外文翻譯--網(wǎng)路控制信號的優(yōu)化設計

1、<p>　　網(wǎng)路控制信號的優(yōu)化設計</p><p><b>　　摘要：</b></p><p>　　考慮到整個使用信號的遲滯,網(wǎng)路信號的優(yōu)選設計使平衡流的最小化。這個問題可以由用戶平衡交通任務作為限制,對優(yōu)化設計進行公式化。在本文中, 一個突出的共軛梯度方法被提出,解決全球集中的網(wǎng)路優(yōu)化設計問題。數(shù)例用于研究簡單柵格網(wǎng)路。據(jù)顯示, 在解決網(wǎng)路優(yōu)化設計的平衡流

2、,教傳統(tǒng)的方法,被提出的方法更能出色的完成。</p><p>　　關鍵詞: 共軛梯度;網(wǎng)路信號; 平衡約束; 優(yōu)化</p><p><b>　　1. 介紹</b></p><p>　　當考慮到使用者的路線選擇，網(wǎng)路信號的優(yōu)化設計是需要考慮的。這個問題可以由用戶平衡交通任務作為限制,對優(yōu)化設計進行公式化。在過去的幾十年中, 許多研究者通過優(yōu)化技術

3、研究了這個問題[ 1,6,9,2 ] 。在本文里, 一個雙層規(guī)劃設計可以用網(wǎng)路控制信號來闡述。在上層規(guī)劃中, 性能指標可定義為遲滯率的有利的線性綜合，以及在整個交通流中每段時間停止的數(shù)量,可以用TRANSYT[ 7 ] 交通模擬來評估。工作性能相應的數(shù)學近似值和在下游匯合處的媒介物的平均延遲，在TRANSYT 模型中已經(jīng)獲得。在底層規(guī)劃中，用戶的平衡交通任務服從Wardrop第一原則，可以作為最小化問題來闡明。由于用戶的均衡分配約束是非

4、線性的, 導致網(wǎng)路信號化的設計問題是不突出的, 因此創(chuàng)建了唯一最佳方案。</p><p>　　在本文里, 一個計劃的共軛梯度(PCG)提議確定優(yōu)選的信號設置和全球集中的網(wǎng)絡流程。柵格網(wǎng)絡的數(shù)值計算在明顯的交接處提出的PCG 方法勝過傳統(tǒng)方法在各種各樣的初始需求量。本文的其他章節(jié)安排如下。在下一章,表明網(wǎng)路的信號化考慮到使用者的路線選擇。在第三章中，提出的共軛梯度方法在全球集中開發(fā)。在第四章, 一個柵格網(wǎng)絡以信號控

5、制的連接點在各種初始環(huán)節(jié)之下被考慮提出的PCG的數(shù)值計算和傳統(tǒng)方法被提出。第五章是本文的結論與討論。</p><p><b>　　2. 問題公式化</b></p><p><b>　　2.1. 符號</b></p><p>　　這篇論文的符號概述如下：</p><p>　　G(N, l) 表示一個指

6、定的網(wǎng)路, N 是信號控制連接并且L 是套鏈接</p><p>　　表示套信號設置可變物, 各自地為相互綠色周期、開始和期間, 那里和代表開始傳染媒介綠色的期間為信號小組j 在連接點m 如同共同的周期的比例</p><p>　　代表有效的綠色的期間為鏈接a</p><p>　　代表極小值綠色為信號小組j 在連接點m</p><p>　　代表清

7、除時間在綠色的結尾為小組j 和綠色之間開始為不相容的小組l 在連接點m</p><p>　　代表飽和流速在鏈接a</p><p>　　代表第號0 和1 的一件收藏品為各對不相容的信號小組在連接點m;</p><p>　　如果綠色開始為信號小組j 進行那l 和否則</p><p>　　代表延遲的率在鏈接a</p><p>

