ORIE 3300/5300 ASSIGNMENT 4 SOLUTION Fall 2008
Problem 2
(b) If we consider the minimum number of employees needed for each day
as OUTPUT in the general blending model, the INPUT are diﬀerent types
of employees with respect to diﬀerent schedules. There are a total of 10
diﬀerent types as we choose four days out of ﬁve weekdays, and one day out
of two weekend day (see data ﬁle
schedule.dat
for details). The minimum
input of each type is 0 and the maximum is any number larger than the total
sum of requirement. The minimum output for Monday through Sunday is:
45, 45, 40 ,50, 65, 35, 35, and the maximum is also any number larger than
the total sum of requirement. Then the output j produced by input i is 1 if
input i work on that output day j, and 0 otherwise. Last, the cost for each
input is 1 as we are counting the total number of employees needed.
(a) Using the model ﬁle
blend.mod
and the data ﬁle
schedule.dat
to solve
using AMPL, we get the minimum total employees needed is 70. The num
ber of employees needed for each type is (see data ﬁle for details of each
type): fn=0, rn=15, wn=20, tn=0, mn=0, fs=5, rs=5, ws=0, ts=0, ms=25.
(Note: it’s easy to verify minimality of this solution since one needs at least
35 + 35 = 70 to satisfy Sat and Sun requirements.)
Problem 3
(a) Since ¯
c
3
= 1
>
0, ¯
c
5
= 2
>
0, and ¯
c
1
=

2
<
0, only
x
3
and
x
5
can
enter.
If
x
3
enters,
¯
t
=
min(
¯
b
4
¯
a
43
,
¯
b
6
¯
a
63
)
=
min(
2
4
,
1
2
)
=
min(
1
2
,
1
2
)
=
1
2
so either
x
4
or
x
6
can leave if
x
3
enters.
If
x
5
enters,
¯
t
=
min(
¯
b
2
¯
a
52
,
¯
b
4
¯
a
54
)
=
min(
3
2
,
2
1
)
=
3
2
1
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x
2
must leave if
x
5
enters.
Therefore, all possible pairs are (3,4), (3,6), and (5,2).
(b) In general, increasing or decreasing of the objective value depends on
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 Fall '08
 TODD

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