Lecture 8 Network Flow Models

Transcription

1 Lecture 8 Andrew Nunekpeku / Charles Jackson Fall 2011

2 Outline 1

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4 Assignment 4 When we ask for the solution of an LP, we typically mean tell me the values of the decision variables or what is the plan? more than we are asking how much money will the plan make for me? or the objective function at the solution, although you could give both. Be prepared to perform some simple calculations using shadow prices or reduced costs as compared to the appropriate limits (range) to answer what-if questions, without necessarily re-solving the LP. You cannot reject out of hand a proposed sale of resources to an outsider that exceeds the lower limit on the sensitivity report on that resource. You may have to re-run the Solver to check the new optimum solution.

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6 Generalized Network

7 Terminology

8 Terminology Network: An arrangement of paths connected at various points, through which items move.

9 Terminology Network: An arrangement of paths connected at various points, through which items move. Nodes: Junction points represented as circles on the network diagram. E.g. street intersections, cities, rail/road terminals.

10 Terminology Network: An arrangement of paths connected at various points, through which items move. Nodes: Junction points represented as circles on the network diagram. E.g. street intersections, cities, rail/road terminals. Arcs: Branches, paths or lines connecting the nodes. E.g. roads, air routes, pipelines, cables, etc.

11 Terminology Network: An arrangement of paths connected at various points, through which items move. Nodes: Junction points represented as circles on the network diagram. E.g. street intersections, cities, rail/road terminals. Arcs: Branches, paths or lines connecting the nodes. E.g. roads, air routes, pipelines, cables, etc. upper limit or capacity on the amount of material which can flow down the arc.

12 Terminology Network: An arrangement of paths connected at various points, through which items move. Nodes: Junction points represented as circles on the network diagram. E.g. street intersections, cities, rail/road terminals. Arcs: Branches, paths or lines connecting the nodes. E.g. roads, air routes, pipelines, cables, etc. upper limit or capacity on the amount of material which can flow down the arc. a cost per unit of material sent down the arc.

13 Terminology

14 Terminology Directed Arcs: Arcs with arrows indicating the direction of flow from source to sink.

15 Terminology Directed Arcs: Arcs with arrows indicating the direction of flow from source to sink. Sources: Supply or sending nodes, each with an upper limit on amount of material it can supply.

16 Terminology Directed Arcs: Arcs with arrows indicating the direction of flow from source to sink. Sources: Supply or sending nodes, each with an upper limit on amount of material it can supply. Sinks: Demand or receiving nodes, each with an associated number indicating the amount of material required. Intermediate (Transshipment Nodes): Between sources and sinks. Allow flow through them to other intermediate nodes or sinks.

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18 Shortest Route To determine the shortest or least costly route or path through a network between a starting node and several destination nodes. A shipping company transports oranges by six trucks from a city 1 to six other cities. The different routes between city 1 and the destination cities and the length of time, in hours, required by a truck to travel each route is shown in the figure below. The shipping manager wants to determine the best routes (in terms of minimum travel time) for the trucks to take to reach their destinations.

19 Example

20 Rules Select a node with the shortest direct route from the origin. Establish a permanent set with the origin node and the node that was selected in step 1. Determine all nodes directly connected to the permanent set of nodes. Select the node with the shortest route (branch) from the group of nodes directly connected to the permanent set nodes. Repeat steps 3 and 4 until all nodes have joined the permanent set.

21 The Solution

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23 Maximal Flow Objective: To maximize the amount of flow of items from an origin to a destination through a network given the limited flow capacities of the branches. Applications include:

24 Maximal Flow Objective: To maximize the amount of flow of items from an origin to a destination through a network given the limited flow capacities of the branches. Applications include: Flow of water, oil, gas through a network of pipelines.

25 Maximal Flow Objective: To maximize the amount of flow of items from an origin to a destination through a network given the limited flow capacities of the branches. Applications include: Flow of water, oil, gas through a network of pipelines. Flow of forms through a paper processing system.

26 Maximal Flow Objective: To maximize the amount of flow of items from an origin to a destination through a network given the limited flow capacities of the branches. Applications include: Flow of water, oil, gas through a network of pipelines. Flow of forms through a paper processing system. Flow of traffic through a road network.

27 Maximal Flow Objective: To maximize the amount of flow of items from an origin to a destination through a network given the limited flow capacities of the branches. Applications include: Flow of water, oil, gas through a network of pipelines. Flow of forms through a paper processing system. Flow of traffic through a road network.

28 Maximal Flow Objective: To maximize the amount of flow of items from an origin to a destination through a network given the limited flow capacities of the branches. Applications include: Flow of water, oil, gas through a network of pipelines. Flow of forms through a paper processing system. Flow of traffic through a road network. Flow of products through a production line.

29 Rules Arbitrarily select a path in the network from origin to destination. Adjust the capacities at each node by subtracting the maximal flow for the path selected in step 1. Add the maximal flow along the path in the opposite direction at each node. Repeat steps 1, 2, and 3 until there are no more paths with available flow capacity.

30 Example A company ships tractor parts from city 1 to city 7 by railroad. However the contract limits the number of railroad cars the company can secure on each branch during a week. Given these limiting conditions, the company want to know the maximum number of railroad cars containing tractor parts that can be shipped from 1 to 7 during a week.

31 Railway Network

32 Solution

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34 Minimal Spanning Tree Objective: To connect all the nodes in the network so that the total branch lengths are minimized. Resulting network spans (connects) all the points in the network at a minimum total distance. Select any starting node. Select the node closest to the starting node to join the spanning tree. Select the closest node not presently in the spanning tree. Repeat step 3 until all nodes have joined the spanning tree.

35 Example The Electricity Company of Ghana (E.C.G.) is to connect seven districts to the national grid. E.C.G. wants to minimize total length of cable that must be installed. The possible paths available (by consent of the district assembly) and the feet of cable (in thousands of feet) required for each path are shown on next slide.

36 Example

37 Solution 72,000 feet of electrical cable will be required.

38 Reading Taylor Chapter 7