Ordinary Linear Programming Research Articles

Intuitionistic fuzzy multi-objective milk-run modelling under time window constraints

Milk-run is a delivery method allowing to move small quantities of a large number of different items with predictable lead times from various suppliers to a customer. The main goal is to minimize the transportation cost by minimizing the travel distance and by maximizing vehicle capacities. The effects of uncertainties in arrival times of vehicles and loading times of shipments should also be considered in modeling the milk-run problems. In this paper, a multi-objective linear programming model, an ordinary fuzzy multi-objective linear programming model and an intuitionistic fuzzy multi-objective linear programming model are proposed for the milk-run modeling under time window constraints. The proposed approaches are applied on the real-life data of Borusan Logistics which is one of the largest logistics firms in Turkey and the results are presented.

Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

The paper concerns multiobjective linear optimization problems in R^n that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific directions. This idea is formalized by means of the so-called epigraphical multifunction, which is defined by adding a fixed cone to the images of the original mapping. Through the epigraphical feasible and Pareto front mappings we describe the corresponding vector subdifferentials, and employ them to verifying Lipschitzian stability of the perturbed mappings with computing the associated Lipschitz moduli. The particular case of ordinary linear programs is analyzed, where we show that the subdifferentials of both multifunctions are proportional subsets. We also provide a method for computing the optimal value of linear programs without knowing any optimal solution. Some illustrative examples are also given in the paper.

DYNAMIC PROBLEM OF FORMATION OF SECURITIES PORTFOLIO UNDER UNCERTAINTY CONDITIONS

The analysis of known methods for solving the problem of forming a portfolio of securities in the face of uncertainty is carried out. Traditionally, the problem is solved under the assumption that for each type of asset, the values of the main statistical characteristics of the random value of their profitability (mathematical expectation and variance) are known. At the same time, the variance of portfolio returns, which is minimized, is used as a criterion for portfolio optimization. Two alternative approaches to solving the formulated problem are proposed. The first of them provides a decision on the criterion of the probability that the random total portfolio return will not be lower than the given. It is assumed that the random return for each type of asset is distributed normally and the statistical characteristics of the respective densities are known. The original problem is reduced to the problem of maximizing the quadratic fractional criterion in the presence of linear constraints. To solve this non-standard optimization problem, a special iterative algorithm is proposed that implements the procedure for sequential improvement of the plan. The method converges and the computational procedure for obtaining a solution can be stopped by any of the standard criteria. The second approach considers the possibility of solving the problem under the assumption that the distribution densities of random asset returns are not known, however, based on the results of preliminary statistical processing of the initial data, estimates of the values of the main numerical characteristics for each of the assets are obtained. To solve the problem, a new mathematical apparatus is used – continuous linear programming, which is a generalization of ordinary linear programming to the case when the task variables are continuous. This method, in the considered problem, is based on solving an auxiliary problem: finding the worst-case distribution density of a random total portfolio return at which this total return does not reach an acceptable threshold with maximum probability. Now the main minimax problem is being solved: the formation of the best portfolio in the worst conditions. The resulting computational scheme leads to the problem of quadratic mathematical programming in the presence of linear constraints. Next, a method is proposed for solving the problem of forming a portfolio of securities, taking into account the real dynamics of the value of assets. The problem that arises in this case is formulated and solved in terms of the general theory of control, using the Riccati equation.

English

Managing the limited resources of money, time and space poses a real problem in the livestock business in general and the poultry sector in particular. The stochastic linear programming with recourse will permit the formulation of a mathematical model that could easily be solved in order to attain the objective of maximizing profit but this will be hindered by the presence of some parameters like demand and supply which have no predetermined probability distributions. In an attempt to address the problem of unavailable probability distribution functions, this research is aimed at proposing a robust linear programming model which can handle problems and others in similar circumstances with the need of probability distributions.The robust model thus constructed is based on a modification of the stochastic model by Soyster. An application of this model on a real life data produced results showing an increase in profit made by a local poultry farmer from 241,485 FCFA to 362,580FCFA representing an increase in profit of more than 50% in over that obtained when using the ordinary Linear Programming Technique. The belief therefore is that, if models like this are implemented, not only would the livelihood of the poultry farmers be improved, but it will go a long way to better the economy and satisfy the ever increasing need of poultry products by the communities. Key words: Maximization, livestock, stochastic, robust, linear programming, optimization.

Solving Linear Programming Problems with Grey Data

A Grey Linear Programming problem differs from an ordinary one to the fact that the coefficients of its objective function and / or the technological coefficients and constants of its constraints are grey instead of real numbers. In this work a new method is developed for solving such kind of problems by the whitenization of the grey numbers involved and the solution of the obtained in this way ordinary Linear Programming problem with a standard method. The values of the decision variables in the optimal solution may then be converted to grey numbers to facilitate a vague expression of it, but this must be strictly checked to avoid non creditable such expressions. Examples are also presented to illustrate the applicability of our method in real life applications.

