Chapter 6 – Distribution and Network Models
True / False
1. Whenever total supply is less than total demand in a transportation problem, the LP model does not determine how the
unsatisfied demand is handled.
a.
True
b.
False
True
Transportation problem
2. Converting a transportation problem LP from cost minimization to profit maximization requires only changing the
objective function; the conversion does not affect the constraints.
a.
True
b.
False
True
Transportation problem
3. A transportation problem with 3 sources and 4 destinations will have 7 decision variables.
a.
True
b.
False
False
Transportation problem
4. If a transportation problem has four origins and five destinations, the LP formulation of the problem will have nine
constraints.
a.
True
b.
False
True
Transportation problem
5. The capacitated transportation problem includes constraints which reflect limited capacity on a route.
a.
True
b.
False
True
Transportation problem
6. When the number of agents exceeds the number of tasks in an assignment problem, one or more dummy tasks must be
introduced in the LP formulation or else the LP will not have a feasible solution.
a.
True
b.
False
False
Chapter 6 – Distribution and Network Models
7. A transshipment constraint must contain a variable for every arc entering or leaving the node.
a.
True
b.
False
8. The shortest-route problem is a special case of the transshipment problem.
a.
True
b.
False
True
9. Transshipment problem allows shipments both in and out of some nodes while transportation problems do not.
a.
True
b.
False
True
10. A dummy origin in a transportation problem is used when supply exceeds demand.
a.
True
b.
False
False
11. When a route in a transportation problem is unacceptable, the corresponding variable can be removed from the LP
formulation.
a.
True
b.
False
True
12. In the LP formulation of a maximal flow problem, a conservation-of-flow constraint ensures that an arc’s flow
capacity is not exceeded.
a.
True
b.
False
False
Chapter 6 – Distribution and Network Models
13. The maximal flow problem can be formulated as a capacitated transshipment problem.
a.
True
b.
False
True
Maximal flow problem
14. The direction of flow in the shortest-route problem is always out of the origin node and into the destination node.
a.
True
b.
False
True
Shortest-route problem
15. A transshipment problem is a generalization of the transportation problem in which certain nodes are neither supply
nodes nor destination nodes.
a.
True
b.
False
16. The assignment problem is a special case of the transportation problem in which all supply and demand values equal
one.
a.
True
b.
False
True
Assignment problem
17. A transportation problem with 3 sources and 4 destinations will have 7 variables in the objective function.
a.
True
b.
False
False
Assignment problem
18. Flow in a transportation network is limited to one direction.
a.
True
b.
False
True
19. In a transportation problem with total supply equal to total demand, if there are four origins and seven destinations,
and there is a unique optimal solution, the optimal solution will utilize 11 shipping routes.
Chapter 6 – Distribution and Network Models
a.
True
b.
False
False
Transportation problem
20. In the general assignment problem, one agent can be assigned to several tasks.
a.
True
b.
False
True
Assignment problem
21. In a capacitated transshipment problem, some or all of the transfer points are subject to capacity restrictions.
a.
True
b.
False
Multiple Choice
22. The problem which deals with the distribution of goods from several sources to several destinations is the
a.
maximal flow problem
b.
transportation problem
c.
assignment problem
d.
shortest-route problem
Transportation problem
23. The parts of a network that represent the origins are
a.
the capacities
b.
the flows
c.
the nodes
d.
the arcs
c
Transportation problem
24. The objective of the transportation problem is to
a.
identify one origin that can satisfy total demand at the destinations and at the same time minimize total
shipping cost.
b.
minimize the number of origins used to satisfy total demand at the destinations.
c.
minimize the number of shipments necessary to satisfy total demand at the destinations.
Chapter 6 – Distribution and Network Models
d.
minimize the cost of shipping products from several origins to several destinations.
Transportation problem
25. The number of units shipped from origin i to destination j is represented by
a.
xij.
b.
xji.
c.
cij.
d.
cji.
Transportation problem
26. Which of the following is not true regarding the linear programming formulation of a transportation problem?
a.
Costs appear only in the objective function.
b.
The number of variables is (number of origins) x (number of destinations).
c.
The number of constraints is (number of origins) x (number of destinations).
d.
The constraints’ left-hand side coefficients are either 0 or 1.
Transportation problem
27. The difference between the transportation and assignment problems is that
a.
total supply must equal total demand in the transportation problem
b.
the number of origins must equal the number of destinations in the transportation problem
c.
each supply and demand value is 1 in the assignment problem
d.
there are many differences between the transportation and assignment problems
Assignment problem
28. In the general linear programming model of the assignment problem,
a.
one agent can do parts of several tasks.
b.
one task can be done by several agents.
c.
each agent is assigned to its own best task.
d.
one agent is assigned to one and only one task.
Assignment problem
29. The assignment problem is a special case of the
a.
transportation problem.
b.
transshipment problem.
