Review Questions

6.1-1A supply node is a node where the net amount of flow generated is a fixed positive number. A demand node is a node where the net amount of flow generated is a fixed negative number. A transshipment node is a node where the net amount of flow generated is fixed at zero.

6.1-2The maximum amount of flow allowed through an arc is referred to as the capacity of that arc.

6.1-3The objective is to minimize the total cost of sending the available supply through the network to satisfy the given demand.

6.1-4The feasible solutions property is necessary. It states that a minimum cost flow problem will have a feasible solution if and only if the sum of the supplies from its supply nodes equals the sum of the demands at its demand nodes.

6.1-5As long as all its supplies and demands have integer values, any minimum cost flow problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its flow quantities.

6.1-6Network simplex method.

6.1-7Applications of minimum cost flow problems include operation of a distribution network, solid waste management, operation of a supply network, coordinating product mixes at plants, and cash flow management.

6.1-8Transportation problems, assignment problems, transshipment problems, maximum flow problems, and shortest path problems are special types of minimum cost flow problems.

6.2-1One of the company’s most important distribution centers (Los Angeles) urgently needs an increased flow of shipments from the company.

6.2-2Auto replacement parts are flowing through the network from the company’s main factory in Europe to its distribution center in LA.

6.2-3The objective is to maximize the flow of replacement parts from the factory to the LA distribution center.

6.3-1Rather than minimizing the cost of the flow, the objective is to find a flow plan that maximizes the amount flowing through the network from the source to the sink.

6.3-2The source is the node at which all flow through the network originates. The sink is the node at which all flow through the network terminates. At the source, all arcs point away from the node. At the sink, all arcs point into the node.

6.3-3The amount is measured by either the amount leaving the source or the amount entering the sink.

6.3-41. Whereas supply nodes have fixed supplies and demand nodes have fixed demands, the source and sink do not.

2. Whereas the number of supply nodes and the number of demand nodes in a minimum cost flow problem may be more than one, there can be only one source and only one sink in a standard maximum flow problem.

6.3-5Applications of maximum flow problems include maximizing the flow through a distribution network, maximizing the flow through a supply network, maximizing the flow of oil through a system of pipelines, maximizing the flow of water through a system of aqueducts, and maximizing the flow of vehicles through a transportation network.

6.4-1The origin is the fire station and the destination is the farm community.

6.4-2Flow can go in either direction between the nodes connected by links as opposed to only one direction with an arc.

6.4-3The origin now is the one supply node, with a supply of one. The destination now is the one demand node, with a demand of one.

6.4-4The length of a link can measure distance, cost, or time.

6.4-5Sarah wants to minimize her total cost of purchasing, operating, and maintaining the cars over her four years of college.

6.4-6When “real travel” through a network can end at more that one node, a dummy destination needs to be added so that the network will have just a single destination.

6.4-7Quick’s management must consider trade-offs between time and cost in making its final decision.

Problems

6.1In this study, flight delay and cancellation problems faced by United Airlines (UA) are...