Linear Programming Solution Research Articles

Approximating Min-Mean-Cycle for Low-Diameter Graphs in Near-Optimal Time and Memory

We revisit Min-Mean-Cycle, the classical problem of finding a cycle in a weighted directed graph with minimum mean weight. Despite an extensive algorithmic literature, previous work failed to achieve a near-linear runtime in the number of edges $m$. We propose an algorithm with near-linear runtime $\tilde{O}(m (W_{\max}/\epsilon)^2)$ for computing an $\epsilon$ additive approximation on graphs with polylogarithmic diameter and weights of magnitude at most $W_{\max}$. In particular, this is the first algorithm whose runtime scales in the number of vertices $n$ as $\tilde{O}(n^2)$ for the complete graph. Moreover---unconditionally on the diameter---the algorithm uses only $O(n)$ memory beyond reading the input, making it “memory-optimal". Our approach is based on solving a linear programming (LP) relaxation using entropic regularization, which reduces the LP to a Matrix Balancing problem---à la the popular reduction of Optimal Transport to Matrix Scaling. We then round the fractional LP solution using a variant of the classical Cycle-Canceling algorithm that is sped up to near-linear runtime at the expense of being approximate, and implemented in a memory-optimal manner. The algorithm is simple to implement and is competitive with the state-of-the-art methods in practice.

Sums of Squares Polynomial Program Reformulations for Adjustable Robust Linear Optimization Problems with Separable Polynomial Decision Rules

We show that adjustable robust linear programs with affinely adjustable box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) polynomial reformulations. These problems share the same optimal values and admit a one-to-one correspondence between the optimal solutions. A sum of squares representation of non-negativity of a separable non-convex polynomial over a box plays a key role in the reformulation. This reformulation allows us to find adjustable robust solutions of uncertain linear programs under box data uncertainty by numerically solving their associated equivalent SOS polynomial optimization problem using semi-definite linear programming. We illustrate how the quality of the adjustable robust solution of a robust optimization problem with polynomial decision rules improves as the degree of the polynomial increases. Our results demonstrate that the adjustable robust solutions approach the actual optimal solution as the degree of the polynomial increases from one to fifteen.

Deconstruction of Road Logistics Transportation Cost Management Evaluation Based on Optimal Solution of Linear Programming

Due to the rapid development of the logistics industry, the business volume of the road transportation industry is also growing rapidly and its transportation costs are also increasing. Therefore, it is particularly important to control the transportation cost. This paper aims to evaluate and analyze the cost management of road logistics transportation based on the optimal solution of linear programming. This paper introduces the basic concept of highway logistics in general and analyzes its components and characteristics. Road logistics transportation has the characteristics of fast speed, low investment, strong flexibility, and easy to adapt to local conditions and does not require much infrastructure. We then analyzed the general linear programming model, made a brief introduction to its mathematical model, and proposed the construction of the optimization model to explore the transportation model of production and sales balance. Finally, taking W logistics company as a case to analyze and discuss, the influence of the transportation route in transit on the transportation cost is studied. And its structure and the corresponding transportation cost of logistics enterprises are analyzed in detail. The experimental results of this paper show that by comparing the calculation results of the road segment method and the linear programming model, the total freight calculated by the segment method is 5610 yuan, while the total freight calculated by the linear programming method is 5490 yuan. Comparing the piecewise methods, it can be found that the linear programming method is more economical in transportation cost than the piecewise method. Therefore, when logistics companies optimize road transportation costs, it is more feasible and effective to use linear programming methods.

