CD-ROM MODULE 2
Dynamic Programming
TRUE/FALSE
M2.1
Dynamic programming can be applied to a professional tennis player’s serving strategy.
M2.2
Each item in a knapsack problem will be a stage of the dynamic programming problem.
M2.3
For knapsack problems, sn-1 = an  sn + bn  dn + cn is a typical transformation expression.
M2.4
The problem that NASA has in determining what types of cargo may be loaded on the space
shuttle is an example of a knapsack problem.
M2.5
Both dynamic programming and linear programming take a multi-stage approach to solving
problems.
M2.6 The second step in solving a dynamic programming problem is to solve the last stage of
the problem for all conditions or states.
M2.7
Subproblems in a dynamic programming problem are called stages.
M2.8
In a shortest-route problem, the nodes represent the destinations.
M2.9
In a shortest-route problem, the limit on the number of allowable decision variables from one
node to another is the number of possible nodes to which one might yet travel.
M2.10
Your local paperperson could make use of the shortest-route technique.
*M2.11 Linear programming is typically applied to problems wherein one must make a decision at a
specified point (or points) in time. Dynamic programming is typically applied to problems
wherein one must make a sequence of decisions.
*M2.12 Dynamic programming can only be used to solve network-based problems.
*M2.13 In dynamic programming, the decision rules defining an optimal policy give optimal decisions
for any entering condition at any stage.
*M2.14 In dynamic programming, there is a state variable defined for every stage.
*M2.15 A transformation changes the identities of the state variables.
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Dynamic Programming  CD-ROM MODULE 2
MULTIPLE CHOICE
M2.16
There are six cities (City 1 City 6) serviced by a particular airline. Limited routes are
available, and the distance for each of these routes is presented in the table below.
From
City
1
1
2
2
3
3
4
5
To
City
2
3
4
5
4
5
6
6
Distance
(100s miles)
4
2
6
4
4
7
3
2
If dynamic programming were used to solve for the minimum distance from City 1 to City 6,
how many stages would there be?
(a)
(b)
(c)
(d)
(e)
6
5
4
3
2
M2.17 There are three items (A, B, and C) that are to be shipped by air freight. The weights
of these are 4, 5, and 3 tons, respectively, and the plane can carry 13 tons. The profits (in
thousands of dollars) generated by these are 3 for A, 4 for B, and 2 for C. There are four units
of each available for shipment. If this were to be solved as a dynamic programming problem,
how many stages would there be?
(a)
(b)
(c)
(d)
(e)
M2.18
1
2
3
4
none of these
There are three items (A, B, and C) that are to be shipped by air freight. The weights of these
are 4, 5, and 3 tons, respectively. The profits (in thousands of dollars) generated by these are 3
for A, 4 for B, and 2 for C. There are four units of each available for shipment. Only 12 tons
may be loaded on the plane. The maximum possible profit for this would be
(a)
(b)
(c)
(d)
(e)
7.
8.
9.
10.
none of these
560
Dynamic Programming  CD-ROM MODULE 2
M2.19
There are three items (A, B, and C) that are to be shipped by air freight. The weights of these
are 4, 5, and 3 tons, respectively. The profits (in thousands of dollars) generated by these are 6
for A, 7 for B, and 5 for C. A total of 14 tons may be carried by the plane. There are four
units of each available for shipment. What is the maximum possible profit for this situation?
(a)
(b)
(c)
(d)
(e)
14
20
21
22
none of these
561
Dynamic Programming  CD-ROM MODULE 2
M2.20
The following information describes a shortest-route problem with the distance in miles. How
many stages will this dynamic problem have?
From
1
1
1
1
2
2
3
(a)
(b)
(c)
(d)
(e)
To
2
3
4
5
6
7
6
Distance
40
22
66
39
18
24
23
From
3
4
4
5
6
7
To
7
6
7
7
8
8
Distance
20
43
33
66
72
58
8
4
3
2
1
The data below is a dynamic programming solution for a shortest route problem.
Stage 1
s1
6
5
Stage 2
s2
4
3
2
Stage 3
M2.21
s0
7
7
f0
0
0
f1
12
8
d2
46
36
35
26
25
r2
5
4
13
9
4
s1
6
6
5
6
5
f1
12
12
8
12
8
f2
17
16
21
21
12
d3
14
13
12
r3
7
7
10
s2
4
3
2
f2
17
16
12
f3
24
23
22
Using the data in Table M2-1, determine the minimum distance from point 1 to point 7.
(a)
(b)
(c)
(d)
(e)
M2.22
s3
1
Table M2-1
d1
r1
12
67
8
57
21
22
23
24
75
Using the data in Table M2-1, determine the distance of stage 1 for the optimal route.
