On some aspects of the matrix data perturbation in linear program

In this paper, we study general linear programs in which right hand sides are interval numbers. This model is relevant when uncertain and inaccurate factors make difficult the assignment of a single value to each right hand side. When objective function coefficients are interval numbers in a linear program, classical criteria coming from decision theory (like the worst case criterion) are usually applied to determine robust solutions. When the set of feasible solutions is uncertain, we identify a class of linear programs for which these classical approaches are no longer relevant. However, it is possible to compute the worst optimum solution. We study the complexity of this optimization problem when each right hand side is an interval number. Then, we exhibit some duality relationships between the worst optimum solution problem and the best optimum solution to the dual problem.

The traditional linear programming model is deterministic. The way that uncertainty is handled is to compute the range of optimality. After the optimal solution is obtained, typically by the simplex method, one considers the effect of varying each objective function coefficient, one at a time. This yields the range of optimality within which the decision variables remain constant. This sensitivity analysis is useful for helping the analyst get a sense for the problem. However, it is unrealistic because objective function coefficients tend not to stand still. They are typically profit contributions from products sold and are subject to randomly varying selling prices. In this paper, a realistic linear program is created for simultaneously randomizing the coefficients from any probability distribution. Furthermore, we present a novel approach for designing a copula of random objective function coefficients according to a specified rank correlation. The corresponding distribution of objective function values is created. This distribution is examined directly for central tendency, spread, skewness and extreme values for the purpose of risk analysis. This enables risk analysis and business analytics, emerging topics in education and preparation for the knowledge economy.

In constrast to “ordinary” sensitivity analysis in linear programming, the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right-hand side terms. This approach yields a maximum tolerance percentage such that, as long as selected coefficients or terms are accurate to within that percentage of their estimated values, the same basis is optimal. In particular, if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values, then the same solution is optimal.

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company’s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company’s activity. The paper is accompanied by a practical example.

In this paper, a linear fractional programming problem with interval coefficients in both objective function and constraints is solved. The uncertain objective function is transformed into two deterministic objective functions. The two objective functions are converted into a single objective problem through the linear combination method. Through a modified possibility degree, the uncertain inequality and equality constraints are transformed into deterministic inequality constraints. The existing results are concerning the qualitative and quantitative analysis of basic notions in parametric linear fractional programming problem. These notions are the set of feasible parameters, the solvability set and the stability set of the first kind. A numerical example is investigated to demonstrate the effectiveness of the present approach.

The various robust linear programming models investigated so far in the literature essentially appear to be based either on what is referred to as rowwise uncertainty models or on columnwise uncertainty models (these basically assume that the rows, or the columns, of the constraint matrix are subject to changes within a well-specified uncertainty set). In this chapter, we discuss a special case of columnwise uncertainty, namely the subclass of robust linear programming (LP) models with uncertainty limited to the right hand side (RHS) only (this subclass does not appear to have been significantly investigated so far). In this context we introduce the concept of a ‘two-stage robust LP model’ as opposed to the standard case (which might be referred to as a ‘single-stage robust LP model’) and we address the question of whether LP duality can be used to convert a LP problem with RHS uncertainty into a robust LP problem with uncertainty on the objective function. We show how to derive both statements of (a) the dual to the robust model and (b) the robust version of the dual. The resulting expressions of the objective function to be optimized in both cases, appear to be clearly distinct. Moreover, from a complexity point of view, one appears to be efficiently solvable (it reduces to a convex optimization problem), whereas the other, as a nonconvex optimization problem, is expected to be computationally difficult in the general case. As an application of the two-stage robust LP model introduced here, we next investigate the PERT (program evaluation and review technique) scheduling problem, considering two possible natural ways of specifying the uncertainty set for the task durations: the case where the uncertainty set is a scaled ball with respect to the L ∞ norm; the case where the uncertainty set is a scaled Hamming ball of bounded radius (which, though leading to a quite different model, bears some resemblance to the well-known Bertsimas–Sim approach to robustness). We show that in both cases the resulting robust optimization problem can be efficiently solved in polynomial time.

In recent years, many researchers in the field of fuzzy mathematics and operations research in the world are interested to solve the interval type-2 fuzzy linear programming (IT2FLP) problems. In IT2FLP problems all parameters, including coefficients in objective function, coefficients in the constraints and right hand sides of the constraints are interval type-2 fuzzy numbers (IT2FN). Hence, the IT2FLP model is one of the most important and the most appropriate models to interpret and determine the optimal decision. In this paper, a new ranking function method is proposed for solving these problems. Moreover, an example is used to study and describe the proposed method.