8、;　　代表中止的數(shù)量每單位時間在鏈接a</p><p><b>　　表示套OD 對</b></p><p>　　表示對OD 對的旅行需求</p><p>　　表示套道路在OD 對w 之間</p><p>　　表示道路流程傳染媒介</p><p>　　表示鏈接流程傳染媒介</p>&l

9、t;p>　　表示鏈接道路發(fā)生矩陣, 如果道路p 在OD 對w 之間使用鏈接a 和</p><p>　　c 表示鏈接旅行時間</p><p>　　2.2. 信號化的公路網(wǎng)問題</p><p>　　信號化的公路網(wǎng)設計問題可能被公式化至于</p><p><b>　　依于</b></p><p>

10、　　和是各自鏈接具體衡量的因素為延遲的率和數(shù)字中止每單位時間被使用在TRANSYT 。第一限制在在共同的周期并且constraints(3)-(5) 在在綠色階段、鏈接容量和清除時間在各個連接點。并且平衡流程由解決發(fā)現(xiàn)以下交通分配問題。</p><p><b>　　分鐘 </b></p><p><b>　　依于</b></p>&

11、lt;p>　　3. 一個解答方法為信號化的公路網(wǎng)設計問題</p><p>　　在這個部分, 一次有效的查尋解決問題(1)-(9) 被開發(fā), 為哪些下降的查尋方向引起并且新重復被創(chuàng)造。查尋過程將被終止在KKT 點或a 新查尋方向可能引起。在以下, 一個計劃的共軛梯度方法提議獲得下降查尋方向。</p><p>　　3.1. 一計劃的共軛梯度methodIn 以下, 一個計劃的共軛梯度方法

12、提議獲得下降查尋方向。</p><p>　　題詞1 (Fletcher & 穿過了共軛梯度方法) ?？紤]一個連續(xù)能區(qū)分的作用</p><p><b>　　引起序列重復根據(jù)</b></p><p><b>　　每當</b></p><p>　　然后為點{xk} 序列由共軛梯度方法引起</

13、p><p>　　方向引起由(11) 為一個跌宕的非線性問題是嚴密地的下降方向減少目標函數(shù)價值在對應的梯度價值不是零條件下。</p><p>　　那里優(yōu)先處理的衍生物談到信號設置和流程從Chiou [ 3 ] 被獲得并且第二個項目是從靈敏度分析為網(wǎng)絡流程在Patriksson [ 5 ] 。讓A 表示系數(shù)恒定的傳染媒介壓抑的矩陣和B (2)-(5) 問題(1)-(9) 可能被重寫</p&g

14、t;<p>　　在追隨者, 我們應用Fletcher & 依照被給作為穿過了共軛梯度方法對一個線性限制被設置在(14) 和(15) 由介紹一個矩陣在射出目標函數(shù)的梯度活躍限制空空間(2)-(5) 以平等為了有效地尋找implementable 點。</p><p>　　定理1 (計劃的共軛梯度(PCG) 方法) ?？紤]問題在(14) 和(15) 序列可行重復{星期} 能引起根據(jù)</p&

15、gt;<p>　　那里 是共軛梯度方向由(11) 確定和 是步長度使減到最小 是在可行的區(qū)域之內的定義了(2)-(5) 。假設, 有充分的等級在星期, 是活躍限制梯度以平等(2)-(5) 并且投射矩陣 是以下形式:</p><p>　　一個修改過的查尋方向sk+1 可能被確定以以下形式:</p><p>　　然后可行的點 序列由計劃的共軛梯度方法單調地引起減少表現(xiàn)價值,&l

16、t;/p><p>　　每當 和是從(13) 。</p><p>　　證明。在題詞以后1 的結果, 我們有</p><p>　　倍增Eq 。(20) 由投射矩陣 它成為</p><p>　　因而為充足地小 我們有</p><p>　　由于由定義使 減到最小沿 從的 是步長度, 它暗示</p><p>

