Properties of Max-Plus Algebraic Determinants Derived from Duality Theorem

Elementary Linear Programming with Applications

Applied mathematics plays an important role in solving real-world problems, including in the agricultural product processing industry. This study aims to apply the Modified Ghadle-Munot method to solve employee assignment problems in a private company operating in chocolate production and to connect its implementation with applied mathematics learning. The core issue addressed is the imbalance between the number of employees and the number of production stages, which creates an unbalanced assignment problem. This study uses an applied quantitative research approach with assignment optimization as the main variable. Data were collected through interviews and documentation related to the production process and employee task distribution. The results showed that the Modified Ghadle-Munot method was able to reduce the total work time from an initial average of 970 minutes to 660 minutes. The application of this algorithm not only improves operational efficiency but also carries educational value. It can serve as a contextual case study in teaching applied mathematics, particularly in linear programming and optimization. Therefore, this research contributes both to practical problem-solving in small-scale industry settings and to strengthening real-world, problem-based learning in mathematics education.

An efficient cost scaling algorithm is presented for the independent assignment problem of Iri and Tomizawa, which is equivalent to the weighted matroid intersection problem of Edmonds. Our algorithm in general can be viewed as a generalization of Orlin and Ahuja's scaling algorithm for the ordinary assignment problem. On a bipartite graph with n vertices and integer arc costs bounded by C, an optimal r-independent assignment can be found in O(√<rgt:n^2 log(rC)) time by our algorithm under an independence oracle for matorids.

We apply column generation with branch‐and‐price optimization to a multi‐target, multi‐task assignment problem, with precedence constraints. Column generation transforms the nonlinear program with separable costs and constraints into a linear program. This reformulation divides the original problem into a number of smaller problems, where one of these smaller problems accounts for the coupling constraints between agents and must be known by every agent. All other divisions consider local constraints affecting only one agent; these smaller problems are known by only one corresponding agent. Because of this reformulation, the assignment problem can be solved in a distributed manner. A theorem is proven which details the central analytical result of the paper, allowing a nonlinear program to be reformulated as a linear program. Simulation results for a multi‐target, single‐task assignment problem, as well as a multi‐target, multi‐task assignment problem with precedence constraints are presented. Published 2011. This article is a US Government work and is in the public domain in the USA.

Recently, the installation of the battery for the power system has been investigated. This study focuses on the optimal battery assignment method which assigns the batteries to contribute to the system stabilization against N-1 line accident. In this method, the optimal battery assignment and operation problem whose objective is the minimization of the phase difference fluctuation on all nodes caused by N-1 line accident at each branch is formulated. Then, the optimal battery assignment and the power supplies of batteries are determined by solving the formulated problem. However, this method requires a lot of computation time, because this method requires a large number of power flow calculations with high computational cost. This study proposes to apply the response surface method with the radial basis function networks to the optimal battery assignment and operation problem in order to reduce the number of the power flow calculations. This paper confirms the effectiveness of the proposed method through numerical experiments of the application to the IEEJ EAST 10-machine system model.

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling and transshipment problems, where a set of jobs needs to be assigned to the available set of machines or personnel (resources), in a way that all jobs have assigned resources, and no two jobs share the same resource. In its nominal form, the resulting computational problem becomes the assignment problem. This paper deals with the Robust Assignment Problem (RAP) which models situations in which certain assignments are vulnerable and may become unavailable after the solution has been chosen. The goal is to choose a minimum-cost collection of assignments (edges in the corresponding bipartite graph) so that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all jobs. We develop algorithms and hardness results for RAP and establish several connections to well-known concepts from matching theory, robust optimization, LP-based techniques and combinations thereof.

This paper describes the method of optimization of frequency assignment in satellite communication, aiming at reducing cochannel interference. The method is based on the formulation of the assignment problem, so‐called mathematical programming, and is widely applicable. The basic model is defined, and is composed of a pair of systems sharing the same frequency band. A discrete frequency assignment is made with the small segment as the unit, which is obtained by equi‐partitioning the common channel, and the model is formulated as an assignment problem. The lexicographical minimization is employed as the criterion of optimization, by which the interference is reduced for all carriers. A method is presented in which the optimum solution is obtained by the numerical conversion of the cost matrix and the Hungarian method for the assignment problem. To determine efficiently the optimum assignment for carriers with various bandwidths, an algorithm is presented which is based on the branch‐and‐bound method used frequently in the mathematical programming. By this method, the solution can be determined with a small amount of computation for a problem of practical size. Examples are shown which are the assignment of FM carriers by the basic model and the optimum frequency assignment for eight satellites. As a result, it is shown that the frequency assignment can be determined with sufficient accuracy, and the interference is reduced drastically by sufficiently fine segmentation.