562
Dynamic Programming  CD-ROM MODULE 2
(a) 0
(b) 8
(c) 12
(d) 16
(e) 24
ANSWER: b
M2.23
Using the data in Table M2-1, determine the distance of stage 2 for the optimal route.
(a)
(b)
(c)
(d)
(e)
M2.24
Using the data in Table M2-1, determine the distance of stage 3 for the optimal route.
(a)
(b)
(c)
(d)
(e)
M2.25
5
6
7
67
57
Using the data in Table M2-1, determine the optimal arc of stage 2.
(a)
(b)
(c)
(d)
(e)
M2.27
22
23
24
7
10
Using the data in Table M2-1, determine the optimal arc of stage 1.
(a)
(b)
(c)
(d)
(e)
M2.26
0
4
8
12
21
46
36
35
26
25
What is the optimal arc of stage 3?
(a)
(b)
(c)
(d)
1
14
13
12
563
Dynamic Programming  CD-ROM MODULE 2
(e) none of the above
M2.28
What is the optimal travel path from point 1 to 7?
(a)
(b)
(c)
(d)
(e)
M2.29
According to Table M2-2, which gives a solution to a shortest route problem solved with
dynamic programming, which cities would be included in the best route?
(a)
(b)
(c)
(d)
(e)
M2.30
5, 7
6, 7
1, 2, 6, 7
1, 2, 5, 7
1, 3, 6, 7
1,2,3,4,5,6
1,4,6,7
1,2,5,6,7
6,7
none of the above
According to Table M2-2, which gives a solution to a shortest route problem solved with
dynamic programming, the total distance from City 1 to City 7 is 14. What is the shortest
distance from City 3 to City 7?
(a)
(b)
(c)
(d)
(e)
7
10
13
25
none of these
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Dynamic Programming  CD-ROM MODULE 2
M2.31
There are six cities (City 1 City 6) serviced by a particular airline. Limited routes are
available, and the distances for each of these routes are presented in the table below.
From
City
1
1
2
2
3
3
4
5
M2.32
To
City
2
3
4
5
4
5
6
6
Distance
(100s miles)
4
2
6
4
4
7
3
2
What is the minimum distance that must be traveled to get from City 1 to City 6?
(a) 9
(b) 10
(c) 11
(d) 12
(e) none of these
M2.33
A stage is a(n)
(a)
(b)
(c)
(d)
(e)
alternative.
policy.
condition at the end of the problem.
subproblem.
none of the above
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Dynamic Programming  CD-ROM MODULE 2
M2.34 A transformation describes
(a)
(b)
(c)
(d)
(e)
the relationship between stages.
the initial condition of the system.
a stage.
a state variable.
none of the above
The following information is to be used for questions M2.35 - M2.38:
There are seven cities (City 1 -- City 7) served by Acme Trucking. Route availability is
limited. The distances, in hundreds of miles, are given in the table below for each route.
From
City
1
1
2
2
2
3
M2.35
Distance
5
7
4
6
9
6
From
City
3
4
4
5
5
6
To
City
5
5
6
6
7
7
Distance
9
8
12
10
15
8
What is the minimum distance a load being moved from City 1 to City 7 must travel?
(a)
(b)
(c)
(d)
(e)
M2.36
To
City
2
3
3
4
5
4
3000 miles
2900 miles
1500 miles
2700 miles
none of the above
What route should the truck from City 1 to City 7 take?
(a)
(b)
(c)
(d)
(e)
1-2, 2-5, 5-7
1-3, 3-4, 4-6, 6-7
1-2, 2-4, 4-5, 5-6, 6-7
1-3, 3-5, 5-6, 6-7
none of the above
566
Dynamic Programming  CD-ROM MODULE 2
M2.37
If the truck was required to take the route from City 4 to City 5, what would be the shortest
distance from City 1 to City 7?
(a)
(b)
(c)
(d)
(e)
M2.38
2900 miles
3200 miles
3700 miles
3400 miles
none of the above
If the truck was required to take the route from City 4 to City 5, what would be the overall
route?
(a)
(b)
(c)
(d)
(e)
1-3, 3-4, 4-5, 5-6, 6-7
1-2, 2-4, 4-5, 5-7
1-2, 2-3, 3-4, 4-5, 5-6, 6-7
1-3, 3-5, 5-6, 6-7
none of the above
The following information is to be used for questions M39 – M40:
GATRA, the Greater Attleboro-Taunton Regional Transit Authority serves six cities (City 1
City 6). While there are many restrictions (primarily roads on which they may not travel), they
do have some choice of routes. The distances between cities, along permitted routes, are
presented below.
From
1
1
2
2
2
M2.39
To
2
3
3
4
5
Distance
3
4
7
12
28
From
3
3
4
4
5
To
4
5
5
6
6
Distance
8
22
12
15
2
What is the minimum distance that must be traveled to get from City 1 to City 6?
