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SIMPLEX液压扳手扭矩技术—评价SIMPLEX液压扭矩扳手及其竞争对手综合指南上海舜诺机械有限公司内部资料创造“扭矩”的技术在工业应用中，施加“扭矩”于紧固件时，完成这些工作的工具必须满足两（2）个关键指标，它们是：精度和重复性！在“液压扭矩扳手”工业领域，所有竞争对手无数次的声称他们是如何完成了工作。
但是他们能够从机械原理上证明吗？它们能证明那是事实吗？SIMPLEX能够证明！首先，作为一名客户，您必须完全懂得液压扳手是如何输出扭矩的。
这一系统组成为：1．液压扭矩扳手（方驱-套筒扳手或薄型动力头-六方中空工作头扳手）2．液压动力装置（电动或气动）3．液压软管4．工业级冲击套筒（配方驱液压扳手）利用以上所列设备，您可以将扭矩作用于螺纹紧固件。
但是实际是怎样工作的呢？一旦系统连接好（即，扳手通过液压软管连接到泵），泵产生压力油作为液压扳手的动力。
液压扳手内有一活塞，活塞产生力来推动机械部件，部件转动方驱或六方中空工作头的棘轮。
这一工作方式与汽车发动机转动轮胎并推动汽车一样。
发动机和轮胎之间的部件确定功率。
关键一点是，“发动机”和轮胎之间部件越少越好，就如“活塞”与转动点“方驱”之间部件越少越好。
力产生点（活塞）和力作用点（方驱）之间产生“松配”或“拖拽”越多。
我们把它叫做。
摩擦！！部件越多。
意味摩擦越多。
功率越小。
精度越差。
重复性越差！！在一部SIMPLEX液压扳手中，从力产生点（活塞）到转动点（方驱或棘轮）只经过三（3）个运动机械部件！！力从活塞（#1）传递到驱动盘（#2)到棘爪（#3)到棘轮（#4）到方驱（#5)。
总体上讲，WT液压扳手只有五个运动部件，如果你包括活塞和方驱的话。
1．更少的部件=更少摩擦2．更少摩擦=更少功率损失3．更少功率损失=精度增加4．精度增加=为什么您选SIMPLEX液压扳手接下来您会问“什么是重复性？”当您锁紧十颗螺栓到一特定要求扭矩时，您想它们都得到相同扭矩值，对吗？那就用一部SIMPLEX液压扳手，利用我们专利的一体成型机身而实现吧！众所周知，液压扳手机身具有“弯曲”。
新普利斯（SIMPLEX）网络系统功能简介苏金明1．网络系统的容量：以令牌方式进行点对点通讯的对等式网络，网络系统最多可支持7个复合网络环，每个网络环可附加99个节点，每个节点可连接2000个地址设备，从而网络系统的单环容量就可达198000个地址点，且中文图形控制中心（IMS）单机的容量也可达50000个地址点。
网络上的节点均可通过软件编程来实现各种网络功能。
2．网络系统的组成结构：网络系统组网灵活，方式多样。
主/从式环型网络：所有的网络通讯都经过主机，主机控制全系统，执行所有的网络功能及操作，分机并不需要独立运行，只有在与主机通讯中断时实现独立的工作方式。
主/从式环型网络结构只是从网络管理的角度来定义的而不是从控制器的功能上定义。
网络上的每个节点即互相独立，又互为补充，每个控制器都可以检测和控制本身的系统，也可以控制其他控制器的点，并可以向网络发布它本身所处的状态及其它信息。
无主式环型网络：网络上的每个控制器都可以传送信息，也可以接收其它控制器发送的信息，灵活的编程软件可以设定任意的控制器接收或忽略网络上的信息，每个控制器的地位相同并可自我控制。
环型网络的重要特性：环型网络（STYLE7）功能，为网络的构成与通讯提供更高的可靠性，就像在网络中创建了两个路径，即便一个路径有故障，网络马上可通过第二个路径进行通讯，绝对保证了网络系统的可靠性。
比如：网络线路中有开路、短路故障时，网络可以通过另一条路径来进行完整的网络通讯。
而在网络中有多处故障时，网络会进行重组，使网络功能尽可能完整（系统会自动组成若干个子网络）。
而当系统中有一个控制器有故障时，只有该控制器的监控部分失去功能，而其它部分仍可正常工作，增强了系统功能和抗风险的能力，使风险降至最低，因此环型网络系统是生命力极强的网络，它综合了多种网络连接方法的优点，祛除了各种致命的缺点，为此很多重特大的项目考虑到系统的稳定性及可靠性，广泛采用了环型网络，而且得到了良好的使用效果。
单双工的原理及应用1. 什么是单工和双工单工（Simplex）和双工（Duplex）是指在通信系统中，信息传输的方向和方式。
在单工通信中，数据仅仅在一个方向上进行传输，而在双工通信中，数据可以同时在两个方向上进行传输。
2. 单工的原理和应用2.1 单工的原理单工通信是指数据在通信双方之间只能单向传输，无法进行双向传输。
在单工通信中，通信设备可以充当发送方或接收方，但不能同时充当两者。
在单工通信中，发送方将数据传输到接收方，但接收方无法给发送方发送回复。
这种通信方式适用于只需要一方向传输数据的场景。
2.2 单工的应用单工通信适用于一些场景，如：•电视广播：电视广播信号只能从电视台传输到用户的电视机上，用户不能通过电视机发送信号给电视台。
•AM/FM广播：传统广播接收器只能接收广播信号，无法发送信号。
•遥控器：遥控器只能向电视、空调等设备发送信号，无法接收信号。
3. 双工的原理和应用3.1 双工的原理双工通信是指数据可以在两个方向上同时进行传输。
在双工通信中，通信的双方既可以充当发送方，也可以充当接收方。
这种通信方式允许双方可以同时发送和接收数据。
在双工通信中，发送方和接收方通过共享的通信通道进行交互。
发送方将数据传输到通道中，接收方可以实时接收到发送方传输的数据，并且也可以将自己的数据传输到通道中。
3.2 双工的应用双工通信适用于一些场景，如：•电话通信：在电话通信中，双方可以同时进行通话，双方可以交替进行发言和听取对方的发言。
•对讲机：对讲机可以实现双方之间的实时语音交流，双方可以同时发送和接收语音消息。
•视频会议：在视频会议中，与会人员可以同时进行语音和视频交流。
4. 单工和双工的对比单工和双工具有以下几点区别：•数据传输方向：单工只能在一个方向上传输数据，而双工可以同时在两个方向上进行数据传输。
•通信能力：单工通信只有发送数据的能力，而双工通信具有双向传输数据的能力。
•通信效率：由于单工通信只能在一个方向上传输数据，所以比双工通信的传输效率低。
simple的词根词缀1. “Simple”这个词的词根是“sim -”，它有“一、单一”的意思。
就像我们说“similar”，这个词表示相似，好像是从同一个模子里出来的，有着单一的源头一样。
比如说，“These two pictures are similar. They both have a single theme of nature.”（这两张图片很相似。
它们都有一个关于自然的单一主题。
）2. 词缀“ - ple”有“充满、填满”的意思呢。
你看“ample”这个词，意思是充足的。
这就好像一个杯子，被满满的东西填满了一样。
例如，“There is ample food for everyone at the party.”（聚会上有足够的食物供每个人吃。
）3. 从“simple”还能联想到“simplify”，这是个动词，加上“ - ify”这个词缀就表示“使……简单化”。
这就好比整理杂乱的房间，把东西都归位，让房间变得简单有序。
像“Let's simplify this complex problem.”（让我们把这个复杂的问题简单化。
）4. 再说说“simplicity”，这是“simple”的名词形式。
它就像是一股清流，在这个复杂的世界里显得尤为珍贵。
比如，“The simplicity of her dress made her stand out in the crowd.”（她裙子的简约使她在人群中脱颖而出。
）5. 有个词叫“simulacrum”，词根也是“sim -”，它表示模拟物、假象。
这有点像那些山寨货，看起来像真的，但其实不是。
例如，“That cheap watch is just a simulacrum of the famous brand.”（那块便宜的手表只是那个著名品牌的仿制品。
）6. “simpleton”这个词可有点贬义了，指的是头脑简单的人。
简单多面体的定义
简单多面体（Simplex）是指由n个面组成的多边形，其中n > 3。
简单多面体可以定义为一个空间中的n个点，每个点都与其他点相关联，而且n个点之间不存在任何共同边。
一般来说，简单多面体的n 个点必须在同一个平面上。
简单多面体也可以在一个超平面上定义，其中超平面是一个超过n个维度的空间。
简单多面体的面数可能从3到无穷多，但它们的特点一定是简单多面体中的每个面都与另一个面的距离都不相等。
此外，简单多面体也要求每个面上的角度要与其他面相同，或者尽量相近。
