sma_netwrk_desig

sma_netwrk_desig - Network Design: Network Network Loading...

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Unformatted text preview: Network Design: Network Network Loading and Pup Matching Thomas L. Magnanti Thomas L. Magnanti Today’s Agenda Today’s Network design in general Network loading Solution approaches Polyhedral Polyhedral combinatorics combinatorics Heuristics Heuristics Pup matching Network Design: Basic Issue Network Total (Fixed) Cost on Each Arc Total (Fixed) Cost on Each Arc Commodity k: Commodity k: Origin O(k) Origin O(k) Destination D(k) Destination D(k) Flow req. rk Flow req. rk Link (i,j): Link (i,j): Fixed cost Fij Fixed cost Fij Flow cost cij Flow cost cij Possibly: Possibly: Capacity C per Capacity C per unit installed unit installed on any edge on any edge Network Design Applications Network Telecommunications systems Airline route maps Chip design Facility location Even TSP! Multicommodity Flow Model with Complex Costs minimize c( f ) subject to Nf k = b k for k = 1, 2,..., K f = ( f 1 , f 2 ,..., f K ) ≥ 0 r kk if i = O(k ) biikk = − r kk if i = D(k ) 0 otherwise k ij (possible flow bounds on f ) c( f ) = ∑ ( i , j )∈A cij ( fij ) separability c( f ) = ∑ ( i , j )∈A lij c ( fij ) proportionality fij = ∑ k fijk ( fij = ∑ k wk f ijk ) Basic Cost Structures cij(fij) cij (fij) fij fij Network Loading Cost Network LP Approximation cij(fij) LP Gap C 2C fij Other Cost Structures Other cij (fij) cij (fij) fij fij Integer Programming Model minimize ∑ k c k f k + ∑ ( i , j )∈E Fij yij subject to Nf k = b k k = 1, 2,..., K ∑ (f k k ij +f k ji ) ≤ Cy ij {i, j} ∈ E fijk ≤ r k yij {i, j} ∈ E , all k f = ( f 1 , f 2 ,..., f K ) ≥ 0 yij ≥ 0 and integer all {i, j} ∈ E (configuration constraints and y ) (1) (2) Cuts for Lower Bounds Cuts LP relaxations yield LP relaxations yield lower bounds lower bounds Addition of cuts can Addition of cuts can tighten bounds tighten bounds Cut away solutions to Cut away solutions to the LP relaxation but the LP relaxation but leave all feasible leave all feasible integer points integer points Network Loading Model Network min ∑ {i , j}∈E Fij yij subject to 1, i = O(k ) k k ∑∈A} fij − { j:(∑∈A} f ji = −1, i = D(k) all k ∈ K, i ∈ N j ,i ) { j:(i , j ) 0, otherwise ∑( k∈K ) k fijk + f ji ≤ Cyij all {i,j}∈ E fijk ≥ 0, yij ≥ 0 and integer Improved Modeling Improved (Cutset Inequalities) Fjj = 1 all edges, all cij = 0 F = 1 all edges, all cij = 0 C = 24 C = 24 Designs Designs 1 1 Demands Demands Y>3 1 Y>2 24 24 1 6 6 Y>1 2 30 30 4 12 12 1 1/4 1 1/4 1/4 4 1/4 Y>3 3 12 12 2 2 1 2 1 4 1/2 1/2 3 LP = LP = 3½ 3½ 1/2 1/2 1 1 1 1 Opt = 3 Opt = 5 5 General Cutset Inequality General DST = total demand (nodes in S to nodes in T) Set S of nodes Set T of nodes DST YST ≥ C Cutset Inequalities Aren’t Sufficient Cutset Capacity C = 1 Demand = 1 between all non-adjacent nodes non-adjacent Loading 1 unit on all visible edges satisfies cutset inequalities, but not feasible Pup Matching Pup Example Example optimal solution is 9 Pup Matching Pup IInstance: A directed network G = (N,A), a nstance: A directed network G = (N,A), a set of K pairs of elements from N, and a set of K pairs of elements from N, and a cost function c: A→R+. cost function c: A→R+. Problem: Find the minimum cost loading Problem: Find the minimum cost loading of G permitting unit flow from the first to of G permitting unit flow from the first to the second node of each of the K pairs the second node of each of the K pairs such that 1 unit or 2 units together can such that 1 unit or 2 units together can traverse an arc for each unit of loading. traverse an arc for each unit of loading. One unit of loading on a ∈A costs c(a). One unit of loading on a ∈A costs c(a). Example on City Blocks Example Solution with cost 196 Solution Several days computation can prove only that the objective is at least 184 (LP lower bound = 182). Can we do better? Heuristics for Upper Bounds Heuristics Matching Heuristic Matching Heuristic permits each pup to be paired with at most permits each pup to be paired with at most one other pup one other pup solved with a weighted matching routine solved with a weighted matching routine Shortest Path Heuristics Shortest Path Heuristics three variations three variations Each heuristic provides a 2-approximation Each heuristic provides a 2-approximation to the NLP formulation to the NLP formulation Odd Flow Inequality If the flow on an arc is odd, one unit of loaded capacity will be unused If the net demand of a node is odd, the total inflow or total outflow is odd 1 1 k + y ji ) − ∑∑ ( fijk + f ji ) ≥ ij 2 k j 2 ∑( y j Odd Flows on the City Block Odd Each of the 56 nodes Each of the 56 nodes must be incident to at must be incident to at least one arc with least one arc with 1 unit of spare capacity unit of spare capacity 2 solution requires at solution requires at least least 56 1 ⋅ = 14 cabs’ cabs’ 2 2 worth of empty worth of empty capacity capacity LP relaxation of 182 LP relaxation of 182 gives lower bound on gives lower bound on required used required used capacity capacity 196 is optimal 196 is optimal Gap Reductions on Gap City Block Problems Trials Using Realistic Data Trials Node set given in (latitude, longitude) Node set given in (latitude, longitude) format based on a real logistics network format based on a real logistics network Defined problems by choosing a subset of Defined problems by choosing a subset of nodes, calculating arc lengths, and nodes, calculating arc lengths, and randomly selecting O-D pairs randomly selecting O-D pairs 30 problems, about half single origin 30 problems, about half single origin Complete graphs, 12-25 nodes, 6-50 pups Complete graphs, 12-25 nodes, 6-50 pups Results Results Branch and Bound limited to 2 hours CPU Branch and Bound limited to 2 hours CPU time and a 220M tree time and a 220M tree With all 3 cut families, 67% were solved to With all 3 cut families, 67% were solved to optimality with an average gap reduction optimality with an average gap reduction of 18.8% to 6.4% of 18.8% to 6.4% Without odd flow cuts, 30% were solved, Without odd flow cuts, 30% were solved, and the gap was reduced to 7.8% on and the gap was reduced to 7.8% on average average With no cuts, 17% were solved With no cuts, 17% were solved Among solved problems, average Among solved problems, average heuristic error was 1.3% heuristic error was 1.3% Conclusions and Extensions Conclusions Extensions apply to compartmentalized Extensions apply to compartmentalized problems problems Cuts seem critical to provably solving the Cuts seem critical to provably solving the PM problem PM problem Odd flow inequalities define what seems Odd flow inequalities define what seems an important set of facets an important set of facets generalize to arbitrary capacity generalize to arbitrary capacity can generalize to several facilities? can generalize to several facilities? Are there other cuts based on even-odd Are there other cuts based on even-odd type arguments? type arguments? Many, Many Network Design Variants Many, Network loading for compartmentalized capacity airline capacity planning, tanker trucks airline capacity planning, tanker trucks Network survivability Network restoration Hierarchical designs … Today’s Lessons Today’s Network design arises in numerous applications Problem is a large-scale integer program Introduction to cutting planes (polyhedral combinatorics) Cutting planes valuable in tightening formulations and in problem solving Module on Large-Scale Module Integer Programming & Combinatorial Optimization Three Lectures Traveling salesman problem Facility location Network design Games/Challenges Applications, Models, and Solution Methods ...
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