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Network Optimization

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  1. Network Optimization Network optimization models: Special cases of linear programming models Important to identify problems that can be modeled as networks because: (1) network representations make optimization models easier to visualize and explain (2) very efficient algorithms are available

  2. Example of (Distribution) Network

  3. Terminology • Nodes and arcs • Arc flow • Upper and lower bounds • Cost • Gains and losses • External flow • Optimal flow

  4. Network Flow Problems

  5. Transportation Problem We wish to ship goods (a single commodity) from m sources to n destinations at minimum cost. Warehouse i has si units available i = 1, . . . ,m and destination j has a demand of dj, j = 1, . . . ,n. Goal - Ship the goods from sources to destinations at minimum cost. Plants Supply Markets Demand Example San Francisco New York 350 325 Los Angeles Chicago 600 300 Austin 275 From/To NY Chi Aus Unit Shipping Costs SF 2.5 1.7 1.8 LA -- 1.8 1.4

  6. Total supply = 950, total demand = 900 Transportation problem is defined on a bipartite network Arcs only go from supply nodes to destination nodes To handle excess supply create a dummy destination with a demand of 50. The min-cost flow network for this transportation problem is given by: NY [-325] (2.5) (1.7) SF [350] (1.8) CHI [-300] (0) (M) (1.8) AUS [-275] (1.4) [600] LA (0) DUM [-50]

  7. · Costs on arcs to dummydestination = 0 (In some settings it would be necessary to include a nonzero warehousing cost.) ® · The objective coefficient on the LA NY arc is M. This denotes a large value and effectively prohibits use of this arc (could eliminate arc). · We are assured of integer solutions because technological matrix A is totally unimodular. (important in some applications)

  8. The LP formulation of the transportation problem with m sources and n destinations is given by: m n å å Min c x ij ij j=1 i=1 n å s.t. x = s i = 1,…,m ij i j=1 m å x = d j = 1,…,n ij j i=1 x ³ 0 i = 1,…,m; j = 1,…,n ij

  9. Shortest Path Problem • Given a network with “distances” on the arcs our goal is to find the shortest path from the origin to the destination. • These distances might be length, time, cost, etc, and the values can be positive or negative. (A negative cij can arise if we earn revenue by traversing an arc.) • The shortest path problem may be formulated as a special case of the pure min-cost flow problem.

  10. Example (cij) cost or length (2) 2 4 (3) (4) (1) (1) (2) [-1] 6 [1] 1 (6) (7) (2) 3 5 • We wish to find the shortest path from node 1 to node 6. • To do so we place one unit of supply at node 1 and push it through the network to node 6 where there is one unit of demand. • All other nodes in the network have external flows of zero.

  11. Network Notation A = set of Arcs, N = set of nodes Forward Star for node i: FS(i) = { (i,j) : (i,j) Î A } Reverse Star for node i: RS(i) = { (j,i) : (j,i) Î A } RS(i) FS(i) i i

  12. In general, if node s is the source node and node t is the termination node then the shortest path problem may be written as follows. å x c Min ij ij (i,j)ÎA { 1, i = s –1, i = t 0, i Î N \ {s,t} å å x x s .t. = - ij ji (i,j)ÎFS(i) (j,i)ÎRS(i) xij³ 0, " (i,j)ÎA

  13. Maximum Flow Problem • In the maximum flow problem our goal is to send the largest amount of flow possible from a specified source node to a specified destination node subject to arc capacities. • This is a pure network flow problem (i.e., gij = 1) in which all the (real) arc costs are zero (cij = 0) and at least some of the arc capacities are finite. Example (2) 2 4 (3) (4) (uij) arc capacities (1) (1) 6 (2) 1 (7) (6) (2) 3 5 1

  14. 6 Goal for Max Flow Problem Send as much flow as possible from node 1 to node 6 Solution [2](2) [2](3) [3](4) 2 4 [xij] (uij) flow capacities [1](1) 1 [3](7) [2](6) 3 5 [2](2) Maximum flow = 5

  15. Cut: A partition of the nodes into two sets S and T. The origin node must be in S and the destination node must be in T. Examples of cuts in the network above are: {2,3,4,5,6} S1 = {1} T1 = {4,5,6} S2 T2 = {1,2,3} = S3 {2,4,6} T3 = {1,3,5} = The value of a cut V(S,T) is the sum of all the arc capacities that have their tails in S and their heads in T. V(S2,T2) = 5 V(S3,T3) = 12 V(S1,T1) = 10

  16. Max-Flow Min-Cut Theorem The value of the maximum flow is equal to the value of the minimum cut. • In our problem, S = {1,2,3} / T = {4,5,6} is a minimum cut. • The arcs that go from S to T are (2,4), (2,5) and (3,5). • Note that the flow on each of these arcs is at its capacity. As such, they may be viewed as the bottlenecks of the system.

