Deck 4: Network Models

Question 1

In network flow models, all arcs are generally assumed to be directed.In connectivity models, all arcs are generally assumed to be undirected.

Answer 1

True

Question 2

In a problem formulated as a standard transportation model, if all the source/destination quantities are integer-valued, the set of solution variables is guaranteed to be integer-valued as well.

Answer 2

True

Question 3

A bi-directional arc in a shortest path problem can be modeled as two unidirectional arcs.

Answer 3

True

Question 4

The objective function of a transportation model is to minimize the total shipping costs.

Question 5

The optimal total shipping cost in a minimization transportation model is $622, and the reduced cost from source node 1 to destination node 6 is $6.This implies that if one were to deviate from the optimal solution by shipping one unit from node 1 to node 6, the new minimum total shipping cost would be $628.

Question 6

You can solve the traveling salesman problem by selecting a cycle with the lowest cost.

Question 7

In a problem formulated as a standard transportation model, where there are m sources and n destinations, the number of arcs (routes) in an optimal solution need be:

A)m * n.
B)m + n.
C)no less than (m * n) - 1.
D)no greater than (m + n) - 1.

Question 8

Transshipment, assignment, shortest path, and traveling salesman problems are network flow models.

Question 9

Minimal spanning tree problems do not lend themselves to a convenient linear programming formulation.

Question 10

For a network of n nodes, a spanning tree contains n - 1 arcs.

Question 11

A tree cannot contain a cycle.

Question 12

An optimal maximum flow solution will involve positive flow on every arc.

Question 13

In the standard assignment problem where, for example, workers are being assigned tasks, which of the following situations would preclude the application of the assignment approach (the Hungarian algorithm)?

A)A worker may be assigned to more than one job.
B)A worker may be unable to perform a certain job.
C)The number of workers does not equal the number of jobs.
D)Workers may perform fractional parts of jobs.

Question 14

In addressing and solving a maximal flow problem, flow capacities between adjacent nodes must be equal in both directions.

Question 15

The assumptions underlying the application of the basic transportation model do not include:

A)any node may represent either a source or a destination.
B)unit shipping costs may be constant or declining (i.e., quantity discounts) but not increasing with volume.
C)all shipments must occur in the same time frame.
D)shipments may not be sent back from a destination to a source.

Question 16

A city needs to design and build a completely new storm drainage system.The excess rainwater runoff in each neighborhood is to be channeled to a suburban reservoir.The appropriate network model would most likely be:

A)transportation/transshipment.
B)shortest path.
C)minimal spanning tree.
D)maximal flow.

Question 17

A map of the United States can be drawn as a network with each state represented by a node and arcs connecting each pair of states that share a boundary.

Question 18

A minimal spanning tree connects all nodes in a minimal total distance or cost.

Question 19

In a standard transportation model, each objective function coefficient represents the:

A)total cost of shipping from a source to a destination node.
B)fixed cost of utilizing the route between a source and destination node.
C)unit cost of shipping from a source to a destination node.
D)negative "unit cost" of not using the route between a source and a destination node.

Question 20

If a general linear programming problem can also be formulated as a network problem, the network formulation is generally preferable because of easier model input and a quicker solution technique.

Question 21

An environmental impact report for a proposed new industrial center complex must include the total amount of sewage that can be pumped through existing pipes.Which network model would be the most appropriate representation?

A)Shortest path
B)Minimal spanning tree
C)Traveling salesman
D)Maximal flow

Question 22

Network connectivity models solve what type of problem?

Question 23

Which statement is not true?

A)The Ford-Fulkerson method solves the maximal flow problem by finding the minimum cut.
B)The minimum cut will provide an optimal solution to the maximal flow problem.
C)The maximal flow equals the sum of the capacities of the minimum cut.
D)The flows on all arcs on the minimum cut will be at their upper bounds.

Question 24

In a shortest path model:

A)The arc between nodes A and B must have the same value as the arc between arc B and A.
B)The goal is to find the minimum distance between the start node and the terminal node.
C)Negative arc distances are not prohibited.
D)Unidirectional arcs are permitted.

Question 25

A car dealership uses its van to drive its service customers to their jobs.If the dealership van has five people who need to be taken to work, what network algorithm should be used in order to minimize the distance the van must travel in order to accomplish this task?

A)Shortest path
B)Minimal spanning tree
C)Traveling salesman
D)Transportation

Question 26

Network flow models are integer linear programming models.Why, then, would we prefer using a linear programming model? Does the linear programming model give an optimal solution?

Question 27

A transportation model has been selected to represent the production schedule of a product to meet demand over a twelve month planning horizon with 12 production nodes, on each of which is defined the production capacity for each month, and 12 demand nodes, on each of which is defined the demand for each month.If goods can be stored a maximum of 3 months, which of the following statements is not true?

A)There should be no arc from production node 1 to demand node 5.
B)There should be no arc from production node 5 to demand node 1.
C)There will be no feasible solution if demand in any one month cannot exceed the production capacity for that month.
D)There will be no feasible solution if the total demand of the first m months cannot exceed the total production capacity for the first m months.

Question 28

In which of the following network models is it not possible to "ship" both in and out of a given node?

A)Transshipment
B)Traveling salesman
C)Maximal flow
D)Assignment

Question 29

Two nodes are said to be connected if what exists?

A)An arc between them.
B)A path between them.
C)A network containing both nodes.
D)A cycle between them.

Question 30

In a minimal spanning tree problem:

A)If node 1 is connected to node 3, which is connected to node 5, the shorted path from node 1 to node 5 is 1-3-5.
B)The optimal solution is found by adding arcs of least distance until all nodes are connected.
C)The arcs between nodes 1 and 5, between nodes 1 and 9, between nodes 3 and 5, and between nodes 5 and 9 cannot all be included in the optimal solution.
D)The solution will include n arcs.

