- Email: [email protected]
Cost-minimal preemptive scheduling of independent jobs with release and due dates on open shop under resource constraints
Volume 9, number 5
INFORMATION PROCESSINGLETTERS
COST-MINIMALPREEMPTIVES DATESONOPENSHOPUNDER
16 December 1979
ENTJOBSWITHRELEASEAN
Roman. SiX~WIfiSKI Institute of Control Engineering, Technical Universityof Poznari, 60965 Poznafi,Poland
Received 20 February 1979; revised version received 19 September 1979
Open shop, resource allocation, renewable and non-renewable resources, preemptive sclhedule,cost minimization
1. Introduction The open shop [3] is an ordered set {M,, ._, 14,) Of m 2 1 machines (in particular processors); n > 1 independent jobs are to be scheduled on these machines. Each job j consists of a set of tusks {Tlj, ..,, Tmj}, j = 1, .... n. Tii has to be processed on Mi in pij time units, but the-order in which the tasks are executed is immaterial (as opposed to flow and job shops). Each machine can only work on one task at a time, and a job can only be processed by one machine at a time. A review [4] of previous studies on open shop scheduling showed that all of them used the maximum completion time C,,, as an optimality criterion. In [3] it is demonstrated that preemptive and non-preemptive Cma,-optimal schedules can be obtained in linear time when m = 2. When m > 2, &,-optimal preemptive sc”sdules can still be obtained in polynomial time [2,51, whereas the problem of finding C,,,aoptimal nonpreemptive schedules becomes NP-hard [La]mThe latter problem with m = 2 and two distinct release dates is also NP-hard [4]. Recently, Cho and Sahni [ 11 investigated the problem of obtaining feasible schedules, called DD-schedules, for open shops. A DD-schedule is an assignment of tasks to machines such that the processing of every job finishes by its due date nd no job begins p:oc.:sown that the nonsing before its release date. p!:eemptive DD-scheduling problem for open shops is I!#-hard, even when m = 2, however for preemptive
case, a linear programming formulation may be obtained for m > 2 [l]. The solution to the linear program is then used in conjunction with the polynomial-in-time preemptive scheduling algorfthm of [2,3] to obtain a preemptive DD-schedule. This paper deals with an augmented open shop model where in addition to the set of machines, the set of resources is considered. It hicludes two categories of resources: renewable (or non-storable), such as facilities, personnel, input/output channels, etc., for which only total usage at every moment is constrained, and non-renewable (or storable), such as money, energy, raw .material, etc., which are consumed over the period of processing. The resources are also classified into types depending upon the functions which they perform. We shall assume the presence of p 2 1 renewable resource types G , . .. . RL, and v B 1 non-renewable resource types RY, . .. . R:. The usage of each Ri is constrained to Bk G m units at any time. Each task, besides the appropriate machine, requkes for its processing the following resources: - one unit of a.specified 1- amounts of non-rene arbitrary elements of
resource types. The tasks of one job require the same renewable resource type. In tirder to express the amounts of 233
INFORMATION PROCESSING L!ETTERS
Volume 9, number 5
non-renewable resources which may be required by Tij- let US introduce tht:(gij X V) matrix Sii = [Fii**-’ FijvJ, where the Qthrow of Sij, [rijla **rijva],is composed of feasible numbers of units of RE, k = 1, . . . . v, which may take part together in the performance of Tiiq Q= 1, . ..) gij; i = 1, . .. . m; j = l,* .... n. The Qth row of matrix Sij determines the Qthvariantfor the allocation of non-renewable resources to Tii- Associated with each task is the vector pijn], where pijn, Q= 1, .... , represents the total processing time required to mpfete task Ttj, if the task is processed exclusively by machine Mi with the required renewable resource unit and the amount of non-renewable resources cified in variant Q. In addition to task times, job j o has associated with it a releasedate aj, at which j becomes available for processing, and a due date 4; {or deadiine) by which j must be completed. The processing of each task may be arbitrarily interrupted and restarted later with no time penalty (preemption), possibly with another variant of non-renewable resource allocation. The fractions of a task require the same fractions of the amount of non-renewable resources allotted to the whole task, that is, each task consumes non-renewable resources at the constant rate rijkP/pije, k = 1, . . . . V, for any variant Q. AS has been stated, for each task Tij there are known gij 2 1 different variants of non-renewable resource allocation with corresponding task processing times. It is assumed that the allocation of a larger amount of non-renewable resources leads to a shorter time being required for task execution, The problem we wish to consider is that of finding a feasible schedule for which the consumption (or rather its cost) of non-renewable resources is;minimized. In the following we shal show how the linear ,programming (LP) formulation of [l ] can be extended to handle the above problem. We shall also outline a new polynomial-in-time algorithm which should be used in conjunction with the LP solution to obtain a feasible schedule. The LP formulation of the considered scheduling problem provides a means for esta&ishing the upper bound on the m.rmber of preemptions required for an optimal schedule. The complexity of remains open, since up to now there is lynomial-bounded algorithm for the neral LP problem (even though experience indicates e LP algoritlm is usualiy very practical), nor problem been shown to be NP-hard.
