Zero-one Variables Research Articles

A Sequential Approach to the $0 - 1$ Linear Programming Problem

The paper gives a method of minimizing a linear function of zero-one variables subject to linear constraints. The method is based on transforming an n-variable m-constraint problem into a sequence of n-variable 2-constraint problems which are easier to solve. The algorithm has been tested on test problems found in the literature and on a set of 80 random problems. Good results have been obtained, both in terms of the number of iterations involved in solving this sequence of subproblems, and in terms of the total computation time required to reach the final solution. The method is most effective when the coefficients of the problem are small integer numbers.

Estimating Broiler Supply and Demand as a Simultaneous System

The production and consumption of broiler meat varies seasonally and is accompanied by seasonally fluctuating prices. In addition to seasonal factors, the marketing process occurs in the short-run, giving rise to the need for a simultaneous model of supply and demand. Therefore, a two-equation simultaneous model was utilized which included zero-one (dummy) variables as monthly intercept shifters for both supply and demand.January represented the base intercept period for both equations. Ten monthly demand curves and eight monthly supply curves were estimated whereby the intercepts were significantly different from the January intercept value.The estimated coefficients indicate that as broiler quantity changes by one percent, price changes inversely by 1.66 percent. Pork price, price expectations (as measured by lagged broiler price), and consumer income have a direct influence on the demand (price) for broilers.Supply is directly influenced by price, whereby a one cent price change induces a corresponding quantity change of 73.47 million pounds. Supply is also positively influenced by pork price while changes in input costs, as reflected by farm value, has a negative influence on supply.

A Simplified Algorithm for Obtaining Approximate Solutions to Zero-One Programming Problems

This paper is intended to present a simple and quick method for obtaining approximate solutions to large scale zero-one programming problems. The method does not use enumeration. Instead, it assigns measures of preferability to zero-one variables that change the values of the variables from zero to one. The method yields very good approximate solutions to zero-one programming problems in dramatically short computation time. Even for problems involving more than a thousand zero-one variables the computation time is of little concern. The method is applicable not only to those problems associated with obtaining the optimal package of variables with the value one but also to a great variety of binary choice (“Yes-No”) problems.

The use of linear programming to select an optimal set of prime implicants

This study shows how the selection of an optimal subset of prime implicants from a prime implicant table, as encountered in digital logic design and similar applications, may be formulated as a linear programming problem in zero-one variables.

An Imbedding Approach to Additive Value Zero-One Problems

Existing approaches to the determination of the parameters for zero-one additive value functions can involve a large amount of work. This paper considers an alternative imbedding approach which allows the parameters to be determined to within an error range 1/K within K steps. In considering the method itself specific elements of theory are developed relating to the existence of additive values for imbedding sets X · Y*, X · Ỹ where X is a subset of zero-one vectors, Y* is a countable subset of numbers in the range −∞ < y < ∞, and Ỹ is a connected open interval of the range −∞ < y < ∞. In seeking further approximation methods, it is necessary to investigate properties of the component value, u(·) over Y* and Ỹ. The coefficients of the zero-one variables are shown to be unique for Y* once two values have been preset. Under certain circumstances it can be demonstrated that a continuous u(·) over Ỹ exists, and that considerable variations in u(·) may be compatible with the specified preference structures, although in special circumstances u(·) is unique over Ỹ.

Evaluation and Measurement of Characteristics Affecting Fresh Market Blueberry Demand1

Abstract Fresh market demand relationships were estimated for several major markets for North Carolina, New Jersey, and Michigan fresh blueberries. Demand relationships were estimated from daily price and unload statistics for the 1965–1971 seasons by regression analysis using daily price as the dependent variable. Zero-one variables were used to account for the seasonal and within season effects. The relationships accounted for 50 to 70% of the variation in daily prices during the 7-year period. The remaining unexplained variation resulted in standard errors of estimate of around 15 cents per flat for most markets. Large standard errors associated with the quantity coefficients appeared to be related in part to the relatively large and discrete intervals used in price reporting by the Market News Service. The quantity coefficients were relatively small. These indicated rather elastic demands for fresh blueberries in the major markets.

