急急急求關(guān)于運籌學的最小費用最大流的英文文獻，有中英文翻譯更佳！！！！！！！！！！ 求運籌學的中英文對照文章
下面先是漢語-----
【MCMF問題及數(shù)學模型】
在介紹最大流問題時，我們列舉了一個最大物資輸送流問題。如果這個問題的已知條件還包括每條邊運送單位物資的費用，那么怎樣運送，才能得到最大運輸量，并且輸送費用最少？這便是所謂最小費用最大流問題。
在最大流的有關(guān)定義的基礎(chǔ)上，若每條有向邊除權(quán)數(shù)c(e)（表示邊容量）外還有另外一個權(quán)數(shù)w(e)（表示單位流所需費用），并且已求得該網(wǎng)絡(luò)的最大流值為F， 那么最小費用最大流問題，顯然可用以下線性規(guī)劃模型加以描述：
Min ∑ w(e)f(e)
e∈E
滿足 0≤f(e)≤c(e) ，對一切e∈E
f+(v)=f-(v) ，對一切v∈V
f+(x)=F （最大流約束）
（或f-(y)=F )
【算法思路】
解決最小費用最大流問題，一般有兩條途徑。一條途徑是先用最大流算法算出最大流，然后根據(jù)邊費用，檢查是否有可能在流量平衡的前提下通過調(diào)整邊流量，使總費用得以減少？只要有這個可能，就進行這樣的調(diào)整。調(diào)整后，得到一個新的最大流。
然后，在這個新流的基礎(chǔ)上繼續(xù)檢查，調(diào)整。這樣迭代下去，直至無調(diào)整可能，便得到最小費用最大流。這一思路的特點是保持問題的可行性（始終保持最大流），向最優(yōu)推進。另一條解決途徑和前面介紹的最大流算法思路相類似，一般首先給出零流作為初始流。這個流的費用為零，當然是最小費用的。然后尋找一條源點至匯點的增流鏈，但要求這條增流鏈必須是所有增流鏈中費用最小的一條。如果能找出增流鏈，則在增流鏈上增流，得出新流。將這個流做為初始流看待，繼續(xù)尋找增流鏈增流。這樣迭代下去，直至找不出增流鏈，這時的流即為最小費用最大流。這一算法思路的特點是保持解的最優(yōu)性（每次得到的新流都是費用最小的流），而逐漸向可行解靠近（直至最大流時才是一個可行解）。
由于第二種算法和已介紹的最大流算法接近，且算法中尋找最小費用增流鏈，可以轉(zhuǎn)化為一個尋求源點至匯點的最短路徑問題，所以這里介紹這一算法。
在這一算法中，為了尋求最小費用的增流鏈，對每一當前流，需建立伴隨這一網(wǎng)絡(luò)流的增流網(wǎng)絡(luò)。例如圖 1 網(wǎng)絡(luò)G 是具有最小 費用的流，邊旁參數(shù)為c(e) , f(e) , w(e)，而圖 2 即為該網(wǎng)絡(luò)流 的增流網(wǎng)絡(luò)G′。增流網(wǎng)絡(luò)的頂點和原網(wǎng)絡(luò)相同。 按以下原則建 立增流網(wǎng)絡(luò)的邊：若G中邊(u，v)流量未飽，即f(u,v) ＜ e(u,v)，則G ' 中建邊(u,v)，賦權(quán)w ' (u,v)=w(u,v)；若G中邊(u, v)已有流量，即f(u,v)〉0，則G′中建邊(v,u)，賦權(quán)w′(v,u) =-w(u,v)。建立增流網(wǎng)絡(luò)后，即可在此網(wǎng)絡(luò)上求源點至匯點的最短路徑，以此決定增流路徑，然后在原網(wǎng)絡(luò)上循此路徑增流。這里，運用的仍然是最大流算法的增流原理，唯必須選定最小費用的增流鏈增流。
計算中有一個問題需要解決。這就是增流網(wǎng)絡(luò)G ′中有負權(quán)邊，因而不能直接應(yīng)用標號法來尋找x至y的最短路徑，采用其它計算有負權(quán)邊的網(wǎng)絡(luò)最短路徑的方法來尋找x至y的最短路徑，將 大大降低計算效率。為了仍然采用標號法計算最短路徑，在每次建立增流網(wǎng)絡(luò)求得最短路徑后,可將網(wǎng)絡(luò)G的權(quán)w(e)做一次修正，使再建的增流網(wǎng)絡(luò)不會出現(xiàn)負權(quán)邊，并保證最短路徑不至于因此而改變。下面介紹這種修改方法。
當流值為零，第一次建增流網(wǎng)絡(luò)求最短路徑時，因無負權(quán)邊，當然可以采用標號法進行計算。為了使以后建立增流網(wǎng)絡(luò)時不出現(xiàn)負權(quán)邊，采取的辦法是將 G中有流邊（f(e)＞0）的權(quán)w(e)修正為0。為此， 每次在增流網(wǎng)絡(luò)上求得最短路徑后，以下式計算G中新的邊權(quán)w " (u,v)：
w " (u,v)=L(u)-L(v)+w(u,v) (*)
式中 L(u),L(v) -- 計算G′的x至y最短路徑時u和v的標號值。第一次求最短徑時如果(u,v)是增流路徑上的邊， 則據(jù)最短 路徑算法一定有 L(v)=L(u)+w ' (u,v)=L(u)+w(u,v)， 代入(*)式必有
w〃(u,v)=0。