A Model for Optimal Economic Lockdown to Contain Epidemic

This paper is concerned with the application of operations research in defining the optimal lockdown of economic activities to contain epidemic. The problem of optimal lockdown consists in deciding as best economic sectors can be lockdown with respect to fundamental sectors (essential goods and services) while disruptive impacts are minimized on the economy as a whole. Many countries around the world are currently implementing the lockdown of most economic activities to contain the spread of the Covid-19 pandemic. The lockdown brings health benefits for the society as it contains the spread of the virus, reducing the number of infections and allowing the health system to treat those infected better. This paper describes a Boolean linear programming model to deal with the problem of selecting several economic sectors to be shutdown. The objective function is linear and the constraints are linear inequalities related to the Leontief’s input-output table. The model permits to analyze the feasibility of national economic system in which some elements of the input-output table are set equal to zero. The mathematical approach to the shutdown problem permits to identify the greatest number of economic sectors that can be closed without destroying the fundamental sectors. Since solution of the shutdown problem is the greatest number of lockdown economic sectors, author believes the model allows to oppose effectively the spread of virus. Once spread of the virus decreases, another feature of the model is to support decision makers in assigning priorities to the economic sectors to be gradually unlocked. Standard input-output models are able to reveal how different sectors of an economy are interconnected and how changes in one sector affect all other sectors. Besides the use of the input-output data for descriptive analyses of economic interrelations, this table also provides the empirical fundament for a wide scale of impact analysis. input- output models greatly differ in size and possible applications. Simple static input-output models are used for comparative-static impact (scenario) analysis. With the help of input-output quantity models, statements can be made about direct and indirect effects based on exogenous changes in demand. More complex dynamic input-output models largely resolve the limitations and inherent assumptions of static input-output models. Time is considered explicitly; quantity and price reactions are modelled endogenously in a holistic approach and feedback effects are captured. Last models are not only suitable for scenario analysis and for forecasting as well. In “A Linear Programming Solution to Dynamic Leontief type Models” Harvey M. Wagner presents a general dynamic model of an economy and investigates a number of questions related to the feasibility of certain time profiles of demand and the rate of substitution between economic activities. The Inoperability Input-Output Model [1-4]. was developed to understand better the infrastructure interdependencies. Based on the Leontief Input- Output Model, the model is demand-driven, wherein perturbations to the final demand levels are considered the initiating event, and the impact to sectors’ production outputs are the direct and indirect effects resulting from sector interdependencies. Case study shows that the different perturbation classes (demand or supply, quantity or price) yield different rank orders of sectors impacted by the initiating event, thus providing various perspectives of impacts. The present study differs from most previous studies in two aspects. First, linear programming techniques have been widely used in Input-Output model: from the primary and dual formulation of the problem to the analysis of Lagrange multipliers in order to assess the mutual relative importance of sectors. Second, the impact of lockdown is measured in terms of annihilated interdependencies rather than in terms of canceled economic values such as revenues and monetary exchanges. For this reason, the proposed model takes into account the adjacency matrix got from input-output Table rather than the Table itself. This analysis is static in character (it is a static I-O model) because it does not take care explicitly of time, quantity and price reactions, consumption and production lags, growth of final demand (these ones are known as dynamic I-O models). The study simply considers the interruption of several activity as a consequence of a government measure adopted to reduce the risk of contagion. The proposed model answers to the basic question if the annihilation of N sectors is feasible, secondary the model identifies in the Lagrange multipliers the means to sequencing the sectors to be reopened.

Advanced distribution transformer asset management embracing its relocation: A cost saving approach

Change in customers' consumption patterns due to the introduction of new electricity consuming technologies into human consumption practices has impacted the existing optimal loading profile of the grid facilities. Although some MV/LV distribution transformers (DTs) are operating under a light load profile, this does not guarantee that they are in a healthy condition, since necessary maintenance procedures are required. On the other hand, many unplanned overloading conditions occur on the other DTs and causes accelerated failures which contributes to very high operational costs. This paper presents the outcome of a study on how the planned DTs relocation can save asset managers against their huge DTs replacement costs developed due to the highly overloaded DTs failures in a utility, and how it will improve the overall reliability of the DTs. A mixed-integer linear programming (MILP) solution is proposed to formulate the introduced economic relocation-based asset management (AM) framework while minimizing total cost (TC) of the network. The results show that incorporating relocation techniques into the AM formulation ends in reduction of the distribution network TC and a decrease in asset failure rates if compared with application of traditional techniques.