Chapter 6 – Distribution and Network Models
c.
maximal flow problem.
d.
shortest-route problem.
Assignment problem
30. Which of the following is not true regarding an LP model of the assignment problem?
a.
Costs appear in the objective function only.
b.
All constraints are of the ≥ form.
c.
All constraint left-hand side coefficient values are 1.
d.
All decision variable values are either 0 or 1.
Assignment problem
31. The assignment problem constraint x31 + x32 + x33 + x34 ≤ 2 means
a.
agent 3 can be assigned to 2 tasks.
b.
agent 2 can be assigned to 3 tasks.
c.
a mixture of agents 1, 2, 3, and 4 will be assigned to tasks.
d.
there is no feasible solution.
Assignment problem
32. Arcs in a transshipment problem
a.
must connect every node to a transshipment node.
b.
represent the cost of shipments.
c.
indicate the direction of the flow.
d.
All of the alternatives are correct.
33. Constraints in a transshipment problem
a.
correspond to arcs.
b.
include a variable for every arc.
c.
require the sum of the shipments out of an origin node to equal supply.
d.
All of the alternatives are correct.
34. In a transshipment problem, shipments
a.
cannot occur between two origin nodes.
Chapter 6 – Distribution and Network Models
b.
cannot occur between an origin node and a destination node.
c.
cannot occur between a transshipment node and a destination node.
d.
can occur between any two nodes.
35. Consider a shortest route problem in which a bank courier must travel between branches and the main operations
center. When represented with a network,
a.
the branches are the arcs and the operations center is the node.
b.
the branches are the nodes and the operations center is the source.
c.
the branches and the operations center are all nodes and the streets are the arcs.
d.
the branches are the network and the operations center is the node.
Shortest-route problem
36. The shortest-route problem finds the shortest-route
a.
from the source to the sink.
b.
from the source to any other node.
c.
from any node to any other node.
d.
from any node to the sink.
Shortest-route problem
37. Consider a maximal flow problem in which vehicle traffic entering a city is routed among several routes before
eventually leaving the city. When represented with a network,
a.
the nodes represent stoplights.
b.
the arcs represent one way streets.
c.
the nodes represent locations where speed limits change.
d.
None of the alternatives is correct.
Maximal flow problem
38. We assume in the maximal flow problem that
a.
the flow out of a node is equal to the flow into the node.
b.
the source and sink nodes are at opposite ends of the network.
c.
the number of arcs entering a node is equal to the number of arcs exiting the node.
d.
None of the alternatives is correct.
Maximal flow problem
Chapter 6 – Distribution and Network Models
39. If a transportation problem has four origins and five destinations, the LP formulation of the problem will have
a.
5 constraints
b.
9 constraints
c.
18 constraints
d.
20 constraints
Transportation problem
40. Which of the following is not a characteristic of assignment problems?
a.
costs appear in the objective function only
b.
the RHS of all constraints is 1
c.
the value of all decision variables is either 0 or 1
d.
the signs of constraints are always <
Assignment problem
41. The network flows into and out of demand nodes are what makes the production and inventory application modeled in
the textbook a
a.
shortest-route model.
b.
maximal flow model.
c.
transportation model
d.
transshipment model
A production and inventory application
Subjective Short Answer
42. Write the LP formulation for this transportation problem.
Chapter 6 – Distribution and Network Models
X1A + X1B ≤ 100
X2A + X2B ≤ 200
X3A + X3B ≤ 150
X4A + X4B ≤ 50
X1A + X2A + X3A + X4A = 250
X1B + X 2B + X3B + X4B = 250
all Xij ≥ 0
43. Draw the network for this transportation problem.
Min
2XAX + 3XAY + 5XAZ+ 9XBX + 12XBY + 10XBZ
s.t.
XAX + XAY + XAZ ≤ 500
X BX + XBY + XBZ ≤ 400
XAX + XBX = 300
XAY + XBY = 300
XAZ + XBZ = 300
Xij ≥ 0
44. Canning Transport is to move goods from three factories to three distribution centers. Information about the move is
given below. Give the network model and the linear programming model for this problem.
Source
Supply
Destination
Demand
A
200
X
50
B
100
Y
125
C
150
Z
125
Shipping costs are:
Destination
Source
X
Y
Z
Chapter 6 – Distribution and Network Models
A
3
2
5
B
9
10
—
C
5
6
4
(Source B cannot ship to destination Z)
45. The following table shows the unit shipping cost between cities, the supply at each source city, and the demand at
each destination city. The Management Scientist solution is shown. Report the optimal solution.
