Network Optimization Problems Chapter 6: Hillier and Hillier

Agenda Minimum-Cost Flow Problems Case Study: Distribution Unlimited Co. Maximum Flow Problems Case Study: BMZ Company Shortest Path Problems Case Study: Littletown Fire Department Minimum Spanning Tree Problems Case Study: The Modern Corp. Problem

Minimum-Cost Flow Problems These problems are also known as distribution network problems which were seen in chapter 3. Your objective is to distribute goods from a supply source to an ultimate demand destination with potential distribution centers in-between.

Needed Terminology A network in this case is a representation of the model for a minimum-cost flow model that has flow passing through it. Circles in the network are known as nodes. These nodes can be broken-up into supply nodes, demand nodes, and transshipment nodes. A supply node is a node where the net amount of flow generated (outflow minus inflow) is a fixed positive number.

Needed Terminology Cont. A demand node is a node where the net amount of flow generated (outflow minus inflow) is a fixed negative number. A transshipment node is a node where the net amount of flow generated (outflow minus inflow) is a fixed at zero. Having the outflow equal to the inflow is known as conservation of flow.

Needed Terminology Cont. Arcs are lines from node to node which represent the direction in which flow can occur. Capacity of an arc represents the maximum amount of flow that can be sent through an arc. Flow is defined as an amount that is sent through an arc.

Assumptions of the Minimum-Cost Flow Problem There is at least one supply node. There is at least one demand node. Any node that is not a supply node or a demand node is a transshipment node. Flow through an arc is only allowed in the direction the arc is pointing, where the maximum amount of flow is dictated by its capacity.

Assumptions of the Minimum-Cost Flow Problem The network has enough arcs with sufficient capacity to move all the flows from the supply nodes to the demand nodes. The cost of the flows through the arc are proportional to the amounts sent through the arc. The cost of each flow is known. The objective is to minimize the cost of the flows from the supply nodes to the demand nodes.

Feasible Solution and Integer Properties Feasible Solution Property A minimum-cost flow problem is said to have a feasible solution if and only if the sum of the supplies is equal to the sum of the demands. Integer Solution Property A minimum-cost flow problem will have an integer solution as long as all its supplies, demands, and arc capacities have integer values.

Case Study: Distribution Unlimited Co. Distribution Unlimited Co. (DUC) has two factories producing a product that needs to be shipped to two warehouses. Factory 1 produces 80 units Factory 2 produces 70 units Warehouse 1 needs 60 units Warehouse 2 needs 90 units DUC has one distribution center that both factories can ship to.

Case Study: Distribution Unlimited Co. Cont. DUC has the ability for factory 1 to ship to warehouse 1 or the distribution center. DUC has the ability for factory 2 to ship to warehouse 2 or the distribution center. The distribution center can ship to either warehouses. Shipments to and from distribution centers are limited to a 50 unit maximum amount for each route.

Case Study: Distribution Unlimited Co. Cont. Unit Cost For Shipments Distribution Center Warehouse 1 Warehouse 2 Factory 1 $300 $700 N/A Factory 2 $400 $900 $200

Case Study: Distribution Unlimited Co. Cont. Maximum Amount of Shipments to Each Source Distribution Center Warehouse 1 Warehouse 2 Factory 1 50 N/A Factory 2

Case Study: Distribution Unlimited Co. Cont. DUC wants to find the cheapest shipping plan to transport is supply from the factories to the demand of the warehouses.

Representation of Distribution Unlimited Co. Problem

Representation of Distribution Unlimited Co. Problem Cont.

Mathematical Model of Distribution Unlimited Co.

Mathematical Model of Distribution Unlimited Co. Cont.

SUMIF Function in Excel The SUMIF function takes a range and compares it to a some condition. If the condition is true, then the function will add to the running sum the corresponding value in another range. If the condition is false, then the function will not add to the running sum the corresponding value in another range. SUMIF(Range A, condition, Range B) If you put in a value or a number, the function checks for “equality.”

A Note on Solutions While the Simplex Method can solve minimum-cost flow problems, it is not the most efficient algorithm to use. The most efficient algorithm is the the Network Simplex Method. Unfortunately with Excel’s Solver, the network Simplex cannot be utilized because Solver does not currently have the algorithm.

