up of only 0 s and 1 s. It is not easy to determine this double 0–1 LP by mere inspection but can be obtained in stages as given in the numerical illustration presented in this paper. The 0–1 integer programming models have applications in so many areas of business. These include large economic/financial models, marketing strategy models, production scheduling and labor force planning models, computer design and networking models, military operations, agriculture, wild fire fighting, vehicle routing and health care and medical models

A Goal Programming Model for Staff Scheduling at a Telecommunications Center

In this paper, a mathematical model is developed to solve a staff scheduling problem for a telecommunications center. Currently, weekly schedules are manually produced. The manual nature of the process and the large number of constraints and goals lead to a situation where the used schedules are both inefficient and unfair. A zero-one linear goal programming model is suggested to find an optimized cyclical schedule. The center objectives as well as the engineers’ preferences are taken into account. The developed model had to produce balanced schedules that provide the required coverage while satisfying fairness considerations, in terms of weekends off, working night shifts, isolated days on, and isolated days off. A staffing analysis and mathematical properties have been developed to find the optimal parameters of the staff scheduling model. A 6-week scheduling period has been suggested instead of the current weekly period. Work patterns have been suggested to improve schedules quality. These work patterns have been mathematically formulated as a set of soft constraints. The suggested mathematical model has been implemented using Lingo software. The optimal cyclical schedule has been found. It significantly increases both efficiency and staff satisfaction. The suggested approach can be used for any similar staff scheduling problem.

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.

Water allocation and optimization of minor canal operation

In this study a mathematical simulation of canal operation is presented for optimizing water allocation within a multiple crop rotation canal. Zero-one linear programming algorithm was made for optimal flow regulation and to optimize irrigation water allocation and sequencing of different outlets in the irrigation canal under the constraints of fixed canal capacity and irrigation interval. The model was coded in a personal computer using Excel – visual basic application language (VBA) and designed to serve as a decision-making tool for operating an existing canal or for appropriate sizing of canal capacity (flow rate and cross-sectional area) of a new one. The model is verified statistically in comparison with published models. The allocation model is applied to the real cases of wet and dry regions of Gezira Scheme – Sudan for sequencing canal outlets for the cases of early and peak season at Sunni Minor canal and Ugud Minor canal. The model sensitivity to changes in outlet inflow rate (working time per outlet) was made for the said two cases. This indicated the model capabilities to effectively provide a constant flow rate into the canal during the operating period, and consequently minimize the need to frequently adjust the settings of canal head regulator. The model capabilities to optimize water scheduling and allocation process can be used to save irrigation water during early stages of crop growth and to upgrade the operation of minor canals in the irrigated schemes in the Sudan.

Linear and quadratic programming approaches for the general graph partitioning problem

The graph partitioning problem is to partition the vertex set of a graph into a number of nonempty subsets so that the total weight of edges connecting distinct subsets is minimized. Previous research requires the input of cardinalities of subsets or the number of subsets for equipartition. In this paper, the problem is formulated as a zero-one quadratic programming problem without the input of cardinalities. We also present three equivalent zero-one linear integer programming reformulations. Because of its importance in data biclustering, the bipartite graph partitioning is also studied. Several new methods to determine the number of subsets and the cardinalities are presented for practical applications. In addition, hierarchy partitioning and partitioning of bipartite graphs without reordering one vertex set, are studied.

Robust portfolio modeling with incomplete cost information and project interdependencies

Abstract Robust portfolio modeling (RPM) [Liesio, J., Mild, P., Salo, A., 2007. Preference programming for robust portfolio modeling and project selection. European Journal of Operational Research 181, 1488–1505] supports project portfolio selection in the presence of multiple evaluation criteria and incomplete information. In this paper, we extend RPM to account for project interdependencies, incomplete cost information and variable budget levels. These extensions lead to a multi-objective zero-one linear programming problem with interval-valued objective function coefficients for which all non-dominated solutions are determined by a tailored algorithm. The extended RPM framework permits more comprehensive modeling of portfolio problems and provides support for advanced benefit–cost analyses. It retains the key features of RPM by providing robust project and portfolio recommendations and by identifying projects on which further attention should be focused. The extended framework is illustrated with an example on product release planning.

