外文翻譯--數(shù)學(xué)規(guī)劃的席位分配模型

1、<p>　　中文2000字，1300英文單詞，6800英文字符</p><p><b>　　附 錄</b></p><p><b>　　英文文獻</b></p><p>　　Allocation of Seats Mathematical Programming Model</p><p>

2、;<b>　　Abstract</b></p><p>　　Several methods of allocation of seats are studied. Based on these methods six mathematical programming models of allocation of seats have been given and their solutio

3、ns have been given also. These mathematical models attempt to minimize a number of different measures of the deviation between the actual percentage of votes received and the percentage of seats allocated to a certain pa

4、rty. These methods are compared with foregone methods through the example.The six mathematical programming models</p><p>　　Keywords: Allocation of Seats; Mathematical Programming; Equivalent Forms</p>

5、<p>　　1. Introduction</p><p>　　Proportional representation (PR) systems are a family of voting systems used in multiple-winner elections. The principle behind PR elections is that every vote deserves i

6、ts representation in government and each political party involved should be represented in the legislature in proportion to its strength in the electorate. Essentially, this means that each party should receive the same

7、percentage of representation as the percentage of votes received. All PR systems set out to achieve this objec</p><p>　　2. A Short Overview of Existing Seat Allocation Methods</p><p>　　2.1. High

8、est Averages Methods</p><p>　　Suppose that is the number of votes cast for party i , P = + +...+ is the total number of votes cast, m is the number of party. is the number of seats allocated to party i

9、and N is the total number of seats. Different methods of allocating seats after votes have been counted are in use all over the democratic world. The vast majority of democratic countries use some form of proportional re

10、presentation. Some of the few exceptions are France, the UK and some of its former colonies, such as th</p><p>　　A highest average method requires that the number of votes for each party is divided successiv

11、ely by a series of divisors. Seats are then allocated to parties with the highest resulting quotient until all the seats available are allocated. Two highest average methods, namely the d’Hondt method and the Sainte-Lagu

12、e method, are the most commonly used andare considered in this paper. Gao put up follow the Maximum entropy method of allocating seats.</p><p>　　2.2. Largest Remainder Methods</p><p>　　The large

13、st remainder methods (LR methods) are the other class of allocation methods. This method requires that each party’s votes are divided by a quota, which represents the number of votes required for a seat. A notional numbe

14、r of seats is given to each party. This seat allocation typically includes an integer part and a remainder part. Each party receives the number of seats equal to the integer value. Generally this will leave some seats un

15、allocated. The parties are then ranked on the basis</p><p>　　2.3. Mathematical Programming Methods</p><p>　　Yan Yusong gave an O-1 programming model on allocation of seats, the results have impr

16、oved the traditional Q-value method and the new Q-value method. Two restrictive reasonable condition are put forward for fair allotment of representative quota．Mathematical models with least variance and their solutions

17、respectively meeting these conditions are given, and the mathematical model and the solution simultaneously meeting these conditions are also brought forward．Three mathematical models for seats </p><p>　　3.

18、Mathematical Programming Model</p><p>　　The mathematical programming models that we present here use mixed integer programming to minimize the deviation caused by the discrete nature of the seats to be alloc

19、ated. Thus, the objective is to ensure that the percentage of the seats allocated is as close as possible to the actual percentage of votes received.</p><p>　　A seat of party i represents , and perfect numbe

20、r is , so deviation for party i is -. This task is formulated as the optimization problem:</p><p>　　The principle behind this method is to minimize, over all parties, the total of absolute deviation and the

21、largest deviation instead of the total square of deviation. The mathematical programming formulations are given by</p><p>　　According to percentage rule, perfect number of seats of party i is , and the actu

22、al seats allocated to party i is , then the deviation between them is-. Similarly to mathematical programming model I, II and III, the mathematical programming formulations are got</p><p>　　4. Algorithms fo

23、r Mathematical Programming Models</p><p>　　Yan Yusong translated mathematical programming model I into an O-1 programming model on allocation of seats. The mathematical programming model I will not be transl

24、ated into an 0-1 programming model, and it will be solved directly. Initially is set to for all the parties. The rest seats is k = N ? （++...+）（0 ≤ k ≤ m ?1）.The mathematical programming model I is how to allocate the