如果(u,v)不是增流路徑上的邊，則一定有：
L(v)≤L(u)+w(u,v)，
代入(*)式則有 w(u,v)≥0。
可見第一次修正w(e)后，對任一邊，皆有w(e)≥0， 且有流 的邊（增流鏈上的邊），一定有w(e)=0。以后每次迭代計算，若 f(u,v)＞0，增流網(wǎng)絡(luò)需建立(v,u)邊，邊權(quán)數(shù)w ' (v,u)=-w(u,v) =0，即不會再出現(xiàn)負權(quán)邊。
此外，每次迭代計算用(*)式修正一切w(e)， 不難證明對每一條x至y的路徑而言，其路徑長度都同樣增加L(x)-L(y)。因此，x至y的最短路徑不會因?qū)(e)的修正而發(fā)生變化。
【計算步驟】
1. 對網(wǎng)絡(luò)G=[V,E,C,W]，給出流值為零的初始流。
2. 作伴隨這個流的增流網(wǎng)絡(luò)G′=[V′,E′,W′]。
G′的頂點同G：V′=V。
若G中f(u,v)＜c(u,v)，則G′中建邊(u,v)，w(u,v)=w(u,v)。
若G中f(u,v)＞0，則G′中建邊(v,u)，w′(v,u)=-w(u,v)。
3. 若G′不存在x至y的路徑，則G的流即為最小費用最大流，
停止計算；否則用標號法找出x至y的最短路徑P。
4. 根據(jù)P，在G上增流： 對P的每條邊(u,v)，若G存在(u,v)，則(u,v)增流；若G存在(v,u)，則(v,u)減流。增（減）流后，應(yīng)保證對任一邊有c(e)≥ f(e)≥0。
5. 根據(jù)計算最短路徑時的各頂點的標號值L(v)，按下式修 改G一切邊的權(quán)數(shù)w(e):
L(u)-L(v)+w(e)→w(e)。
6. 將新流視為初始流，轉(zhuǎn)2。
-----------------
======================
下面是英文-----
【MCMF problems and the mathematical model】
Maximum flow problem in the introduction, we listed one of the largest flow of goods delivery. If this issue also includes the known conditions of delivery of each unit while the cost of goods, then how to transport, to get the most traffic, and transportation costs to a minimum? This is the so-called maximum flow problem minimum cost.
The maximum flow based on the definition, if each side of a first-priority claim to the number of c (e) (that the edge capacity) but also have another weights w (e) (that the unit cost flow), and has been seeking a maximum flow of the network value of F, then the minimum cost maximum flow problem, it is clear the following linear programming model can be used to describe:
Min ∑ w (e) f (e)
e ∈ E
Satisfy 0 ≤ f (e) ≤ c (e), for all e ∈ E
f + (v) = f-(v), for all v ∈ V
f + (x) = F (maximum flow constraints)
(Or f-(y) = F)
】 【Algorithm ideas
Solve the minimum cost maximum flow problem, there are two general ways. Way is to use a maximum flow algorithm to calculate the maximum flow, and then based on the cost side, check whether it is possible to balance the flow by adjusting the flow side, so that to reduce the total cost? As long as there is a possibility, on such adjustments. After adjusting for a new maximum flow.
