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15.081J/6.251J Introduction to Mathematical Programming Lecture 16: Network Flows, I1 Networks • Electrical & Power Networks • Road Networks • Airline Routes • Internet Backbone • Printed Circuit Board • Social Networks 2 Common Thrust Slide 1 Slide 2 Move some entity (electricity, a consumer product, a pers on, a vehicle, a mes-sage, . . . ) from one point to another in the underlying network , as efficiently as possible. 1. Learn how to model application settings as network flow problems. 2. Study ways to solve the resulting models. 3 Shortest Path 3.1 Description Slide 3 • Identify a shortest path from a given source node to a given sink node. • Finding a path of minimum length. • Finding a path taking minimum time. • Finding a path of maximum relia bility. 4 Maximum Flow 4.1 Description Slide 4 • Determine the maximum flow that can be sent from a given source node to a sink node in a capacitated network. • Determining maximum steady-state flow of – petroleum pr oducts in a pipeline network – cars in a r oad network – messages in a telecommunication network – electricity in an electric al networ k 15 Min-Cost Flow 5.1 Description Slide 5 • Determine a least cost shipment of a commodity through a network in order to sa tisfy demands at certain nodes from available supplies at other nodes. Arcs have capacities and cost associated with them • Distribution of products • Flow of items in a production line • Routing of cars through stree t networks • Routing of telephone calls 5.2 In LOP Form Slide 6 • Network G = (N, A). • Arc costs c : A → R. • Arc capacities u : A → N . • No de balances b : N → R. min � cij xij (i,j)∈A s.t. � xij − � xji = bi for all i ∈ N j:(i,j)∈Aj:(j,i)∈A xij ≤ uij for all (i, j) ∈ A xij ≥ 0 for all (i, j) ∈ A 6 Outline Slide 7 • Shortest path applications • Maximum Flow applications • Minimum cost flow applications 7 Shortest Path 7.1 Interword Spacing in LATEX Slide 8 The spacing between words and characters is normally set automatically by LATEX. Interword spacing within one line is uniform. LATEX also attempts t o keep the word spacing for different lines as nearly the same as possible. 2The spacing between words and characters is normally set auto- matically by LATEX. Interword spacing within one line is uniform. LATEX also attempts to keep the word spacing for different lines as nearly the same as possible. 7.2 Interword Spacing in LATEX (2) Slide 9 • The p aragraph consists of n words, indexed by 1, 2, . . . , n. • cij is the attractiveness of a line if it begins with i and ends with j − 1. • (LATEX uses a formula t o compute the value of each cij .) For instance, c12 = −10, 000 c13 = −1, 000 c14 = 100 c1,37 = −100, 000 . . . 7.3 Interword Spacing in LATEX (3) Slide 10 • The problem of decomposing a paragraph into several lines of text to max- imize tota l attractiveness c an be for mulated as a shortest path problem. • Nodes? Arcs? Costs? 7.4 Project Management Slide 11 • A project consists of a set of jobs and a s e t of precedence relations • Given a set A of job pairs (i, j) indicating that job i cannot start before job j is co mpleted. • ci duration of job i • Find the least possible duration o f the project 7.4.1 Formulation Slide 12 • Introduce two artificial jobs s and t, of zer o duration, that signify the beginning and the completion of the project • Add (s, i) and (i, t) to A • pi time that job i begins • (i, j) ∈ A: pj ≥ pi + ci • Project duration: pt − ps 3Slide 13 • min pt − ps s.t pj − pi ≥ ci, ∀ (i, j) ∈ A. • Dual max � cifij (i,j)∈A s.t. � fji − � fij = bi {j|(j,i)∈A}{j|(i,j)∈A} fij ≥ 0 Slide 14 • bs = − 1, bt = 1, and bi = 0 for i �= s, t. • Shortest path problem, where each precedence relation (i, j) ∈ A corre-sp onds to an arc with cos t of −ci. Slide 15 Activity s A B C D E t 0 0 S Immediate Predecessor S S A,B A C,D E 14 14 35 A D B C Time(ci) 0 14 3 5 7 10 0 Slide 16 7 10 T E Slide 17 7.5 DNA Sequencing • Given two sequences of letters, say B = b1 · · · bp and D = d1 · · · dq • How similar are the two sequences? • What is the min cost of transforming B to D? 47.5.1 Transformation costs Slide 18 • α = cost of inserting a letter in B • β = cost of deleting a letter from B • g(bi, dj ) = cost of mutating a letter bi into dj 7.5.2 Transformation steps Slide 19 1. Add or delete letters fro m B so as to make |B ′ | = |D|. 2. Align B ′ and D 3. Mutate letters of B ′ so that B ′′ = D. 7.5.3 Algorithm Slide 20 • f(b1 · · · bp, d1 · · · dq ): the min cost of transforming B into D by the three steps above. We obtain this cost by a recursive way. • f(∅ · · · ∅, d1 · · · dj ) = jα, j = 1, ..., q f(b1 · · · bi, ∅ · · · ∅) = iβ, i = 1, ..., p. Slide 21 Substitution B ′ = b1 · · · bi D = d1 · · · dj f(b1 · · · bi, d1 · · · dj ) = f (b1 · · · bi−1, d1 · · · dj−1) + g(bi, dj ) Slide 22 • Addition of dj B ′ = b1 · · · bi · · · ∅ D = d1 · · · · · · dj f(b1 · · · bi, d1 · · · dj ) = f(b1 · · · bi, d1 · · · dj−1) + α. • Deletion of bi: f(b1 · · · bi, d1 · · · dj ) = f(b1 · · · bi−1, d1 · · · dj ) + β Slide 23 Recursion f(b1 · · · bi, d1 · · · dj ) = min{f(b1 · · · bi−1, d1 · · · dj−1) + g(bi, dj ), f(b1 · · · bi, d1 · · · dj−1) + α, f(b1 · · · bi−1, d1 · · · dj ) + β} Slide 24 The shortest path from 00 to 32 5� 8 Maximum Flow 8.1 The tournament problem Slide 25 • Each of n teams plays against every other …


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