Production Planning of LCDs: Optimal Linear Programming and Sensitivity Analysis?

Aim: This research takes into the production of Flat Panel Monitor of four sizes and will point more the products that contribute the main function of profit. Methodology: For the optimization of the profit of LCDs manufacturing company, the linear programming and sensitivity analysis methods were applied. The four constraints of the LCDs production planning are (1) acquire of line space for production, (2) the assembly of products, (3) Quality control and assurance Hours (4) and packaging of material. Results: In all three scenarios the total profit is optimum and increases from scenario 1 to scenario 3. The difference between the profit of scenario 1 and scenario 2 is 257625, and gap between scenario 2 and scenario 3 is 171750. Conclusion: the three scenarios for the production of the LCDs present the varying consequences of the maximum profit for the company. However, the third scenario is the most optimal solution for the maximization of the objective function. Keywords: Production Planning, Linear Programming, Sensitivity Analysis, Simplex Method, Operations Research, LCD Monitor

Harnessing Certainty to Speed Task-Allocation Algorithms for Multi-Robot Systems

Some problems are best solved by systems of multiple robots, in which each robot is assigned one task. A multi-robot system can, upon the start of a series of tasks, compute the optimal task allocation for best performance of the team. For certain systems, during runtime, changes in the environment, tasks, and state of individual robots might change which allocation of tasks to robots is optimal, and the performance of the team would improve if the robots switched tasks. Because communication between robots is expensive, in some cases it is better to calculate an interval in which changes in the environment, tasks, and robots are not significant enough to render the original allocation suboptimal. This way, robots only initiate communication and correction if the system is likely to switch tasks, which limits the costs of communication and computation in the system. In the problem of task allocation of single robot, single task cases where environments and thus optimal assignments are expected to vary over time, some knowledge of the system might help reduce computation and make possible a more scalable algorithm for determining cost changes. In some systems, some costs may be known not to vary over time. This research proposes creating and analyzing cost matrices of assignments to examine if taking advantage of the certainty of some variables will improve performance. If 2 successful, models for exploiting certainty of task allocation will take less computation than calculating ranges for all variables, and will save resources during runtime

TOLERANCE SENSITIVITY ANALYSIS: THIRTY YEARS LATER

Tolerance sensitivity analysis was conceived in 1980 as a pragmatic approach to effectively characterize a parametric region over which objective function coefficients and right-hand-side terms in linear programming could vary simultaneously and independently while maintaining the same optimal basis. As originally proposed, the tolerance region corresponds to the maximum percentage by which coefficients or terms could vary from their estimated values. Over the last thirty years the original results have been extended in a number of ways and applied in a variety of applications. This paper is a critical review of tolerance sensitivity analysis, including extensions and applications

New conditions for testing necessarily/possibly efficiency of non-degenerate basic solutions based on the tolerance approach

In this paper, a specific type of multiobjective linear programming problem with interval objective func- tion coefficients is studied. Usually, in such problems, it is not possible to obtain an optimal solution which optimizes simultaneously all objective functions in the interval multiobjective linear programming (IMOLP) problem, requiring the selection of a compromise solution. In conventional multiobjective pro- gramming problems these compromise solutions are called efficient solutions. However, the efficiency cannot be defined in a unique way in IMOLP problems. Necessary efficiency and possible efficiency have been considered as two natural extensions of efficiency to IMOLP problems. In this case, necessarily ef- ficient solutions may not exist and the set of possibly efficient solutions usually has an infinite number of elements. Furthermore, it has been concluded that the problem of checking necessary efficiency is co- NP-complete even for the case of only one objective function. In this paper, we explore new conditions for testing necessarily/possibly efficiency of basic non-degenerate solutions in IMOLP problems. We show properties of the necessarily efficient solutions in connection with possibly and necessarily optimal solu- tions to the related single objective problems. Moreover, we utilize the tolerance approach and sensitivity analysis for testing the necessary efficiency. Finally, based on the new conditions, a procedure to obtain some necessarily efficient and strictly possibly efficient solutions to multiobjective problems with interval objective functions is suggested.This research was partly supported by the Spanish Ministry of Economy and Competitiveness (project ECO2017-88883-R ) and by the Fundação para a Ciência e a Tecnologia (FCT) under project grant UID/Multi/00308/2019 . This work has been also partly sup- ported by the Consejería de Innovación, Ciencia y Empresa de la Junta de Andalucía (PAI group SEJ-532 ). Carla Oliveira Henriques also acknowledges the training received from the University of Malaga PhD Programme in Economy and Business [Programa de Doctorado en Economía y Empresa de la Universidad de Malaga]. José Rui Figueira acknowledges the support from the FCT grant SFRH/BSAB/139892/2018 under POCH Program and to the DOME (Discrete Optimization Methods for Energy management) FCT Re- search Project (Ref: PTDC/CCI-COM/31198/2017)

Assignment Algorithms for Multi-Robot Task Allocation in Uncertain and Dynamic Environments

Multi-robot task allocation is a general approach to coordinate a team of robots to complete a set of tasks collectively. The classical works adopt relevant theories from other disciplines (e.g., operations research, economics), but oftentimes they are not adequately rich to deal with the properties from the robotics domain such as perception that is local and communication which is limited. This dissertation reports the efforts on relaxing the assumptions, making problems simpler and developing new methods considering the constraints or uncertainties in robot problems. We aim to solve variants of classical multi-robot task allocation problems where the team of robots operates in dynamic and uncertain environments. In some of these problems, it is adequate to have a precise model of nondeterministic costs (e.g., time, distance) subject to change at run-time. In some other problems, probabilistic or stochastic approaches are adequate to incorporate uncertainties into the problem formulation. For these settings, we propose algorithms that model dynamics owing to robot interactions, new cost representations incorporating uncertainty, algorithms specialized for the representations, and policies for tasks arriving in an online manner. First, we consider multi-robot task assignment problems where costs for performing tasks are interrelated, and the overall team objective need not be a standard sum-of costs (or utilities) model, enabling straightforward treatment of the additional costs incurred by resource contention. In the model we introduce, a team may choose one of a set of shared resources to perform a task (e.g., several routes to reach a destination), and resource contention is modeled when multiple robots use the same resource. We propose efficient task assignment algorithms that model this contention with different forms of domain knowledge and compute an optimal assignment under such a model. Second, we address the problem of finding the optimal assignment of tasks to a team of robots when the associated costs may vary, which arises when robots deal with uncertain situations. We propose a region-based cost representation incorporating the cost uncertainty and modeling interrelationships among costs. We detail how to compute a sensitivity analysis that characterizes how much costs may change before optimality is violated. Using this analysis, robots are able to avoid unnecessary re-assignment computations and reduce global communication when costs change. Third, we consider multi-robot teams operating in probabilistic domains. We represent costs by distributions capturing the uncertainty in the environment. This representation also incorporates inter-robot couplings in planning the team’s coordination. We do not have the assumption that costs are independent, which is frequently used in probabilistic models. We propose algorithms that help in understanding the effects of different characterizations of cost distributions such as mean and Conditional Value-at-Risk (CVaR), in which the latter assesses the risk of the outcomes from distributions. Last, we study multi-robot task allocation in a setting where tasks are revealed sequentially and where it is possible to execute bundles of tasks. Particularly, we are interested in tasks that have synergies so that the greater the number of tasks executed together, the larger the potential performance gain. We provide an analysis of bundling, giving an understanding of the important bundle size parameter. Based on the qualitative basis, we propose multiple simple bundling policies that determine how many tasks the robots bundle for a batched planning and execution