Then, on the basis of the new flow to continue to check and adjust. This iteration continues until no adjustment may be, they will have the minimum cost maximum flow. The characteristics of this line of thought is to maintain the feasibility of the problem (always maintain maximum flow), to promote optimal. Solution to another and in front of the maximum flow algorithm, introduced a similar line of thought, first of all, given the general flow as the initial flow of zero. The cost of the flow to zero, of course, is the smallest cost. And then find a source to the Meeting Point by flow chain, but by the requirements of this chain must be a stream flow of all chain costs by a minimum. If we can find out by flow chain, the chain in the flow by increasing flow, a new flow. The flow will be treated as the initial flow, continue to search for links by increasing stream flow. This iteration continues, until found by flow chain, then the flow is the minimum cost maximum flow. Idea of the characteristics of this algorithm is to maintain the optimal solution of (each of the new fees are the smallest stream flow), but gradually close to the feasible solution (up to maximum flow is a feasible solution when).
As a result of the second algorithm and has introduced close to the maximum flow algorithm and the algorithm by finding the minimum cost flow chain, can be turned into a source to find the shortest path to the Meeting Point, so this algorithm here.
In this algorithm, in order to seek to increase the minimum cost flow chain, the current flow of each, accompanied by the need to establish a network flow by the flow network. For example, Figure 1 is a network G of minimum cost flow, next to parameters c (e), f (e), w (e), and Figure 2 is the network flow by the flow network G '. By the peak-flow network and the same as the original network. By the following principles in accordance with the establishment of the network edge flow: If G in the edge (u, v) is not enough traffic, that is, f (u, v) <e (u, v), then G 'in the building edge (u, v), Empowering w '(u, v) = w (u, v); edge of G if (u, v) has been the flow, that is, f (u, v)> 0, then G' in the building edge (v, u ), to empower the w '(v, u) =- w (u, v). The establishment of the network by streaming, you can seek in this network to the Meeting Point source shortest path, as decided by flow path, and then in the original network by flow in this path. Here, the use of maximum flow algorithm is still the principle of increasing flow, but the cost must be selected by the smallest chain by stream flow.
Calculation, there is a need to address the problem. This is the stream network by G 'the right to have a negative side, thus labeling law can not be directly applied to find x to y of the shortest path, using the right of other negative side computing network approach to the shortest path x to y to find the shortest path, will greatly reduce the computational efficiency. In order to still use the labeling method to calculate the shortest path, each flow set up by the network to achieve the shortest path, the network G can be the right of w (e) an amendment to do so by the stream to build the network will not be a negative right side, and guarantee the shortest path does not change. This modified method described below.
When the flow value is zero, the first built by the shortest path for flow network, the result of non-negative right side, of course, can be used to calculate labeling law. In order to increase flow network after the establishment of a negative time is not right side of the approach taken is to have stream G edge (f (e)> 0) the right to w (e) amendment to 0. To this end, each time a flow network obtained by the shortest path, the following computing G in the right side of the new w "(u, v):
w "(u, v) = L (u)-L (v) + w (u, v) (*)
Where L (u), L (v) - calculation of G 'of the shortest path x to y when u and v the value of the label. For the first time if the shortest path (u, v) is the flow path by the edge, then, according to the shortest path algorithm must have L (v) = L (u) + w '(u, v) = L (u) + w (u, v), substituting into (*) type must
w "(u, v) = 0.
If (u, v) rather than by the side of flow path, it must have:
L (v) ≤ L (u) + w (u, v),
Into the (*)-type, there w (u, v) ≥ 0.
Shows that the first amendment to w (e), against either side, there are w (e) ≥ 0, and a stream side (by side chain flow), there will be w (e) = 0. Calculated after each iteration, if f (u, v)> 0, by the need to establish the network flow (v, u) edge, edge weights w '(v, u) =- w (u, v) = 0, that is, the right will not be a negative side.
