Deck 5: Network Modeling

Question 1

What happens to the solution of a network flow model if side constraints are added that do not obey the balance of flow rules?

A) The model solution is not guaranteed to be integer.
B) The model solution will more accurately reflect reality.
C) The model solution will be integer but more accurate.
D) The model solution is not guaranteed to be feasible.

Answer 1

The model solution is not guaranteed to be integer.

Question 2

The arcs in a network indicate all of the following except?

A) routes
B) paths
C) constraints
D) connections

Answer 2

constraints

Question 3

For minimum cost network flow problems where Total Supply > Total Demand, the balance of flow rule to use at each node is
Inflow - Outflow >= Supply or Demand.

Answer 3

True

Question 4

A maximal flow problem differs from other network models in which way?

A) arcs are two directional
B) multiple supply nodes are used
C) arcs have limited capacity
D) arcs have unlimited capacity

Question 5

​Any shortest path problem can be modeled as a transshipment problem.

Question 6

What is the constraint for node 2 in the following maximal flow problem? ​ ​

A) X₁₂ − X₂₃ − X₂₄ = 0
B) X₁₂ + X₂₃ + X₂₄ = 0
C) X₁₂ ≤ 4
D) X₁₂ + X₁₃ − X₂₃ = 0

Question 7

The equipment replacement problem is a common type of business problem that can be modeled as a shortest path problem.

Question 8

Transportation/assignment problems are sparse or not fully interconnected, meaning all the supply nodes have arcs connecting them to all the demand nodes.

Question 9

What is the correct constraint for node 2 in the following diagram? ​ ​

A) X₁₂ + X₂₃ = 100
B) X₁₂ − X₂₃ ≤ 100
C) −X₁₂ + X₂₃ ≥ −100
D) X₁₂ − X₂₃ ≥ 100

Question 10

An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is $11 and the yield is 85 percent. The following network representation depicts this problem. What is the balance of flow constraint for node 3 (Refinery 1)? ​ ​

A) X₁₃ + X₂₃ − .95 X₃₅ − .90 X₃₆ − .90 X₃₇ = 0
B) .80 X₁₃ + .95 X₂₃ − X₃₅ − X₃₆ − X₃₇ = 0
C) .80 X₁₃ + .95 X₂₃ − .90 X₃₆ − .90 X₃₇ ≥ 0
D) X₁₃ + X₂₃ − X₃₅ − X₃₆ − X₃₇ ≥ 0

Question 11

The supply nodes in the graphical representation of the transshipment problem:

A) have all directed arcs originating at them
B) have all directed arcs terminating at them
C) have some unidirectional arcs
D) are greater than the demand nodes

Question 12

In a network flow problem, the value +80 next to a node indicates that the number of units needs to decrease by 80.

Question 13

The idea that the total flow into a node must be consumed at a node and the remainder must flow out of a node is referred to as:

A) the conservation of flow principle
B) the node-arc incidence matrix
C) a directed chain
D) integrality constraint

Question 14

In a transshipment problem, which of the following statements is a correct representation of the balance-of-flow rule if Total Supply < Total Demand?

A) Inflow − Outflow ≥ Supply or Demand
B) Inflow + Outflow ≥ Supply or Demand
C) Inflow − Outflow ≤ Supply or Demand
D) Inflow + Outflow ≤ Supply or Demand

Question 15

The transshipment nodes in the graphical representation of the transshipment problem:

A) have total demanded quantities expressed as positive numbers
B) have all directed arcs originating at them
C) have some unidirectional arcs
D) have all directed arcs terminating at them

Question 16

​Most of the other types of network flow problems can be viewed as simple variations of the transshipment problem.

Question 17

When the lines connecting the nodes in a network are arrows that indicate a direction, the arcs in the network are called directed arcs.

Question 18

The constraint X₁₃ + X₂₃ − X₃₄ ≥ 50 indicates that

A) 50 units are required at node 3.
B) 50 units will be shipped from node 3.
C) 50 units will be shipped in from node 1.
D) 50 units must pass through node 3.

