1
Homework #2 solutions / IEOR4405
3.1. (a) By Theorem 3.1.1 in the textbook, the WSPT (weighted shortest processing time) rule is
optimal for 1| wj Cj . In order to apply this rule, we have to calculate the values of wj /pj .
These values are given below
1
Homework #7 solutions /
1. Consider the following network.
(a) The unique critical path is 1-4-8-9. It has length 17. So Cmax = 17.
(b) The minimal cuts in the graph are cfw_2, 3, 4, cfw_2, 8, cfw_3, 4, 5, 6, cfw_3, 4, 6, 7, cfw_5, 8, cfw_7, 8.
(c) The
1
Homework #1 solutions / IEOR4405
3.1. (a) By Theorem 3.1.1 in the textbook, the WSPT (weighted shortest processing time) rule is
optimal for 1| wj Cj . In order to apply this rule, we have to calculate the values of wj /pj .
These values are given below
Homework #5 solutions / IEOR4405
1
1. Let f (j, t) denote the maximum weight set of jobs that can be completed by time t, using jobs 1, 2, ., j (with j = 1, 2, ., 4 and t = 1, 2, ., 10). We use the recurrence f (j, t) = max(f (j 1, t), f (j, t 1), f (j 1,
Homework #7 solutions / IEOR4405
1
7.1. Time 0: We have:
Machine
M1
M2
M3
M4
Workload
10 + 3 + 4 = 17
8 + 8 + 7 = 23
4 + 6 = 10
5+3=8
Hence we begin by scheduling Machine 2 at time 0. The only available job is Job 2, so we
schedule Job 2 on Machine 2. The
Homework #1 solutions / IEOR4405
1
1. Problem 2.4
(a) For the problem 1|Tj , one needs to minimize the total tardiness. Tj is dened as max(Cj
dj ,0). The optimal value (minimum) is Tj = 22. This can be achieved by the schedule 1-2-4-3.
Since, the number
IEOR E4405 Homework 4
Prof. Cliff Stein
1. First notice that no optimal schedule will schedule jobs 4,5,6, or 7 first, due to their release
dates. We compute lower bounds on Lmax using preemptive EDD in the cases when we
schedule jobs 1, 2, or 3 first:
Pa
IEOR E4405 Homework 5
Prof. Cliff Stein
1. We use the DP relation shown in class: f (j, t) = maxcfw_f (j 1, t), f (j, t 1), f (j 1, t
pj ) + wj . Using this relation, we get the following values:
For j = 1, we have:
f (1, 1) = . . . = f (1, 10) = 3.
For
Homework 7
Prof. Cliff Stein
1. Problem 5.4). Without the due date constraint, the optimal schedule is by SRPT. Here we
follow the same procedure but we must schedule a job when it becomes tight, meaning
scheduling it later would make it miss its deadline
Homework 8
Prof. Cliff Stein
1. 6.2). Let pij be given as in Exercise 6.1. Our decision variables are xjk , Iik and Wik where
xjk = 1 if job j is the k th job in the sequence and 0 otherwise, Iik is the idle time on
machine i between the processing times
IEOR E4405 Homework 3
Prof. Cliff Stein
P
1. The proof is analogous to 1|chains| wj Cj .
2. The preemptive EDD rule is the optimal algorithm for 1|rj , prmp|Lmax . Every time a job
is released, we find the job with the earliest due date and schedule it.
T
Midterm Solutions
Prof. Cliff Stein
1. (a) NPC
(b) LCL
(c) LRPT-FM
(d) NPC
2. (a) Shortest processing time produces the optimal schedule. Assume not. Then there
must be some optimal schedule S which does not follow SPT. It follows there must
be two consec
IEOR E4405 Homework 1
Prof. Cliff Stein
1. We want to even out the workload of each machine as much as possible. Note that we
cant achieve a Cmax better than the average workload per machine rounded up (since all
processing times are integers).
For two ma
IEOR E4405 Homework 2
Prof. Cliff Stein
1.
jobs
wj
pj
wj
pj
1
0
3
0
2
18
6
3
3
12
6
2
4
8
5
1.6
5
8
4
2
6
17
8
2.125
7
16
9
1.778
(a) The optimal sequence is by WSPT. In this case, since jobs 3 and 5 have the same
value for wj /pj , there are two optimal
Homework #4 solutions / IEOR4405
1
4.11 Let us assume that S is an optimal schedule. Let us also assume that at time t0 , S starts to
process the rst job and that it has a rst idle period during period of time [t1 , t2 ]. Let us assume
t0 < t1 < t2 . Give
Chapter 20
Auctions for the Safe, Efficient and Equitable Allocation of
Airspace System Resources
Michael Ball, George L. Donohue, Karla Hoffman
1. Introduction
Most countries attempt to design their air transportation system so that it is
economically vi
1
Homework #6 solutions /
1. Consider the following network.
(a) The unique critical path is 1-4-8-9. It has length 17. So Cmax = 17.
(b) The minimal cuts in the graph are cfw_2, 3, 4, cfw_2, 8, cfw_3, 4, 5, 6, cfw_3, 4, 6, 7, cfw_5, 8, cfw_7, 8.
(c) The
1
Homework #9 solutions / IEOR4405
1. We have, for j = 1, 2, 3,
3 w.p. 2/3
6 w.p. 1/3
Xj =
and
dj =
4
10
w.p. 1/4
w.p. 3/4.
(a) If we use a static list policy, we need to decide at time t = 0 in what order we will run the
jobs. As the jobs are indistingui
Homework #8 solutions /
1
6.1 Draw the graph as in page 153 of the text book. Optimal Makespan is 48. critical path:
p1,j1 > p2,j1 > p2,j2 > p3,j2 > p3,j3 > p3,j4 > p4,j4 > p4,j5
6.2 Let xjk , Iik and Wik have its usual meaning. Then the formulation is
mi