Linear Programming 3: The Tolerance Approach

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.

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.

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

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.

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.

Purpose – Real estate development appraisal is a quantification of future expectations. The appraisal model relies upon the valuer/developer having an understanding of the future in terms of the future marketability of the completed development and the future cost of development. In some cases the developer has some degree of control over the possible variation in the variables. However, other variables are totally dependent upon the vagaries of the market at the completion date. To try to address the risk of a different outcome to the one expected the developer will often carry out a sensitivity analysis on the development. This paper aims to look at the processes for so doing and the role of the decision maker in the analysis.Design/methodology/approach – Traditional sensitivity analysis has generally only looked at the best and worst scenarios and has focused on the anticipated or expected outcomes. This does not take into account uncertainty and the range of outcomes that can happen. A fuller analysis...

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.

Low-Cost, Ready-to-Use Therapeutic Foods Can Be Designed Using Locally Available Commodities with the Aid of Linear Programming ,

Optimizing the production can be obtained by regulating the use of limited company resources. The purpose of this study is to know the optimization of corporate resources to generate maximum profit. The case study was conducted on Terry Kripik Small Business in Nyanglan Kaja Village, Tembuku Subdistrict, Bangli District. Based on the results of linear programming analysis with the help of POM-QM software, optimum production is 9 sacks of Ladrang Chips, 71 sacks of Chicken Chips, 46 sacks of Spinach Chips and 74 sacks of Limo Stick Leaf Chips. BEP value is Rp 30.708.228,00 or equal to 72 sacks. The projection of net profit if production is on demand is Rp 8,293,323.00 while net profit if produced according to the optimal product combination is Rp 11,718,143.00. Companies are encouraged to combine linear programming analysis, break even point, and cost analysis as input for management in making decisions related to optimization.
 Keywords: optimization of production, maximum profit, terry chips, and linear programming

This study investigates the impact of melting/binding rates (referred to hereafter as the parameters) over the polymers and monomers on the dynamics of carbon-monoxide-mediated sickle cell hemoglobin (HbS) de-polymerization. Two approaches, namely the traditional sensitivity analysis (TSA) and the multi-parameter sensitivity analysis (MPSA), have been developed and applied to the mathematical model system to quantify the sensitivities of polymers and monomers to the parameters. The Runge-Kutta method and the Monte-Carlo simulation are employed for the implementation of the sensitivity analyses. The TSA utilizes the traditional sensitivity functions (TSFs). The MPSA enumerates the overall effect of the model input parameters on the output by perturbing the model input parameters simultaneously within large ranges. All four concentrations (namely, de-oxy HbS monomers, CO-bound HbS monomers, de-oxy Hbs polymer and CO-bound HbS polymer) as model outputs, and all four binding/melting rates (namely, the CO binding and melting rates for polymers and monomers) as input parameters are considered in this study. The sensitivity results suggest that TSA and MPSA are essentially consistent.

The utility of transient sensitivity for wildlife management and conservation: Bison as a case study

To design an anti-aircraft gun with high hit-probability, the effect of each design parameter on the hit-probability needs to be estimated. For this purpose, the sensitivity analysis is widely used. However, the whole firing process of the anti-aircraft gun is too complicated to use the traditional sensitivity analysis that uses partial derivatives of the output function regarding the specific design parameter. The inaccuracy of the parts of the anti-aircraft gun that exists at different stage of the firing process will result in decrease of hit-probability. In this paper, the firing process of the anti-aircraft gun is described and each step of the firing process is formulated with the equation that includes the inaccuracies of parts. The anti-aircraft gun is modeled using the transformation matrix from design parameter to the traverse and the elevation angle considering inaccuracy of design parameters. Then by disturbing the each design parameter with random numbers whose average and standard deviation are known and calculating the resultant hit performance, the sensitivity of the design parameter on the hit probability can be estimated.