简单多面体有许多种，其中最简单的形式是三角形，而最常用的是正多边形。
由此可见，简单多面体可以是一个多边形或者是一个立方体。
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4100ESiIntegrated. Ideal. Intuitive. Informative. Inspiring.Australia Level3,95CoventryStreetSouthbankVIC3006Tel:1300725688Tel:+61393139700Email:*******************************SIMPLEX is a leading global brand in the fire detection industry. The SIMPLEX product line includes a wide range of AS-listed, UL-listed and FM-approved life safety products for fire detection, emergencycommunications, voice evacuation, security, and related solutions. Many other product approvals and listings are also available including Chinese, Korean, and Russian certifications. The product line is availablethrough a variety of Johnson Controls companies and SIMPLEX authorised dealers around the world.© 2017 Johnson Controls. All rights reserved. All specifications and other information shown were current as of document revision date and are subject to change without notice.Building Safety4100ESibroSFP1706 August 2017 4100ESiIntegrated. Ideal. Intuitive. Informative. Inspiring.The fire detection system of choice for medium to large facilitiesWe’ve been working on the evolution of the Simplex 4100 for over 20 years, and we truly believe we have now achieved a unique and robust approach to fire detection in medium to large facilities – including tunnels, hospitals, universities, prisons, airports, shopping centres and the list goes on.Our latest development allows this model to control up to 500 zones within its core software, whileretaining the flexibility to navigate your individual and unique programming requirements. The model’s high level of technical capability equips you to perform non-standard setups by simply allocating commands to your chosen zones.At Simplex, we’re always striving to beat our personal bestOur aim to increase fire detection and safety in your facilities is why we are always improving and developing our tried and trusted products. We strive for the best and safest – and that’s a benchmark we aim to redefine.Simplex panels have always had a strong reputation for being reliable, dependable, and withstanding the test of time.With Simplex 4100ESi, we’ve raised the bar and made them even better, yet simple.Cutting edge technology torevolutionise your fire detectionThe latest addition to the Simplex family allows you to use your PC to program a basic system in two simple steps:1. Enter devices and descriptions.2. Allocate and group devices to zones. That’s it!Once programmed, enjoy fast, effective controls at the tips of your fingers. Use the intuitive colourtouchscreen to easily operate the AS 7240.2 panel and the zone disable/enable controls.The Simplex 4100ESi greatly reduces the time required for programming.We’re redefining theclassics, taking the hassle out of large fire detection requirements.Welcome to the future of fire safety technology.30000 7240 3500 2000 1000 500 99 2010Fully compliant to latest standardsUser-friendly AS7240.2-certified panel conforms to the latest Australian Standards to offer you peace of mindBetter capacity, greater connectivity2,000-point addressable device capacity – ideal for facilities of any sizeConnect up to 99 panels on a singlenetwork ring – up to 3,500 metres apart in copper, or a stunning 30,000 metres apart in single mode fibreGenerous 10A power supply – Reduces the need for extra power supplies or battery boxesIntuitive and intelligentEasy to read and navigate interactive touch-screen displayRegularly updated e-manuals, accessible on any internet-connected deviceEasier installation and upgradesThe backward compatible Simplex 4100ESi uses intuitive