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Depending on the nature, objectives, and constraints of the decision variables; linear programming, nonlinear programming, integer programming, mixed integer programming etc. can be classified. Extensive research has been conducted to solve all types of these problems in a parametric context. In this paper, to solve optimization problems having uncertainties represented by a single parameter on the objective function, a systematic linearization approach is developed considering the parametric expression as nonlinear. In the proposed approach, the objective function is considered as nonlinear which is converted into linear by using first order Taylor series expansion at the points making the parametric costs zero. Thus, the optimal solution is obtained from the constructed linear programming problem. In this way, by determining the intervals in which the optimal solution changes, the solution of the parametric linear programming problem is obtained. A numerical experiment is illustrated to present the effectiveness of the proposed approach.

Резюме. Методите на линейното оптимиране са добре известни в литературата и практиката като едни от най-ефективните методи за оптимизиране на икономически явления и процеси. Линейните модели с детерминирани коефициенти са задълбочено изследвани, но все още има насоки, в които могат да бъдат доразвити изследванията, свързани с параметричните линейни оптимизационни модели. В настоящото изследване се използва графичният метод за решаване на двуфакторна линейна оптимизационна задача, за да се направи анализ на решението на задача за оптимално разпределение на ресурси при наличието на два параметъра в коефициентите пред неизвестните в едно от ограничителните условия.

Imbalanced learning is a challenging task in predictive modeling and machine learning that has inspired many researchers to attempt to improve the existing algorithms for more accurate predictions of imbalanced data sets. Due to the nature of rare events, developing reliable and efficient classification models for imbalanced data has not been easily accomplished, and over the past two decades, various methods have been proposed. To this end, we propose a Linear Programming Support Vector Machine (LP-SVM) model to address the issue of imbalanced learning in weather applications. To further improve the model’s predictive accuracy, we have implemented a parameter selection method based on the multi-objective parametric simplex approach for parameter tuning of LP-SVM. For numerical tests, we have used a real data set consisting of weather observations made by the Bureau of Meteorology’s (BM) system in Australia. The results obtained from training and testing the model demonstrate the effectiveness of our proposed model on the tested examples.

Purpose The purpose of this paper is twofold: first, to estimate the water demand elasticities using a parametric linear programming (LP) model to provide an insight into the accurate and flexible pricing policy of irrigation water in the Jordan Valley; and second, to highlight key risk aspects, related to water demand, which are likely to impact the community. Design/methodology/approach A parametric LP model was used in this research. Primary and secondary data were collected. Findings Results revealed that the demand elasticity is high in Spring and Summer than in Fall and Winter, meaning that during Spring and Summer farmers are willing to forgo larger amounts of water than in other months. This is because of areas planted during Spring seasons are much less than those of Autumn and Winter. Practical implications The Jordan Valley suffers from water scarcity risk, and consequently the area to be planted is not fully utilized, leading to lower cropping intensities. Responsible authorities in Jordan need to address these issues and propose proper solutions in order to reduce further escalation of this risk and subsequent impact on local communities. Insight into the value of water demand elasticities is essential to support and mitigate policy decision making under risk conditions, concerning investments in water supply systems; investments in the water distribution and irrigation systems; efficient allocation of water with competing sectors; setting water pricing and tariffs; setting cost recovery mechanisms, and the risks encountered under lack of mitigated policy decision making. Originality/value This is one of few studies that addresses in detail using a parametric LP model the issue of water scarcity, related risks and subsequent impact on society in Jordan. It is expected to help policy and decision makers better formulate future estimates and demand which subsequently reduce related risks.

The article describes a model to optimal control and adapt the hierarchic structure of the company's production subsystem in an unstable external environment, the results of numerical solution of dynamic optimization of the hierarchic structure of the company's production subsystem objectives of the company's in the absence of the uncertainty of dynamic programming method in the conditions of uncertainty simulation method for different values of a parameter which characterizes the uncertainty.

Although parametric optimization with uncertainties on the objective function (OF) or on the so-called “right-hand-side” (RHS) of the constraints has been addressed successfully in recent papers, very little work exists on the same with uncertainties on the left-hand-side (LHS) of the constraints or in the coefficients of the constraint matrix. The goal of this work has been to develop a systematic method to solve such parametric optimization problems. This is a very complex problem and we have begun with the simplest of optimization problems, namely the linear programming problem with a single parameter on the LHS. This study reviews the available work on parametric optimization, describes the challenges and issues specific to LHS parametric linear programming (LHS-pLP), and presents a solution algorithm using some classic results from matrix algebra.