Maximum Flow Problems The objective of a maximum flow problem is to find a plan that maximizes the flow through the network. The characteristics of the maximum flow problem are similar to those of the minimum-cost flow problem.

Assumptions of the Maximum Flow Problem All flow originates from a single node known as the source. All flow terminates at a single node known as the sink. All remaining nodes are transshipment nodes. Flow through the arc is only allowed by the direction of the flow arrow.

Assumptions of the Maximum Flow Problem Maximum flow through an arc is dictated by the capacity of the arc. All arcs extending from the source node point away from the source node. All arcs coming to the sink node point towards the sink node. The objective is to maximize the total flow from the source to the sink. The total flow can be either measured by the amount entering the sink or the amount leaving the source.

Differences Between Minimum-Cost and Maximum Flow In the minimum-cost model, the supply nodes and demand nodes have fixed amounts, whereas the maximum flow problem has a variable amount. In the minimum-cost model, there can be multiple demand and supply nodes, whereas the maximum flow model has only one supply node and one demand node.

Case Study: BMZ Company BMZ is a car company that is trying to get as many cars from its factory in Stuttgart to the Los Angeles market. BMZ can send cars across rail to three shipping points to the US: Rotterdam Bordeaux Lisbon

Case Study: BMZ Company Cont. From the Rotterdam terminal, cars can be shipped to a port in New York. From Bordeaux, cars can be shipped to either New York or New Orleans. From Lisbon, cars can be shipped to New Orleans. Once the cars are in New York and New Orleans, the cars will be shipped to the LA market. Each route for shipment has a limited amount of cars that it can handle.

Case Study: BMZ Company Cont. Maximum Shipments Allowed Rotterdam Bordeaux Lisbon New York New Orleans Los Angeles Stuttgart 50 70 40 N/A 60 30 80

BMZ Company Visual Diagram

Mathematical Formulation

Mathematical Formulation Cont.

Shortest Path Problems In the shortest path problem the objective is to find the shortest path from an origin node to a destination node.

Assumptions of the Shortest Path Problem You need to choose a path that starts at an origin node and ends at a destination node and follows a particular path. The key to these types of problems is that ordering of the solution matters. The lines connecting nodes in the shortest path problem are known as links and usually are bi-directional. Arcs are also allowed.

Assumptions of the Shortest Path Problem Cont. Associated with each link is a nonnegative number known as the length. Lengths can be considered time, costs, distance, etc. The objective is to find the shortest path from the origin node to the destination node.

Case Study: Littletown Fire Department Littletown Fire Department is trying to find the fastest route through the county to reach a specific farming community. The fire department would like to know what is the shortest path to get from the fire department to the farming community.

Diagram of Littletown’s Problem

Minimum Spanning Tree Problems (Not in textbook) Minimum spanning tree problems are problems where you are trying to find the most efficient network. You start out with nodes that have potential links and you must decide whether a link from two nodes should exist. The objective in spanning tree problems are to link all the nodes directly or indirectly to each other with the smallest amount of resources used.

Needed Terminology Tree Spanning Tree A tree is a network that does not have any paths that begin and end at the same node. Spanning Tree It is a tree that provides a path between every pair of nodes, i.e., it spans the tree.

Assumptions of the Minimum Spanning Tree Problem You are given the nodes of the network, but not the links. You are given all the costs related to linking nodes together. You must put enough links into the network to link every pair of nodes so that the nodes are either directly or indirectly linked. The objective is to choose the links that satisfy the above at the cheapest cost.

Algorithm for a Minimum Spanning Tree Problem The Greedy Algorithm Step 1: Select the cheapest potential link. Step 2: Select the cheapest potential link between a node that already has a link with one that does not have a link. Step 3: Continue step two until all nodes have been directly or indirectly linked. Note: Ties may be broken arbitrarily.

Notes on Minimum Spanning Trees The number of links in a spanning tree are one less than the number of nodes. Each node is directly connected by a single link to at least one other node. It is possible to have multiple minimum spanning trees.

Case Study: The Modern Corp. Problem Modern Corp is trying to provide fiber-optic network to all its major centers. The company only needs to make sure that the centers are directly or indirectly linked to each other. They want to link the centers with the cheapest amount of links. The potential links are given on the next slide.

The Modern Corp. Problem