Windows-style programming softwareProgramming templates for common functions (including 1668 controls, day-night sensing, alarm acknowledgement, delay and investigation)Enjoy total flexibilityNon-proprietary – Y our freedom of choice to select any trained Service Company to service Simplex fire detection products Available off the shelf with expansion box options and a further three sizes on demandINTEGRATED Simplicity and ease – that’s what we’ve achieved with the Simplex 4100ESi.Backwards-forwards compatibility allows you to integrate the new system with your existing Simplex 4100 without having to commit a significant amount of time and labour to complete the transition. We aim to facilitate a fusion between your Simplex 4100 system and the improved Simplex 4100ESi, in a way that supports and enhances your processes rather than disrupts them, and increases the lifespan of your existing 4100 safety system.IDEAL The Simplex 4100ESi is the ideal flexible system for your challenging fire detectionrequirements. Our scalable programming options suit all sizes of network application. Y our panels can operate 3,500 metres apart using copper, and as far as 30,000 metres apart using single mode fibre – and you can have as many as 99 panels on a single network with up to 7 rings connectable to one TrueSite Workstation (TSW). Y our Simplex 4100ESi system comes complete with on-going advice and support for many years – and for a system that is built to last and to seamlessly evolve through backward-forward compatibility, that’s the kind of service you can rely on.INTUITIVE The extra-large touchscreen display provides an elite user interface with impeccablecontrol and monitoring capabilities at your fingertips. The easy-to-use screen allows you to view and control faults, enable and disable points and zones, isolate specific issues and run detector status and service reports. In the unlikely event of any issues with your detection system, the diagnostics features and onboard help screen allow you to pinpoint exactly what needs to be altered in order to resume full working order of your fire safety operations – without needing to close your facility.INFORMATIVE Use a network display unit (NDU) to see all zones in your network – withouthaving to add separate panels. When configured as an NDU, the Simplex 4100ESi can control up to 1000 zones. Supervise – and control – all zones from a single, cost-effective screen. Reliability and connectivity is increased thanks to the availability of several High Level Interface options such as MODBUS, BACnet and SafeLINC.INSPIRING We’ve tried it, we’ve tested it, and we’ve been guiding the evolution of the Simplex4100ESi since 1992. Y ou’ve seen what a Simplex panel can do – and now it can do even more.We’ve eliminated manual programming, shortened internal processes and created a rounded solution that can keep up with your fast-paced work environment – no matter how large your facility. In fact, the Simplex 4100ESi has a faster network response time enabling it to react quickly even in a complex orlarge network environment.4100ESiIntegrated. Ideal. Intuitive. Informative. Inspiring.metres between nodes using single mode fibremetres between nodes using copper networkaddressable deviceszones can be controlled by a 4100ESi NDUzones controlled by the standard configpanels can be connected up on a single ringyears of evolution of the 4100Amp power supply as standard fitmentpart 2 standards complianceAt a Glance:“I need a time-efficient solution.”The Simplex 4100ESi is a quick and simple choice for your facility – there’s no need to account for lead-time when ordering your system, because the standard panel is ready off the shelf. And intuitive, automated programming cuts down hours of set up.