Point-Based Neighborhoods for Sharp Calmness Constants in Linear Programming

From a computational point of view, this paper provides a significant advance in the study of the calmness property of ordinary (finite) linear programs under canonical perturbations (i.e., perturbations of the objective function coefficient vector and the right-hand side of the constraint system). In the recent literature we find, for both the feasible and the optimal set (argmin) mappings, computable expressions for the corresponding calmness moduli. These expressions are called point-based as far as they only depend on the nominal data. In this paper we show that both calmness moduli are indeed (sharp) calmness constants, and provide point-based expressions for neighborhoods where such constants work. We show that our results cannot be extended to general convex inequality systems under right-hand side perturbations.

An efficient irrigation water allocation model under uncertainty

An interval linear fractional irrigation water allocation (ILFIWA) model is developed in response to the complexity of errors in estimating crop yields, fluctuating hydrological elements as well as varying economic profits in an irrigation system. The model is capable of quantitatively solving multi-objective problems, i.e. to obtain the maximum system net benefit and the minimum irrigation water use. Particularly, it can handle multi-objective functions expressed as ratios, such as irrigation water productivity. Moreover, it can reflect the uncertainties of the variables/parameters and functions involved. The potential of the developed model is shown by applying to a case study in northwest China. Results of the model can help make irrigation water allocation decisions for different time periods under varying flow levels. Comparison between ILFIWA model and ordinary interval linear programming model shows that the developed model is conducive to improving irrigation water productivity and saving irrigation water, and helping decision makers formulate desired irrigation water resources management policies under uncertainty.

Fractional transportation problem with fuzzy parameters

The fractional transportation problem (FTP) plays an important role in logistics and supply management for reducing cost and improving service. In the real world, however, the parameters in the models are seldom known exactly and have to be estimated. This paper investigates the FTP where the cost coefficients and right-hand sides are represented by fuzzy parameters. Intuitively, when the parameters in the FTP are fuzzy numbers, the derived objective value should be also a fuzzy number. Based on Zadeh's extension principle, a pair of two-level mathematical programs is formulated to calculate the fuzzy objective value of the FTP with fuzzy parameters. By applying the dual formulation of linear fractional programming and variable substitution techniques, the two-level mathematical programs are transformed into ordinary one-level linear programs to solve. At a specific $$\alpha $$ź-cut, solving the pair of linear programs produces the bounds of the objective value of the fuzzy FTP. By collecting the bounds from different $$\alpha $$ź levels, one can depict the shape of the membership function. An example illustrates how to apply the concept of this paper to solve the fuzzy FTP problem.

Solving Linear and Bilinear Problems with Interval Uncertainty

Two kinds of optimization problems, which can be used in production planning, have been considered: interval linear programming (ILP), finding equilibrium position interval for the Von Neumann model (bilinear problem). Initial data for these models commonly have interval uncertainty. The paper presents definitions of different solution types and methods for finding these solutions. These methods imply reducing interval optimization problems to exact ordinary linear programming problems.

On the integration of row and column uncertainty in robust linear programming

The usual formulation of a linear program is max $$c\cdot x{:}Ax \le b, x \ge 0$$c·x:Ax≤b,xź0. The core part of this linear program is the $$A$$A matrix since the columns define the variables and the rows define the constraints. The $$A$$A matrix is constructed by populating columns or populating rows, or some of both, depending on the nature of the data and how it is collected. This paper addresses the construction of the $$A$$A matrix and solution procedures when there are separate data sources for the columns and for the rows and, moreover, the data is uncertain. The $$A$$A matrices which are "realizable" are only those which are considered possible from both sources. These realizable matrices then form an uncertainty set $$U$$U for a robust linear program. We show how to formulate and solve linear programs which provide lower and upper bounds to the robust linear program defined by $$U$$U. We also show how to use ordinary linear programming duality to share and divide the "credit/responsibility" of the optimal value of the robust linear program between the two alternative data sources.

Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.

On Solving Optimization Problems with Inexact Data

There are considered two kinds of optimization problems with interval uncertainty. The first kind is interval linear programming (ILP), the second kind is finding equilibrium position for interval von Neumann's model (bilinear problem). Definitions of different types of solutions for both kinds of problems and methods for finding these solutions are given. These methods imply using matrices of upper and lower bounds of initial interval data and reducing interval optimization problems to exact (ordinary) linear programming problems.

A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming

A simplicial branch and bound duality-bounds algorithm is presented to globally solving the linear multiplicative programming (LMP). We firstly convert the problem (LMP) into an equivalent programming one by introducingpauxiliary variables. During the branch and bound search, the required lower bounds are computed by solving ordinary linear programming problems derived by using a Lagrangian duality theory. The proposed algorithm proves that it is convergent to a global minimum through the solutions to a series of linear programming problems. Some examples are given to illustrate the feasibility of the present algorithm.