In addition, the calculation of each iteration with (*) fixes all the w (e), it is not difficult to prove that to each path x to y, its all the same to increase the path length L (x)-L (y). Therefore, x and y will not be the shortest path to w (e) the amendment changes.
】 【Calculation steps
1. On the network G = [V, E, C, W], given the initial value is zero flow streams.
2. Be accompanied by this stream flow network G '= [V', E ', W'].
G 'the vertex-G: V' = V.
If G in f (u, v) <c (u, v), then G 'in the building edge (u, v), w (u, v) = w (u, v).
If G in f (u, v)> 0, then G 'in the building edge (v, u), w' (v, u) =- w (u, v).
3. If G 'does not exist the path x to y, then G is the minimum cost flow of maximum flow,
To stop calculation; otherwise labeling method used to find x to y of the shortest path P.
4. According to P, the increased flow in G: each edge of P of (u, v), if G exists (u, v), then (u, v) by flow; if G exists (v, u), while (v, u) by flow. Increase (decrease) in flow should be on either side to ensure that there is c (e) ≥ f (e) ≥ 0.
5. According to the calculation of the shortest path at the time of peak value of the label L (v), press the G-type modification of all the edge weights w (e):
L (u)-L (v) + w (e) → w (e).
6. The new stream as the initial flow to 2.
=========
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【MCMF問題及數(shù)學模型】
在介紹最大流問題時，我們列舉了一個最大物資輸送流問題。如果這個問題的已知條件還包括每條邊運送單位物資的費用，那么怎樣運送，才能得到最大運輸量，并且輸送費用最少？這便是所謂最小費用最大流問題。
在最大流的有關(guān)定義的基礎(chǔ)上，若每條有向邊除權(quán)數(shù)c(e)（表示邊容量）外還有另外一個權(quán)數(shù)w(e)（表示單位流所需費用），并且已求得該網(wǎng)絡(luò)的最大流值為F， 那么最小費用最大流問題，顯然可用以下線性規(guī)劃模型加以描述：
Min ∑ w(e)f(e)
e∈E
滿足 0≤f(e)≤c(e) ，對一切e∈E
f+(v)=f-(v) ，對一切v∈V
f+(x)=F （最大流約束）
（或f-(y)=F )
【算法思路】
解決最小費用最大流問題，一般有兩條途徑。一條途徑是先用最大流算法算出最大流，然后根據(jù)邊費用，檢查是否有可能在流量平衡的前提下通過調(diào)整邊流量，使總費用得以減少？只要有這個可能，就進行這樣的調(diào)整。調(diào)整后，得到一個新的最大流。
然后，在這個新流的基礎(chǔ)上繼續(xù)檢查，調(diào)整。這樣迭代下去，直至無調(diào)整可能，便得到最小費用最大流。這一思路的特點是保持問題的可行性（始終保持最大流），向最優(yōu)推進。另一條解決途徑和前面介紹的最大流算法思路相類似，一般首先給出零流作為初始流。這個流的費用為零，當然是最小費用的。然后尋找一條源點至匯點的增流鏈，但要求這條增流鏈必須是所有增流鏈中費用最小的一條。如果能找出增流鏈，則在增流鏈上增流，得出新流。將這個流做為初始流看待，繼續(xù)尋找增流鏈增流。這樣迭代下去，直至找不出增流鏈，這時的流即為最小費用最大流。這一算法思路的特點是保持解的最優(yōu)性（每次得到的新流都是費用最小的流），而逐漸向可行解靠近（直至最大流時才是一個可行解）。
由于第二種算法和已介紹的最大流算法接近，且算法中尋找最小費用增流鏈，可以轉(zhuǎn)化為一個尋求源點至匯點的最短路徑問題，所以這里介紹這一算法。