This study deals with an aircraft-to-target assignment (ATA) problem considering the modern air operation environment, such as the strike package concept, multiple targets for a sortie, and the strike packages’ survivability. For the ATA problem, this study introduces a novel mathematical model in which a heterogeneous vehicle routing problem (HVRP) and a weapon-to-target assignment (WTA) problem are conceptually integrated. The HVRP generates the flight routes for strike packages because this study confirms that the survivability of a strike package depends on the path, and the WTA problem evaluates the likelihood of successful target destruction of assigned weapons. Although the first version of the model is developed as a mixed-integer nonlinear programming (MINLP) model, this study attempts to convert it to a mixed-integer linear programming (MILP) model using the logarithmic transformation and piecewise linear approximation methods. For an ATA problem, this activity could provide an opportunity to use the excellent existing algorithms for searching the optimal solution of LP models. To maximize the operational effectiveness, the MILP model simultaneously determines the following for each strike package: (a) composition type, (b) targets, (c) flight route, (d) types, and (e) quantity of weapons for each target.

Linear assignment problems are well-known combinatorial optimization problems involving domains such as logistics, robotics and telecommunications. In general, obtaining an optimal solution to such problems is computationally infeasible even in small settings, so heuristic algorithms are often used to find near-optimal solutions. In order to attain the right assignment permutation, this study investigates a general-purpose learning strategy that uses a bipartite graph to describe the problem structure and a message passing Graph Neural Network (GNN) model to learn the correct mapping. Comparing the proposed structure with two existing DNN solutions, simulation results show that the proposed approach significantly improves classification accuracy, proving to be very efficient in terms of processing time and memory requirements, due to its inherent parameter sharing capability. Among the many practical uses that require solving allocation problems in everyday scenarios, we decided to apply the proposed approach to address the scheduling of electric smart meters access within an electricity distribution smart grid infrastructure, since near-real-time energy monitoring is a key element of the green transition that has become increasingly important in recent times. The results obtained show that the proposed graph-based solver, although sub-optimal, exhibits the highest scalability, compared with other state-of-the-art heuristic approaches. To foster the reproducibility of the results, we made the code available at https://github.com/aircarlo/GNN_LSAP.

3 - Further Topics in Linear Programming

The problem of state feedback optimal pole assignment is to design a feedback gain such that the closed-loop system has desired eigenvalues and such that certain quadratic performance index is minimized. Optimal pole assignment controller can guarantee both good dynamic response and well robustness properties of the closed-loop system. With the help of a class of linear matrix equations, necessary and sufficient conditions for the existence of a solution to the optimal pole assignment problem are proposed in this paper. By properly choosing the free parameters in the parametric solutions to this class of linear matrix equations, complete solutions to the optimal pole assignment problem can be obtained. A numerical example is used to illustrate the effectiveness of the proposed approach.

The purpose of the work is to develop models and algorithms for optimizing matching in dynamically generated graphs of asymmetric relations in coordinated open systems of interacting agents with centralized and collective control. The dynamic asymmetric matching optimization problem arises here as a result of a compromise approximation of the mapping of the dynamic programming method onto a stream of known open assignment problems or several traveling salesmen. However, the branching alternatives presented in this way for independent tasks do not take into account the interdependence of real relationships between agents and their tasks, including their relationship to time. Ignoring the dependence of branching alternatives leads to a delay in the moment or to a loss in the quality of assignment of tasks to coordinated agents. The main idea of the proposed implementation of the principle known for effective control is to postpone the moment the final decision is made to the latest moment, taking into account the susceptibility of the system to local changes in state variables. The interdependence of states is revealed on the basis of the analysis of the correspondence of the graph of the current matching with the optimal solution on the subgraph of perfect matching. The transition between states is implemented by the incremental version of the reoptimization algorithm for solving linear problems of assigning the shortest replenishing path using the method. The space of search states is a dynamically generated bipartite sparse graph of alternatives for a combination of agents and tasks, represented by a list of arcs. To highlight the sets of changed arcs, it is proposed to supplement the weight of the arcs with the boundaries of the stability intervals of the solution, optionally formed in the background. By default, the weight of the modified arc matches the boundary of the stability interval. On each correction cycle of the lists of agents, tasks, and their associations, subsets of elements are selected for which reconsideration of matching is required. An enhanced condition for the selection of such elements is to go beyond the boundaries of the stability interval. In this case, the asymmetry of the assignment problem is taken into account by choosing the adjacency structure for the fraction of the graph with a minimum of vertices. As a result, the reaction time of procedures for solving the assignment problem is reduced by an order of magnitude.

Chapter 26 - Duality in nonlinear programming

Genetic algorithm for the personnel assignment problem with multiple objectives

Recently, Daganzo introduced the cell transmission model—a simple approach for modeling highway traffic flow consistent with the hydrodynamic model. In this paper, we use the cell transmission model to formulate the single destination System Optimum Dynamic Traffic Assignment (SO DTA) problem as a Linear Program (LP). We demonstrate that the model can obtain insights into the DTA problem, and we address various related issues, such as the concept of marginal travel time in a dynamic network and system optimum necessary and sufficient conditions. The model is limited to one destination and, although it can account for traffic realities as they are captured by the cell transmission model, it is not presented as an operational model for actual applications. The main objective of the paper is to demonstrate that the DTA problem can be modeled as an LP, which allows the vast existing literature on LP to be used to better understand and compute DTA. A numerical example illustrates the simplicity and applicability of the proposed approach.