This paper critically examines parametric linear programming problem (PLPP) with interval in the coefficients of the objective function and the composition of Plus, Beauty and Nova soap. The finding of soap production in NASCO Company in Jos, Nigeria will be studied and the problem formulated. The formulated problem is tested using the developed computer program by varying the parameter at regular interval and obtaining the corresponding values of the objective function. The interpretation of the results will be analysized.

A procedure relying on linear programming techniques is developed to compute (regression) quantile regions that have been defined recently. In the location case, this procedure allows for computing halfspace depth regions even beyond dimension two. The corresponding algorithm is described in detail, and illustrations are provided both for simulated and real data. The efficiency of a Matlab implementation of the algorithm is also investigated through extensive simulations.

This study intends to develop easy-to-implement and effective approaches to help decision makers cope with the challenges of designing their optimal systems, and determine the corresponding optimal budgets. The approaches are realized by constructing optimal system design (OSD) data envelopment analysis (DEA) models that build on the concepts and techniques of DEA, de novo programming and parametric linear programming with a parametric right-hand side. The proposed models explicitly take the important issue of congestion in system design into account. Therefore, the OSD DEA models can help decision makers not only optimally design their systems, but also determine their optimal budgets. Numerical examples are used to evaluate the strengths of the proposed models.

This article is concerned with the computational aspect of l1 regularization problems with a certain class of piecewise linear loss functions. The problem of computing the l1 regularization path for a piecewise linear loss can be formalized as a parametric linear programming problem. We propose an efficient implementation method of the parametric simplex algorithm for such a problem. We also conduct a simulation study to investigate the behavior of the number of “breakpoints” of the regularization path when both the number of observations and the number of explanatory variables vary. Our method is also applicable to the computation of the regularization path for a piecewise linear loss and the blockwise l∞ penalty. This article has supplementary material online.

We estimate Japanese banking inefficiency and the shadow price of problem loans by treating problem loans as a jointly produced undesirable by-product of the loan production process. Our method uses the directional output distance function of Fare et al. [R. Fare, S. Grosskopf, D.-W. Noh, W.L. Weber, Characteristics of a polluting technology: Theory and practice, Journal of Econometrics 126 (2005) 469-492] and seeks the maximum expansion of desirable outputs, such as loans and securities investments, and the simultaneous contraction in undesirable outputs, such as problem loans. The directional output distance function is estimated using data envelopment analysis and a parametric linear programming method. While the two methods give similar estimates of inefficiency, the shadow price estimates for the two methods diverge.

Based on previous research on two-stage cooperative CCR efficiency, this article discusses the effects of scale and technical factors on the efficiency of the system and the sub-stages, and proposes a two-stage BCC model for cooperative efficiency using a geometric mean method. Various efficiency evaluation models are presented to calculate the efficiencies of the two sub-stages in a way that is most preferable by one of the sub-stages, wherein the upper and lower bounds of the efficiencies are explicitly determined. A transformation is proposed to transfer a non-linear cooperative model into a linear one by setting one of the two-stage efficiencies as a parameter. A heuristic parametric linear programming is suggested to solve the cooperative model.

Joint forest management (JFM) has emerged as the most promising institutional arrangement for sustainable forest management, and village-level organizations (production units) are responsible for all the productive activities of JFM. Employing a parametric linear programming approach, a deterministic input distance function is estimated characterizing the production structure of JFM organizations, in the Gujarat state of India, using the production data from 50 organisations. The distance function includes economic, biological and social outputs, and neo-classical – land, labour, and capital – as well as non-neo-classical – social, political, institutional, and organisational – factors. The results are used to calculate the production efficiency and shadow prices of neo-classical as well as non-neo-classical factors of different village-level JFM organizations. Policy and management implications of production efficiency and shadow prices are discussed.

Parametric programming has received a lot of attention in the control literature in the past few years because model predictive controllers (MPC) can be posed in a parametric framework and hence pro-solved offline, resulting in a significant decrease in on-line computation effort. In this paper we survey recent work on parametric linear programming (pLP) from the point of view of the control engineer. We identify three types of algorithms, two arising from standard convex hull paradigms and one from a geometric intuition, and classify all currently proposed methods under these headings. Through this classification, we identify a third standard convex hull approach that offers significant potential for approximation of pLPs for the purpose of control. We present the resulting algorithm, based on the beneath/beyond paradigm, that computes low-complexity approximate controllers that guarantee stability and feasibility.

