Network Optimization Models Chapter 10 Network Optimization Models

10.1 Prototype Example The road system for Seervada Park Location O: park entrance Location T: a scenic wonder Trams transport sightseers from park entrance to location T and back

Prototype Example Park management faces three problems Determine the route with the smallest total distance A shortest-path problem Determine where telephone lines should be laid A minimum spanning tree problem Determine how to route tram to maximize number of trips during peak season A maximum flow problem

10.2 The Terminology of Networks Network consists of a set of points and a set of lines connecting points Node: point (vertex) in the network Lines: links, arcs, edges, or branches Labeled by naming the node at each end From node precedes the to node Have a flow of some type through them Directed arcs have unidirectional flow Undirected arcs (links) allow bidirectional flow

The Terminology of Networks Directed network Network has only directed arcs Undirected network Network has only undirected arcs Path between two nodes A sequence of distinct arcs connecting the nodes Directed path from node i to node j Sequence of connecting arcs toward node j

The Terminology of Networks Undirected path from node i to node j Sequence of connecting arcs whose direction can be with toward or away from node j Connected network Every pair of nodes in the network has at least one undirected path between them Tree (spanning tree) Connected network with no undirected cycles

The Terminology of Networks

The Terminology of Networks

The Terminology of Networks Arc capacity Maximum amount of flow that can be carried on a directed arc Supply node Flow out exceeds flow in Demand node Flow in exceeds flow out Transshipment node Flow in equals flow out

10.3 The Shortest-Path Problem Consider an undirected, connected network Contains origin and destination nodes Each link has a nonnegative distance The problem Find the shortest path from origin to destination

The Shortest-Path Problem Algorithm Objective of nth iteration: find the nth nearest node to the origin Repeat for n = 1, 2… until destination is reached Input for nth iteration: n − 1 nearest nodes to the origin, including shortest path and distance from the origin These are called the solved nodes

The Shortest-Path Problem Algorithm (cont’d.) Candidates for nth nearest node: unsolved node with shortest connecting link to the solved node Calculation of nth nearest node For each solved node and its candidate, add the distance between them and the distance of the shortest path from the origin to this solved node Candidate with smallest total distance is the nth nearest node

The Shortest-Path Problem Shortest path for the Seervada park problem Looking at last column in Table 10.2, two potential shortest paths exist from the destination to the origin T→ D → E → B → A → O or T → D → B → A → O Total of 13 miles on either path

The Shortest-Path Problem Network simplex method An alternate option for solving shortest-path problems Three categories of applications Minimize total distance traveled Minimize total cost of a sequence of activities Minimize total time of a sequence of activities

10.4 The Minimum Spanning Tree Problem Given: nodes of a network, potential links, and positive length of each link if it is inserted into the network Design the network by inserting links A path must exist between every pair of nodes Problem: minimize total length of links inserted into the network Network of n nodes requires only n−1 links Choose the links to form a spanning tree

The Minimum Spanning Tree Problem Applications Design of telecommunications networks Design of a lightly-used transportation network to minimize cost of providing links Design network of power transmission lines Electrical equipment wiring Piping systems

The Minimum Spanning Tree Problem Algorithm Select any node arbitrarily and then add a link to connect it to its nearest node Identify the unconnected node that is closest to a connected node, and add a link between them Repeat until all nodes have been connected Ties may be broken arbitrarily There may be multiple optimal solutions

The Minimum Spanning Tree Problem Example of graphical approach to implementing the algorithm Problem: installing telephone lines in Seervada park See Pages 384-386 in the text for solution

10.5 The Maximum Flow Problem General problem description All flow through a directed, connected network originates at a source, and terminates at a sink Remaining nodes are transshipment nodes Flow through an arc is allowed in only one direction (indicated by the arrowhead) Maximum flow is given by arc capacity Objective: maximize total flow from source to sink

The Maximum Flow Problem Applications Maximize flow through company’s distribution network from factories to customers Maximize flow through company’s supply network from vendors to factories Maximize oil flow through a system of pipelines Maximize water flow through aqueducts Maximize flow of vehicles through a transportation network

The Maximum Flow Problem Algorithms Simplex method can be used Augmenting path algorithm is more efficient Residual network Remaining arc capacities after some flows have been assigned Augmenting path Directed path from source to sink in residual network such that every arc on path has positive residual capacity

The Maximum Flow Problem Algorithm (each iteration follows these steps) Identify an augmenting path If none exists, net flows already constitute an optimal flow pattern Identify the residual capacity, c* of this augmenting path It will equal the minimum residual capacity of the arcs on this path Increase the flow in this path by c*