Intuitionistic fuzzy optimization approach in optimal irrigation planning of Ukai-Kakrapar irrigation project, India

ABSTRACT This article addresses the formulation of the Intuitionistic Multiobjective fuzzy linear programming (IFO MOFLP) model for optimal allocation of crops in the command area of the Ukai-Kakrapar Irrigation Project, India. Initially, the crisp linear programming (LP) model is applied to acquire the optimal results of three conflicting objectives, namely, maximization of net irrigation benefits (NIBs), minimization of cost of cultivation (CC) and maximization of revenue generation from industrial and municipal supplies (MI). Subsequently, the LP solutions were used to develop the IFO MOFLP, IFO MOFLP with a hesitation index and two-phase IFO MOFLP (TPIFO MOFLP) models for the complete command area. The performance of the aforesaid models is assessed in terms of irrigation intensity, degree of acceptance (α), rejection (β) and hesitation index (π) for inflows at 75% probability of exceedance. The irrigation intensity from the proposed TPIFO MOFLP model has been found to be 101.28%, while NIB, CC and MI from the proposed model are 10,575.38, 5438.42 and 2638.20 million Rs, respectively, with α = 0.64, β = 0.18 and π = 0.18, respectively. The proposed TPIFO MOFLP model has been compared better with MOFLP as the former gives additional uncertainty controlling parameters like α, β and π, which would help the planner to take better decisions for real-world problems.

Exact solution of network flow models with strong relaxations

We address the solution of Mixed Integer Linear Programming (MILP) models with strong relaxations that are derived from Dantzig–Wolfe decompositions and allow a pseudo-polynomial pricing algorithm. We exploit their network-flow characterization and provide a framework based on column generation, reduced-cost variable-fixing, and a highly asymmetric branching scheme that allows us to take advantage of the potential of the current MILP solvers. We apply our framework to a variety of cutting and packing problems from the literature. The efficiency of the framework is proved by extensive computational experiments, in which a significant number of open instances could be solved to proven optimality for the first time.

Application of fuzzy random-based multi-objective linear fractional programming to inventory management problem

This research article aims to study a multi-objective linear fractional programming (FMOLFP) problem having fuzzy random coefficients as well as fuzzy pseudorandom decision variables. Initially, the FMOLFP model is converted to a single objective fuzzy linear programming (FLP) model. Secondly, we show that a fuzzy random optimal solution of an FLP problem is resolved into a class of random optimal solution of relative pseudorandom linear programming (LP) model. As a result, some of theorems show that a fuzzy random optimal solution of a fuzzy pseudorandom LP problem is combined with a series of random optimal solutions of relative pseudorandom LP problems. As an application, the developed approach is implemented to an inventory management problem by taking the parameters as trapezoidal fuzzy numbers, ultimately resulting in a new initiative for modelling real-world problems for optimization. In the last, some numerical examples are introduced to clarify the obtained results and their applicability.

Interactive visualization for diagnosis of industrial Model Predictive Controllers with steady-state optimizers

Model Predictive Controllers (MPCs) are widely used in the process industries and are typically implemented with an integrated Linear Program (LP) optimizer in the form of two-stage LP-MPC systems. Despite significant control-theoretic advances in MPC design and performance evaluation in academia, there is still a gap in addressing operational issues in real-world MPC controllers. In particular, engineers and operators responsible for sustaining MPCs often need to interpret the LP solution to understand the controller’s actions. Without easy interpretability, it is difficult to troubleshoot MPCs especially for large-dimensional controllers. To alleviate this difficulty, a systematic approach that facilitates LP solution diagnostics using tools from data visualization and process control is developed. The ‘partial pivoting’ operation – an industrial practice that has seen limited exposure in academic literature – is discussed in detail with regards to its role in LP solution diagnosis. Typical workflows for diagnosing problematic controllers are used in conjunction with data visualization principles to guide the design of new tools focused on visualizing variable constraint data that facilitate the diagnosis process. These proposed tools are designed using Munzner’s “Nested Model” as a guiding framework for visualization design and evaluation. The use of these tools is demonstrated in multiple industrial examples, with comparison to current industrial methodologies.