Question 19

What is the objective function in the following maximal flow problem? ​ ​

A) MIN X₄₁
B) MAX X₁₂ + X₁₃
C) MAX X₁₄
D) MAX X₄₁

Question 20

A number of practical decision problems in business fall into a category known as network flow problems. These problems can be described or displayed in a graphical form known as a network.

Question 21

What is the constraint for node 2 in the following shortest path problem? ​ ​

A) −X₁₂ − X₁₃ = 0
B) −X₁₂ − X₂₄ = 1
C) X₁₂ + X₁₃ = 0
D) −X₁₂ + X₂₄ = 0

Question 22

What is the interpretation of units "shipped" along arcs from dummy supply nodes to demand nodes?

A) Indicates unmet demand at demand nodes
B) Indicates unmet supply at demand nodes
C) Indicates unmet demand at supply nodes
D) Indicates unmet supply at supply nodes

Question 23

The minimal spanning tree solution algorithm works by defining a subnetwork and

A) adding the least expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork.
B) adding the most expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork.
C) adding the least expensive arc which connects unconnected nodes in the current subnetwork.
D) adding the least expensive arc which connects the most recently added node in the current subnetwork to the closest node not in the current subnetwork.

Question 24

How many arcs are required to make a spanning tree in a network with n nodes and m arcs?

A) n
B) n − 1
C) m
D) m − 1

Question 25

Consider the equipment replacement problem presented in the chapter. Recall that in the network model formulation of this problem a node represents a year when the equipment was purchased. An arc from node i to node j indicates that the equipment purchased in year i can be replaced at the beginning of year j. How could the network model below be modified to depict an equipment purchase in year 4 and operating costs only through the remainder of the planning window? ​ ​

A) Modify the cost on arc 4-5 to account for only operating costs.
B) Add a second arc 4-5 to represent just the operating costs.
C) Add a dummy node, 6, so that arc 4-6 represents just the operating costs.
D) Add a dummy node, 6, so that arc 4-5 represents operating costs and 5-6 represents new equipment purchase.

Question 26

A factory which ships items through the network would be represented by which type of node?

A) demand
B) supply
C) random
D) decision

Question 27

Which balance of flow rule should be applied at each node in a network flow problem when Total Supply > Total Demand?

A) Inflow − Outflow ≤ Supply or Demand
B) Inflow − Outflow ≥ Supply or Demand
C) Inflow − Outflow = Supply or Demand
D) Inflow − Supply ≥ Outflow or Demand

Question 28

A node which can both send to and receive from other nodes is a

A) demand node.
B) supply node.
C) random node.
D) transshipment node.

Question 29

The assignment problem is equivalent to a transportation problem with

A) the sums of all rows and columns equal to one.
B) binary decision variables.
C) non-negativity constraints removed.
D) all of the above.

Question 30

Which formula should be used to determine the Net Flow values in cell K6 in the following spreadsheet model?

A) SUMIF($C$6:$C$16,I6,$B$6:$B$16)−SUMIF($E$6:$E$16,I6,$B$6:$B$16)
B) SUMIF($I$6:$I$12,B6,$B$6:$B$16)−SUMIF($I$6:$I$12,I6,$B$6:$B$16)
C) SUMIF($E$6:$E$16,I6,$B$6:$B$16)−SUMIF($C$6:$C$16,I6,$B$6:$B$16)
D) SUMPRODUCT(B6:B16,G6:G16)

Question 31

Decision variables in network flow problems are represented by

A) nodes.
B) arcs.
C) demands.
D) supplies.

Question 32

In the generalized network flow problem solver could not find a feasible solution. This means that:

A) the total supply is not capable of meeting the total demand
B) the total supply is capable of meeting the total demand
C) the total demand is not capable of meeting the total supply
D) dummy demand is needed in the formulation

Question 33

What is missing from transportation problems compared to transshipment problems?

A) arcs
B) demand nodes
C) transshipment nodes
D) supply nodes

Question 34

What formula would be entered in cell G18 in this Excel model?