“I need a powerful fire alarm panel that is easy to design.”Fully compliant to the latest AS 7240.2 standard, the Simplex 4100ESi panel is not only easy to program but also the preferred choice for many fire system designers. The ability to control up to 500 zones and connect up to 99 panels on a single network ring offers designers a less complex or contradictory experience, while achieving greater integrity in the overall design. Fire system design is now a breeze.“I need a fire detection system that works efficiently in a large facility.”The Simplex 4100ESi is great for facilities of any size – with its fast network response time, ability to connect up to 99 panels on a single network ring and ability to cover a massive distance of up to 30kmbetween panels using single mode fibre.“I need a system that is built to last.”The reliable Simplex 4100ESi is not only designed to be long-lasting and hard-working, it’s also created with lifespan in mind. Not only will the base model work hard for many years, it’s also backwards-forwards compatible – so as technology advances, so can you.“I need a versatile product that I can customise to my facility’s particular requirements now and in the future.”The Simplex 4100ESi is the most flexible Simplex panel to date. Not only is it backwards-forwards compatible, you can also configure the panel as an NDU, and control double the amount of zones – without having to add extra panels or cumbersome zone switch controls.4100ESiIntegrated. Ideal. Intuitive. Informative. Inspiring.Control at your fingertipsOur high specification, technologically advancedand unique touchscreen makes managing firesafety simpler than it’s ever been.From a single screen, and at the touch of yourfingers, you can:• View and monitor all fault points, supervisorypoints and Pri2 alarms• Disable and enable points and zones• Conduct alarm tests on points and zones• Access level changes• View and upload previous alarm and faultsupervisory logs• Print and upload reports• Inspect – and respond to – servicediagnosticsOnce you’ve easily programmed the panel on aPC, everything is accessible from the panel itself– where and when you need it.Here when you need usSupporting you to make the most of your Simplexproducts is our priority. Contact our team oftechnicians during business hours to answer any andall questions you have about your Simplex 4100ESi– we’ll work step-by-step with you to resolve anyissues.Simplex Technical Support Hotline1300 552 559 (toll free)Learn with usSimplex’s dedicated training department providescomplete product training – in all major cities ofAustralia. Undertake training support to ensure youare utilising the most up-to-date information whenit comes to using our products and ensuring fireprotection.4100ESiIntegrated. Ideal. Intuitive. Informative. Inspiring.4100ESiIntegrated. Ideal. Intuitive. Informative. Inspiring.Freedom of choice, power to youRemain in control: it’s not only how you upgrade your systems, but also who upgrades them. The versatile compatibility of the Simplex 4100ESi enables a simple and low-cost add-on to an existing system – and can be periodically upgraded with ease, as and when your individual requirements command such a need.Not only that, our promise to you with provision of all Simplex Fire Detection products, is to save you valuable time and money. With this in mind, we make all our fire detection systems non-proprietary, so you’re free to choose who you like for technical installation andsupport throughout the lifetime of your system. You’re not bound tous in any way. We allow you to access the best services to suit yourrequirements – and that means the best costs to suit your budget.SolutionsFor All MarketsWhy choose Simplex?• Peace of mind and exceptional product quality from industry leaders you can trust.• Our constantly evolving range of project solutions means you’re ahead of the game.• Our products are developed by the world’s most well regarded fire technology experts.•With Simplex products, costs are low, problems are solved, and lives are saved.。