This paper presents an improved algorithm for solving the sum of linear fractional functions (SOLF) problem in 1-D and 2-D. A key subproblem to our solution is the off-line ratio query (OLRQ) problem, which asks to find the optimal values of a sequence of m linear fractional functions (called ratios), each ratio subject to a feasible domain defined by O(n) linear constraints. Based on some geometric properties and the parametric linear programming technique, we develop an algorithm that solves the OLRQ problem in O((m+n)log (m+n)) time. The OLRQ algorithm can be used to speed up every iteration of a known iterative SOLF algorithm, from O(m(m+n)) time to O((m+n)log (m+n)), in 1-D and 2-D. Implementation results of our improved 1-D and 2-D SOLF algorithm have shown that in most cases it outperforms the commonly-used approaches for the SOLF problem. We also apply our techniques to some problems in computational geometry and other areas, improving the previous results.

In this paper we investigate the sensitivity analysis of the parameterized central path. First, a complete marginal analysis of the central optimal solution is developed. This analysis explains the differential properties of the central optimal solution with respect to both the cost coefficients and the right-hand side components. We also show that the marginal derivatives are uniformly bounded. Second, we present three conditions for which the parameterized central path converges. Two of these results allow the difficult situation of simultaneous perturbations in the cost coefficients and right-hand side levels.

Jenkins developed an heuristic algorithm for performing the right-hand-side parametric analysis of a Mixed Integer Linear Programming (MILP) problem with an approach completely different from all previous work in the area: his method can use any software capable of solving MILP problems and able to perform parametric linear programming (PLP). There is no proof of completeness by using Jenkins's algorithm. In this paper we present results that allow us to verify the completeness of the parametrical analysis. Like Jenkins's algorithm our procedure is theoretically independent of the solution method to solve a point-value problem.

Many important algorithms in signal and image processing can be described by uniform recurrences. A common method for the synthesis of systolic arrays from uniform recurrences is based on space-time transformations each of which consisting of two linear mappings, an allocation and a timing function. In this paper, we address the problem of finding space-time transformations which are time-optimal or at least nearly time-optimal. For a given allocation function, a continuous relaxation of this problem is studied by passing from linear to quasi-linear timing functions. A parametrized linear programming formulation is provided for finding quasi-linear timing functions. The solution of each such linear problem, however, depends on the basis representation of the null space of the allocation function. Therefore, a branching approach is proposed for finding quasi-linear timing functions which are optimal or have at least low latency. It will be demonstrated by several large test examples that branching into hundreds or even thousands of linear subproblems can be computed with reasonable effort and often leads to an optimum linear timing function.

In the present paper we prove that if the data of a parametric linear optimization problem are smooth, the solution map admits a local smooth selection “almost” everywhere. This in particular shows that the set of points where the marginal function of the problem is nondifferentiable is nowhere dense.

The production-transportation problem (PTP) is a generalization of the transportation problem. In PTP, we decide not only the level of shipment from each source to each sink but also the level of supply at each source. A concave production cost function is associated with the assignment of supplies to sources. Thus the objective function of PTP is the sum of the linear transportation costs and the production costs. We show that this problem is generally NP-hard and present some polynomial classes. In particular, we propose a polynomial algorithm for the case in which the transportation cost matrix has the Monge property and the number of sources is fixed. The algorithm generalizes a polynomial algorithm of Tuy, Dan, and Ghannadan [Oper. Res. Lett., 14 (1993), pp. 99–109] for the problem with two sources.

This paper proposes one-dimensional spacing algorithms that minimize [maximize] the degree of separation [proximity] among the specified elements of a given integrated circuit layout. Number of problems on chip and MCM systems such as poor performance, higher crosstalk, lower yield etc. are related to the degree of separation [proximity]. The proposed algorithms utilize the attractive [repulsive] constraints to shrink [expand] the distances among the specified elements while keeping the layout free of any design rule errors. The spacing problem is reduced to a parametric linear programming problem and a network simplex algorithm is proposed for solving it. The proposed algorithms are implemented into a system called PERFECT for performance enhancement and crosstalk reduction. Given a routed design, PERFECT minimizes delay due to the coupled capacitance, which could contribute as much as 50-75% to the interconnect delay in near future. PERFECT utilizes the repulsive constraints between interconnect segments for the purpose. The experimental results show a significant delay improvement and the crosstalk reduction for submicron technologies. >

In this paper, the concept of possibility is further considered, fuzzy optimal value of the linear programming with fuzzy numbers is given, and the continuity theorems of optimal value in parametric linear programming are proposed.

For stochastic programs with complete (linear) recourse we present easily verifiable sufficient conditions for the strong convexity of the expected-recourse function. Both, programs with random right-hand side and with random technology matrix are considered. Among the implications of strong convexity those with respect to the stability of stochastic programs are worked out in detail. In this way, former results on the quantitative stability of optimal solutions are extended.

