dvir shabtay moshe kaspi the department of ie&m ben-gurion university of the negev, israel
TRANSCRIPT
Dvir Shabtay
Moshe Kaspi
The Department of IE&M
Ben-Gurion University of the Negev, Israel
Outline
Problem Description
Motivation
Main Results
Problem description
The classical TSP can be stated as follows:
Given n cities and a cost (distance) matrix
C=(cij) which describes the cost of traveling
from one city to the other (the changeover cost),
the objective is to find an optimal tour, i.e., to
visit all the cities and to return to the home city
at a minimal total changeover cost.
We study a special case of the TSP where the
cost matrix is constructed by two vectors:
and , and the
changeover cost is given by
We refer to this special matrix structure as a
root cost matrix.
),...,,( 21 naaaA ),...,,( 21 nbbbB
.ijij bac
Motivation
Application to scheduling
A set of n independent nonpreemptive jobs,
, are available for processing at time
zero.
The jobs are to be processed on a set of two
machines in a flow-shop scheduling system.
The jobs are not allowed to delay between the
two machines.
},...,2,1{ nJ
The operation processing time of job j in
machine i, pij , is depicted by the following
convex decreasing function,
,
(1)
where wij is the processing parameter
(workload) and uij is the amount of continuous
non-renewable resource that is allocated for the
operation.
The total amount of resource consumption
is limited to U units, .
ijij uwpij /
Uui
n
jij
2
1 1
The Objective
To determine simultaneously
1. The optimal resource allocation for each job on each machine and
2. the optimal job sequence,
in order to minimize the makespan (Cmax).
The makespan is defined as ,
where is the completion time of job j.
jnj
CCmax,...,1
max
jC
The Optimization Method
First, we determine the optimal resource
allocation for any given arbitrary job sequence
and thereby reduce the problem to a
combinatorial (sequencing) one.
Then, we determine the optimal job sequence.
Optimal Resource Allocation for Any Given Arbitrary Job Sequence
The makespan in the no-wait two-machine
flow-shop scheduling problem is calculated as
the longest path within the following series-
parallel (s-p) graph (Figure 1), where [j] is the
job in the jth position of the sequence.
p
p2[3]p2[2]
p1[3]
p2[1]
p1[4]p1[1]p1[n]
............... p2[n]
...........Figure 1. The series-parallel (s-p) graph
representing the job order.
Optimal Resource Allocation within a
Series Parallel Graph
Definition: An s-p graph is a special case of a
directed acyclic graph which is recursively
defined as follow: Given a set of disjoint s-p
graphs, :
A series-connection of these K s-p graphs results
in a new s-p graph, which is constructed by
adding an arc from each node in with
outdegree zero to each node in with
indegree zero.
KGGG ,...,, 21
kG
1kG
A parallel-connection of these K s-p graphs
results in a new s-p graph and is defined as their
union, namely no additional arc is added, and
the result is a new s-p graph that remains
disjointed.
A s-p graph can be a single node, a series-
connection, or a parallel-connection of several
disjoint s-p graphs.
The optimal resource allocation to minimize
the longest path within an s-p graph is derived
from the equivalence property (Monma et al.
(1990)) as follows:
Let and be the equivalent load of two s-
p graphs, and , respectively.
The equivalent load of a parallel-connection
is and the equivalent load of a
series-connection is .
1w 2w
1G 2G
21 GG 21 ww
221 ww
The optimal resource allocation for Gj, defined as Uj,
for the parallel-connection is
and for the series-connection it is
As a result, under an optimal resource allocation any
s-p graph can be collapse to a single node with an
equivalent workload of , and the minimal longest
path is .
By applying this method we obtain that the equivalent
workload of the s-p graph presented in Figure 1 is:
Gw
UwG /
.)( 21
UUww
w jj
.21
Uww
wU j
j
, (2)
where , and the optimal
resource allocation is
(3)
(4)
Thus, the minimal makespan as a function of the
job permutation is:
. (5)
21
1]1[2][1
n
jjjG www
0]0[2]1[1 ww n
nkwww
Uwu
kkG
kk ,...,2,1 ,
)( ]1[2][1
][1][1
nkwww
Uwu
kkG
kk ,...,2,1 ,
)( ][2]1[1
][2][2
UwC G /max
The Reduced Combinatorial Problem
Our problem is therefore reduced to finding the
optimal job sequence that minimizes eq. (2) or
equivalently to find the job permutation that
minimizes .
The reduced problem is equivalent to the TSP with
n+1 cities and a root cost matrix where, and
.
1
1]1[2][1
n
jjj ww
jj wa 1
jj wb 2
Main Results
A root cost matrix is a special case of the
Permuted Distribution (Monge) cost matrix
family.
The TSP for root cost matrices is NP-hard
(Partition Graph Spanning Tree TSP for root
cost matrices).
Let be an optimal tour. Then,
for any arbitrary tour, .
*
)(2)( * CC
Main Results
We suggested a heuristic algorithm which is
based on the theory of subtour patching to solve
the problem.
We found some properties for which the
heuristic solution is necessarily an optimal
solution.
We formulated a branch-and-bound optimization
algorithm to the problem.
References
(1)Gilmore, P.C., and Gomory, R.E., 1964, Sequencing a
One-State Variable Machine: A Solvable Case of the
Traveling Salesman Problem, Operations Research, 12(5),
655-679.
(2) Monma, C.l., Schrijver, A., Todd, M.J., and Wei, V.K.,
1990, Convex Resource Allocation Problems on Directed
Acyclic Graphs: Duality, Complexity, Special Cases and
Extensions. Mathematics of Operations Research, 15, 736-
748.