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 #5 solutions / IEOR4405
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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,
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 #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
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
Homework #1 solutions / IEOR4405
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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
Application Example
In order to make the PTAS heuristic more concrete, we came up with one example that would fit the
requirements for using the PTAS and still be a realistic representation of real emergency room situations
m=10 doctors
= 0.5
45 jobs
5 d
Scheduling
Parallel Machine Scheduling
Tim Nieberg
Parallel machine models: Makespan Minimization
Problem P |Cmax :
m machines
n jobs with processing times p1 , . . . , pn
Parallel machine models: Makespan Minimization
Problem P |Cmax :
m machines
n jobs
Department of Computer & Information Science
Technical Reports (CIS)
University of Pennsylvania
Year 2000
A PTAS for Minimizing Average
Weighted Completion Time with Release
Dates on Uniformly Related Machines
Chandra Chekuri
Bell
Sanjeev Khanna
Laborato
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
Homework #4 solutions / IEOR4405
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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
Homework #6 solutions / IEOR4405
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1 Consider the following jobs (a, b, c, d) with the processing times (7, 8, 9, 11). The algorithm to solve is LRPT-FM. Denote the machines with speeds 1,2,4 as M1, M2 and M3 respectively. Machine/Job present at each time
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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 /
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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
1
Homework #9 solutions / IEOR4405
6.12. Suppose that SPT(1) LPT(m) is not optimal and another type of schedule is optimal. In this
optimal schedule, there must be a pair of adjacent jobs, say job j followed by job k , that satises
p1,j p1,k and so pm,k p
Homework #9 solutions / IEOR4405
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9.5. There are a total of 23 = 8 scenarios, but they can be combined into four groups with the following probabilities: (i) (ii) (iii) (iv)
1 All three processing times are 0.5. This happens with probability 8 . Two proc
Homework #2 solutions / IEOR4405
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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 #4 solutions / IEOR4405
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3.5. We solve problem 3.4 initially and then infer the result for problem 3.5 We use Algorithm 3.2.1 for this problem. Notice that each of the functions hj (Cj ) is increasing in Cj , and hence it is appropriate to use t
Homework #4 solutions / IEOR4405
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3.5. We solve problem 3.4 initially and then infer the result for problem 3.5
We use Algorithm 3.2.1 for this problem. Notice that each of the functions hj (Cj ) is increasing in
Cj , and hence it is appropriate to use t