The Maximum Flow Problem Algorithm (cont’d.) Decrease by c* the residual capacity of each arc on this augmenting path Increase by c* the residual capacity of each arc in the opposite direction on this augmenting path Return to the first step Example: Seervada park transportation problem See Pages 390-392 in the text

10.6 The Minimum Cost Flow Problem General description of the minimum cost flow problem The network is directed and connected At least one of the nodes is a supply node, and one of the other nodes is a demand node All remaining nodes are transshipment nodes Flow is only allowed in direction of the arrowhead Arc capacity gives maximum allowable flow

The Minimum Cost Flow Problem General description (cont’d.) Network has enough arcs with sufficient capacity to enable all flow generated at supply nodes to reach all demand nodes Cost of flow through each arc is proportional to the amount of flow Objective: minimize total cost of sending available supply through the network to meet the given demand

The Minimum Cost Flow Problem

The Minimum Cost Flow Problem Linear programming problem formulation

The Minimum Cost Flow Problem Feasible solutions property Integer solutions property For minimum cost flow problems where every bi and uij have integer values, all the basic variables in every basic feasible solution also have integer values

The Minimum Cost Flow Problem Special cases that fit the minimum cost flow problem The transportation problem The assignment problem The transshipment problem The shortest-path problem The maximum flow problem

The Minimum Cost Flow Problem Network simplex method An alternative method to solving the special cases when the special-purpose algorithms are not available

10.7 The Network Simplex Method Streamlined version of the simplex method Same basic steps Finding the entering basic variable Determining the leaving basic variable Solving for the new BF solution General concepts of the method are covered in the text

The Network Simplex Method Incorporate the upper bound technique: To deal with the arc capacity constraints Network representation of BF solutions Basic arcs: arcs corresponding to basic variables Key property: they never form undirected cycles Nonbasic arcs: arcs corresponding to nonbasic variables

The Network Simplex Method BF solutions can be obtained by solving spanning trees For arcs not in the spanning tree, set the corresponding variables (xij or yij) equal to zero For arcs in the spanning tree, solve for the corresponding variables (xij or yij) in the system of linear equations provided by the node constraints

The Network Simplex Method Feasible spanning tree Spanning tree whose solution from the node constraints also satisfies all the other constraints Fundamental theorem for the network simplex method Basic solutions are spanning tree solutions (and conversely) BF solutions are solutions for feasible spanning trees (and conversely)

10.8 A Network Model for Optimizing a Project’s Time-Cost Trade-off Network based OR techniques developed in the 1950s PERT (Program Evaluation Review Technique) CPM (Critical Path Method) Both are used in project management Concepts have merged into PERT/CPM CPM method for time-cost tradeoff Addresses a project with a specific deadline

A Network Model for Optimizing a Project’s Time-Cost Trade-off CPM method for time-cost trade-off (cont’d.) Problem: find optimal plan for expediting activities to minimize the total cost of completing the project within the deadline General approach Use a network to display the various activities And the order in which they need to be performed Form optimization model Solve using linear programming

A Network Model for Optimizing a Project’s Time-Cost Trade-off Prototype example The Reliable Construction Co. won the contract to construct a new plant within a time period of 40 weeks See Table 10.7 Project network options Activity-on-arc (AOA) Each activity is represented by an arc Nodes separate activities from predecessors Used by original versions of PERT and CPM

A Network Model for Optimizing a Project’s Time-Cost Trade-off Project network options (cont’d.) Activity-on-node (AON) Each activity is represented by a node Arcs show precedence relationships between activities Has several advantages over AOA May become the standard format for project networks

A Network Model for Optimizing a Project’s Time-Cost Trade-off Path One of the routes following the arcs from start to finish The critical path Relevant: length of each path through the network Sum of estimated durations of activities on the path

A Network Model for Optimizing a Project’s Time-Cost Trade-off Estimating the critical path (project duration) for the Reliable Construction Co. example See Pages 415-417 in the text Crashing an activity Taking special costly measures to reduce an activity’s duration Crashing the project involves crashing a number of activities

A Network Model for Optimizing a Project’s Time-Cost Trade-off Example problem: determine least expensive way to crash activities to reduce overall duration to 40 weeks Solution methods Marginal cost analysis See Table 10.10 and Table 10.11 on Pages 419 and 420 of the text Linear programming Follow steps on Pages 420-424 of the text

10.9 Conclusions Problems addressed with network models Optimizing an existing network Designing a new network Minimum spanning tree problem CPM method of time-cost trade-offs Powerful way of applying network optimization to project management