Energy balance optimization of sugarcane crop residual biomass

This work presents a mathematical model for helping mills choose sugarcane varieties for planting. It maximizes crop residual biomass energy balance by considering the difference between generated and consumed energy in the process of transferring this biomass from the field to the processing center; it takes into account enterprise demand restrictions and cane planting area. For this full zero-one linear programming techniques were proposed. The model is viable for choosing sugarcane varieties that would benefit sugarcane production and industrial systems, by reducing crop residue and increasing final energy production.

OPTIMAL SELECTION OF COLLIERIES FOR RECONSTRUCTION: AN APPLICATION OF FUZZY 0-1 LINEAR PROGRAMMING

Zero-one linear programming is a well-accepted approach for decision making for many real world problems. Optimal selection of collieries from a set of collieries for reconstruction is a policy of the concerned authority. The selected collieries are funded for reconstruction based on certain criteria. No colliery will be partially funded, as it is not possible to reconstruct the colliery partially so it is required either to select or reject a colliery for reconstruction. Further, in the real field generally the data available are imprecise in nature. The present paper is concerned with the application of fuzzy 0-1 linear programming to select optimally a set of collieries from a given set of colliery projects for reconstruction with an optimal planning under existing constraints and fuzzy goals. The reconstruction problem is posed as multi objective decision making problem. The application potentiality of the model has been examined by using it in the context of a group of collieries need reconstruction investments in the Jharia Coalfield, Jharkhand, INDIA.

Efficient Solutions for Special Zero-One Programming Problems

A zero-one linear programming is under consideration. It has been proved that for special structures and values of the parameters, the solution of the linear relaxation of the problem is integral and can be either predetermined or computed efficiently. In general, a tight upper bound is provided in order to establish an efficient procedure for solving the problem. The results may have practical implementations in knowledge—management, data-mining, network flow, graph theory, reliability and statistical studies.

Least trimmed squares regression, least median squares regression, and mathematical programming

In this paper, we study LTS and LMS regression, two high breakdown regression estimators, from an optimization point of view. We show that LTS regression is a nonlinear optimization problem that can be treated as a concave minimization problem over a polytope. We derive several important properties of the corresponding objective function that can be used to obtain algorithms for the exact solution of LTS regression problems, i.e., to find a global optimum to the problem. Because of today's limited problem-solving capabilities in exact concave minimization, we give an easy-to-implement pivoting algorithm to determine regression parameters corresponding to local optima of the LTS regression problem. For the LMS regression problem, we briefly survey the existing solution methods which are all based on enumeration. We formulate the LMS regression problem as a mixed zero-one linear programming problem which we analyze in depth to obtain theoretical insights required for future algorithmic and computational work.

The optimal location of replicas in a network using a READ-ONE-WRITE-ALL policy

We consider the problem of locating replicas in a network to minimize communications costs. Under the assumption that the READ-ONE-WRITE-ALL policy is used to ensure data consistency, an optimization problem is formulated in which the cost function estimates the total communications costs. The paper concentrates on the study of the optimal communications cost as a function of the ratio between the frequency of the read and write operations. The problem is reformulated as a zero-one linear programming problem, and its connection to the p-median problem is explained. The general problem is proved to be NP-complete. For path graphs a dynamic programming algorithm for the problem is presented.

A continuous approch for globally solving linearly constrained quadratic

In a continuous approach we propose an efficient method for globally solving linearly constrained quadratic zero-one programming considered as a d.c. (difference of onvex functions) program. A combination of the d.c. optimization algorithm (DCA) which has a finite convergence, and the branch-and-bound scheme was studied. We use rectangular bisection in the branching procedure while the bounding one proceeded by applying d.c.algorithms from a current best feasible point (for the upper bound) and by minimizing a well tightened convex underestimation of the objective function on the current subdivided domain (for the lower bound). DCA generates a sequence of points in the vertex set of a new polytope containing the feasible domain of the problem being considered. Moreover if an iterate is integral then all following iterates are integral too.Our combined algorithm converges so quite often to an integer approximate solution.Finally, we present computational results of several test problems with up to 1800 variables which prove the efficiency of our method, in particular, for linear zero-one programming

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.

