The One Machine Scheduling Problem

4. Scheduling Problems

4.3 The One Machine Scheduling Problem

In the one machine scheduling problem n jobs have to be sequenced on one below, where t_j is the variable representing the starting instant of processing job j .

)

4. Scheduling

the set of jobs N and two other fictitious nodes, the origin 0 representing the beginning of the system, and the sink n+1, representing the end of the system. The set of edges in G_one is ^E⁼

{ ( )

ⁱ^,^j ^∀ⁱ^,^j^∈^N^,ⁱ^≠ ^j

}

connecting all pairs of nodes representing jobs; and there are two sets of arcs ^Ao=

{ ( )

0, ^j ∀^j∈^N

}

, the arcs connecting the origin node to the nodes of all the jobs and ^An₊₁=

{ (

^j,ⁿ+1

)

∀^j∈^N

}

, the arcs from each one of the nodes of the jobs to the sink node. An edge

( )

ⁱ^,^j ^∈^E

will have weight p or _i p_j, depending on job i being processed before or after job j . Arcs in A have weight ₀ r_j since the machine can not start to process job j before rj. Arcs in A_n₊₁ have weight p_j +q_j, since a job j has to spend an amount of time qj in the system after being processed by the machine for p_j units of time. Finding a solution to the one machine scheduling problem means choosing an orientation for every edge in E , constructing an acyclic directed graph. The subgraph ^C⁼

(

^N^,^E

)

^of

Gone is a fully connected graph, named a clique, which means that for every possible pair of nodes in N , there is an edge in E connecting them.

As we will see in the following sections, many authors have worked on this problem; as the knowledge of their properties is crucial for addressing more complicated shop environment systems.

4.3.1 Algebraic Structure of the One Machine Problem

Based on the theory on disjunctive programming, Balas (Balas 1985) derives results on the characterisation of the facets of convex hull of the feasible solutions to the one machine scheduling problem. The facets are defined by valid inequalities in the variables t_j, j∈N, involving the parameters representing the release dates of the jobs r_j, the processing times p_j and the queue of the jobs q_j. Balas presents facet defining inequalities with one, two and three nonzero coefficients on the variables t_j.

Inequalities with one nonzero coefficient These inequalities with one nonzero coefficient are the ones related to the release date of the jobs: t_j ≥r_j ∀j∈N. These are obvious inequalities present in the formulation.

Inequalities with two nonzero coefficients The facet defining inequalities having exactly two coefficients different from zero are of the form

(

pi +ri−rj

) (

ti + pj +rj −ri

)

tj ≥ pipj +ripj+rjpi (4.3)

for every pair of distinct jobs i and j of N such that r_j <r_i +p_i ∧r_i <r_j +p_j. This condition implies that the coefficients of the variables t and _i t_j are non negative. To verify that these inequalities are valid please not that: if i is scheduled before j then the smallest possible values for t and _i t_j will be r and _i r_i + p_i respectively; if j is scheduled before i then the smallest possible values for t_j and t _i will be r_j and r_j +p_j. Inequalities (4.3) are derived from the solution of the system solved to find the vertices of the polyhedron correspondent to the subproblem considering only jobs i and j .

Inequalities with three nonzero coefficients To present the facet defining inequalities with three nonzero coefficients we need first to introduce the following matrix V with three columns, each corresponding to a job ( i , j and k ), and each row corresponding to a permutation of the three jobs. V will have six rows. Let us assume the rows are ordered this way: row 1 corresponds to permutation

(

ⁱ^,^j^,^k

)

^{, row}

2 to

(

^j^,^k^,ⁱ

)

; row 3 to

(

^k^,ⁱ^,^j

)

; row 4 to

(

ⁱ^,^k^,^j

)

; row 5 to

(

^j^,ⁱ^,^k

)

and row 6 to

(

^k^,^j^,ⁱ

)

Each element v_rc of matrix V will be the earliest possible date to start processing job c in the permutation correspondent to row r .

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4. Scheduling

The facet defining inequalities with three nonzero coefficients are of the form

≥1

for some triplets of rows

(

^{a ,}^,^b ^c

)

. These inequalities are the generalisation of the ones with two nonzero coefficients. They are the solution to the system that defines the vertices of the polyhedron correspondent to the subproblem considering only jobs i , j and k . The expression defining the α is just the Cramer Rule to solve systems of linear equations.

Balas proves that there are at most four distinct inequalities (4.4) that define facets of the convex hull of the feasible solutions to the one machine problem.

For further details on the inequalities of Balas please refer to his paper (Balas 1985).

Another author that has studied the algebraic structure of the one machine scheduling problem and proved several properties is Jacques Carlier. In his work (Carlier 1982) he developed a branch-and-bound algorithm to solve the one machine problem, building an initial feasible solution with the priory rule algorithm of Schrage (Schrage 1970) and performing branching based on a proposition presented next. The algorithm of Schrage schedules available jobs giving priority to the one job j with bigger queue q_j. It builds the list schedule associated with the most work remaining priority dispatching rule of Jackson (Jackson 1955). Carlier proves that

( )

j subset of jobs J ⊆N. He also shows that given a Jackson’s scheduled with makespan mk one of two situations occurs. Or the schedule is optimal and there is a set of jobs J such that ^h

( )

^J ⁼^mk. Either the schedule is not optimal and then there are a critical

set of jobs J and a critical job c such that h

( )

J >mk−pc. This means that in this case the distance to the optimum from the Jackson’s schedule is less than p and _c implies that in an optimal schedule, either job c is processed before all the jobs in set J or job c is processed after all the jobs in set J . This property is used to define the branching scheme of the branch-and-bound algorithm.

Carlier also proves some other propositions that given an upper bound

( )

^UB ^on

then on an optimal schedule k is sequenced either before or after all other jobs in set J ; if both conditions (4.6) and (4.7) are verified, then on an optimal schedule k will be the last job of set J in the schedule and k is called the output of J ; if both conditions (4.6) and (4.8) are verified, then on an optimal schedule k will be the first job of set

J in the schedule and k is called the input of J .

4. Scheduling

In document Optimised search heuristics: combining metaheuristics and exact methods to solve scheduling problems (Page 65-70)