Question 31

Which of the following is true about an intermediate node in a transshipment problem?

A)The net flow out must be zero.
B)The node must connect to at least one supply node and at least one demand node.
C)The number of arcs connected to supply nodes must be less than or equal to the number of arcs connected to demand nodes.
D)The total capacity of outbound arcs must equal the total capacity of inbound arcs.

Question 32

In formulating a transportation model using a balanced format in which total supply must equal total demand, indicate how to model each of the following unusual circumstances:

Question 33

Which of the following is not an algorithm for solving the assignment model?

A)Transportation model
B)Dynamic programming
C)The Hungarian algorithm
D)The Dijkstra algorithm

Question 34

Which of the following cannot model a blocked route on a transportation model?

A)A very large objective coefficient.
B)A constraint on shipment equal to zero.
C)Excluding the cell from the list of changing cells in the Excel Solver parameters.
D)Assigning an objective function coefficient of zero.

Question 35

Name the two types of nodes in a network flow model.

Question 36

A B-1 bomber must fly through a choice of "safe areas," deliver its bombing payload, and return to its home base using minimum fuel.Which network model would be the most appropriate representation?

A)Shortest path
B)Minimal spanning tree
C)Traveling salesman
D)Transportation

Question 37

In a maximal flow problem:

A)Multiple sources may be accommodated, but multiple sinks may not.
B)Arc capacities must be the same in either direction.
C)There are several different linear programming formulations for the model.
D)The connected arcs may contain no cycles.

Question 38

The optimal solution to an assignment model can be obtained by solving the problem as each of the following models except:

A)a traveling salesman model.
B)a transportation model.
C)a binary integer programming model.
D)a linear programming model.

Question 39

A model which matches salespeople to retail customers based on estimated profit potential of each salesperson-retail customer match, where each salesperson is to be matched with at most 25 retail customers, could be modeled as which network models?

A)Either as a shortest path or a traveling salesman model.
B)Either as a minimal spanning tree or a maximal flow model.
C)Either as a maximization assignment model or a maximization transportation model.
D)Either as a capacitated transshipment model or a direct enumeration model.

Question 40

The optimal solution to a standard linear programming problem need have no more non-zero-valued decision variables than there are linear constraints.The standard balanced transportation model, in which total supply equals total demand, has one linear constraint restricting the amount shipped from each of the m source nodes and one linear constraint restricting shipments into each of the n demand nodes.For this model, which is simply a special case of linear programming, the number of non-zero-valued decision variables in the optimal solution need be no more than one less than the total number of linear constraints.Why?

Question 41

Consider the problem faced by SnowDevil in problem 1.Suppose the next week Tahoe can produce only 15 units.

Question 42

Cajun Oil Company operates pipelines for transporting oil from its source in the Gulf of Mexico to storage facilities located on the mainland in Louisiana.Before reaching the storage facilities, the oil must pass through one or more pumping stations.The network below gives the maximum amount that can be transported in pipes between sites in thousands of gallons per hour.

Question 43

In a given transportation problem involving cost minimization, there are three sources, with outputs of 10, 18, and 17, respectively.There are four destinations, with demands of 14, 11 9 and 6, respectively.

Question 44

The sensitivity report from problem 9 is given below.The "Reduced Gradient" column contains reduced costs, and the "Lagrange Multiplier" column contains shadow prices.

Question 45

As part of a work-study program, Winslow Graphics has agreed to have 5 internship students from the Communications Department of the local university spend one day each during the next week in its labs participating in the development of magazine layouts for a variety of products.Since these internship students are in their final year of a master's degree program, they possess significant skills, and Winslow feels it would benefit by having the maximum total number of student hours during the week.Each student was asked for his availability in terms of maximum number of continuous hours for each day during the coming week.

Question 46

The Greedy algorithms for solving the minimal spanning tree problem require selecting the remaining arc with the smallest cost unless what is true?

Question 47

SnowDevil produces snowboards at its factories in Reno and Tahoe with weekly outputs of 30 and 20 units, respectively.This commodity is to be transported to three regional warehouse locations, Warehouses 1, 2, and 3, which have weekly requirements of 10, 15, and 25 units, respectively.Unit shipping costs are:

Question 48

Suppose in problem 3 that over the 15 week semester each of the five interns must spend 12 dates each at Winslow Graphics with no two interns being at Winslow on the same day.Assume that student availability will not change from week to week over the semester.

Question 49

The Department of Defense (DOD) is conducting weapons tests in an isolated part of the Nevada desert.At one particular site, the DOD has six structures that are in proximity to the blast area and must be evacuated to an observation area at times when testing is done.The blast area lies in a combination of flat and mountainous terrain, and the actual blast site, the DOD sites, and the observation area are connected by various dirt paths over which only 4-wheel drive vehicles are permitted.The network below shows these paths and the approximate driving times in minutes on these paths using 4-wheel drive vehicles.

Question 50

Describe two Greedy algorithms for solving the minimal spanning tree problem.

Question 51

What is the term for a path from a node that returns to the same node without including any arc more than once?

Question 52

Consider the DOD blast area represented by the network for problem 5.The DOD would like to eliminate the need for 4-wheel drive vehicles and is considering paving some of the paths between sites.Paving a path will cost taxpayers about $125,000 per mile.

Question 53

In a transportation model, how are shadow prices interpreted for supply nodes? For demand nodes?

Question 54

Identify the four parts of a maximal flow problem.