Iterative decoding of linear codes by a modulo-2 linear program

We present a new form of linear programming that minimizes the real sum of zero-one variables under constraints that are linear in the field of binary numbers and thus nonlinear in the field of real numbers. This problem is motivated by coding theory where binary sequences are transmitted over noisy communication links. The decoding process which corrects some of the errors caused by noise requires the solution of the stated problem. Simulation results are also presented and they show that the algorithms converge to the correct solution.

Measuring the Impacts of Solid Waste Disposal Site Location on Property Values

T HE large and growing quantity of solid wastes poses an increasingly complex disposal problem for rural and urban areas. One important effect of the location of disposal sites is the external effect on adjacent property values. This paper attempts to estimate external effects on property prices resulting from the location of landfill types of solid waste disposal sites. This has important implications for public policy on site location and possible compensation to internalize the resultant external effects. A single equation linear regression model for price estimation is hypothesized. The data sample is 182 observations of single-unit house sales over the period 1962-1970 in the neighborhoods around five solid waste disposal sites in Fort Wayne, Indiana. Date and price of sale, as well as location of the property and detailed information about dwelling characteristics, were obtained. Three general categories of predetermined variables are used: (1) Physical attributes of residential property-size of house and lot, number of bathrooms and bedrooms, age of house, amount of encumbrance and ownership-tenant occupancy (the key features of the quantity and quality of living service provided by a piece of residential property); (2) year of sale-the best available measure of the general level of cost of housing; (3) factors representing amenities and disamenities associated with solid waste disposal sites and neighborhood characteristics. Based on a diffusion phenomenon, the impacts of solid waste disposal sites on residential property prices were measured by distance from the nearest disposal site and the absolute degrees that the resiential property is away from downwind (prevailing) of the disposal site. Zero-one variables were used to represent differences among disposal sites. The predetermined variables in the linear model explained approximately 78 percent of the variation in residential property prices. Coefficients of the physical attribute variables were all s g ificantly different from zero at the .05 level and all had the hypothesized sign except the number of bedrooms. The distance and angle v riables are key policy variables in the model. Both coefficients were positive as hypothesized, suggesting that a premium is placed on being a g eater distance away and being away from downwind of disposal sites. The estimated coefficients indicate that on the average the price of residential property increases 61 cents per foot of distance and $10.30 per degree away from downwind of a disposal site. This first attempt to measure external effects f solid waste disposal sites on adjacent property values has important implications for community development and policy. The empirical results suggest magnitudes of costs imposed and indicate the nature of their distribution around solid waste disposal sites. In decisions concerning disposal site location, external social cost and benefit are an integral and possibly major factor. Further research is therefore warranted in refining the model and its variables to more accuately measure these effects.

Inference about the change-point in a sequence of random variables

Abstract : The report discusses the problem of making inference about the point in a sequence of zero-one variables at which the binomial parameter changes. The asymptotic distribution of the maximum likelihood estimate of the change-point is derived in computable form using random walk results. The asymptotic distributions of likelihood ratio statistics are obtained for testing hypotheses about the change-point. Some exact numerical results for these asymptotic distributions are given and their accuracy as finite sample approximations is discussed. (Author)

Inference about the change-point in a sequence of binomial variables

The problem of making inferences about the point in a sequence of zero-one variables at which the binomial parameter changes is considered. The asymptotic distribution of the maximum likelihood estimate of the change-point is derived in computable form using random walk- results. The asymptotic distributions of likelihood ratio statistics for testing hypothesis about the change-point are also obtained. Some exact numerical results for these asymptotic distributions are given and their accuracy as finite sample approximations is discussed.