A) SUMPRODUCT(K6:K12,L6:L12)
B) SUMPRODUCT(B6:B16,G6:G16)
C) SUMPRODUCT(G6:G16,K6:K12)
D) SUMPRODUCT(B6:G16,L6:L12)

Question 35

The demand nodes in the graphical representation of the transshipment problem:

A) have all directed arcs originating at them
B) have all directed arcs terminating at them
C) have some unidirectional arcs
D) are greater than the demand nodes

Question 36

The demand nodes in the graphical representation of the transshipment problem:

A) have total demanded quantities expressed as positive numbers
B) have all directed arcs originating at them
C) have some unidirectional arcs
D) have total available quantities expressed as negative numbers

Question 37

The right hand side value for the ending node in a shortest path problem has a value of

A) −1
B) 0
C) 1
D) 2

Question 38

The right hand side value for the starting node in a shortest path problem has a value of

A) −1
B) 0
C) 1
D) 2

Question 39

How could a network be modified if demand exceeds supply?

A) add extra supply arcs
B) remove the extra demand arcs
C) add a dummy supply
D) add a dummy demand

Question 40

For a network with n nodes, a spanning tree is

A) a set of (n-1) arcs that connects all nodes and contains no loops
B) a set of dummy arcs
C) a set of n arcs that connects all nodes
D) a random subset of arcs covering all nodes

Question 41

How many constraints are there in a transshipment problem which has n nodes and m arcs?

A) n
B) m
C) (n + m)
D) (m − n)

Question 42

What is the objective function for the following shortest path problem? ​ ​

A) −X₁₂ − X₁₃ = 0
B) MIN −50 X₁₂ − 200 X₁₃ + 100 X₂₄ + 35 X₃₄
C) MIN 50 X₁₂ + 200 X₁₃ + 100 X₂₄ + 35 X₃₄
D) MAX −50 X₁₂ − 200 X₁₃ + 100 X₂₄ + 35 X₃₄

Question 43

Maximal flow problems are converted to transshipment problems by

A) connecting the supply and demand nodes with a return arc
B) adding extra supply nodes
C) adding supply limits on the supply nodes
D) requiring integer solutions

Question 44

In the assignment problem:

A) the sums of all rows and columns must be equal to one.
B) the number of rows is greater than the number of columns
C) the number of rows is smaller than the number of columns
D) there is no limit on the sum of all rows

Question 45

A network flow problem that allows gains or losses along the arcs is called a

A) non-constant network flow model.
B) non-directional, shortest path model.
C) generalized network flow model.
D) transshipment model with linear side constraints.

Question 46

The street intersections in a city road network represent

A) nodes.
B) arcs.
C) resources.
D) expenses.

Question 47

Which method is preferred for solving minimal spanning tree problems?

A) linear programming
B) transshipment models
C) simulation
D) manual algorithms

Question 48

Which property of network flow models guarantees integer solutions?

A) linear constraints and balance of flow equation format
B) linear objective function coefficients
C) integer objective function coefficients
D) integer constraint RHS values and balance of flow equation format

Question 49

An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is $11 and the yield is 85 percent. The following flowchart depicts this problem. What is the balance of flow constraint for node 7 (Diesel)? ​ ​

A) X₃₅ + X₃₆ + X₃₇ = 75
B) X₃₇ + X₄₇ ≥ 75
C) .90 X₃₇ + .95 X₄₇ = 75
D) X₃₇ + X₄₇ −X₃₆ − X₃₅ − X₄₅ − X₄₆ ≥ 75

Question 50

A minimum or maximum flow restriction in the network flow problem can be modeled by

A) adding dummy nodes
B) adding dummy arcs
C) adding additional flow restrictions on affected arcs
D) all of the above

Question 51

In the shortest path problem, the originating and terminating network nodes are called:

A) source and sink
B) source and end
C) beginning and end
D) beginning and sink

Question 52

When might a network flow model for a transportation/assignment problem be preferable to a matrix form for the problem?

A) When an integer solution is required.
B) When the problem is large and not fully connected.
C) When the problem is large and fully connected.
D) When supply exceeds demand.

Question 53

If a side constraint for a network flow model cannot be avoided, and non-integer solutions result, how can the solution be expressed as an integer solution?