Destination
Source
Terre Haute
Indianapolis
Ft. Wayne
South Bend
Supply
St. Louis
8
6
12
9
100
Evansville
5
5
10
8
100
Bloomington
3
2
9
10
100
Demand
150
60
45
45
TRANSPORTATION PROBLEM
*****************************
OBJECTIVE: MINIMIZATION
SUMMARY OF ORIGIN SUPPLIES
********************************
ORIGIN
SUPPLY
———-
———–
Chapter 6 – Distribution and Network Models
1
100
2
100
3
100
SUMMARY OF DESTINATION DEMANDS
***************************************
DESTINATION
DEMAND
——————-
————-
1
150
2
60
3
45
4
45
SUMMARY OF UNIT COST OR REVENUE DATA
*********************************************
FROM
TO DESTINATION
ORIGIN
1
2
3
4
———-
—–
—–
—–
—–
1
8
6
12
9
2
5
5
10
8
3
3
2
9
10
OPTIMAL TRANSPORTATION SCHEDULE
****************************************
SHIP
FROM
TO DESTINATION
ORIGIN
1
2
3
4
———-
—–
—–
—–
—–
1
0
10
45
45
2
100
0
0
0
3
50
50
0
0
TOTAL TRANSPORTATION COST OR REVENUE IS 1755
1
Transportation problem
46. After some special presentations, the employees of the AV Center have to move projectors back to classrooms. The
table below indicates the buildings where the projectors are now (the sources), where they need to go (the destinations),
and a measure of the distance between sites.
Destination
Source
Business
Education
Parsons Hall
Holmstedt Hall
Supply
Baker Hall
10
9
5
2
35
Tirey Hall
12
11
1
6
10
Arena
15
14
7
6
20
Demand
12
20
10
10
a. If you were going to write this as a linear programming model, how many decision variables would there be, and how
many constraints would there be?
The solution to this problem is shown below. Use it to answer the questions b – e.
Chapter 6 – Distribution and Network Models
TRANSPORTATION PROBLEM
*****************************
OPTIMAL TRANSPORTATION SCHEDULE
****************************************
FROM
TO DESTINATION
FROM ORIGIN
1
2
3
4
——————-
——
——
——
——
1
12
20
0
3
2
0
0
10
0
3
0
0
0
7
TOTAL TRANSPORTATION COST OR REVENUE IS 358
NOTE: THE TOTAL SUPPLY EXCEEDS THE TOTAL DEMAND BY 13
ORIGIN
EXCESS SUPPLY
———-
———————–
3
13
b. How many projectors are moved from Baker to Business?
c. How many projectors are moved from Tirey to Parsons?
d. How many projectors are moved from the Arena to Education?
e. Which site(s) has (have) projectors left?
a.
12 decision variables, 7 constraints
b.
12
c.
10
d.
0
e.
Arena
1
Transportation problem
47. Show both the network and the linear programming formulation for this assignment problem.
Task
Person
A
B
C
D
1
9
5
4
2
2
12
6
3
5
3
11
6
5
7
Chapter 6 – Distribution and Network Models
48. Draw the network for this assignment problem.
Min
10x1A + 12x1B + 15x1C + 25x1D + 11x2A + 14x2B + 19x2C + 32x2D
+ 18x3A + 21x3B + 23x3C + 29x3D + 15x4A + 20x4B + 26x4C + 28x4D
s.t.
x1A + x1B + x1C + x1D = 1
x2A + x2B + x2C + x2D = 1
x3A + x3B + x3C + x3D = 1
x4A + x4B + x4C + x4D = 1
x1A + x2A + x3A + x4A = 1
x1B + x 2B + x3B + x4B = 1
x1C + x 2C + x3C + x4C = 1
Let
Min
+ 12X2A + 6X2B + 3X2C + 5X2D
+ 11X3A + 6X3B + 5X3C + 7X3D
s.t.
X 3A + X3B + X3C + X3D ≤ 1
Chapter 6 – Distribution and Network Models
x1D + x2D + x3D + x4D = 1
1
Assignment problem
49. A professor has been contacted by four not-for-profit agencies that are willing to work with student consulting teams.
The agencies need help with such things as budgeting, information systems, coordinating volunteers, and forecasting.
Although each of the four student teams could work with any of the agencies, the professor feels that there is a difference
in the amount of time it would take each group to solve each problem. The professor’s estimate of the time, in days, is
given in the table below. Use the computer solution to see which team works with which project.
Projects
Team
Budgeting
Information
Volunteers
Forecasting
A
32
35
15
27
B
38
40
18
35
C
41
42
25
38
D
45
45
30
42
ASSIGNMENT PROBLEM
************************
OBJECTIVE: MINIMIZATION
SUMMARY OF UNIT COST OR REVENUE DATA
*********************************************
TASK
AGENT
1
2
3
4
———-
—–
—–
—–
—–
1
32
35
15
27
2
38
40
18
35
3
41
42
25
38
4
45
45
30
42
OPTIMAL ASSIGNMENTS COST/REVENUE
************************ ***************
ASSIGN AGENT 3 TO TASK 1
41
ASSIGN AGENT 4 TO TASK 2
45
ASSIGN AGENT 2 TO TASK 3
18
ASSIGN AGENT 1 TO TASK 4
27