Dvir Shabtay

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A New Special Case of the TSP with an
Application to Scheduling
Dvir Shabtay Moshe Kaspi
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• Outline
• Problem Description
•  Motivation
• Main Results

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• 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.

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• 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.

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• 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.

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• 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, .

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• 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.

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• 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.

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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.
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Figure 1. The series-parallel (s-p) graph
representing the job order.
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• 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.

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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.
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• 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 .

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• 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
  
  

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,
(2) where
, and the optimal resource allocation
is
(3)


(4) Thus, the minimal makespan as a
function of the job permutation is
.
(5)
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• 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 n1 cities and a root cost matrix where,
  and .

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• 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, .

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• 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.

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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.