Simplex4100ES产品手册10096*****+86 21-***-*****北京138 3 7 ***** +86 10-***-*****广州161 A 1701 ***** +86 20-***-*****武汉568 49 17 ***** +86 27-***-*****Form No. S - BRO *****CN1. 新普利斯简介 (2)2. 4100ES火灾报警控制器 (3)2.1 系统构成2.2 系统容量2.3 主要技术特点2.4 产品认证2.5 主要部件2.5.1 CPU中央处理器单元(Master Controller)2.5.2 用户操作面板2.5.3 系统主电源(SPS)2.5.4 扩展电源(XPS)2.5.5 电源分配模块2.5.6 回路卡2.5.7 控制器机箱2.5.8 微型打印机组件2.5.9 多线手动控制盘2.5.10 网络卡(NIC)2.5.11 基本配置2.5.12 扩展模块安装参考8.6 四输入/两输出模块8.7 通讯隔离模块8.8 电源隔离模块9. 报警输入输出设备 ............. 319.1 手动报警按钮9.2 *****x防暴型手动按钮9.3 消火栓按钮9.4 声光报警器10. 三频红外火焰探测器........ 3311. 消防电话 ........................ 3511.1 多线制电话主机11.2 多线制电话分机11.3 多线制电话插孔11.4 多线制电话手柄11.5 总线制电话总机11.6 总线制电话分机11.7 总线制电话插孔12. 消防应急广播设备 .. (38)12.1 多线广播控制盘12.2 广播录放盘12.3 功率放大器12.4 音箱3. 4100ES网络 ........................... 163.1 概述3.2 环型和星型架构配置3.3 网络回路3.4 系统功能附录．接线图 (40)4. 火灾显示盘 (18)5. 气体灭火控制器 (19)6. 图文显示终端 (20)7. 现场外围设备 (21)7.1 智能光电感烟探测器7.2 智能感温探测器7.3 智能型探测器通用底座7.4 非地址式感烟探测器7.5 非地址式感温探测器7.6 非地址式探测器通用底座8. 智能模块 (26)8.1 区域适配模块8.2 可编址监视模块8.3 可编址信号模块(IAM)8.4 单输出模块8.5 单输入单输出模块目录1.新普利斯简介2. 4100ES 型火灾报警控制器4100ES系列火灾报警控制器（联动型）是Simplex公司专门针对中、大型及超大型项目开发和推出的消防报警控制器。
simplex 单纯形法单纯形法（Simplex Algorithm）是一种用于线性规划问题求解的有效算法。
它由美国运筹学家Dantzig于1947年提出，被广泛应用于工业生产优化、资源分配、物流管理等领域。
本文将介绍单纯形法的基本原理、步骤与应用，并探讨其优缺点。
一、基本原理单纯形法是通过不断地在可行解空间中移动来逼近最优解的方法。
该方法从一个初始可行解出发，通过一系列迭代操作，每次改变一个基本变量以达到更优的目标函数值。
最终，算法将找到一个全局最优解或者判断问题无界或无可行解。
二、基本步骤1. 线性规划标准形式化：将线性规划问题转化为标准形式，即目标函数最小化，约束条件为线性等式。
2. 初始可行解：找到一个满足约束条件的初始可行解，并将其称为基本可行解。
3. 进行迭代操作：通过改变基本变量来改善目标函数值，直到达到最优解或者判断问题无界或无可行解。
4. 基本变量的选择：在每一次迭代中，选择一个非基本变量作为入基变量，并选取一个基本变量作为出基变量。
5. 确定迭代终止条件：判断是否终止迭代，若目标函数值无法继续改善或者判断问题无界或无可行解，则终止迭代。
6. 输出最优解：若找到了最优解，输出最优解及最优目标函数值。
若判断问题无界或无可行解，则给出相应的判断结果。
三、应用领域单纯形法广泛应用于工业生产优化、资源分配、物流管理等领域。
以下是一些典型应用案例：1. 生产计划优化：通过使用单纯形法，可以优化生产计划以最大化产出，同时考虑资源约束和成本限制。
这对于提高生产效率和降低成本非常重要。
2. 物流网络优化：单纯形法可以帮助优化物流网络的设计和运作，以最小化物流成本、最大化物流效率，并满足客户需求。
3. 能源系统调度：单纯形法可以应用于能源系统的调度问题，包括电力系统、天然气输送网络等，以最大化供应效率，并解决资源分配和运营问题。
4. 金融投资组合优化：通过单纯形法，可以优化金融投资组合以最大化收益或最小化风险，并满足投资者的需求。
Linear Programming: The Simplex Method⏹An Overview of the Simplex Method⏹Standard Form⏹Tableau Form⏹Setting Up the Initial Simplex Tableau⏹Improving the Solution⏹Calculating the Next Tableau⏹Solving a Minimization Problem⏹Special CasesOverview of the Simplex Method⏹Steps Leading to the Simplex MethodFormulate Put In Put In Execute Problemas LPStandardFormTableauFormSimplexMethodExample: Initial Formulation ⏹ A Minimization ProblemMIN 2x1-3x2-4x3ts. t. x1+ x2+ x3<302x1+ x2+ 3x3>60x1-x2+ 2x3= 20x1, x2, x3>0Standard Form⏹An LP is inAn LP is in standard formstandard form when:All i bl ti•All variables are nonAll variables are non--negative•All constraints are equalitiesP tti LP f l ti i t t d d f⏹Putting an LP formulation intoPutting an LP formulation into standard formstandard form involves:to “to “<“ constraints •AddingAdding slack variablesslack variables to < constraints •SubtractingSubtracting surplus variablessurplus variables from “from “>>” constraints.Example: Standard Form⏹Problem in Standard FormMINMIN 22x1-3x2-4x3s. t. x1+ x2+ x3+ s1= 302x+ x+ 3x-s= 601232x1-x2+ 2x3= 20x1, x2, x3, s1, s2>0Tableau Form⏹ A set of equations is inA set of equations is in tableau formtableau form if for each equation:•its right hand side (RHS) is nonits right hand side (RHS) is non--negative, and(Athere is a basic variable (A basic variable •there is a basic variable. (Athere is a basic variable. (A basic variablebasic variable for an equation is a variable whose coefficient in theequation is +1 and whose coefficient in all otherqequations of the problem is 0.)