在這一算法中，為了尋求最小費用的增流鏈，對每一當前流，需建立伴隨這一網(wǎng)絡(luò)流的增流網(wǎng)絡(luò)。例如圖 1 網(wǎng)絡(luò)G 是具有最小 費用的流，邊旁參數(shù)為c(e) , f(e) , w(e)，而圖 2 即為該網(wǎng)絡(luò)流 的增流網(wǎng)絡(luò)G′。增流網(wǎng)絡(luò)的頂點和原網(wǎng)絡(luò)相同。 按以下原則建 立增流網(wǎng)絡(luò)的邊：若G中邊(u，v)流量未飽，即f(u,v) ＜ e(u,v)，則G ' 中建邊(u,v)，賦權(quán)w ' (u,v)=w(u,v)；若G中邊(u, v)已有流量，即f(u,v)〉0，則G′中建邊(v,u)，賦權(quán)w′(v,u) =-w(u,v)。建立增流網(wǎng)絡(luò)后，即可在此網(wǎng)絡(luò)上求源點至匯點的最短路徑，以此決定增流路徑，然后在原網(wǎng)絡(luò)上循此路徑增流。這里，運用的仍然是最大流算法的增流原理，唯必須選定最小費用的增流鏈增流。
計算中有一個問題需要解決。這就是增流網(wǎng)絡(luò)G ′中有負權(quán)邊，因而不能直接應(yīng)用標號法來尋找x至y的最短路徑，采用其它計算有負權(quán)邊的網(wǎng)絡(luò)最短路徑的方法來尋找x至y的最短路徑，將 大大降低計算效率。為了仍然采用標號法計算最短路徑，在每次建立增流網(wǎng)絡(luò)求得最短路徑后,可將網(wǎng)絡(luò)G的權(quán)w(e)做一次修正，使再建的增流網(wǎng)絡(luò)不會出現(xiàn)負權(quán)邊，并保證最短路徑不至于因此而改變。下面介紹這種修改方法。
當流值為零，第一次建增流網(wǎng)絡(luò)求最短路徑時，因無負權(quán)邊，當然可以采用標號法進行計算。為了使以后建立增流網(wǎng)絡(luò)時不出現(xiàn)負權(quán)邊，采取的辦法是將 G中有流邊（f(e)＞0）的權(quán)w(e)修正為0。為此， 每次在增流網(wǎng)絡(luò)上求得最短路徑后，以下式計算G中新的邊權(quán)w " (u,v)：
w " (u,v)=L(u)-L(v)+w(u,v) (*)
式中 L(u),L(v) -- 計算G′的x至y最短路徑時u和v的標號值。第一次求最短徑時如果(u,v)是增流路徑上的邊， 則據(jù)最短 路徑算法一定有 L(v)=L(u)+w ' (u,v)=L(u)+w(u,v)， 代入(*)式必有
w〃(u,v)=0。
如果(u,v)不是增流路徑上的邊，則一定有：
L(v)≤L(u)+w(u,v)，
代入(*)式則有 w(u,v)≥0。
可見第一次修正w(e)后，對任一邊，皆有w(e)≥0， 且有流 的邊（增流鏈上的邊），一定有w(e)=0。以后每次迭代計算，若 f(u,v)＞0，增流網(wǎng)絡(luò)需建立(v,u)邊，邊權(quán)數(shù)w ' (v,u)=-w(u,v) =0，即不會再出現(xiàn)負權(quán)邊。
此外，每次迭代計算用(*)式修正一切w(e)， 不難證明對每一條x至y的路徑而言，其路徑長度都同樣增加L(x)-L(y)。因此，x至y的最短路徑不會因?qū)(e)的修正而發(fā)生變化。
【計算步驟】
1. 對網(wǎng)絡(luò)G=[V,E,C,W]，給出流值為零的初始流。
2. 作伴隨這個流的增流網(wǎng)絡(luò)G′=[V′,E′,W′]。
G′的頂點同G：V′=V。
若G中f(u,v)＜c(u,v)，則G′中建邊(u,v)，w(u,v)=w(u,v)。
若G中f(u,v)＞0，則G′中建邊(v,u)，w′(v,u)=-w(u,v)。
3. 若G′不存在x至y的路徑，則G的流即為最小費用最大流，
停止計算；否則用標號法找出x至y的最短路徑P。
4. 根據(jù)P，在G上增流： 對P的每條邊(u,v)，若G存在(u,v)，則(u,v)增流；若G存在(v,u)，則(v,u)減流。增（減）流后，應(yīng)保證對任一邊有c(e)≥ f(e)≥0。
5. 根據(jù)計算最短路徑時的各頂點的標號值L(v)，按下式修 改G一切邊的權(quán)數(shù)w(e):
L(u)-L(v)+w(e)→w(e)。
6. 將新流視為初始流，轉(zhuǎn)2。
-----------------
======================
下面是英文-----
【MCMF problems and the mathematical model】
Maximum flow problem in the introduction, we listed one of the largest flow of goods delivery. If this issue also includes the known conditions of delivery of each unit while the cost of goods, then how to transport, to get the most traffic, and transportation costs to a minimum? This is the so-called maximum flow problem minimum cost.