17、<b>　　哪些完成這證明。</b></p><p>　　定理2 () 。在定理1, 當, 如果所有拉格朗日乘算器對應于活躍限制梯度以平等(2)-(5) 是正面或零,它暗示當前的是KKT 點。否則選擇一個消極拉格朗日乘算器, 說, 和修建新活躍限制梯度由刪除列, 對應于消極組分, 并且做投射矩陣以下形式:</p><p>　　查尋方向由(18) 和定理1 舉行的結果

18、然后確定。</p><p>　　證明。讓是拉格朗日乘算器的一個消極組分并且 被定義(17), 我們顯示。由矛盾, 假設</p><p><b>　　并且讓 </b></p><p>　　然后(25) 可能被重寫 </p><p>　　為任何, 那里存在對應的列, , 活躍限制(2)-(5) 并且這樣</p>

19、<p>　　我們減去(27) 從(26) 并且它隨后而來:</p><p>　　自從抗辯假定的有充分的等級。因而。</p><p>　　推論1 (停止狀態(tài)) 。 </p><p>　　如果是KKT 點為問題在(14) 并且(15) 查尋過程然后可以中止; 否則一個新下降方向在能引起根據(jù)定理1 和2 。</p><p>　　3.2.

20、 PCG 解答計劃</p><p>　　認為信號化的公路網(wǎng)優(yōu)選設計問題(1)-(9), PCG 解答計劃被給如下:</p><p>　　步驟1: 開始時,設置索引k =0 。</p><p>　　步驟2: 解決有信號設置的交通分配問題,發(fā)現(xiàn)優(yōu)先處理的衍生物(13) 。</p><p>　　步驟3: 使用計劃的共軛梯度方法確定查尋方向(18)

21、。進入步驟4 。</p><p>　　步驟4: 如果發(fā)現(xiàn)一新在(16) 和讓進入步驟2 。如果和所有拉格朗日乘算器對應于活躍限制梯度non-negative, 是KKT 點和中止。否則, 發(fā)現(xiàn)最消極的拉格朗日乘算器和取消對應限制和發(fā)現(xiàn)一個新投射矩陣和進入步驟3。</p><p>　　4. 數(shù)字例子和計算比較</p><p>　　在這個部分, 數(shù)字試驗做兩重。首先,

22、數(shù)字計算被執(zhí)行為顯示提出的PCG 的有效率和強壯與那些傳統(tǒng)方法, 即LCA [ 4 ] 并且SAB [ 8 比較] 在一個信號控制的柵格網(wǎng)絡以分明套最初。第二,數(shù)字比較用各種各樣的壅塞連續(xù)做由單調地增加旅行需求標量。</p><p>　　4.1. 柵格網(wǎng)絡以信號控制的連接點</p><p>　　一個5x5 柵格大小網(wǎng)絡被顯示在無花果。1 被考慮為說明提出的PCG 方法的有效率在4 對OD

23、旅行和9 個信號化的連接點是信號化的網(wǎng)絡的優(yōu)選設計考慮到。旅行率被設置在100 veh/h 并且鏈接容量被設置1800 veh/h 。數(shù)字結果以二套分明最初被總結在表1 。如同它被看見在表1, 提出的PCG 方法達到在是短的為系統(tǒng)優(yōu)化的全球性最宜40 veh 附近在2.6% 如此之內。提議 PCG 方法極大勝過傳統(tǒng)方法LCA 按表現(xiàn)給定值(PI) 近似地根據(jù)39% 和31% 和根據(jù)11% 和9% SAB 方法。關于計算時代由提出的PCG

24、 方法需要作為它明顯地看從表1, 它要求較不計算努力或在CPU 時間或在解決對應的交通任務如同它做為LCA 或SAB 。</p><p>　　4.2. 柵格信號化了網(wǎng)絡隨著壅塞的增加</p><p>　　第二次調查將測試提出的PCG 方法的強壯在越來越被充塞的柵格信號化的網(wǎng)絡當與常規(guī)方法比較。增長的交通壅塞導致連續(xù)增長的標量對基本的旅行要求。計算結果為提出的PCG 方法被總結在表2, 結果

