Homework 9 Solution
April 14, 2009
Exercise 3, page 352
Suppose you have 7 full water bottles, 7 half-full, and 7 empty. You would like to
divide the 21 bottles among three individuals so that each receives 7 bottles with
the same quantity of water. Express the problem as integer programming and find
an optimal solution.
Solution
: Let
x
ij
denote the number of bottles of type
i
assigned to individual
j
= 1
,
2
,
3, where
i
= 1 denotes a full bottle,
i
= 2 denotes a half-full and
i
= 3
denotes an empty bottle. We have
x
ij
≥
0 and integer for all
i
and
j
.
The fact that there are 7 bottles of each type is modeled as follows:
x
i
1
+
x
i
2
+
x
i
3
= 7
for all
i
= 1
,
2
,
3
.
The total available amount of water is 10 and 1
/
2, which gives a share of 3 and
1
/
2 bottle with water to each. The fact that we want to divide the amount of water
equally among the individuals is modeled as
x
1
j
+
x
2
j
/
2 = 3
.
5
for all
j
= 1
,
2
,
3
.
The fact that each individual receives 7 bottles is modeled as follows:
x
1
j
+
x
2
j
+
x
3
j
= 7
for all
j
= 1
,
2
,
3
.
To complete the formulation, we may introduce the dummy objective of minimizing
∑
3
i,j
=1
0
x
ij
, so the problem is actually integer programming feasibility problem.
The solution is no unique. One possible solution is that individual 1 receives 3
full bottles, 1 half-full and 3 empty bottles. Individual 2 receives the same, i.e., 3
full bottles, 1 half-full and 3 empty bottles. Individual 3 receives 1 full, 5 half-full
and 1 empty bottle.
Exercise 7, page 352
You have the following three-letter words: AFT, FAR, TVA, ADV, JOE, FIN, OSF,
and KEN. Each letter in the alphabet is assigned a value, staring with A=1 to Z=26.
Each word is scored by summing the numeric values of its letters.
For example,
AFT has score of 1+6+20=27. You are to select five of the given eight words that
yield the maximum total score. The selected five words should satisfy the following
constraints:
sum of letter 1 scores
<
sum of letter 2 scores
<
sum of letter 3 scores
.
1