Obtaining coteries that optimize the availability of replicated databases

Techniques for implementing the coterie scheme and for obtaining optimal coteries for a system are presented. Central to the techniques is the notion of an acceptance set, which is an alternative representation of the information contained in a coterie. Using this concept, the coterie scheme can be implemented efficiently, and an optimal coterie for a system can be obtained more directly. The problem of determining an optimal acceptance set is formulated as a sparse zero-one linear programming problem. Hence, the optimization problem can be handled using the very rich class of existing techniques for solving such problems. Experimental results indicate that the optimization approach is feasible for up to eight nodes at least. The ways in which the scheme and the optimization approach can be used for systems that distinguish between read and write operations are indicated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new global router using zero-one integer linear programming techniques for sea-of-gates and custom logic arrays

A global routing method is proposed for sea-of-gates (SOG) arrays and custom logic arrays based on a zero-one linear integer programming technique. This method overcomes the various difficulties due to numerous physical constraints faced by many heuristic routing algorithms. In the proposed approach, all the physical constraints are rigorously modeled in the constraint equations. Alternatives for all the nets are considered simultaneously with the routing environment, instead of being routed in a particular net-ordering. An optimal set of global paths routing all the nets is found with well established zero-one linear integer programming methods. The global router has run successfully on industrial SOG circuits, including 108,000 transistor circuits with 9400 nets, achieving 100% routing and high packing densities, which compare favorably with recently reported results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut

We present various techniques for automatically improving the LP-representation of general zero-one linear programming problems. These include detection of redundant rows and blatant infeasibilities, coefficient reduction using the Euclidean algorithm, optimality fixing and variable elimination. Extensions to the case where special-ordered-set constraints are present are discussed as well. A summary of the branch-and-cut approach to general zero-one problems (including flowcharts) is given. We report numerical experiments to test the effect of such preprocessing within a branch-and-cut algorithm for eleven large-scale real-world zero-one linear-programming problems. An illustrative example is included in the Appendix. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A Hierarchy of Relaxations between the Continuous and Convex Hull Representations for Zero-One Programming Problems

In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. The reformulation technique readily extends to produce a similar hierarchy of linear relaxations for zero-one polynomial programming problems. A characterization of the convex hull in the original variable space is also available through a projection process. The structure of this convex hull characterization (or its other relaxations) can be exploited to generate strong or facetial valid inequalities through appropriate surrogates in a computational framework. The surrogation process can also be used to study various classes of facets for different combinatorial optimization problems. Some examples are given to illustrate this point.

A Branch-and-Cut Approach to a Traveling Salesman Problem with Side Constraints

A problem posed by O. L. Deutsch (Deutsch, O. L. 1988. Artificial intelligence design challenge—Background, analysis, and relative performance of algorithms. J. Guidance, Control, and Dynamics 11 386–393.) as the Artificial Intelligence Design Challenge for the 1987 American Institute of Aeronautics and Astronautics (AIAA) Conference on Guidance, Navigation &amp; Control (Monterey, CA, August 17–19, 1987) is formulated as a zero-one linear program. We show that the associated (relaxed) linear programming problem can be solved in polynomial time despite an exponential size of the proposed constraint system in terms of the underlying parameter n of the number of cities considered, when a nonlinear constraint of the problem is either ignored or approximated by linearization. We describe a software system AIAA/SOLVER that we have implemented to solve the problem to optimality under an apparently weak assumption about its stochastic cost structure using branch-and-cut.

The use of a mathematical modelling method to determine the optimal retrofitting option of an oil refinery steam system

In this paper, as a practical example, the use of mathematical modelling procedure to determine the optimal retrofitting option of an oil refinery steam system is presented. The basis for setting up a mathematical model is to use a network diagram to describe all potential practicable alternatives, therefore, in principle this method is appropriate to optimal design or optimal modification of any complex energy flow or material flow system. The nature of this model is a multi-objective zero-one linear mixed integer programming problem.