Shortest Path 吳仲軒(9526709) 前言 • 本系列章節主要利用圖解法探討最小花費之網路流通問題 • 同時轉換其他模型使其成為網路流通模型簡介 • Let G=(N,A) be a directed network defined by a set N of n node and A of m directed arcs • Each arc (i,j)€A has associated cost Cij per unit flow on that arc Min ∑Cij Xij (i,j)€A ST ∑ Xij – {J:(i,j)€A} ∑Xji =b(i)……..(1b) {I:(j,i)€A} (Mass balance constraints) Lij ≤Xij≤Uij………………..(1C) (The flow Bound constraint) Shortest Path problem • We wish to minimum cost (length) form the starting node to the ending Applications-1 System of difference constraints Example • X(ik)-X(jk)≦Xb(k) for k=1………….m • X(3)-X(4) ≦5 •X(4)-X(1) ≦-10 •X(1)-X(3) ≦8 •X(2)-X(1) ≦-11 •X(3)-X(2) ≦-2 • 1-最短路徑d(i)從Source Node 至任何 Node i 皆滿足Optomality conditions d(j)d(i) ≦Cij • 2.最短路徑只在沒有Negative Cycle時成立 • Fig 1 • • • • Because 1->2->3->1=-1 It’s not a feasible solution Using a label correcting algorithm Fig 2 Telephone operator Scheduling 23 Min∑ Yi I=0 S.T Yi-7+Yi-6…….+Yi ≥ b(i) (For i=8 to 23) Y17+i+….+Y23+….Yi ≥ b(i) (For i=0 to 7) Yi ≥0 • Shortest path has a special structure. • The associated constraint only 0’s and 1’s and 0’s or 1’s in each row consecutively ≥ • X(i)-X(i-8) ≥b(i) • X(23)-X(16+i)+X(i)=P-X(16-i)+X(i) ≥b(i) • X(i)-X(i-1) ≥0 Application 3 Production planning problems • 問題概敘 • See p11 Fig(a) Application 4 Approximating piecewise linear functions • 嘗試將一線圖簡化如下圖 3 簡化時產生的Cost 可轉換為以下的公式 P β∑[F1(Xk)-F2(Xk)]^2 K=1 • P Cij= α + β∑[F1(Xk)-F2(Xk)]^2 K=1