(a)
(b)
(c)
(d)
(e)
26
9
11
3
none of these
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Dynamic Programming  CD-ROM MODULE 2
M2.40
What is the shortest route?
(a)
(b)
(c)
(d)
(e)
1-3, 3-5, 5-6
1-2, 2-3, 3-4, 4-5, 5-6
1-3, 3-4, 4-5, 5-6
1-2, 2-3, 3-5, 5-6
none of the above
*M2.41 There are four items (A, B, C, and D) that are to be shipped by truck. The weights of these are
3, 7, 4, and 5 tons, respectively, and the plane can carry 13 tons. The profits (in thousands of
dollars) generated by these are 3 for A, 4 for B, 2 for C, and 5 for D. There are four units of
each available for shipment. If this were to be solved as a dynamic programming problem, how
many stages would there be?
(a)
(b)
(c)
(d)
(e)
1
2
3
4
none of these
*M2.42 There are four items (A, B, C, and D) that are to be shipped by truck. The weights of these are
3, 7, 4, and 5 tons, respectively, and the plane can carry 13 tons. The profits (in thousands of
dollars) generated by these are 3 for A, 4 for B, 2 for C, and 5 for D. There are three units of
each available for shipment. The maximum possible profit for this would be
(a)
(b)
(c)
(d)
(e)
$7
$11
$9
$10
none of these
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Dynamic Programming  CD-ROM MODULE 2
*M2.43 The following information describes a shortest-route problem with the distance in miles. How
many stages will this dynamic programming problem have?
Table M2-3
From
To
1
1
1
2
2
2
3
(a)
(b)
(c)
(d)
(e)
2
3
5
4
5
6
5
Distance
in miles
40
12
39
17
24
33
20
From
To
3
4
4
5
6
7
6
7
6
8
8
8
Distance
in miles
49
23
76
27
53
35
8
4
3
2
1
*M2.44 For the shortest route problem described in Table M2-3, what is the distance for the shortest
route?
(a)
(b)
(c)
(d)
(e)
155 miles
66 miles
59 miles
114 miles
none of the above
*M2.45 For the shortest route problem described in Table M2-3, which arcs comprise the shortest
route?
(a)
(b)
(c)
(d)
(e)
1-2, 2-6, 6-8
1-5, 5-8
1-2, 2-6, 6-8
1-3, 3-5, 5-8
none of the above
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Dynamic Programming  CD-ROM MODULE 2
*M2.46 For the shortest route problem described in Table M2-4, what is the length of the shortest
route?
(a) 205 miles
(b) 94 miles
(c) 241 miles
(d) 108 miles
(e) none of the above
*M2.47 For the shortest route problem described in Table M2-4, which arcs comprise the shortest
route?
(a)
(b)
(c)
(d)
(e)
1-2, 2-6, 6-8
1-3, 3-5, 5-8
1-5, 5-8
1-2, 2-4, 4-7, 7-8
none of the above
570
Dynamic Programming  CD-ROM MODULE 2
*M2.48 Using the data in Table M2-5, determine the minimum distance from point 1 to point 7.
(a)
(b)
(c)
(d)
(e)
18
17
23
14
none of the above
*M2.49 Using the data in Table M2-5, determine the optimal distance of stage 1.
(a)
(b)
(c)
(d)
(e)
0
8
7
14
none of the above
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Dynamic Programming  CD-ROM MODULE 2
*M2.50 Using the data in Table M2-5, determine the optimal distance of stage 2.
(a)
(b)
(c)
(d)
(e)
10
7
8
11
none of the above
*M2.51 Using the data in Table M2-5, determine the optimal distance of stage 3.
(a)
(b)
(c)
(d)
(e)
22
17
24
7
none of the above
*M2.52 Using the data in Table M2-5, determine the optimal arc of stage 1.

(a)
(b)
(c)
(d)
(e)
3 
6 

47
none of the above
*M2.53 Using the data in Table M2-5, determine the optimal arc of stage 2.
(a)
(b)
(c)
(d)
(e)
46
56
45
25
none of the above
*M2.54 Using the data in Table M2-5, determine the optimal arc of stage 4.
(a)
(b)
(c)
(d)
(e)
1 
24
13
34
none of the above
*M2.55 Using the data in Table M2-5, determine the optimal travel path from point 1 to 7.
(a) 1 
(b) 1 
(c) 1 
572
Dynamic Programming  CD-ROM MODULE 2
(d) 1 
(e) none of the above
Table M2-6
Stage 1 s1
5
2
d1
56
26
r 1 s0
6 6
9 6
f0
0
0
f1
6
9
Stage 2
s2
4
3
d2
45
35
r 2 s1
8 5
3 5
f1
6
9
f2
14
12
Stage 3
s3
1
d3
14
13
12
r 3 s2
7 4
5 3
7 2
f2
14
12
9
f3
21
17
16
*M2.56 According to Table M2-6, which gives a solution to a shortest route problem solved with
dynamic programming, which routes would comprise the optimal?