We present a new algorithm for the problem of determining the intersection of a half-line \(\Delta_{u}=\{x\in \mathbb{R}^{N}\:|\:x=\lambda u\;\mathrm {for}\;\lambda \geq 0\}\) with a polymatroid. We then propose a second algorithm which generalizes the first algorithm and solves a parametric linear program. We prove that these two algorithms are strongly polynomial and that their running time is O(n 8+γ n 7) where γ is the time for an oracle call. The second algorithm gives a polynomial algorithm to solve the submodular function minimization problem and to compute simultaneously the strength of a network with complexity bound O(n 8+γ n 7).

The multiprocessor scheduling problem(MPSP), P|prec,p j =1|C max , is known to be NP-complete. The problem is polynomially solvable, however, if the precedence relations are of the intree(outtree) type, P|intree(outtree),p j =1|C max , or if the number of processors is two, P2|prec,p j =1|C max . In this paper, we introduce a parametric linear program which gives a lower bound for the makespan of MPSP and retrieves the makespans of the two polynomially solvable problems.

In this paper we consider properties of the central path and the analytic center of the optimal face in the context of parametric linear programming. We first show that if the right-hand side vector of a standard linear program is perturbed, then the analytic center of the optimal face is one-side differentiable with respect to the perturbation parameter. In that case we also show that the whole analytic central path shifts in a uniform fashion. When the objective vector is perturbed, we show that the last part of the analytic central path is tangent to a central path defined on the optimal face of the original problem.

The Calculus of Variations represents the oldest mathematical theory for optimization in the case of a controlled evolution.

In this chapter we present the historical development of the notion, theory, and applications of sensitivity analysis and parametric programming.

An integrated software package designed for solving and analyzing fuzzy and parametric linear programming problems is presented.

In this paper we compare two approaches to the computation of eigenvalues by defect minimization which differ principally in the choice of norm to measure the resp. defect. One of them, basically developed by Kuttler and Sigillito, leads to the consideration of parametric (matrix-) eigenvalue problems, whereas the other requires the treatment of parametric linear programming problems. It will be shown that the latter allows a very flexible choice of trial functions, giving good rates of approximation. Numerical results for rhombical membranes are given.

In the present paper the concept of so-called local stability sets introduced by F. Nožicka in parametric linear programming is extend to the case of convex quadratic programs parametrized in the objective function and in the right-hand side of the linear constraints. The continuity of the optimal-set-map and of the extreme value function holds on these sets. It is shown that the local stability sets are connected. Examples which illustrate the necessity of certain assumptions are given.

A simple linear programming model was formulated to examine the impact of resource alternatives on vegetable cropping patterns. The planning objective was to maximize the total vegetable crop net returns. Florida conditions were used to illustrate techniques. The 1973-74 season was chosen as a base period. The reductions in available resources were simulated by the parametric linear programming technique, and their effects on the changes of cropping patterns were obtained. The effects of curtailed resource supplies on crop acreage were varied. Crops which required large amounts of energy were sensitive to reductions in energy. When subject to water reduction only, the response showed a relatively balanced cropping pattern. Crop acreages generally declined when both energy and water levels were reduced. Crops with medium resource requirements were not as adversely affected by reduced inputs; crops with low resource inputs showed marked increases in acreage when resources were limited.

This paper attempts to analyze, using linear programming techniques, production planning of S.A. Silva & Sons, Ltd., the largest processor of coconuts in Sri Lanka. This was the first attempt of its kind to use linear programming in production planning in Sri Lanka. Thus the major objective was to develop a model useful in acquainting the management with linear programming methods, data requirements, and total systems analysis, while demonstrating the usefulness of the model. This study generated several rules of thumb for maximizing profits.

In the present paper an extended transportation problem is considered. It is characterized by the possibility of transportations using different variants (railway, ship, tnick, etc,) from the production center to the customers. Evary.variant bas bounded capacity and the transportation cost differ from one to the other. It is relatively simple to determine the solvability regions of such a problem by irisking use of the soiviity region of the corresponding transportation problem with capacity bounds. Starting with the theory of parametric linear programming an approach is presented to determine the stability region for a fixed paramoter value. Proesduetera are given for the determination of the optimal solution: for a fixed parameter value us wel1 as for the corresponding stsbility regon.They are based on the potential method.

This paper generalizes the class of design problems for water supply systems which can be solved by mathematical programming. The goal is to design a least cost water supply system to satisfy minimal pressure head requirements at specified rates of usage for multiple customers. The generalization shows how parametric linear programming can be used to analyze subnetworks of a large supply system and to decompose the large problem into a series of smaller problems. An example is included.