A) Apply integer programming techniques.
B) Round off all the non-integer arc flow decision variables.
C) Increase the supply until the solutions are all integer using a dummy supply node.
D) Increase the demand until the solutions are all integer using a dummy demand node.

Question 54

Demand quantities for demand nodes in a transshipment problem are customarily indicated by

A) positive numbers.
B) negative numbers.
C) imaginary numbers.
D) either positive or negative numbers.

Question 55

The supply nodes in the graphical representation of the transshipment problem:

A) have total available quantities expressed as negative numbers
B) have all directed arcs terminating at them
C) have some unidirectional arcs
D) have total demanded quantities expressed as positive numbers

Question 56

Which method is preferred for solving fully connected transportation problems?

A) linear programming
B) network flow methods
C) trial and error
D) simulation

Question 57

Consider modeling a warehouse with three in-flow arcs and three outflow arcs. The warehouse node is a transshipment node but has a capacity of 100. How would one modify the network model to avoid adding a side constraint that limits either the sum of in-flows or the sum of the out-flows to 100?

A) Place a limit of 34 on each in-flow arc.
B) Add a side constraint limiting the out-flow arcs sum to 100.
C) Separate the warehouse node into two nodes, connected by a single arc, with capacity of 100.
D) It cannot be accomplished, a side constraint must be added.

Question 58

The number of constraints in network flow problems is determined by the number of

A) nodes.
B) arcs.
C) demands.
D) supplies.

Question 59

Supply quantities for supply nodes in a transshipment problem are customarily indicated by

A) positive numbers.
B) negative numbers.
C) imaginary numbers.
D) either positive or negative numbers.

Question 60

The equipment replacement problem is an example of which network problem?

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

Question 61

A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs $25 per hour while contracted labor costs $45 per hour. The following network flow model captures this problem.
​ ​
Write out the LP formulation for this problem.

Question 62

Clifton Distributing has three plants and four distribution centers. The plants, their supply, the distribution centers, their demands, and the distance between each location is summarized in the following table: Draw the balanced transportation network for Clifton's distribution problem.

Question 63

In generalized network flow problems

A) solutions may not be integer values.
B) flows along arcs may increase or decrease.
C) it can be difficult to tell if total supply is adequate to meet total demand.
D) all of these.

Question 64

The following network depicts a transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.
​

Question 65

The following network depicts a balanced transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.
​

Question 66

An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem. What formula should be entered in cell E6 (and copied to cells E7:E15) in this spreadsheet?

Question 67

A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs $25 per hour while contracted labor costs $45 per hour. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. Draw the network flow model that captures this problem.

Question 68

Draw the network and indicate how many units are flowing along each arc based on the following Analytic Solver Platform solution.

Question 69

Solve the following minimal spanning tree problem starting at node 1.
​

Question 70

A company needs to ship 100 units from Roanoke to Washington at the lowest possible cost. The costs associated with shipping between the cities are: ​
Draw the network representation of this problem.

Question 71

Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is: Draw the balanced network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule.

Question 72

A railroad needs to move the maximum amount of material through its rail network. The numbers on arcs represent maximum flow. Formulate the LP model to determine this maximum amount based on the following network diagram.
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Question 73

The following network depicts a balanced assignment/transportation problem for Joe Fix's repair scheduling problem. Formulate the LP for Joe assuming he wishes to maximize the total repairperson to plane assignment preferences.
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Question 74

Draw the network and show the solution for the maximal flow problem represented by the following Excel spreadsheet.

Question 75

Almost all network problems can be viewed as special cases of the

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

Question 76

Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is: Draw the network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule.

Question 77

A company wants to determine the optimal replacement policy for its photocopier. The company does not keep photocopiers longer than 4 years. The company has estimated the annual costs for photocopiers during each of the 4 years and developed the following network representation of the problem.
​
Write out the LP formulation for this problem.
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Question 78

Draw the network representation of this LP model. What type of problem is it?

Question 79

A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are listed below. Also, the decision variable associated with each pair of cities is shown next to the cost. ​
Write out the LP formulation for this problem.

Question 80

A company wants to manage its distribution network which is depicted below. Identify the supply, demand and transshipment nodes in this problem.
​