Interior-point algorithms for a generalization of linear programming and weighted centring

In this paper, we consider an extension of ordinary linear programming (LP) that adds weighted logarithmic barrier terms for some variables. The resulting problem generalizes both LP and the problem of finding the weighted analytic centre of a polytope. We show that the problem has a dual of the same form and give complexity results for several different interior-point algorithms. We obtain an improved complexity result for certain cases by utilizing a combination of the volumetric and logarithmic barriers. As an application, we consider the complexity of solving the Eisenberg–Gale formulation of a Fisher equilibrium problem with linear utility functions.

Global optimization for generalized geometric programming problems with discrete variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management, but most existing methods for solving GGP actually only consider continuous variables. This article presents a new branch-and-bound algorithm for globally solving GGP problems with discrete variables. For minimizing the problem, an equivalent monotonic optimization problem (P) with discrete variables is presented by exploiting the special structure of GGP. In the algorithm, the lower bounds are computed by solving ordinary linear programming problems that are derived via a linearization technique. In contrast to pure branch-and-bound methods, the algorithm can perform a domain reduction cut per iteration by using the monotonicity of problem (P), which can suppress the rapid growth of branching tree in the branch-and-bound search so that the performance of the algorithm is further improved. Computational results for several sample examples and small randomly generated problems are reported to vindicate our conclusions.

A fuzzy modeling for fuzzy portfolio optimization

Conventional portfolio optimization models have an assumption that the future condition of stock market can be accurately predicted by historical data. However, no matter how accurate the past data is, this premise will not exist in the financial market due to the high volatility of market environment. This paper discusses the fuzzy portfolio optimization problem where the asset returns are represented by fuzzy data. A mean-absolute deviation risk function model and Zadeh’s extension principle are utilized for the solution method of portfolio optimization problem with fuzzy returns. Since the parameters are fuzzy numbers, the gain of return is a fuzzy number as well. A pair of two-level mathematical programs is formulated to calculate the upper bound and lower bound of the return of the portfolio optimization problem. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level linear programs so they can be manipulated. It is found that the calculated results conform to an essential idea in finance and economics that the greater the amount of risk that an investor is willing to take on, the greater the potential return. An example, which utilizes the data from Taiwan stock exchange corporation, illustrates the whole idea on fuzzy portfolio optimization problem.

Uniqueness of integer solution of linear equations

We consider the system of m linear equations in n integer variables Ax = d and give sufficient conditions for the uniqueness of its integer solution x ∈ {−1, 1}n by reformulating the problem as a linear program. Necessary and sufficient uniqueness characterizations of ordinary linear programming solutions are utilized to obtain sufficient uniqueness conditions such as the intersection of the kernel of A and the dual cone of a diagonal matrix of ±1’s is the origin in Rn. This generalizes the well known condition that ker(A) = 0 for the uniqueness of a non-integer solution x of Ax = d. A zero maximum of a single linear program ensures the uniqueness of a given integer solution of a linear equation.

Solution Algorithms for a Class of Continuous Linear Programs with Fuzzy Valued Objective Functions

This paper discusses a class of continuous linear programs with fuzzy valued objective functions. A member of this class is called a fuzzy separated continuous linear program (FSCLP). Such problems have applications in a number of domains, including, production and inventory systems, communication networks, and pipeline systems for transportation. The discretization approach is used to construct two ordinary fuzzy linear programming problems, which give a lower and an upper bound on the optimal value of FSCLP. It is then shown how to construct an improved feasible solution for FSCLP starting from a nonoptimal one. This leads to the development of a class of algorithms based on a sequence of discrete approximations to FSCLP. Numerical examples in the context of continuous-time networks are presented to show the applicability of the proposed method.

Guessing Facets: Polytope Structure and Improved LP Decoder

We investigate the structure of the polytope underlying the linear programming (LP) decoder introduced by Feldman, Karger, and Wainwright. We first show that for expander codes, every fractional pseudocodeword always has at least a constant fraction of nonintegral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword. We further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as LP decoding that provably performs better. The method is very simple: it first applies ordinary LP decoding, and when it fails, it proceeds by guessing facets of the polytope, and then resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that the proposed algorithm, when applied to suitable expander codes, succeeds unless there exists a certain number of pseudocodewords, all adjacent to the ML codeword on the LP decoding polytope, and with higher likelihood than the ML codeword. We then describe an extended algorithm, still with polynomial complexity, that succeeds as long as there are at most polynomially many pseudocodewords above the ML codeword.

Matrix games with interval data

The conventional game theory is concerned with how rational individuals make decisions when they are faced with known payoffs. In the real world, sometimes the payoffs are not known and have to be estimated, and sometimes the payoffs are only approximately known. This paper develops a solution method for the two-person zero-sum game where the payoffs are imprecise and are represented by interval data. Since the payoffs are imprecise, the value of the game should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the value of the game. Based on the duality theorem and by applying a variable substitution technique, the pair of two-level mathematical programs is transformed to a pair of ordinary one-level linear programs. Solving the pair of linear programs produces the interval of the value of the game. It is shown that the two players in the game have the same upper bound and lower bound for the value of the imprecise game. An example illustrates the whole idea and sheds some light on imprecise game.