Linear programming problems with coefficients of the objective function, depending non linear from one parameter, may be solved using the theory of multiparametric linear programming (cf. Sokolova [3]). The condition under which this theory can be used is a special kind of dependence for each coefficient that consists practically in representing each coefficient as a linear combination with constant and variable factors with a known variation (e.g. whether, variation of different economic factors etc.). One gets subsets of the set on which the curve appearing in such a manner in the parameter space is defined by intersecting this curve with certain maximal polyhedrons, arising by computing the associated multiparametric problem. All elements of such a subset have either the same optimal solution or an unbounded objective function. At least one gives a computing procedure for the described problems.

Most of the current methods for solving linear fractional programming (LFP) problems depend on the simplex type method. In this paper, we present a new approach for solving linear fractional programming problem in which the objective function is a linear fractional function, while constraint functions are in the form of linear inequalities. This approach does not depend on the simplex type method. Here first we transform this LFP problem into linear programming (LP) problem and hence solve this problem algebraically using the concept of duality. Two simple examples to illustrate our algorithm are given. And also we compare this approach with other available methods for solving LFP problems.

Linear bilevel programming with interval coefficients

This paper proposes the fuzzy mean-entropy portfolio models with transaction costs based on credibility theory. In the paper, entropy is used as the measurement of risk. Furthermore, sensitivity analysis is discussed for objective function coefficients and constraint coefficients on the right sides in our proposed models. In addition, two numerical examples are given to illustrate the effectiveness of our proposed models and the practicability of sensitivity analysis. More importantly, the obtained results also show that when certain coefficient changes in some value range, we still can obtain the unchanged optimal solutions or unchanged objective function values. Compared with Huang (IEEE Trans Fuzzy Syst 16:1096–1101, 18; Fuzzy Optim Decis Mak 10:71–89, 19), our paper not only proposes the mean-entropy models, but also does research work on sensitivity analysis about objective function coefficients and constraint coefficients in depth in maximizing return model and minimizing risk model. Our results can provide more choices for investors in the practical financial market.

Multivariate probability distributions with given marginals are considered, along with linear functionals, to be minimized or maximized, acting on them. The functionals are supposed to satisfy the Monge or inverse Monge or some higher order convexity property, and they may be only partially known. Existing results in connection with Monge arrays are reformulated and extended in terms of linear programming dual feasible bases. Lower and upper bounds are given for the optimum value as well as for unknown coefficients of the objective function, based on the knowledge of some dual feasible bases and corresponding objective function coefficients. In the two- and three-dimensional cases dual feasible bases are obtained for the problem, where not only the univariate marginals but also the covariances of the pairs of random variables are known.

Many problems appear to be arising in the present as a result of variations in transportation networks. The stochastic fixed-charge transportation problem (SFCTP) is one such problem. The SFCTP is transformed into a chance-constrained programming (CCP) problem where supply and demand are stochastic and objective functions are in a rough interval. In this model, to analyze the multi-objective rough interval stochastic fixed-charge transportation problem (MORISFCTP), where the objective function coefficients are represented by rough intervals and the supply and destination factors are probabilistic constraints. This model operates an expected value operator to deal with uncertainty, in which the coefficient of the objective functions in the fuzzy is changed to a crisp form, and the probabilistic constraints are converted to a deterministic form by the Weibull distribution. To produce the optimal compromise solutions of the proposed model, three distinct methods are used: the fuzzy programming approach, the method of a linear weighted sum, and the €-constraint method. Lastly, the paper delivers a practical illustration of a MORISFCTP to demonstrate the usefulness and feasibility of the suggested methodology.

The increasingly rapid development of the business world requires companies engaged in industry, trade, and services to continue to optimize their business activities to win the market competition. Mikaila Bakery is one of the businesses producing birthday cakes in Gorontalo Province. In the birthday cake business at Mikaila Bakery, so far, the amount of production is only determined by estimates, so profits do not reach optimal profits. This study aimed to determine the optimal profit obtained in the production of birthday cakes at Mikaila bakery using the branch and bound method and to determine the results of sensitivity analysis to changes in the objective function coefficients and constraint functions in the branch and bound method. The analysis results show that using the branch and bound method, Mikaila Bakery's cake shop gets an optimal profit of Rp. 38,433,000 per month or an increase of Rp. 3.4% of the previous profit. The results of the sensitivity analysis show that profits will remain in optimal conditions if the birthday cake business at Mikaila Bakery follows the changes that occur in the coefficient of the objective function and the constraint function based on the upper bound and lower bound according to the output. Which has been generated using POM-QM software.

Fuzzy preference relations in models of decision making