The paper deals with discrete elastic-plastic structures subjected to variable repeated or cyclic loads, which are able to undergo only a limited amount of permanent deformation. By the methods of mathematical programming, the two main problems of shakedown theory, i.e. the evaluation of the shakedown safety factor of a given structure and the optimal shakedown design for a given load multiplier, are reformulated to take into account the limited ductility behaviour of the structure. The appropriate bounding technique used plays an essential role in obtaining a fairly unified approach to both problems, also in relation to the various types of deformation constraints which may be required in practice. To solve these constrained problems numerically, a solution method based on parametric linear programming is proposed and illustrated by means of some simple examples.

Ration formulation is an important aspect of the operation of many beef cattle enterprises. The method for beef cattle ration formulation depends on the system used to specify the nutrient requirements of the animals. With the system which is recommended by the Ministry of Agriculture, Fisheries and Food in the U.K., difficulties arise in ration formulation because the energy requirements of the animal depend on the energy concentration of the ration. A method for formulating rations to satisfy the recommended nutrient standards is developed. The approach involves using parametric linear programming to obtain a piecewise representation of the cost of the ration as a function of the energy concentration of the ration, and then using differential calculus to determine the least cost ration.

The ideas of consistency and stability of aggregation in parametric optimization problems are introduced and discussed. Stability of consistent aggregation is proved in the class of parametric linear programming problems.

A path-following philosophy (continuation method, global Newton method) is used to compute equilibria for piecewise linear economies while taking advantage of the linear structure of the model. The existence of a path leading through certain faces of a polyhedral set to an equilibrium point is demonstrated. Computational experience is reported which indicates that this method is promising for models dealing with many commodities and relatively few consumers.

The paper represents an immediate continuation of part I [1], where the independent twofold one parametric linear programming problem was treated. The author deals with a linear programming problem with as objective function as capacity rector depending each on one parameter. Both parameters may depend too in a fixed manner. This depe~ldenoe can be considered as tt curve being contained in parameter space. One resolves the problem using t'he corresponding independent problem in forming the intersections of the curve with t,he rectangles resulting as solution of the independent problem. There are obtained again maximal regions for respectively one system of basic variables, yielding the corresponding solution of such a region. The author gives two procedures for solving the problem and three examples illustrating various cases of dependence.

Field peas of cull grade replaced 0, 25, 50, 75 and 100% of the soybean meal-flshmeal component of a swine grower ration without detriment to rate of growth, feed utilization or carcass quality. Methionine supplementation proved unnecessary. When cull peas replaced 10, 20, 30 or 40% of the barley in the ration, pig performance was also maintained. The digestibility of protein and energy in cull peas was 87 and 81% and of good (4CW) peas was 83 and 86%, respectively.The monetary values of cull and 4CW peas when incorporated into swine grower rations at various levels were determined in two different market situations by use of the parametric linear programming technique. Up to 25% peas entered least-cost formulas involving a 17% maximum level of crude protein in the ration. When pea prices equalled corn or barley prices in the respective markets, 12 to 20% peas were indicated in the ration formulas.

In this treatment are considered coefficient perturbations in linear programming problems. For two cases the effects of these perturbations on the elements of the simplex-table are given. The first case consists in the linear dependence of the coefficients of one or several columns on one parameter, the second in the linear dependence of some columns of coefficients each on one parameter. It is shown, how formulas of perturbation according to the case of the polynomial dependence on one parameter and for the case of the simultaneous linear dependebce on some parameters can be formed. Finally and example of parametric linear programming with two columns of coefficients, which depend each on one parameter is presented.

The use of linear programming and computers to formulate rations that meet the nutritional requirements of poultry at lowest cost is a widely accepted practice in the poultry feed industry. One application of linear programming not readily employed by poultry nutritionists is the use of parametric linear programming in the planning and economic evaluation of research on new or improved feedstuffs. Three parametric techniques useful in research are parametric cost ranging, parametric nutrient ranging, and parametric restriction ranging. Parametric cost ranging is useful in evaluating the market potential for new or improved feedstuffs and examining competitive and substitution relationships among ingredients. Parametric nutrient ranging, by analyzing at different levels the nutrient content of a particular feed ingredient, shows the changes in economic value that result from changes in the ingredient’s nutrient composition. Parametric restriction ranging enables one to analyze the effect of changes in dietary requirements on ration cost and composition and to identify those nutrients for which diet requirements need to be more precisely defined.

An efficiency theorem is stated expressing the one-to-one correspondence betwem the linear maximurn-vector problem and the corresponding parametric linear programming problem.