Neutrosophic Handling of the Simplex Direct Algorithm to Define the Optimal Solution in Linear Programming

Mathematical programming has the ability to express competency concepts in a well-defined mathematical model for a particular situation or system, and the ability to derive computational methods to solve this mathematical model, it is also a mathematical tool that allows us to model, analyze and solve a wide range of problems concerned with allocating rare resources of labor, materials, machinery, and capitals. Consequently, using them in the best attainable way in order to minimize costs or maximize profits. In such issues, the linear programming is one of the most widely used types of mathematical programming because it is a method that helps to make good decisions and decide the best program for independent activities, taking into account the available sources. It does not take in consideration the continuous and rapid changes and the state of instability in data. So, this manuscript studies one of the methods to solve linear models, which is the simplex method using the neutrosophic theory that covers all the data in analysing, whether specific or not, determined or not, having consistency or not, as well as it deeming all occurring changes. However, the optimal solution is related to the variables in the objective function, which in turn are affected by the fixed quantities that express the available possibilities. This article presents a study to solve the linear model using the simplex method in which the variables and their coefficients are indeterminate values, and we will explain the effected of the indeterminate values on the optimal solution of the mathematical model. The product mixture problem has been presented as case study to demonstrate the efficiency of the proposed method.

A Branch and Bound Algorithm for Transmission Network Expansion Planning Using Nonconvex Mixed-Integer Nonlinear Programming Models

The branch and bound (BB) algorithm is widely used to obtain the global solution of mixed-integer linear programming (MILP) problems. On the other hand, when the traditional BB structure is directly used to solve nonconvex mixed-integer nonlinear programming (MINLP) problems, it becomes ineffective, mainly due to the nonlinearity and nonconvexity of the feasible region of the problem. This article presents the difficulties and ineffectiveness of the direct use of the traditional BB algorithm for solving nonconvex MINLP problems and proposes the formulation of an efficient BB algorithm for solving this category of problems. The algorithm is formulated taking into account particular aspects of nonconvex MINLP problems, including (i) how to deal with the nonlinear programming (NLP) subproblems, (ii) how to detect the infeasibility of an NLP subproblem, (iii) how to treat the nonconvexity of the problem, and (iv) how to define the fathoming rules. The proposed BB algorithm is used to solve the transmission network expansion planning (TNEP) problem, a classical problem in power systems optimization, and its performance is compared with the performances of off-the-shelf optimization solvers for MINLP problems. The results obtained for four test systems, with different degrees of complexity, indicate that the proposed BB algorithm is effective for solving the TNEP problem with and without considering losses, showing equal or better performance than off-the-shelf optimization solvers.

A constructive method for solving a multi-objective linear program with bounded variables

The search for an optimal solution remains a central objective in optimisation, but in many situations the optimal solution cannot be obtained in polynomial time. For this, it would sometimes be necessary to find a good solution close to the optimum that can be obtained in a reasonable time. The aim of this paper is to find an ε-efficient or efficient solution for a multi-objective linear programming problem with polyhedral constraints and bounded decision variables. For this, we define an ε-efficient solution for which a characterisation theorem is proved, and where the ε-optimality criterion is formulated simultaneously for all the objectives of the problem. By using this concept of ε-efficiency, we develop an algorithm that combines the well known Benson's procedure in multi-objective linear programming and the adaptive method elaborated for finding ε-optimal solutions in mono-objective linear programming, based on the diminution of the suboptimality estimate. The proposed algorithm is illustrated by a numerical example and finally, it is implemented in MATLAB to solve a set of generated test problems. The efficient solutions found are located on the Pareto fronts of the same test problems treated by using the well-known Gamultiobj algorithm of MATLAB optimisation tool.

