The Tolerance Approach to Sensitivity Analysis in Linear Programming

Traditional sensitivity analysis in linear programming usually focuses on variations of one coefficient or term at a time. While methods have been proposed for dealing with simultaneous variations of multiple coefficients or terms, the results are limited or more difficult for decision-makers to use. The tolerance approach was proposed to provide a decision-maker with an effective and easy-to-use method to summarize the effects of simultaneous and independent changes in selected parameters. Herein, we review the basic ideas and results of the tolerance approach and explore various extensions that have been proposed.

The 100% Rule was introduced by Bradley, Hax and Magnanti [1] in Sensitivity Analysis of linear programming theory. It is concerned with the qualitative behavior of an optimal solution as it changes according to the right hand side vector. We extend this 100% Rule to the most generalized form, to deal with changes in all the parameters of any given linear program, including the coefficients of the matrix, the objective function as well as the right hand side vector.

We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as functions of a scalar parameter. We present a unifying geometric framework that subsumes the concept of the optimal partition in linear programming (LP) and semidefinite programming (SDP) and extends it to conic optimization. Similar to the optimal partition approach to sensitivity analysis in LP and SDP, the range of perturbations for which the optimal partition remains constant can be computed by solving two conic optimization problems. Under a weaker notion of nondegeneracy, this range is simply given by a minimum ratio test. We discuss briefly the properties of the optimal value function under such perturbations.

The constraint programming community has recently begun to address certain types of optimization problems. These problems tend to be discrete or to have discrete elements. Although sensitivity analysis is well developed for continuous problems, progress in this area for discrete problems has been limited. This paper proposes a general approach to sensitivity analysis that applies to both continuous and discrete problems. In the continuous case, particularly in linear programming, sensitivity analysis can be obtained by solving a dual problem. One way to broaden this result is to generalize the classical idea of a dual to that of an ’’inference dual,‘‘ which can be defined for any optimization problem. To solve the inference dual is to obtain a proof of the optimal value of the problem. Sensitivity analysis can be interpreted as an analysis of the role of each constraint in this proof. This paper shows that traditional sensitivity analysis for linear programming is a special case of this approach. It also illustrates how the approach can work out in a discrete problem by applying it to 0-1 linear programming (linear pseudo-boolean optimization).

The next generation (NextGen) transport aircraft configurations being investigated as part of the NASA Aeronautics Subsonic Fixed Wing Project have more control surfaces, or control effectors, than existing transport aircraft configurations. Conventional flight control is achieved through two symmetric elevators, two antisymmetric ailerons, and a rudder. The five control surfaces, reduced to three command variables, produce moments along the three main axes of the aircraft and enable the pilot to control the attitude of the aircraft. Next generation aircraft will have additional redundant control effectors to control the three moments, creating a situation where the aircraft is over-actuated and where a simple relationship no longer exists between the required control surface deflections and the desired moments. NextGen flight controllers will incorporate control allocation algorithms to determine the optimal effector commands to attain the desired moments, taking into account the effector limits. Approaches to solving the problem using linear programming and quadratic programming algorithms have been proposed and tested. It is of great interest to understand their relative advantages and disadvantages and how design parameters may affect their properties. In this paper, we investigate the sensitivity of the effector commands with respect to the desired moments and show on some examples the sensitivity of the solutions provided by the linear programming and quadratic programming methods.

Sensitivity analysis, a key linear programming (LP) concept, is often explained in text books using complex problem scenarios to which students have difficulty relating. Consequently, many students do not fully comprehend or appreciate the significance of shadow prices or range of optimality for objective coefficients. This adds to the challenges instructors face in promoting critical thinking, a key goal in operations research and management science courses. Limited student-faculty interactions further exacerbate the problem in online learning environments. These issues can be effectively addressed through use of simple real world examples for instruction, followed by discussion of insights and intuition behind results from these mathematical models. This is the motivation of this study, with a special focus on sensitivity analysis concepts and LP formulations. We then demonstrate its effectiveness by comparing student performance before and after exposure to this innovative teaching supplement and as well as results from a student feedback survey.

Application of mixed-integer linear programming in multiterm expansion planning under multiobjectives

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.

With more wind power permeating into the power grid, one of the issues caused by its random output is the fluctuation of locational marginal price (LMP) which is also affected by the fluctuation of system load. However, the fluctuation of electricity price will affect many aspects of the national economy. In order to cope with this issue, a demand response (DR) allocation mechanism is presented in this paper to hedge the LMP variability which is equal to keep the optimal basis invariant. Whether DR is available to keep the optimal basis optimal when wind power and load fluctuate within certain ranges is estimated from the perspective of sensitivity analysis of linear programming. Finally, the scheme is applied to test case IEEE 31 bus system and its feasibility is proved by the simulation results.

Linear program under changes in the system matrix coefficients has proved to be more complex than changes of the coefficients in objective functions and right hand sides. The most of the previous studies deals with problems where only one coefficient, a row (column), or few rows (columns) are linear functions of a parameter. This work considers a more general case, where all the coefficients are polynomial (in the particular case linear) functions of the parameter tT??R. For such problems, assuming that some non-singularity conditions hold and an optimal base matrix is known for some particular value t of the parameter, corresponding explicit optimal basic solution in the neighborhood of t is determined by solving an augmented LP problem with real system matrix coefficients. Parametric LP can be utilized for example to model the production problem where, technology, resources, costs and similar categories vary with time. .

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.

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.

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.

Duality and sensitivity analysis of multistage linear stochastic programs

When a linear programming problem is found to have no feasible solution, the model-builder generally has no tools for systematically determining why the infeasibility exists and what might be done to eliminate it. The assumption is made, at least implicitly, that if the model has no feasible solution, it is wrong to some fundamental way and must be corrected before it can be of use to the model-builder. In fact, the model may be quite correct as it stands, in the sense that it captures exactly what the model-builder meant for it to capture. When this is the case, discovering the absence of a feasible solution can be useful to the model-builder, and correspondingly, tools for exploring the nature of the infeasiblity can be quite valuable. The purpose of this paper is to develop a set of methods for doing post-infeasibility analysis on linear programming problems. The methods are designed to identify constraints that might be “relaxed” to attain feasibility and estimate the magnitudes of the required changes.