Initialising Geoffrion's implicit enumeration algorithm for the zero-one linear programming problem

This paper describes a method of initialising Geoffrion's algorithm for the solution of linear programs in zero-one variables. The modification proposed involves three modules, one or all of which may be employed in a particular computation. A computational comparison is made between the original and modified algorithms.

A Remark on the Computation of Optimum Media Schedules

The media scheduling problem of Ellis is transformed into an integer linear programming problem in zero-one variables. The transformed problem is recognized as the knapsack problem and exact and approximate algorithms are proposed.

Discrete Programming by the Filter Method

In this paper (section 1) a two-phase procedure, the filter method or accelerated additive algorithm, is proposed for solving linear programs with zero-one variables. In Phase I an auxiliary problem is constructed that, in Phase II, is used to “filter” the solutions to which the tests of the additive algorithm are to be applied. The filter method is then extended (section 2) by J. F. Benders’ partitioning procedure to the mixed-integer zero-one case, as well as to general integer and mixed-integer programs. Finally, a specialized version of this method is used (section 3) to tackle a general machine-sequencing model, formulated as the problem of finding a minimaximal path in a disjunctive graph.

Computational Experience with Variants of the Balas Algorithm Applied to the Selection of R&D Projects

Allocating funds to independent R&D projects is a problem of practical importance for many firms. We formulate the problem as a 0-1 integer programming problem with the objective of selecting projects that will maximize the anticipated dollar contract volume, yet not exceed cost budgets. Our form-mulation accommodates R&D projects extending over several budget periods and permits carryover of unspent funds from one budget period to later periods. Experience in solving such problems by using the Balas [Balas, Egon. 1965. An additive algorithm for solving linear programs with zero-one variables. Oper. Res. 13 (4, July–August) 517–549.] algorithm in its literal form and in reformulated form is summarized. Several other modifications to the algorithm are described and; their effect on efficiency is shown through presentation of computational experience on problems with as many as 50 variables.

Letter to the Editor—Computational Experience with the Algorithm of Balas

This note reports on the application of the Balas algorithm for linear programs with zero-one variables on larger problems involving up to 159 variables. Two modifications of the algorithm are discussed. Some information about the machine program and numerical results are given.

An Additive Algorithm for Solving Linear Programs with Zero-One Variables

An algorithm is proposed for solving linear programs with variables constrained to take only one of the values 0 or 1. It starts by setting all the n variables equal to 0, and consists of a systematic procedure of successively assigning to certain variables the value 1, in such a way that after trying a (small) part of all the 2n possible combinations, one obtains either an optimal solution, or evidence of the fact that no feasible solution exists. The only operations required under the algorithm are additions and subtractions; thus round-off errors are excluded. Problems involving up to 15 variables can be solved with this algorithm by hand in not more than 3–4 hours. An extension of the algorithm to integer linear programming and to nonlinear programming is available, but not dealt with in this article.

Monthly Price-Quantity Relations for Broilers at the Farm Level

Knowledge of seasonal demand patterns can be of value in production planning and orderly marketing of agricultural commodities. This paper presents monthly estimates of broiler price-quantity relations at the farm level for the 1953–63 period. The monthly relations differed significantly from each other. Differences were mainly in the levels of the relations rather than the slopes. A model using zero-one variables was then employed which constrained the slopes of the regression lines to a single value but allowed the levels to vary. The level was lowest in December. The regression line shifted upward each month until July, after which it shifted rapidly back downward during the fall months. Estimates of farm level price elasticity of demand for broilers ranged from around −1.4 during the late winter and early spring months to around unity during the summer and fall months.

Using Zero-One Variables with Time Series Data in Regression Equations

ERO-ONE variablesl are being used with increasing frequency by economists in regression analysis. The purpose of this paper is to review some problems related to the use of these variables.2 Emphasis is placed on applications with quarterly time series observations. Three problem areas are discussed: (1) determining the appropriate circumstances for the use of such variables, (2) the mechanics of use, and (3) the interpretation of results.