⏹To generate an initial tableau form:g•An artificial variable must be added to eachconstraint that does not have a basic variable.Example: Tableau Form⏹Problem in Tableau FormMIN 2x1-3x2-4x3+ 0s1 -0s2+ Ma2+ Ma3s. t. x1+ x2+ x3+ s1= 302x+ x+ 3x-s+ a= 6012322x1 -x2+ 2x3 + a3 = 20x1, x2, x3, s1, s2, a2, a3 >0Simplex Tableau⏹TheThe simplex tableausimplex tableau is a convenient means for performing the calculations required by the simplex method.Setting Up Initial Simplex Tableau⏹Step 1: If the problem is a minimization problem,1multiply the objective function bymultiply the objective function by --1.⏹Step 2:If the problem formulation contains anyhand sides constraints with negative rightconstraints with negative right--hand sides,multiply each constraint bymultiply each constraint by --1.S3Add l k i bl h i⏹Step 3: Add a slack variable to eachAdd a slack variable to each <<constraint.⏹Step 4: Subtract a surplus variable and add anartificial variable to eachartificial variable to each >>constraint.Setting Up Initial Simplex Tableau⏹Step 5: Add an artificial variable to each = constraint. St 6S t h l k d l i bl'⏹Step 6: Set each slack and surplus variable'scoefficient in the objective function equal tozerozero.⏹Step 7: Set each artificial variable's coefficient in thebj i f i l h iobjective function equal toobjective function equal to --M, where M is avery large number.⏹Step 8: Each slack and artificial variable becomes oneof the basic variables in the initial basicf ibl l tifeasible solution.Simplex Method⏹Step 1: Determine Entering Variable•Identify the variable with the most positive valuein the c j-z j row. (The entering column is calledthethe pivot columnpivot column.).)p)⏹Step 2: Determine Leaving Variable•For each positive number in the entering column, compute the ratio of the rightcompute the ratio of the right--hand side valuesdivided by these entering column values.•If there are no positive values in the enteringcolumn, STOP; the problem is unbounded. •Otherwise, select the variable with the minimalratio. (The leaving row is called theratio. (The leaving row is called the pivot rowpivot row.).)Simplex Method⏹Step 3: Generate Next Tableau•Divide the pivot row by theDivide the pivot row by the pivot elementpivot element(the entry at the intersection of the pivot row and pivot column) to get a new row. We denote this new)grow as (row *).•Replace each nonReplace each non--pivot row i with:p p[new row i] = [current row i] ] --[(a ij) x (row *)],where a is the value in entering column of row iijg jSimplex Method⏹Step 4: Calculate z j Row for New TableauF h l lti l th bj ti f ti •For each column j, multiply the objective functioncoefficients of the basic variables by thecorresponding numbers in column j and sum them.Simplex Method⏹Step 5: Calculate c j-z j Row for New Tableausubtract the row •For each column j, subtract the zjrow from the c j row. •If none of the values in the cj-z j row are positive, GO TO STEP 1TO STEP 1.•If there is an artificial variable in the basis with apositive value, the problem is infeasible. STOP.