25、用相對區(qū)別百分比被表達對因此在二集合ofdistinct 最初的地方。</p><p>　　如同它被顯示在表2, 提出的PCG 方法達到了系統(tǒng)最宜在3% 之內一致地產(chǎn)生最少相對區(qū)別百分比在20 套旅行要求。另一方面, 常規(guī)方法, 象LCA 和SAB, 達到了系統(tǒng)最宜以上限值的相對區(qū)別百分比比那些做了提出的PCG 方法。此外, 采取LCA 方法例如, 相對區(qū)別百分比的價值是更大雖然交通壅塞變得嚴厲和價值平均是一樣高

26、的象72 和63 為二套最初?？紤]表現(xiàn) SAB, 它達到了系統(tǒng)最宜與19%, 比那些做了LCA 似乎相對地好但仍然較不健壯比那些做了提出的PCG 方法。</p><p>　　實施為執(zhí)行計算努力在LCA 、SAB 和PCG 方法有被舉辦在SUN SPARC 超II 工作站根據(jù)操作系統(tǒng)Unix SunOS 5.5.1 使用 C++ 編譯器。停著的標準為這些解答被設置當在表現(xiàn)上的相對區(qū)別給定值在連貫疊代少于0.05%

27、之間。</p><p><b>　　5. 結論和討論</b></p><p>　　在本文里, 我們提出了一個最近計劃的共軛投射方法(PCG) 以全球性匯合有效地解決信號化的公路網(wǎng)問題。一個5 個* 5 個柵格公路網(wǎng)與信號控制連接點被使用給常規(guī)數(shù)字上展示提出的方法的強壯和優(yōu)勢方法在解決信號化的公路網(wǎng)優(yōu)選設計。如同它報告了從數(shù)字比較在交通壅塞之下在各種各樣的套最初, 提出

28、的PCG 方法一致地勝過了其它常規(guī)方法以明顯的意義和較不計算努力被采取了。</p><p>　　它被想象測試提出的PCG 方法在大范圍一般路信號化的網(wǎng)絡。更加進一步調查將被做為有鏈接容量擴展的信號化的公路網(wǎng)設計問題并且有彈性旅行要求。</p><p><b>　　鳴謝</b></p><p>　　作者對支持是感恩的從臺灣全國科學委員會通過津貼N

29、SC 95-2416-H-259- 014.</p><p><b>　　參考文獻:</b></p><p>　　[ 1 ] M. Abdulaal, L.J. LeBlanc, 連續(xù)平衡網(wǎng)絡設計模型,運輸研究B 13 (1979) 19-32 。</p><p>　　[ 2 ] H. Ceylan, M.G.H. Bell, 根據(jù)交通信號時間

30、優(yōu)化基因算法方法, 包括司機的發(fā)送, 運輸研究B 38 (2004) 329-342 。</p><p>　　[ 3 ] S-W 。Chiou, TRANSYT 衍生物為區(qū)域交通控制優(yōu)化以網(wǎng)絡平衡流程, 運輸研究B 37 (2003) 263-290 。</p><p>　　[ 4 ] B.G 。Heydecker, T.K. Khoo, 平衡網(wǎng)絡設計問題, 在: AIRO'90

31、關于模型的會議記錄和方法為決策支持, Sorrento 1990 年, 頁587-602 。</p><p>　　[ 5 ] M. Patriksson, 靈敏度分析交通平衡, 運輸科學38 (2004) 258-281 。</p><p>　　[ 6 ] S. Suh, T.J. 金, 解決平衡網(wǎng)絡設計問題的非線性bilevel 編程的模型: 比較回顧, 史冊運籌學34 (1992) 2

