The Complexity of the k-means Method

In this paper, we discusses the problem that involving the conflict fuzzy multi-objective in transportation planning which is one of special case in multi-objective linear programming. To find the simultaneously optimal solution of the problem, we use the Interactive Fuzzy Multi-Objective Linear Programming (IFMOLP) method. This method can reduce the fuzzy multi-objective linear programming problem into deterministic single objective linear programming which can be solved using simplex method. Beside that , with IFMOLP method, decision maker (DM) can establish interactively the goal from the objective function to produce the pareto optimal solution so that.

Omah Jenang is one of the industries engaged in the production of traditional snacks. Currently Omah Jenang does not yet have guidelines for realizing the formation of the composition of production quantities correctly based on calculations that still use forecasting methods. Therefore Omah Jenang often experiences difficulties in providing raw materials and determining maximum profits. The purpose of this study was to de-termine the optimum yield of profits at Omah Jenang by using the simplex method linear program with the help of Lindo Software. The data used is food production data at the Omah Jenang Home Industry in 2022. The method used in this study is the Linear Pro-gramming simplex method and the Lindo Software application. The results of optimization calculations using the linear programming method and the help of Lindo Software are that the maximum profit in a month is IDR 20,000,000 in producing 800 packages of madumongso. Based on the calculation results it can be concluded that the benefits obtained by using the simplex method with the help of Lindo Software can be obtained maximum results.

The simplex method for linear programming is known to be highly efficient in practice, and understanding its performance from a theoretical perspective is an active research topic. The framework of smoothed analysis, first introduced by Spielman and Teng (JACM ’04) for this purpose, defines the smoothed complexity of solving a linear program with d variables and n constraints as the expected running time when Gaussian noise of variance σ2 is added to the LP data. We prove that the smoothed complexity of the simplex method is O(σ−3/2 d13/4log7/4 n), improving the dependence on 1/σ compared to the previous bound of O(σ−2 d2√logn). We accomplish this through a new analysis of the shadow bound, key to earlier analyses as well. Illustrating the power of our new method, we use our method to prove a nearly tight upper bound on the smoothed complexity of two-dimensional polygons.

This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.

A basis-deficiency-allowing variation of the simplex method for linear programming

Chamfer distances on the isometric grid are considered. A new method to compute the chamfer distances based on linear optimization is presented. In the LP model the starting pixel is the Origin, that is a triangle of the grid having co-ordinates (0, 0, 0). The co-ordinates of the end pixel of the path give the right-hand side of the model. The variables are the used numbers of the elementary steps. Each type of an elementary step has a uniquely defined weight. Our operational research approach determines the optimal paths as basic feasible solutions of a linear programming problem. We give directed graphs with feasible bases as nodes and arcs with conditions on the used weights such that the simplex method of linear programming may step from one feasible basis to another feasible basis. Thus, the possible course of the simplex method can be followed and the optimal bases can easily be captured. Thus, the final result of the analysis is an O(1) checking of the feasibility and optimality conditions. The optimal bases are summarized in a theorem which is the consequence of the general theory of linear programming. The method can be applied for other grids, but it needs to be adjusted for the particular grid.

Over the past decade, a pair of instructions called load-linked (LL) and store-conditional (SC) have emerged as the most suitable synchronization instructions for the design of lock-free algorithms. However, current architectures do not support these instructions; instead, they support either CAS (e.g., UltraSPARC, Itanium) or restricted versions of LL/SC (e.g., POWER4, MIPS, Alpha). Thus, there is a gap between what algorithm designers want (namely, LL/SC) and what multiprocessors actually support (namely, CAS or RLL/RSC). To bridge this gap, a flurry of algorithms that implement LL/SC from CAS have appeared in the literature. The two most recent algorithms are due to Doherty, Herlihy, Luchangco, and Moir (2004) and Michael (2004). To implement M LL/SC objects shared by N processes, Doherty et al.'s algorithm uses only O(N + M) space, but is only non-blocking and not wait-free. Michael's algorithm, on the other hand, is wait-free, but uses O(N2 + M) space. The main drawback of his algorithm is the time complexity of the SC operation: although the expected amortized running time of SC is only O(1), the worst-case running time of SC is O(N2). The algorithm in this paper overcomes this drawback. Specifically, we design a wait-free algorithm that achieves a space complexity of O(N2 + M), while still maintaining the O(1) worst-case running time for LL and SC operations.