The maximum flow based on the definition, if each side of a first-priority claim to the number of c (e) (that the edge capacity) but also have another weights w (e) (that the unit cost flow), and has been seeking a maximum flow of the network value of F, then the minimum cost maximum flow problem, it is clear the following linear programming model can be used to describe:
Min ∑ w (e) f (e)
e ∈ E
Satisfy 0 ≤ f (e) ≤ c (e), for all e ∈ E
f + (v) = f-(v), for all v ∈ V
f + (x) = F (maximum flow constraints)
(Or f-(y) = F)
】 【Algorithm ideas
Solve the minimum cost maximum flow problem, there are two general ways. Way is to use a maximum flow algorithm to calculate the maximum flow, and then based on the cost side, check whether it is possible to balance the flow by adjusting the flow side, so that to reduce the total cost? As long as there is a possibility, on such adjustments. After adjusting for a new maximum flow.
Then, on the basis of the new flow to continue to check and adjust. This iteration continues until no adjustment may be, they will have the minimum cost maximum flow. The characteristics of this line of thought is to maintain the feasibility of the problem (always maintain maximum flow), to promote optimal. Solution to another and in front of the maximum flow algorithm, introduced a similar line of thought, first of all, given the general flow as the initial flow of zero. The cost of the flow to zero, of course, is the smallest cost. And then find a source to the Meeting Point by flow chain, but by the requirements of this chain must be a stream flow of all chain costs by a minimum. If we can find out by flow chain, the chain in the flow by increasing flow, a new flow. The flow will be treated as the initial flow, continue to search for links by increasing stream flow. This iteration continues, until found by flow chain, then the flow is the minimum cost maximum flow. Idea of the characteristics of this algorithm is to maintain the optimal solution of (each of the new fees are the smallest stream flow), but gradually close to the feasible solution (up to maximum flow is a feasible solution when).
As a result of the second algorithm and has introduced close to the maximum flow algorithm and the algorithm by finding the minimum cost flow chain, can be turned into a source to find the shortest path to the Meeting Point, so this algorithm here.
In this algorithm, in order to seek to increase the minimum cost flow chain, the current flow of each, accompanied by the need to establish a network flow by the flow network. For example, Figure 1 is a network G of minimum cost flow, next to parameters c (e), f (e), w (e), and Figure 2 is the network flow by the flow network G '. By the peak-flow network and the same as the original network. By the following principles in accordance with the establishment of the network edge flow: If G in the edge (u, v) is not enough traffic, that is, f (u, v) <e (u, v), then G 'in the building edge (u, v), Empowering w '(u, v) = w (u, v); edge of G if (u, v) has been the flow, that is, f (u, v)> 0, then G' in the building edge (v, u ), to empower the w '(v, u) =- w (u, v). The establishment of the network by streaming, you can seek in this network to the Meeting Point source shortest path, as decided by flow path, and then in the original network by flow in this path. Here, the use of maximum flow algorithm is still the principle of increasing flow, but the cost must be selected by the smallest chain by stream flow.
Calculation, there is a need to address the problem. This is the stream network by G 'the right to have a negative side, thus labeling law can not be directly applied to find x to y of the shortest path, using the right of other negative side computing network approach to the shortest path x to y to find the shortest path, will greatly reduce the computational efficiency. In order to still use the labeling method to calculate the shortest path, each flow set up by the network to achieve the shortest path, the network G can be the right of w (e) an amendment to do so by the stream to build the network will not be a negative right side, and guarantee the shortest path does not change. This modified method described below.