CAMDNN: Content-Aware Mapping of a Network of Deep Neural Networks on Edge MPSoCs

Machine Learning (ML) workloads are increasingly deployed at the edge. Enabling efficient inference execution while considering model and system heterogeneity remains challenging, especially for ML tasks built with a network of DNNs. The challenge is to maximize the utilization of all available resources on the multiprocessor system on a chip (MPSoC) at the same time. This becomes even more complicated because the optimal mapping for the network of DNNs can vary with input batch sizes and scene complexity. In this paper, a holistic hierarchical scheduling framework is presented to optimize the execution time for a network of DNN models on an edge MPSoC at runtime, considering varying input characteristics. The framework consists of a local and a global scheduler. The local scheduler maps individual DNNs in the inference pipeline to the best-performing hardware unit while the global scheduler customizes an Integer Linear Programming (ILP) solution to instantiate DNN remapping. To minimize scheduler runtime overhead, an imitation learning (IL) based scheduler is used that approximates the ILP solutions. The proposed scheduling framework (CAMDNN) was implemented on a Qualcomm Robotic RB5 platform. CAMDNN resulted in lower execution time of up to 32% than HEFT, and by factors of 6.67X, 5.6X and 2.17X than the CPU-only, GPU-only and Central Queue schedulers.

Control of Time, Cost and Quality of Construction Project Management

One of the most common methods of projects management of the implementation of construction projects is the Project Management Triangle (PMT), this method give the balance between the three heads of this triangle time, cost and quality. In addition, the main principle is integrating the design and construction phases in a parallel way, where the basic elements of the project time, cost and quality will not be influenced by each other. some of the projects are established by some construction companies in developing countries in the world have been failed to manage and solve the weaknesses points in these projects and sometimes failures due to the increase in cost or the low level of quality, as found by the field survey. Therefore, the research aim is to develop and apply the general planning mechanisms of the project to get the development of projects detailed plans to control the factors time, cost and quality of the executed works. The main results of the study were, depending on the choice of the target function, can give early or late start times by linear programming solutions for all nodes or events in the project domain; so the early start times of the project should be determined first. The supposed project domain includes a number of nodes ranging from (1-m), as node (1) led to represent the opening of the project activity work and node (m) represent the end of the project and that will be found the balance between the (PMT) elements.

On multi-objective extensions of the classical assignment model with fuzzy parameters and fuzzy goals

This study advances the limitations of current fuzzy multi-objective assignment models by exploring some formulations with fuzzy parameters and fuzzy goals. The first formulation expresses the coefficients of the objective functions and constraints as fuzzy sets. Two crucial fuzzy transformations were adopted, along with the computational process of the epsilon-constrained multi-objective optimisation. On the other hand, the second formulation assumes the fuzzy coefficients of objective functions and constraints and extends such fuzziness by introducing fuzzy constraints. Lastly, the coefficients of the objective functions and the constraints are expressed as crisp sets while allowing permissible constraint violations. The symmetric fuzzy linear programming solution concepts in the domain literature were adopted as part of the computational process in arriving at a model solution. Actual case examples aided these formulations to gain insights into their computational complexity, efficiency, scalability, and flexibility.

Lifetime optimization of dense wireless sensor networks using continuous ring-sector model

Wireless sensor networks (WSNs) are becoming increasingly utilized in applications that require remote collection of data on environmental conditions. In particular dense WSNs are emerging as an important sensing platforms for the Internet of Things (IoT). WSNs are able to generate huge volumes of raw data, which require network structuring and efficient collaboration between nodes to ensure efficient transmission. In order to reduce the amount of data carried in the network, data aggregation is used in WSNs to define a policy of data fusion and compression. In this paper, we investigate a model for data aggregation in a dense WSN with a single sink. The model divides a circular coverage region centered at the sink into patches which are intersections of sectors of concentric rings, and data in each patch is aggregated at a single node before transmission. Nodes only communicate with other nodes in the same sector. Based on these assumptions, we formulate a linear programming problem to maximize system lifetime by minimizing the maximum proportionate energy consumption over all nodes. Under a wide variety of conditions, the optimal solution employs two transmissions mechanisms: direct transmission, in which nodes send information directly to the sink; and stepwise transmission, in which nodes transmit information to adjacent nodes. An exact formula is given for the proportionate energy consumption rate of the network. Asymptotic forms of this exact solution are also derived, and are verified to agree with the linear programming solution. We investigate three strategies for improving system lifetime: nonuniform energy and information density; iterated compression; and modifications of rings. We conclude that iterated compression brings the greatest increase in system lifetime, but at the cost of possible information loss.