25、rest k seats to m parties. There are methods what is the numbers of allocate the rest k seats to m parties. The enu</p><p>　　Step1. , k = </p><p>　　Step2. For a=1:k</p><p>　　Step 3

26、. Calculate （i = 1,2,...,m）;</p><p><b>　　Step4. </b></p><p>　　Step5. Endfor</p><p>　　5. Conclusions</p><p>　　In this paper various methods for the allocat

27、ion of seats in a PR election system were reviewed. The objective in each is to minimize, for each party, some measure of the deviation between the actual percentage of votes received and the percentage of seats allocate

28、d to that party. This objective served as a criterion to ascertain whether any fair allocation methods exist. Two categories of allocation methods, namely highest average methods and largest remainder methods, were discu

29、ssed. New mathe</p><p><b>　　譯文：</b></p><p>　　數(shù)學(xué)規(guī)劃的席位分配模型</p><p><b>　　摘 要</b></p><p>　　本文研究了幾種席位分配的方法?；谶@些方法，得出了六種席位分配方法的數(shù)學(xué)模型及其解決方案。這些數(shù)學(xué)模型嘗試最小化某政

30、黨實際所得票數(shù)的比例和席位分配比例之間的誤差。通過例子這些方法與先前的方法進行了比較。本文對這六個數(shù)學(xué)規(guī)劃模型進行了概括，進而提出了另外十二個數(shù)學(xué)規(guī)劃模型。也討論了數(shù)學(xué)規(guī)劃的統(tǒng)一形式以及等價形式。</p><p>　　關(guān)鍵字：席位分配，數(shù)學(xué)規(guī)劃，等價形式</p><p><b>　　1. 介紹</b></p><p>　　比例代表制(PR)系統(tǒng)

31、是一個用于在multiple-winner選舉的選舉投票系統(tǒng)。公關(guān)選舉背后的原理是,每一票都應(yīng)該代表政府以及每個政黨在議會中相應(yīng)比例的選民的力量。實際上,這意味著每個政黨應(yīng)該得到與選票比例相當(dāng)?shù)拇頂?shù)。所有公關(guān)系統(tǒng)都旨在實現(xiàn)這一目標(biāo)。在典型的公關(guān)系統(tǒng)有一些成員地區(qū)，公關(guān)系統(tǒng)所面臨的挑戰(zhàn)是要把選舉人票轉(zhuǎn)化成與實際所得選票比例相同的投票席位。席位分配是整數(shù),與席位的數(shù)量相比，選票的數(shù)量可能被視為連續(xù)量。因此,把得票數(shù)轉(zhuǎn)換成投票席位數(shù)量總是會

32、涉及到調(diào)整方法。從運籌學(xué)的角度來看，它旨在減少實際選票的比例與不同的政黨間席位分配比例衡量上的某種程度的偏差。</p><p>　　2. 現(xiàn)有的座位分配方法的簡單的概述</p><p>　　2.1 最高平均方法</p><p>　　假設(shè) 是i黨得到的選票數(shù)量。 P = + +...+ 。P是總共的選票數(shù)量。M是所有的政黨數(shù)量。是i黨分配到的席位數(shù)，并且n 是總共

33、的席位數(shù)。選票算好后，各個民主國家采用不同方法來進行席位分配。絕大多數(shù)的民主國家使用某種形式的比例代表制。一些為數(shù)不多的例外有法國、英國和它的一些前殖民地國家,如美國、加拿大和印度。今天我們使用的所有的不同的公關(guān)系統(tǒng)被分為三種類別,即最高平均(或因子)方法，最大的剩余部分(或指標(biāo))方法或數(shù)學(xué)規(guī)劃方法。</p><p>　　最高平均方法要求每個政黨的票數(shù)應(yīng)通過一系列的因子來劃分。然后以最高份額為原則，把席位分配給各

34、個黨方，直到把所有的席位分配完。兩個最高平均方法,即d 'Hondt方法和Sainte-Lague方法,它們被認(rèn)為是這篇論文中最常用的方法。Gao提出了遵循的最大熵席位分配方法。</p><p>　　2.2 最大剩余方法</p><p>　　最大的剩余方法(LR方法)是其他類的分配方法。這種方法要求每個黨的選票除以一個配額,得到代表選票需要的席位的數(shù)量。這個席位分配通常包括一個整數(shù)