•Otherwise, an optimal solution has been found. Thecurrent values of the basic variables are optimal. Thepoptimal values of the nonoptimal values of the non--basic variables are all zero. •If any nonnon--basic variable's c-z value is 0, alternatej joptimal solutions might exist. STOP.Example: Simplex Method⏹Solve the following problem by the simplex method:Max 12x1+ 18x2+ 10x3s.t. 2x1+ 3x2+ 4x3<50x1-x2 -x3>0x2 -1.5x3 >0x1, x2, x3>0Example: Simplex Method⏹Writing the Problem in Tableau FormW id i t d i tifi i l i bl tWe can avoid introducing artificial variables to the second and third constraints by multiplying each by --by 1 (making them1 (making them <<constraints). Thus, slack variables s1, s2, and s3are added to the three constraints.Max 12x1+ 18x2+ 10x3+ 0s1+ 0s2+ 0s3t 2 3 4 50s.t. 2x1+ 3x2+ 4x3+ s1= 50-x1+ x2+ x3+ s2= 015 0-x2 + 1.5x3+ s3= 0x1, x2, x3, s1, s2, s3>0⏹Initial Simplex Tableaux1x2x3s1s2s3Basis c B12 18 10 0 0 0s0 2 3 4 1 0 0 501s200 --1 1 1 0 1 0 0 (* row)s0 00 0 --1 1.5 0 0 1 03z j0 0 0 0 0 0 0c j-z j12 18 10 0 0 0 ⏹Iteration 1•Step 1: Determine the Entering VariableThe most positive c j-z j= 18. Thus x2is the enteringvariablevariable.•Step 2: Determine the Leaving VariableTake the ratio between the right hand side andpositive numbers in the x2column:50/3 = 16 2/350/3 16 2/30/1 = 0 minimums2is the leaving variable and the 1 is the pivotelement.Example: Simplex Method⏹Iteration 1 (continued)St 3 G t N T bl•Step 3: Generate New TableauDivide the second row by 1, the pivot element.Call the "new" (in this case unchanged) row the "* Call the new (in this case, unchanged) row therow".Subtract 3 x (* row) from row 1Subtract 3 x ( row) from row 1.SubtractSubtract --1 x (* row) from row 3.New rows 1 2 and 3 are shown in the upcomingNew rows 1, 2, and 3 are shown in the upcomingtableau.Example: Simplex Method⏹Iteration 1 (continued)St 4 C l l t R f N T bl •Step 4: Calculate zjRow for New TableauThe new z j row values are obtained bycolumn by each column element multiplying the c B column by each column, element by element and summing.For example= 5(0) += 5(0) + --1(18) + --1(0) =1(0) = --18 For example, z1 5(0) + 1(18) + 1(0) 18.⏹Iteration 1 (continued)St 5 C l l t R f N T bl •Step 5: Calculate c j -z j Row for New Tableau The new c j -z j row values are obtained bysubtracting z j value in a column from the c j value in the same column. For example -= 12 = 12 ---18) = 30For example, c 1z 1 12 (18) 30.⏹Iteration 1 (continued) Iteration 1 (continued) --New Tableaux 1x 2x 3s 1s 2s 3Basis c B 12 18 10 0 0 00 5 0 1 1 --3 0 50 (* row)s 10 5 0 1 1 3 0 50 ( row)x 218 18 --1 1 1 0 1 0 0s 0 0 --1 0 2.5 0 1 1 03z j -18 18 18 0 18 0 0c j -z j30 0 30 0 --8 0 8 0 --18 0Example: Simplex Method⏹Iteration 2•Step 1: Determine the Entering VariableThe most positive c j -z j = 30. x 1is the entering variable.•Step 2: Determine the Leaving VariableTake the ratio between the right hand side and b h lpositive numbers in the x 1column:10/5 = 2 minimumTh ti f th d d thi d There are no ratios for the second and third rows because their column elements (because their column elements (--1) are negative. Thus, s 1(corresponding to row 1) is the leaving variable and 5 is the pivot element. Example: Simplex Method⏹Iteration 2 (continued)•Step 3: Generate New TableauDivide row 1 by 5, the pivot element. (Call this new row 1 the "* row")row 1 the row ).Subtract (Subtract (--1) x (* row) from the second row.1) x (* row) from the third row Subtract (Subtract (--1) x ( row) from the third row.•Step 4: Calculate z j Row for New Tableau The new z j row values are obtained bymultiplying the c B column by each column, element by element and summing. y gFor example, z 3= .2(12) + 1.2(18) + .2(0) = 24.⏹Iteration 2 (continued)•Step 5: Calculate cj-z j Row for New TableauThe new c j-z j row values are obtained bysubtracting z j value in a column from the c j value inthe same column.