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.