Chapter 10Solutions 1

1
Chapter 10 Solutions 1

• 10.3.1) 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.

Soln
Let xn number crates allocated to a store
pn(xn) expected profit from allocating xn
crates to store n sn number of crates
remaining fn(sn) max pn(xn)
fn1(sn-xn)
2
Chapter 10 Solutions 1

• 10.3.1)

Soln
Let xn number crates allocated to a store
pn(xn) expected profit from allocating xn
crates to store n sn number of crates
remaining fn(sn) max pn(xn)
fn1(sn-xn)
3
Chapter 10 Solutions 1

• 10.3.1)

Soln
Let xn number crates allocated to a store
pn(xn) expected profit from allocating xn
crates to store n sn number of crates
remaining fn(sn) max pn(xn)
fn1(sn-xn)
Store 1 Store 2 Store
3
4
Chapter 10 Solutions 1

• 10.3.1)

Soln
Let xn number crates allocated to a store
pn(xn) expected profit from allocating xn
crates to store n sn number of crates
remaining fn(sn) max pn(xn)
fn1(sn-xn)
Store 1 Store 2 Store
3
5
Chapter 10 Solutions 1

• 10.3.1)

Store 1 Store 2 Store
3
6
Chapter 10 Solutions 1

• 10.3.1)

Store 1 Store 2 Store
3
Optimal Allocation
Optimal path
Optimal path from a given state
7
Chapter 10 Solutions 1

• 10.3.2) 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.

Soln
Let xn number days allocated to a course
pn(xn) grade points earned from allocating xn
days to course n sn number of days
remaining fn(sn) max pn(xn)
fn1(sn-xn)
Course 1 Course 2 Course 3
6
5
5
4
4
3
3
2
8
Chapter 10 Solutions 1

• 10.3.2)

Soln
Let xn number days allocated to a course
pn(xn) grade points earned from allocating xn
days to course n sn number of days
remaining fn(sn) max pn(xn)
fn1(sn-xn)
Course 1 Course 2 Course 3
Course 4
6
5
5
4
4
4
3
3
3
2
2
9
Chapter 10 Solutions 1

• 10.3.2)

Soln
Let xn number days allocated to a course
pn(xn) grade points earned from allocating xn
days to course n sn number of days
remaining fn(sn) max pn(xn)
fn1(sn-xn)
Course 1 Course 2 Course 3
Course 4
6
5
5
4
4
4
3
3
3
2
2
10
Chapter 10 Solutions 1

• 10.3.2)

Soln
Let xn number days allocated to a course
pn(xn) grade points earned from allocating xn
days to course n sn number of days
remaining fn(sn) max pn(xn)
fn1(sn-xn)
Course 1 Course 2 Course 3
Course 4
6
5
5
4
4
4
3
3
3
2
2
11
Chapter 10 Solutions 1

• 10.3.2)

Soln
Let xn number days allocated to a course
pn(xn) grade points earned from allocating xn
days to course n sn number of days
remaining fn(sn) max pn(xn)
fn1(sn-xn)
Course 1 Course 2 Course 3
Course 4
6
5
5
4
4
4
3
3
3
2
2
12
Chapter 10 Solutions 1

• 10.3.7) 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
• The probability that the system will function is
  the product of the probabilities that the
  respective components will function. The cost
  (in 100s) of installing 1, 2, or 3 parallel
  units in the respective components is given by
  the following table (total budget 1,000).

Soln
Let xn number parallel units for component n
pn(xn) probability component will function
with xn parallel units cn(xn) cost of
installing xn units sn number of in
100s remaining fn(sn) max pn(xn)
fn1(sn-cn(xn)
13
Chapter 10 Solutions 1

• 10.3.7)

Soln
Let xn number parallel units for component n
pn(xn) probability component will function
with xn parallel units cn(xn) cost of
installing xn units sn number of in
100s remaining fn(sn) max pn(xn)
fn1(sn-cn(xn)
14
Chapter 10 Solutions 1

• 10.3.7)

Soln
15
Chapter 10 Solutions 1

• 10.3.7)

Soln
16
Chapter 10 Solutions 1

• 10.3.7)

Soln