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Dynamic Programming
Solutions 1
1) The owner of a chain of three grocery stores has purchased 5 crates of fresh
strawberries. The estimated probability distribution of potential sales of the
strawberries before spoilage differs among the three stores. Therefore, the
owner wants to know how to allocate 5 crates to the three stores to maximize
expected profit.
For administrative reasons, the owner does not wish to split crates
between stores. However, he is willing to distribute no crates to any of his
stores. The following table gives estimated expected profit at each store when
it is allocated various numbers of crates.
Crates
0
1
2
3
4
5
Soln:
1
0
5
9
14
17
21
Store
2
0
6
11
15
19
22
3
0
4
9
13
18
20
Dynamic Programming
Solutions 1
2) A college student has 7 days remaining before final examinations begin in
her four courses and she wants to allocate this study time as effectively as
possible. She needs at least 1 day on each course, and she likes to concentrate
on just one course each day, so she wants to allocate 1, 2, 3, or 4 days to each
course. Having recently taken an O.R. course, she decides to use dynamic
programming to make the allocations so as to maximize the total grade points.
Study Days
1
2
3
4
Soln:
1
3
5
6
7
Grade Points per Course
2
3
5
2
5
4
6
7
9
8
4
6
7
9
9
Dynamic Programming
Solutions 1
3) Consider an electronic system consisting of four components, each of
which must work for the system to function. The reliability of the system can
be improved by installing several parallel units in one or more components.
The following table gives the probability that the respective components will
function if they consist of one, two, or three parallel units:
Units
1
2
3
Probability of Functioning
Comp 1
Comp 2
Comp 3
Comp 4
0.5
0.6
0.7
0.5
0.6
0.7
0.8
0.7
0.8
0.8
0.9
0.9
The probability that the system will function is the product of the probabilities
that the respective components will function. The cost (in $100’s) of installing
1, 2, or 3 parallel units in the respective components is given by the following
table: (total budget = $1,000).
Units
1
2
3
Soln:
Comp 1
1
2
3
Cost
Comp 2
Comp 3
2
1
4
3
5
4
Comp 4
2
3
4