Modeling Network Optimization Problems

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1 Modeling Network Optimization Problems John E. Mitchell Mathematical Models of Operations Research MATP4700 / ISYE4770 October 19, 01 Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

2 Outline 1 Single commodity problems Multicommodity problems Mitchell (MATP4700) Network Optimization October 19, 01 / 1

3 Outline 1 Single commodity problems Multicommodity problems Mitchell (MATP4700) Network Optimization October 19, 01 / 1

4 Introduction Flow from supply nodes to demand nodes. Have cost for using an edge could be proportional to the flow, or could be a fixed charge Each edge may have a capacity. Want to meet demand at minimum cost, subject to meeting capacity constraints. Assumption for today: material can flow in either direction along an edge. Mitchell (MATP4700) Network Optimization October 19, 01 4 / 1

5 Alternative paths a c 7 g 6 p b l cost of edge per unit flow f d 6 4 h 4 j 4 k node a supplies 1 unit, node b demands 1 unit Mitchell (MATP4700) Network Optimization October 19, 01 / 1

6 Alternative paths a c 7 g 6 p b l cost of edge per unit flow f d 6 4 h 4 j 4 k cost of sending 1 unit along path adhjb is = 16 Mitchell (MATP4700) Network Optimization October 19, 01 / 1

7 Alternative paths a c 7 g 6 p b l cost of edge per unit flow f d 6 4 h 4 j 4 k cost of sending 1 unit along path acpb is = 11 Mitchell (MATP4700) Network Optimization October 19, 01 / 1

8 An example with multiple destinations a c supply 1 b d 7 g demand 4 4 h p j k 4 demand 7 l cost of edge per unit flow f demand one supply node, three demand nodes Mitchell (MATP4700) Network Optimization October 19, 01 6 / 1

9 An example with multiple destinations a c supply 1 b 14 d 7 g demand 4 4 h p 6 4 l 10 j k demand 7 cost of edge per unit flow f demand cost is Mitchell (MATP4700) Network Optimization October 19, 01 6 / 1

10 An example with multiple destinations a c 6 supply 1 b d 7 6 g demand 4 4 h 4 4 p l 10 j k demand 7 cost of edge per unit flow f demand cost is Mitchell (MATP4700) Network Optimization October 19, 01 6 / 1

11 Optimization model One possible modeling choice is to have different variables to reflect different directions of flow. Use nonnegative variables x ij to denote the flow along edge (i, j) from i towards vertex j. Also have a nonnegative variable x ji if the flow is in the opposite direction, from j to i. Capacity constraints Let u ij be the capacity of the edge between i and j. Need: x ij + x ji u ij. Mitchell (MATP4700) Network Optimization October 19, 01 7 / 1

12 Flow conservation Demand nodes Let node i have demand b i > 0. Need (flow in) - (flow out) equal to demand. Symbolically: ji E Supply nodes Let node i have supply b i < 0. Need: ji E ( xji x ij ) ik E (x ik x ki ) = b i. ( xji x ij ) ik E (x ik x ki ) b i. Transshipment nodes Let node i have net demand b i = 0. Need: ji E ( xji x ij ) ik E (x ik x ki ) = 0. Mitchell (MATP4700) Network Optimization October 19, 01 / 1

13 Flow conservation at node b a c supply 1 b d 7 g demand 4 4 h p j k 4 demand 7 l cost of edge per unit flow f demand Edges (a, b), (b, j), (b, l), (b, p) E. (x ab x ba ) (x bj x jb ) (x bl x lb ) (x bp x pb ) = 4 Mitchell (MATP4700) Network Optimization October 19, 01 9 / 1

14 Flow conservation at node b a c supply 1 b 14 d 7 g demand 4 4 h p 6 4 l 10 j k demand 7 Edges (a, b), (b, j), (b, l), (b, p) E. (x ab x ba ) (x bj x jb ) (x bl x lb ) (x bp x pb ) = 4 x jb = 4 cost of edge per unit flow f demand Mitchell (MATP4700) Network Optimization October 19, 01 9 / 1