32、03-218 。</p><p>　　[ 7 ] R.A. Vincent, A.I. Mitchell, D.I. Robertson, 對TRANSYT 的用戶指南, 版本8 。TRRL 報告、LR888 、運輸和路研究實驗室, Crowthorne 1980 年。</p><p>　　[ 8 ] H. 楊, S. Yagar, 交通任務和信號控制在飽和的公路網(wǎng), 運輸研究A 29 (

33、1995) 125-139 。</p><p>　　[ 9 ] H. 楊, M.G.H. Bell, 模型和算法為公路網(wǎng)設計: 回顧和一些新發(fā)展, 運輸回顧18 (1998) 257-278 。</p><p>　　Optimal design of signal-controlled road network</p><p><b>　　Abstract

34、：</b></p><p>　　An optimal design of signalized road network is considered where total travelers’ delay is minimized subject to user equilibrium flow. This problem can be formulated as an optimization

35、problem by taking the user equilibrium traffic assignment as a constraint. In this paper, a projected conjugate gradient method is presented to solve the signalized road network problem with global convergence. Numerical

36、 examples are investigated on simple grid road network. As it shows, the proposed method achie</p><p>　　Keywords: Conjugate gradient; Signalized road network; Equilibrium constraints; Optimization</p>

37、<p>　　1. Introduction</p><p>　　An optimal design of signalized road network is considered while the route choice of users is taken into account. This problem can be formulated as an optimization proble

38、m by taking the user equilibrium traffic assignment as a constraint. In the past decades, many researchers via the techniques of optimization have investigated this problem [1,6,9,2]. In this paper, a bilevel programming

39、 program is considered to formulate an optimal design of signal-controlled road network. In the upper level pr</p><p>　　In this paper, a projected conjugate gradient (PCG) approach is proposed to determine t

40、he optimal signal settings and network flows with global convergence. Numerical computations are conducted on a grid network with signalized junctions where the proposed PCG method outperforms traditional methods on vari

41、oussets of initials and sets of demand scalars. The rest of the paper is organized as follows. In the next section, a signalized road network is formulated where users’ route choice is taken in</p><p>　　2. P

42、roblem formulation</p><p>　　2.1. Notation</p><p>　　Notation used throughout this paper is summarized below.</p><p>　　G(N, l) denotes a directed road network, where N is the set of s

43、ignal controlled junctions and L is the set of links</p><p>　　denotes the set of signal setting variables, respectively for the reciprocal of cycle time, start and duration of greens, whereandrepresents the

44、vector of startsand durations of greenfor signal group j at junction m as proportions of common cycle time</p><p>　　represents the duration of effective green for link a</p><p>　　represents the

45、minimum green for signal group j at junction m</p><p>　　represents the clearance time between the end of green for group j and the start of green for incompatible group l at junction m</p><p>　　

46、represents saturation flow rate on link a</p><p>　　represents a collection of numbers 0 and 1 for each pair of incompatible signal groups at junction m; whereif the start of green for signal group j proceeds

47、 that of l andotherwise</p><p>　　represents the rate of delay on link a</p><p>　　represents the number of stops per unit time on link a</p><p>　　denotes the set of OD pairs</p>

48、;<p>　　denotes the travel demands for OD pairs</p><p>　　denotes the set of paths between OD pair w</p><p>　　denotes vector of path flow</p><p>　　denotes vector of link flow&l

49、t;/p><p>　　denotes the link-path incidence matrix, whereif path p between OD pair w uses link a andotherwise</p><p>　　C denotes the link travel times</p><p>　　2.2. Signalized road n

50、etwork problem</p><p>　　The signalized road network design problem can be formulated as to</p><p>　　subject to</p><p>　　whereandare respectively link-specific weighting factors for

51、the rate of delay and the number of stops per unit time used in TRANSYT. The first constraint is for the common cycle time and the constraints (3)–(5) are for the green phase, link capacity and clearance time at each jun