When the flow value is zero, the first built by the shortest path for flow network, the result of non-negative right side, of course, can be used to calculate labeling law. In order to increase flow network after the establishment of a negative time is not right side of the approach taken is to have stream G edge (f (e)> 0) the right to w (e) amendment to 0. To this end, each time a flow network obtained by the shortest path, the following computing G in the right side of the new w "(u, v):
w "(u, v) = L (u)-L (v) + w (u, v) (*)
Where L (u), L (v) - calculation of G 'of the shortest path x to y when u and v the value of the label. For the first time if the shortest path (u, v) is the flow path by the edge, then, according to the shortest path algorithm must have L (v) = L (u) + w '(u, v) = L (u) + w (u, v), substituting into (*) type must
w "(u, v) = 0.
If (u, v) rather than by the side of flow path, it must have:
L (v) ≤ L (u) + w (u, v),
Into the (*)-type, there w (u, v) ≥ 0.
Shows that the first amendment to w (e), against either side, there are w (e) ≥ 0, and a stream side (by side chain flow), there will be w (e) = 0. Calculated after each iteration, if f (u, v)> 0, by the need to establish the network flow (v, u) edge, edge weights w '(v, u) =- w (u, v) = 0, that is, the right will not be a negative side.
In addition, the calculation of each iteration with (*) fixes all the w (e), it is not difficult to prove that to each path x to y, its all the same to increase the path length L (x)-L (y). Therefore, x and y will not be the shortest path to w (e) the amendment changes.
】 【Calculation steps
1. On the network G = [V, E, C, W], given the initial value is zero flow streams.
2. Be accompanied by this stream flow network G '= [V', E ', W'].
G 'the vertex-G: V' = V.
If G in f (u, v) <c (u, v), then G 'in the building edge (u, v), w (u, v) = w (u, v).
If G in f (u, v)> 0, then G 'in the building edge (v, u), w' (v, u) =- w (u, v).
3. If G 'does not exist the path x to y, then G is the minimum cost flow of maximum flow,
To stop calculation; otherwise labeling method used to find x to y of the shortest path P.
4. According to P, the increased flow in G: each edge of P of (u, v), if G exists (u, v), then (u, v) by flow; if G exists (v, u), while (v, u) by flow. Increase (decrease) in flow should be on either side to ensure that there is c (e) ≥ f (e) ≥ 0.
5. According to the calculation of the shortest path at the time of peak value of the label L (v), press the G-type modification of all the edge weights w (e):
L (u)-L (v) + w (e) → w (e).
6. The new stream as the initial flow to 2.
=========
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屏南縣二號： ______ 求最小值: 首先確定 負 檢驗數(shù)最小的為主列,由b/aij 中最小的對應(yīng)的 aij 為主元素(aij > 0),這個原則是不變的. 二階段法的時候又出現(xiàn)大的為主元素 原因估計是 小的比值 對應(yīng)的 aij < 0.
屏南縣二號： ______ source對應(yīng)的是A source1-A1 Destination對應(yīng)的是B destination1-B1 shipment是運輸量,也就是從A到B的大小 最優(yōu)值是49,方案如上圖
屏南縣二號： ______ 因為是極大化指派問題,故選取最大的效率值10(丙B),用該值分別減去各效率值,得: 4 8 7 9 3 6 7 8 2 0 3 7 3 3 5 6 求該問題的極小化問題就是求原問題的極大化問題. (1)各行減去該行最小值,得 0 4 3 5 0 3 4 5 2 0 3 7 0 0 2 3 (2)各列減去...
屏南縣二號： ______ 考試科目代碼824考試科目名稱運籌學參考書目 《運籌學》錢頌迪,清華大學出版社(修訂版);《運籌學》(第二版)黨耀國等,科學出版社 2012年 考試大綱一、課程性質(zhì)《運籌學》是南京航空航天大學系統(tǒng)工程、管理科學與工程、工業(yè)工...
屏南縣二號： ______ 1規(guī)劃的目的是( ) 合理利用和調(diào)配人力、物力,以取得最大收益.合理利用和調(diào)配人力、物力,使得消耗的資源最少.合理利用和調(diào)配現(xiàn)有的人力、物力,消耗的資源最少,收益最大.合理利用和調(diào)配人力、物力,消耗的資源最少,收益最大...