15 Flow conservation at node b a c 6 supply 1 b d 7 6 g demand 4 4 h 4 4 p l 10 j k demand 7 Edges (a, b), (b, j), (b, l), (b, p) E. (x ab x ba ) (x bj x jb ) (x bl x lb ) (x bp x pb ) = 4 x bj =, x pb = 6 cost of edge per unit flow f demand Mitchell (MATP4700) Network Optimization October 19, 01 9 / 1

16 Objective function Each edge (i, j) has a cost c ij for each unit of flow on the edge. Objective function is to minimize ( ) c ij xij + x ji (i,j) E Could also incorporate a fixed charge for using the edge, but that would require the use of binary variables. Mitchell (MATP4700) Network Optimization October 19, / 1

17 Modeling with AMPL Given a set of EDGES, one modeling choice is to use two sets of variables, one indicating forward flow on an edge, and one indicating reverse flow: var flow_forward{edges} >= 0; var flow_reverse{edges} >= 0; The constraints are then for capacity: subject to capacity{(i,j) in EDGES}: flow_forward[i,j] + flow_reverse[i,j] <= u[i,j];... Mitchell (MATP4700) Network Optimization October 19, / 1

18 Modeling with AMPL, continued... and for flow conservation for a demand node: subject to meet_demand{i in DEMAND_NODES}: sum{(j,i) in EDGES} ( flow_forward[j,i] - flow_reverse[j,i] ) - sum{(i,k) in EDGES} ( flow_forward[i,k] - flow_reverse[i,k] ) = b[i]; Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

19 Modeling with AMPL, continued AMPL has commands for operations on sets. For example, to get the set of transshipment nodes in the power network, you can use the syntax {u in POWER_NODES diff (DEMAND_POWER_NODES union SUPPLY_POWER_NODES)}: Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

20 Multicommodity problems Outline 1 Single commodity problems Multicommodity problems Mitchell (MATP4700) Network Optimization October 19, / 1

21 Multicommodity problems Multicommodity network flow problems Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

22 Multicommodity problems Multicommodity network flow problems A A Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

23 Multicommodity problems Multicommodity network flow problems B A A B Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

24 Multicommodity problems Multicommodity network flow problems C B A congested A B C Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

25 Multicommodity problems Multicommodity network flow problems C B A A B C Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

26 Multicommodity problems Multicommodity problems Have K commodities, indexed from k = 1,..., K. Each commodity has its own flow conservation constraints. The capacity constraints apply to the sum of flows over all the commodities. Let x k ij denote the flow of commodity k along edge (i, j), in the direction from i to j. Let u ij denote the capacity of edge (i, j). Get constraint K k=1 ( x k ij ) + xji k u ij for each edge (i, j). Mitchell (MATP4700) Network Optimization October 19, / 1

27 Multicommodity problems Modeling multicommodity problems with AMPL Each pair of nodes corresponds to a commodity. If one commodity needs to be shipped from k to l then it is necessary to define variables for this commodity and for each edge. So we need variables with four indices: t_flow_forward [k,l,i,j] >= 0; t_flow_reverse [k,l,i,j] >= 0; The capacity constraint has to link together all the commodities: subject to telecom_capacity_constraint {(i,j) in TELECOM_EDGES}: sum{(k,l) in DEMAND_PAIRS_TELECOM} ( t_flow_forward[k,l,i,j] + t_flow_reverse[k,l,i,j] ) <= capacity_telecom[i,j]; Mitchell (MATP4700) Network Optimization October 19, / 1

28 Multicommodity problems Displaying selected variables with AMPL A multicommodity problem has a large number of flow variables. If you just want to display the flow variables for the commodity routed from node t to node t11, you can use the following command: display {(i,j) in TELECOM_EDGES} t_flow_forward[ t, t11,i,j]; Mitchell (MATP4700) Network Optimization October 19, 01 1 / 1

Strong formulation for fixed charge network flow problem

Strong formulation for fixed charge network flow problem Consider the network topology given by the graph G = (V, E) in the figure, where V = {a, b, c, d}. Each node represents a router of the network,