  17. Max Flow Problem Formulation • There are several different linear programming formulations. • We will use one based on the idea of a “circulation”. • We suppose an artificial return arc from the destination to the origin with uts = + ¥ and cts = 1. • External flows (supplies and demands) are zero at all nodes. s t

  18. Maximum Flow Model Max xts å xij å xji = 0, "iÎN s.t. - (i,j)ÎFS(i) (j,i)ÎRS(i) "(i,j)ÎA 0 £xij£ uij Identify minimum cut from sensitivity report: • If the reduced cost for xij has value 1 then arc (i,j) has its tail (i) in S and its head (j) in T. • Reduced costs are the shadow prices on the simple bound constraint xij£ uij. • Value of another unit of capacity is 1 or 0 depending on whether or not the arc is part of the bottleneck Note that the sum of the arc capacities with reduced costs of 1 equals the max flow value.

  19. Sensitivity Report for Max Flow Problem Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $E$9 Arc1 Flow 3 0 0 1E+30 0 $E$10 Arc2 Flow 2 0 0 0 1 $E$11 Arc3 Flow 0 0 0 0 1E+30 $E$12 Arc4 Flow 2 1 0 1E+30 1 $E$13 Arc5 Flow 1 1 0 1E+30 1 $E$14 Arc6 Flow 2 1 0 1E+30 1 $E$15 Arc7 Flow 0 0 0 0 1E+30 $E$16 Arc8 Flow 2 0 0 0 1 $E$17 Arc9 Flow 3 0 0 1E+30 0 $E$18 Arc10 Flow 5 0 1 1E+30 1 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $N$9 Node1 Balance 0 0 0 0 3 $N$10 Node2 Balance 0 0 0 1E+30 0 $N$11 Node3 Balance 0 0 0 0 3 $N$12 Node4 Balance 0 1 0 0 2 $N$13 Node5 Balance 0 1 0 0 3 $N$14 Node6 Balance 0 1 0 0 3

  20. Minimum Cost Flow Problem Example: Distribution problem • Warehouses store a particular commodity in Phoenix, Austin and Gainesville. • Customers - Chicago, LA, Dallas, Atlanta, & New York Supply [ si ] at each warehouse i Demand [ dj] of each customer j • Shipping links depicted by arcs, flow on each arc is limited to 200 units. • Dallas and Atlanta - transshipment hubs • Per unit transportation cost (cij) for each arc • Problem - determine optimal shipping plan that minimizes transportation costs

  21. Distribution Problem [supply / demand] arc lower bounds = 0 arc upper bounds = 200 (shipping cost) [–200] [700] (6) NY 6 CHIC 2 [–250] PHOE 1 (4) (6) (7) (4) (3) (3) (5) (2) (5) LA 3 DAL 4 [–150] ATL 5 [–200] (7) (2) [–300] (4) (2) (7) (6) (5) GAINS 8 [200] AUS 7 [200]

  22. Solution to Distribution Problem [supply / demand] (flow) [-200] [-250] (200) NY CHIC [700] (50) PHOE (100) (200) (200) [-150] (200) LA ATL DAL [-300] (50) [-200] (200) [200] GAINS AUS [200]

  23. This network flow problem is based on: · Conservation of flow at nodes. At each node flow in = flow out. At supply nodes there is an external inflow (positive) At demand nodes there is an external outflow (negative). · Flows on arcs must obey the arc’s bounds. lower bound & upper bound (capacity) · Each arc has a per unit cost & the goal is to minimize total cost.

  24. [external flow] (cost) lower = 0, upper = 200 Distribution Network [-200] [-250] (6) 2 6 [700] 1 (4) (6) (7) (4) (5) (3) (3) (7) (2) (5) [-150] [-200] 4 3 5 (2) [-300] (4) (6) (2) (5) (7) [200] 8 7 [200]

  25. Linear Program Model for Distribution Problem Minimize z = 6x12 + 3x13 + 3x14 + 7x15 + … + 7x86 Subject to conservation of flow constraints at each node: Node 1: x12 + x13 + x14 + x15 = 700 Node 2: –x12 – x62 – x52 = –200 Node 3: –x13 – x43 – x73 = –200 Node 4: x42 + x43 + x45 + x46 – x14 – x54 – x74 = –300 . . . . . . Node 8: x84 + x85 + x86 = 200 Bounds: 0 ≤ xij ≤ 200 for all (i,j)A