52、ction. Also the equilibrium flowsare found by solving the following traffic assignment problem.</p><p>　　3. A solution method for signalized road network design problem</p><p>　　In this section,

53、 an effective search for solving problems (1)–(9) is developed, for which a search direction of descent is generated and a new iterate is created. The search process will be terminated at a KKT point or a new search dire

54、ction can be generated. In the following, a projected conjugate gradient method is proposed to obtain a descent search direction.</p><p>　　3.1. A projected conjugate gradient method</p><p>　　Lem

55、ma 1 (Fletcher & Reeves’ conjugate gradient method). Consider a continuously differentiable function</p><p>　　to generate a sequence of iterates according to</p><p>　　Then for the sequence o

56、f pointsgenerated by the conjugate gradient method</p><p><b>　　whenever</b></p><p>　　The direction generated by (11) for an unconstrained nonlinear problem is a descent direction whi

57、ch strictly decreases the objective function value provided that the corresponding gradient value is not zero. Suppose that the first-order partial derivatives for the objective function in problems (1)–(9) evaluated ate

58、xist,</p><p>　　which can be expressed as</p><p>　　where the first-order derivatives with respect to signal settings and flows are derived from Chiou [3] and the second item are from the sensitiv

59、ity analysis for network flows in Patriksson [5]. Let A denote the coefficient matrix and B the constant vector in constrains (2)–(5) the problems (1)–(9) can be rewritten as</p><p>　　In the followings, we a

60、pply Fletcher & Reeves’ conjugate gradient method to a linear constraint set as given in (14) and (15) by introducing a matrix in projecting the gradient of the objective function onto a null space of active constrai

61、nts as in (2)–(5) with equality in order to effectively search for implementable points.</p><p>　　Theorem 1 (Projected conjugate gradient (PCG) method). Consider the problem in (14) and (15) a sequence of fe

62、asible iteratescan be generated according to</p><p>　　whereis the conjugate gradient direction determined by (11) andis the step length minimizing</p><p>　　along for whichis within the feasible

63、region defined by (2)–(5). Suppose thathas full rank atwhich is the gradient of active constraints with equalities in (2)–(5) and the projection matrixis of the following form:</p><p>　　A modified search dir

64、ectioncan be determined in the following form:</p><p>　　Then the sequence of feasible pointsgenerated by the projected conjugate gradient method monotonically decreases the performance value,</p><

65、p><b>　　whenever</b></p><p>　　Proof. Following the results of Lemma 1, we have</p><p>　　Multiply Eq. (20) by projection matrixit becomes</p><p>　　Thus for sufficiently

66、 smallwe have</p><p>　　Because by definitionis the step length which minimize Z1 alongfrom</p><p>　　it implies</p><p>　　which completes this proof.</p><p>　　Theorem 2 (

67、PCG method asIn Theorem 1, when</p><p>　　if all the Lagrange multipliers corresponding to the active constraint gradients with equalities in (2)–(5) are positive or zeros, it implies the currentis a KKT poin

68、t. Otherwise choose one negative Lagrange multiplier, sayand construct a newof the active constraint gradients by deleting the jth row of</p><p>　　which corresponds to the negative componentand make the proj

69、ection matrix of the following form:</p><p>　　The search direction then is determined by (18) and the results of Theorem 1 hold.</p><p>　　Proof. Letbe a negative component of the Lagrange multip

70、lier anddefined in (17), we showBy contradiction, suppose</p><p>　　and letthen (25) can be rewritten as</p><p>　　For anythere exists a corresponding jth row, , of the active constraint in (2)–(5

71、) andsuch that</p><p>　　We subtract (27) from (26) and it follows:</p><p>　　sincewhich contradicts the assumption thathas full rank. Thus</p><p>　　Corollary 1 (Stopping condition).