(a)
(b)
(c)
(d)
1-3, 3-5, 5-6
1-2, 2-6
1-4, 4-5, 5-6
none of the above
*M2.57 According to Table M2-6, which gives a solution to a shortest route problem solved with
dynamic programming, the total distance from City 1 to City 6 is 16. What is the shortest
distance from City 3 to City 6?
(a)
(b)
(c)
(d)
(e)
7
14
6
10
none of these
573
Dynamic Programming  CD-ROM MODULE 2
*M2.58 According to Table M2-6, which gives a solution to a shortest route problem solved with
dynamic programming, which cities are visited along the optimal route?
(a)
(b)
(c)
(d)
(e)
1, 2, and 6
1, 5, and 6
1, 3, 5, and 6
1, 4, 5, and 6
none of the above
*M2.59 What is the minimum distance that must be traveled to get from City 1 to City 6?
(a)
(b)
(c)
(d)
(e)
900 miles
700 miles
1,100 miles
1,200 miles
one of these
*M2.60 Which routes should be traveled?
(a)
(b)
(c)
(d)
(e)
1-2, 2-3, 3-5, 5-6
1-2, 2-3, 3-4, 4-6
1-2, 2-5, 5-6
1-4, 4-6
none of the above
574
Dynamic Programming  CD-ROM MODULE 2
PROBLEMS
M2.61
Develop the shortest-route network for the problem below, and determine the minimum
distance from node 1 to node 6.
From
1
1
2
2
3
3
4
5
M2.62
Distance
4
3
6
7
4
5
6
4
Develop the shortest-route network for the problem below, and determine the minimum
distance from node 1 to node 7.
From
1
1
2
2
3
3
3
4
5
6
M2.63
To
2
3
4
5
4
5
6
6
To
2
3
4
5
4
5
6
7
7
7
Distance
4
2
6
4
4
7
6
3
5
4
There are four items (A, B, C, and D) that are to be shipped by air freight. The weights of these
are 3, 4, 5, and 3 tons, respectively. The profits (in thousands of dollars) generated by these are
5 for A, 6 for B, 7 for C, and 6 for D. There are 2 units of A, 1 unit of B, 3 units of C, and 2
units of D available to be shipped. The maximum weight is 16 tons. Use dynamic
programming to determine the maximum possible profits that may be generated.
575
Dynamic Programming  CD-ROM MODULE 2
M2.64
M2.65
The data below details the distances that a delivery service must travel. Use dynamic
programming to solve for the shortest route from City 1 to City 8.
From
To
1
1
1
2
2
3
3
2
3
4
5
6
5
6
Distance
(miles)
18
14
16
9
8
7
6
From
To
4
4
4
5
6
7
5
6
7
8
8
8
Distance
(miles)
8
6
5
17
16
20
Hard D. Head has decided that he wants to climb one of the world's tallest mountains. He has
mapped out a number of routes between various points on the mountain, and rated each route as
to difficulty. His rating scale considers a 1 as being particularly easy, and a 10 as being almost
impossible.
(a) Given the information below, identify the route that would provide the easiest climb.
(b) What would be the average rating of the route?
(c) What is wrong with this approach to Mr. Head's problem?
From
Point
1
1
2
3
3
4
4
4
5
To
Point
2
3
3
4
5
5
6
8
6
Rating
From
Point
5
5
5
6
6
7
7
8
3
4
6
2
6
4
7
8
4
576
To
Point
7
8
9
7
8
8
9
9
Rating
5
8
10
7
7
5
6
9
Dynamic Programming  CD-ROM MODULE 2
SHORT ANSWER/FILL IN THE BLANK
M2.66
In a shortest-route problem, write a typical transformation expression.
M2.67
What is meant by a decision variable in a shortest-route problem?
M2.68
What is the decision criterion for a shortest route problem?
M2.69
What is the decision criterion for a knapsack problem?
M2.70
What are the four steps in dynamic programming?
M2.71
What are the four elements defining each stage in a dynamic programming problem?
M2.72
Identify two types of problems that can be solved by dynamic programming.
M2.73
Discuss, briefly, the role of the transformation function.
M2.74
Discuss, briefly, the difference between a decision variable and a state variable.
M2.75
In the shortest-route problem, circles represent ________, and arrows represent ______.
MATCHING
M2.76
Stage
(a)
Relationship between stages
M2.77
Transformation
(b)
Subproblem
M2.78
Decision variable
(c)
Conditions of a stage
M2.79
Decision criterion
(d)
Alternatives
M2.80
State variable
(e)
Objective
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Dynamic Programming  CD-ROM MODULE 2
578