The Delaunay triangulation of n points in the plane can be constructed in o(n log n) time when the coordinates of the points are integers from a restricted range. However, algorithms that are known to achieve such running times had not been implemented so far. We explore ways to obtain a practical algorithm for Delaunay triangulations in the plane that runs in linear time for small integers. For this, we first implement and evaluate two variants of BrioDC, an algorithm that is known to achieve this bound. We implement the first O(n)-time algorithm for constructing Delaunay triangulations and found that our implementations are practical. While we do not improve upon fast existing algorithms (with non-optimal worst-case running time) for realistic data sets, our BrioDC implementations do give us insight into the optimal time needed for point location. Secondly, we implement and evaluate variants of BRIO, an algorithm which has an O(n log n) worst-case running time on small integers but runs faster for many distributions. Our variants aim to avoid bad worst-case behavior, which is due to high point location time. Our BrioDC implementation shows that point location time can be reduced by 25% and our squarified space-filling curve orders show the first improvement by reducing this by 3%.

This paper presents a very general technique for the implementation of mergeable priority queues. The amortized running time is O(log n) for DELETEMIN and DELETE, and /spl Theta/(1) for all other standard operations. In particular, the operation DECREASEKEY runs in amortized constant time. The worst-case running time is O(log n) or better for all operations. Several examples of mergeable priority queues are given. The examples include priority queues that are particular well suited for external storage. The space requirement is only two pointers and one information field per item. The technique is also used to implement mergeable, double-ended priority queues. For these queues, the worst-case time bound for insertion is /spl Theta/(1), which improves the best previously known bound. For the other operations, the time bounds are the same as the best previously known bounds, worst-case as well as amortized. >

In this paper, we present algorithms for parsing general tree-adjoining languages (TALs). Tree-adjoining grammars (TAGs) have been proposed as an elegant formalism for natural languages. It was an open question for the past ten years as to whether TAL parsing can be done in time $O(n^6 )$. We settle this question affirmatively by presenting an $O(n^3 M(n))$-time algorithm, where $M(k)$ is the time needed for multiplying two Boolean matrices of size $k \times k$ each. Since $O(k^{2.376} )$ is the current best-known value for $M(k)$, the time bound of our algorithm is $O(n^{5.376} )$. On an exclusive-read exclusive-write parallel random-access machine (EREW PRAM), our algorithm runs in time $O(n\log n)$ using ${{(n^2 M(n))} / {\log n}}$ processors. In comparison, the best-known previous parallel algorithm had a run time of $O(n)$ using $n^5 $ processors (on a systolic-array machine). We also present algorithms for parsing context-free languages (CFLs) and TALs whose worst-case run times are $O(n^3 )$ and $O(n^6 )$, respectively, but whose average run times are better. Therefore, these algorithms may be of practical interest.

