The Department of IE&M
Ben-Gurion University of the Negev, Israel
Dvir Shabtay
Moshe Kaspi
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:
A  (a1 , a2 ,..., an ) and B  (b1 , b2 ,..., bn ) , and the
changeover cost is given by cij  a j  bi .
We refer to this special matrix structure as a
root cost matrix.
Motivation
Application to scheduling
A set of n independent nonpreemptive
jobs,
, are available for processing
J {1,2,..., n}
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.
The operation processing time of job j in
machine i, pij , is depicted by the following
convex decreasing
function,
pij  wij / uij
,
(1)
where wij is the processing parameter
(workload) and uij is the amount of continuous
non-renewable resource that is allocated for the
operation.
2 n
The total amount ofresource
 u  Uconsumption
i 1 j 1
ij
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 , Cmax  max C j 
j 1,...,n
where C j is the completion time of job j.
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 seriesparallel (s-p) graph (Figure 1), where [j] is the
job in the jth position of the sequence.
p2[2]
p2[3]
p1[n]
...............
p2[1]
p1[4]
p1[3]
p
...........
p1[1]
Figure 1. The series-parallel (s-p) graph
representing the job order.
p2[n]
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, G1 , G2 ,..., GK :
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 Gk with
outdegree zero to each node in Gk 1 with
indegree zero.
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 seriesconnection, 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 w1 and w2 be the equivalent load of two sp graphs, G1 and G2 , respectively.
The equivalent load of a parallel-connection
G1 G2 is w1  w2 and the equivalent load of a
series-connection is  w1  w2 .
2
The optimal resource allocation for Gj, defined as Uj,
for the parallel-connection is U  w j  U .
j
( w1  w2 )
w
j
and for the series-connection it is U 
U .
j
w1  w2
As a result, under an optimal resource allocation any
s-p graph can be collapse to a single node with an
equivalent workload of wG , and the minimal longest
path is wG / U .
By applying this method we obtain that the equivalent
workload of the s-p graph presented in Figure 1 is:
2
 n1

wG    w1[ j ]  w2[ j 1]  ,
 j 1

where w1[ n1]  w2[0]  0, and the optimal
resource allocation is
u1[ k ] 
u2[ k ] 
w1[ k ]  U
wG  ( w1[ k ]  w2[ k 1] )
w2[ k ]  U
, k  1,2,..., n
(2)
(3)
, k  1,2,..., n (4)
wG  ( w1[ k 1]  w2[ k ] )
Thus, the minimal makespan as a function of the
job permutation is:
C max  wG / U.
(5)
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
n 1
minimizes  w1[ j ]  w2[ j 1] .
j 1
The reduced problem is equivalent to the TSP with
n+1 cities and a root cost matrix where, a j  w1 j and
b j  w2 j .

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, C ( )  2C ( * )
for any arbitrary tour,  .

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 OneState 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, 736748.