Steady-State Planning in Expected Reward Multichain MDPs

The planning domain has experienced increased interest in the formal synthesis of decision-making policies. This formal synthesis typically entails finding a policy which satisfies formal specifications in the form of some well-defined logic. While many such logics have been proposed with varying degrees of expressiveness and complexity in their capacity to capture desirable agent behavior, their value is limited when deriving decision-making policies which satisfy certain types of asymptotic behavior in general system models. In particular, we are interested in specifying constraints on the steady-state behavior of an agent, which captures the proportion of time an agent spends in each state as it interacts for an indefinite period of time with its environment. This is sometimes called the average or expected behavior of the agent and the associated planning problem is faced with significant challenges unless strong restrictions are imposed on the underlying model in terms of the connectivity of its graph structure. In this paper, we explore this steady-state planning problem that consists of deriving a decision-making policy for an agent such that constraints on its steady-state behavior are satisfied. A linear programming solution for the general case of multichain Markov Decision Processes (MDPs) is proposed and we prove that optimal solutions to the proposed programs yield stationary policies with rigorous guarantees of behavior.

An augmenting‐flow algorithm for a class of node‐capacitated maximum flow problems

AbstractWe consider a class of maximum flow problems (MFPs) having node‐ and arc‐capacity constraints, in which each unit of flow on arc consumes a positive amount of capacity at node i. This problem arises in wireless sensor network optimization applications, where node capacities refer to sensor energy limits. This version of the MFP is traditionally solved using linear programming due to the presence of the complicating node‐capacity constraints. As an alternative scheme, we prescribe an approach for this problem based on augmenting flows along paths and cycles, showing why sending flows on augmenting cycles becomes necessary in this class of problems. Although our augmenting flow algorithm ultimately requires the solution of auxiliary linear programs, we demonstrate the computational advantages of our approach on randomly generated test instances.

An Interior Point-Proximal Method of Multipliers for Linear Positive Semi-Definite Programming

In this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in Pougkakiotis and Gondzio (Comput Optim Appl 78:307–351, 2021. https://doi.org/10.1007/s10589-020-00240-9) for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM.

Maximizing Throughput of Delay-Sensitive NFV-Enabled Request Admissions via Virtualized Network Function Placement

Network Function Virtualization (NFV) has attracted significant attention from both industry and academia as an important paradigm change in network service provisioning. Most existing studies on admissions of NFV-enabled requests focused on deploying dedicated Virtualized Network Function (VNF) instances to serve each individual request without exploring the sharing of VNF instances among multiple user requests. However, with ever-growing demands of user services, exclusive usages of VNF instances in most networks drastically degrade the network performance and largely under-utilize the VNF instance resources. In this paper, we jointly explore two different VNF instance scaling techniques, horizontal scaling and vertical scaling techniques, to improve the network throughput while minimizing the operational cost of the network, where horizontal scaling that migrates some existing VNF instances from their current locations to new locations to allow the VNF instances to be shared by multiple requests to reduce the resource consumption and operational cost of the network, and vertical scaling that instantiates new VNF instances to meet the demands of new request admissions if existing VNF instances sharing becomes more expensive or the end-to-end delay requirements of currently executing requests will be violated. We first propose a unified framework of maximizing the network throughput, by admitting as many as NFV-enabled requests while meeting the end-to-end delay requirements of the admitted requests. We then provide an Integer Linear Programming (ILP) solution for the problem when the problem size is small. Otherwise, we devise an efficient algorithm through a non-trivial reduction that reduces the problem to the minimum-weight feedback arc set problem and the generalized assignment problem (GAP). We finally conduct experiments to evaluate the performance of the proposed algorithm. Experimental results demonstrate that the proposed algorithm outperforms a baseline algorithm and achieves a performance on a par with its optimal ILP solution.