Shewhart control charts have previously been suggested as a process control tool for use in routine linear accelerator (linac) output verifications. However, a comprehensive approach to process control has not been investigated for linac output verifications. The purpose of this work is to investigate a comprehensive process control approach to linac output constancy quality assurance (QA). The RBA‐3 dose constancy check was used to verify outputs of photon beams and electron beams delivered by a Varian Clinac 21EX linac. The data were collected during 2009 to 2010. Shewhart‐type control charts, exponentially weighted moving average (EWMA) charts, and capability indices were applied to these processes. The Shewhart‐type individuals chart (X‐chart) was used and the number of data points used to calculate the control limits was varied. The parameters tested for the EWMA charts (smoothing parameter (λ) and the control limit width (L)) were λ=0.05, L=2.492; λ=0.10, L=2.703; and λ=0.20, L=2.860, as well as the number of points used to estimate the initial process mean and variation. Lastly, the number of in‐control data points used to determine process capability (Cp) and acceptability (Cpk) were investigated, comparing the first in‐control run to the longest in‐control run of the process data. Cp and Cpk values greater than 1.0 were considered acceptable. The 95% confidence intervals were reported. The X‐charts detected systematic errors (e.g., device setup errors). In‐control run lengths on the X‐charts varied from 5 to 30 output measurements (about one to seven months). EWMA charts showed in‐control runs ranging from 9 to 33 output measurements (about two to eight months). The Cp and Cpk ratios are higher than 1.0 for all energies, except 12 and 20 MeV. However, 10 MV and 6, 9, and 16 MeV were in question when considering the 95% confidence limits. The X‐chart should be calculated using 8–12 data points. For EWMA chart, using 4 data points is sufficient to calculate the initial mean and variance of the process. The EWMA limits should be calculated with λ=0.10, L=2.703. At least 25–30 in‐control data points should be used to calculate the Cp and Cpk indices.PACS number: 89

The need for cake as a complementary food at various events causes the demand to increase. This increase has an impact on the production level that will be produced with the aim of increasing profits. The production process is related to the allocation of the right raw materials to be efficient, so that maximum profits will be obtained. This study aims to analyze the maximum profit of the Maesa Cake and Bakery Shop, Permata Legenda Branch. There are two types of products sold in this shop, namely Banana Bolu (X1) and Banana Bolen (X2). The research method used is linear programming which is solved by the simplex method and the use of the POM For Windows application. The data obtained is observational data by meeting the shop owner directly to be interviewed. The process of calculating the simplex method is carried out using Ms. Excel with several completion steps, starting from creating a simplex table, entering the coefficients of the objective function and constraints, looking for key columns, key rows, and key numbers, until the last step is achieving positive coefficients generated in the objective function row. The final result obtained is the process of the iterations carried out. Apart from using Ms. Excel, researchers also use the POM for Windows application. Use both software to find the maximum profit to be achieved. So it is found that the maximum profit that can be obtained in one day by the Maesa Cake and Bakery Branch of Permata Legenda is IDR 270,000.00; by producing 5 units of Banana Bolu and 50 units of Banana Bolen.

We study two (classes of) distributed algorithms for power control in a general model of wireless networks. There are n wireless communication requests or links that experience interference and noise. To be successful a link must satisfy an SINR constraint. The goal is to find a set of powers such that all links are successful simultaneously. A classic algorithm for this problem is the fixed-point iteration due to Foschini and Miljanic [8], for which we prove the first bounds on worst-case running times - after roughly O(n log n) rounds all SINR constraints are nearly satisfied. When we try to satisfy each constraint exactly, however, convergence time is infinite. For this case, we design a novel framework for power control using regret learning algorithms and iterative discretization. While the exact convergence times must rely on a variety of parameters, we show that roughly a polynomial number of rounds suffices to make every link successful during at least a constant fraction of all previous rounds.

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following

In the simplex method for linear programming the algorithmic step of checking the reduced costs of nonbasic variables is called the “pricing” step. If these reduced costs are all of the “right sign” the current basis (and solution) is optimal, if not, this procedure selects a candidate vector that looks profitable for inclusion in the basis. While theoretically the choice of any profitable vector will lead to a finite termination (provided degeneracy is handled properly) but the number of iterations until termination depends very heavily on the actual choice (which is defined by the selection rule applied). Pricing has long been an area of heuristics to help make better selection. As a result, many different and sophisticated pricing strategies have been developed, implemented and tested. So far none of them is known to be dominating all others in all cases. Therefore, advanced simplex solvers need to be equipped with many strategies so that the most suitable one can be activated for each individual problem instance. In this paper we present a general pricing scheme. It creates a large flexibility in pricing. It is controlled by three parameters. With different settings of the parameters many of the known strategies can be reproduced as special cases. At the same time, the framework makes it possible to define new strategies or variants of them. The scheme is equally applicable to general and network simplex algorithms.