35、部分和剩余部分。每個黨獲得席位的數(shù)量等于整數(shù)值。通常這也將會導(dǎo)致一些席位得不到分配。然后各個黨按照剩余席位的數(shù)量排名，擁有最多剩余席位的政黨將分配出額外的席位，直到所有的剩余席位被分配完。Hare配額和Droop配額是最常見的。漢密爾頓的分配方法被專門定義為Hare分配應(yīng)用。在納米比亞和香港，他們使用這種方法。</p><p><b>　　2.3 數(shù)學(xué)規(guī)劃法</b></p>&

36、lt;p>　　Yan Yusong提出關(guān)于席位分配的O-1規(guī)劃模型,改善了傳統(tǒng)的經(jīng)典Q值法和新Q值法。針對代表名額的分配，我們提出了兩條合理的限制性條件。那些具有最小方差的數(shù)學(xué)模型而且解決方案也分別滿足這些條件的數(shù)學(xué)模型已經(jīng)被提出。關(guān)于席位分配問題的三個數(shù)學(xué)模型也已經(jīng)被展示出來。所有這些都是有網(wǎng)格搜索算法得出的，并且為了描述公平與席位分布還適當(dāng)?shù)貞?yīng)用了最小二乘法思想。建模過程很簡潔,解決這些模型的運算法則也很有效。Lin Jian

37、liang給出了關(guān)于席位公平分配及其數(shù)學(xué)規(guī)劃模型的差異最小化的方法,并分析了幾種席位數(shù)量分配的方法。在比例代表制(PR)投票系統(tǒng)中，一些席位分配方法的公平性曾被調(diào)查。數(shù)學(xué)模型試圖使某黨得到的實際票數(shù)比例與席位比例之間的差異最小化。本文給出了18種席位分配的數(shù)學(xué)規(guī)劃模型，以及它們相應(yīng)的解決思路。</p><p><b>　　3. 數(shù)學(xué)規(guī)劃模型</b></p><p>　

38、　這里提供的數(shù)學(xué)規(guī)劃模型應(yīng)用了混合整數(shù)規(guī)劃來減少所要分配的席位的離散型所造成的偏差。因此,這個目標(biāo)確保了席位分配的比例盡可能地與實際的選票比例相接近。</p><p>　　i政黨的一個席位代表的人數(shù)由表示， 最佳分配結(jié)果是是，那么這個正當(dāng)?shù)南籭的誤差是-。 這個計算過程可以用下面的公式來表示：</p><p>　　這個方法的原理旨在最小化各個政黨的絕對偏差和最大偏差而不是總平方偏差。這個

39、數(shù)學(xué)規(guī)劃的公式如下：</p><p>　　根據(jù)比例原則，政黨i的最佳席位數(shù)量是，二實際分配的席位是，那么它們之間的誤差是-。與數(shù)學(xué)規(guī)劃模型一，二，三相似，該數(shù)學(xué)規(guī)劃公式如下：</p><p><b>　　數(shù)學(xué)規(guī)劃模型的算法</b></p><p>　　Yan Yusong把數(shù)學(xué)規(guī)劃模型轉(zhuǎn)化為了一個關(guān)于席位分配的O-1規(guī)劃模型。本文不會把數(shù)學(xué)規(guī)劃

40、模型轉(zhuǎn)化為一個0 - 1規(guī)劃模型，而是直接解出。首先， 適用于所有的政黨。剩余的席位k = N ? （++...+）（0 ≤ k ≤ m ?1）。數(shù)學(xué)規(guī)劃模型I旨在分析如何把剩余的k個席位分配給m個政黨。共有種分配方案。枚舉算法計算量太大，因此用一個接一個的分配方法來解決數(shù)學(xué)規(guī)劃模型。</p><p>　　步驟1. , k = </p><p>　　步驟2. For a=1:k</

41、p><p>　　步驟3. Calculate （i = 1,2,...,m）;</p><p><b>　　步驟4. </b></p><p>　　步驟5. 得出結(jié)果</p><p><b>　　總結(jié)</b></p><p>　　本文我們綜述了公關(guān)選舉制度中的各種