-= 10= 10 --(24) =(24) = --For example, c3z3 10 (24) 14.Since there are no positive numbers in the c j-z jrow this tableau is optimal The optimal solutionrow, this tableau is optimal. The optimal solutionis: x1= 10; x2= 10; x3= 0; s1= 0; s2= 0 s3= 10, andthe optimal value of the objective function is 300. ⏹Iteration 2 (continued)Iteration 2 (continued) ––Final Tableaux1x2x3s1s2s3Basis c B12 18 10 0 0 0x12 1 0 .2 .212 1 0 .2 .2 --.6 0 10 (* row)1() x218 0 1 1.2 .2 .4 0 10s30 0 0 2.7 .2 .4 1 10z j12 18 24 6 0 0 300c j-z j0 00 0 --1414 --6 0 0Special Cases ⏹InfeasibilityU b d d⏹Unboundedness⏹Alternative Optimal SolutionD⏹DegeneracyInfeasibility⏹Infeasibility is detected in the simplex methodis detected in the simplex method when.when. an artificial variable remains positive in the final tableauExample: Infeasibility ⏹LP FormulationMAX 2x1+ 6x2s. t. 4x1+ 3x2<122x1+ x2>8x1, x2>0Example: Infeasibility⏹Final Tableaux1x2s1s2a2 Basis C B26000 --Mx12 1 3/41/4003a2 -M00 --1/21/2 --1/2-112z 2 (1/2)M(1/2)M M-M -2M//j+3/2+1/2+6c j-z j00 --(1/2)M-(1/2)M-M09/2+9/2-1/2Example: InfeasibilityIn the previous slide we see that the tableau is the0 However anfinal tableau because all c j-z j<0. However, an artificial variable is still positive, so the problem is infeasible.Unboundedness⏹ A linear program has anA linear program has an unbounded solutionunbounded solution if all entries in an entering column are nonnon--positive.Example: Unboundedness ⏹LP FormulationMAX 2x1+ 6x2s. t. 4x1+ 3x2>122x+ x>812x1, x2 >0Example: Unboundedness⏹Final Tableaux1x2s1s2Basis c B3400x4310-182s10201-13z j1240-432c j-z j-9004Example: UnboundednessIn the previous slide we see that c4-z4= 4 (is but its column is allpositive) but its column is all nonnon This positive), but its column is allpositive), but its column is all nonnon--positivepositive. This. This indicates that the problem isindicates that the problem is unboundedunbounded..Alternative Optimal Solution⏹ A linear program hasA linear program has alternate optimal solutionsalternate optimal solutions if the final tableau has a c j-z j value equal to 0 for a nonvalue equal to 0 for a non--basic variable.Example: Alternative Optimal Solution ⏹Final Tableaux1 x2x3 s1s2 s3s4Basis c B2 4 6 00 00s30 00 2 4-2 10 8x2 4 0 1 2 2-1 00 6x1 2 100 --1 1 2 00 4s40 00 1 3 2 0 1 12z j2 4 6 100 00 32c j–z j00 00 0 --100 00Example: Alternative Optimal SolutionIn the previous slide we see that the optimal solution is:x1= 4, x2= 6, x3= 0, and z= 32N h i b i d i 0 Thi 0Note that x3is nonis non--basic and its c3-z3= 0. This 0 indicates that if x3were increased, the value of the objective function would not changeobjective function would not change.Another optimal solution can be found by choosing x3as the entering variable and performing one iteration of the simplex method. The new tableau on the next slide shows an alternative optimal solution is:x1= 7, x2= 0, x3= 3, and z= 32Example: Alternative Optimal Solution ⏹New Tableaux1 x2 x3s1s2 s3 s4 Basis c B 2 4 60 00 0s3 000 --1 0 22 --1 1 0 2x3 60 .5 1 11 --.5 0 0 3x1 2 1 .5 0 2 1.5 0 0 7 s4 000 --.5 0 2 2.5 0 1 9z j 2 4 6 1000 0 32c j-z j0 0 00 0 0 --1000 0Degeneracy⏹ A A degenerate solutiondegenerate solution to a linear program is one in which at least one of the basic variables equals 0 which at least one of the basic variables equals 0.⏹This can occur at formulation or if there is a tie for the minimizing value in the ratio test to determine the leaving variable.⏹When degeneracy occurs, an optimal solution mayg y,p y have been attained even though some c j–z j> 0.⏹Thus, the condition that c–z<0 is sufficient forj joptimality, but not necessary.Further Reading: Chapter 17 (CD).Exercises Exercises: Question 1: Question 1--6。