72、Ifis a KKT point for problem in (14) and (15) then the search process may stop; otherwise a new descent direction atcan be generated according to Theorems 1 and 2.</p><p>　　3.2. PCG solution scheme</p>

73、<p>　　Consider the signalized road network optimal design problem in (1)–(9), a PCG solution scheme is given below:</p><p>　　Step 1: Start withset index k=0.</p><p>　　Step 2: Solve a traf

74、fic assignment problem with signal settingsfind the first-order derivatives by (13).</p><p>　　Step 3: Use the projected conjugate gradient method to determine a search direction by (18). Go to Step 4.</p&

75、gt;<p>　　Step 4: Iffind a newin (16) and letGo to Step 2. If</p><p>　　and all the Lagrange multipliers corresponding to the active constraint gradients are non-negative, is the KKT point and stop. Oth

76、erwise, find the most negative Lagrange multiplier and cancel the corresponding constraint and find a new projection matrix and go to Step 3.</p><p>　　4. Numerical example and computational comparisons</p

77、><p>　　In this section, numerical experiments are conducted twofold. Firstly, numerical computations are carried out for showing the effectiveness and robustness of the proposed PCG as compared to those of trad

78、itional methods, e.g. LCA [4] and SAB [8] in a signal-controlled grid network with distinct sets of initials. Secondly, numerical comparisons are made continuously with various congestions by monotonically increasing tra

79、vel demand scalars.</p><p>　　4.1. Grid network with signal-controlled junctions</p><p>　　A 5* 5 grid-size network shown in Fig. 1 is considered for illustrating the effectiveness of the proposed

80、 PCG method in optimal design of signalized network where 4-pair OD trips and 9 signalized junctions are taken into account. Trip rates are set at 100 veh/h and link capacity is set 1800 veh/h. Numerical results with two

81、 sets of distinct initials are summarized in Table 1. As it seen in Table 1, the proposed PCG method achieved near global optimum of 40 veh within 2.6% of SO which is short fo</p><p>　　4.2. Grid signalized n

82、etwork with increasing congestions</p><p>　　The second investigation is to test robustness of the proposed PCG method on increasingly congested grid signalized networks when compared to conventional methods.

83、 The increasing traffic congestion is caused by continuously increasing scalars to base travel demands. Computational results for the proposed PCG method are summarized in Table 2, where results are expressed in the rela

84、tive difference percentage to SO at two sets of</p><p>　　distinct initials. As it shown in Table 2, the proposed PCG method achieved the system optimum within 3% by consistently yielding the least relative d

85、ifference percentages at 20 sets of travel demands. On the other hand, the conventional methods, like LCA and SAB, achieved the system optimum with much higher values of the relative difference percentages than those did

86、 the proposed PCG method. Furthermore, taking the LCA method for example, the values of the relative difference percentages were l</p><p>　　The implementations for carrying out the computational efforts on L

87、CA, SAB and PCG methods have been conducted on SUN SPARC Ultra II workstation under operating system Unix SunOS 5.5.1 using C++ compiler. The stopping criterion for these solutions is set when the relative difference in

88、the performance index value between the consecutive iterations less than 0.05%.</p><p>　　5. Conclusions and discussions</p><p>　　In this paper, we presented a newly projected conjugate projectio

89、n method (PCG) with global convergence to effectively solve the signalized road network problem. A 5 * 5 grid road network with signal-controlled junctions is used to numerically demonstrate the robustness and superiorit

90、y of the proposed method to conventional approaches in solving optimal design of signalized road network. As it reported from numerical comparisons under traffic congestion at various sets of initials, the proposed </

91、p><p>　　It is envisaged to test the proposed PCG method on a wide range of general road signalized networks. Further investigations will be made for the signalized road network design problem with link capacity

92、 expansions and elastic travel demands.</p><p>　　Acknowledgement</p><p>　　The author is grateful for support from Taiwan National Science Council via grant NSC 95-2416-H-259-014.</p><

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