16 December 1979
2. LP formulation of the cost-minimal scheduling problem
l
234
Let us introduce the set of cost coefficients {cl, .,.., c,) associated with the consumption of one unit of non-renewable resources Ry, .... Rt. Let bl < bz < a* be the ordered collection of all distinct
vmgI%
minimize K =
ckrijk*x$Z
k=ii=l
j=l
h=l
Q=l
PijQ
(1)
subject to m
Bij
cc i=l
Q=l
x& G Ih,
j = 1, . . .. n, h = 1, . . . . q
(2)
X~CG Ih,
i= l,..., m, h= l,.,,., q,
(3)
gij kc j=l P=l
q
gij
C h=l
C XiQ/pijQ = Q=l
m
gij
cc CX&d&,, i=l jEDk P=l
1,
i = 1, .... m, j - 1, .... n,
k=l,...,
p, h=l,...,
(4)
q, (5)
h
Xijh
20, 0, i =
ifaj < bh anddj > bh+l, otherwise.
(6)
In (1) K represents the cost of non-renewable resources consumed during the processing of all jobs; let us note that in particular, K may be treated as the cost of processing. Inequality (2) requires that no job be scheduled for more than Ih time units in any interval. Inequality (3)‘requires that the amount of processing assigned to any machine be no more than the interval length. Equality (4) requires that each job be completed. Inequality (5) requires that :he constraint on the availability of renewable resources is to be observed in any interval. The constraints on
Volume 9, number 5
INFORMATIONPROCESSING LETTERS
x$~ ensure that no job is assigned to a machine either before its release date or after its due date. It is evident that the values of K and xfin for the feasible schedule constitute a feasible solution to the LP problem (l)-(6). We hold that the converse is also true, that is, for any feasible solution to the LP problem, there is a feasible schedule with the same values of X~Qand K. In the next section we outline an algorithm for finding such a feasible schedule in polynomial time. It is based on the algorithm of [S,6] which has been applied in conjunction with the LB solution for the problem of preemptive scheduling on unrelated parallel machines under resource constraints.
16 December 1979
_
exactly Bk jobs belonging to Dk if Ri is critical, and no more than Bk , otherwise. We shall use the set ED to construct a partial schedule with length 6 > 0. The construction of a feasible schedule for any particular time interval proceeds via a series of partial schedules in the following way [S] : Step 1. Find the set ED. Step 2, Calculate the partial schedule length 6: zmin 9
6 = L
Ih -
Zm,,,
if Ih - zmin > zmax, otherwise,
(7)
where Zmin =
min
[z$],
h
ZijEED
3. Construction of a feasible schedule First, let us aggregate all the fractions resulting from the solution of LP problem (l)-(6) of the same task processed in interval h under different nonrenewable resource allocation variants. The processing time of the aggregated task is xi = #, X$Q.We will construct a feasible schedule for each time interval and the aggregated tasks, and finally we will reassign the appropriate amounts of non-renewable resources to the fractions of tasks resulting from the schedule construction procedure. Let Xh denote the m X m matrix of nonnegative elements which are the optimal values of xh. Column j (job j) of matrix Xh will be called ctitic~2 if e&x8 = Ih. Similarly, resource RL will be called criticalif it is fully utilized in the interval h, that is if EC
xb = BkIh.
i=l jE&
Next, let Yh be the m X m diagonal matrix of nonnegative machine idle times: yt = Ih - x& 1 x8, i= 1, . .. . m. The columns of Yh will represent dummy jobs which do not require additional resources. We shall denote by Zh the m X (n + m) matrix composed of matrices Xh and Yh in the following way: Zh = [Xh 1Yh]. Finally, let us introduce the set ED composed of m positive elements of matrix Zh, with exactly one element of the set in each row and no more than one element in any column. Specifically, the elements of ED should represent : _ each critical job,
zmax =
max
h Z ij$ ED
Step 3. Decrease It, and all elements of ED by 6. if Ih = 0 than stop, otherwise go to Step 1. It can be seen that set ED is constructed such that at the end of each iteration, the elements of Xh fulfil conditions (2), (3), (5) for the present value of Ih. Let us also note that for each set ED, S is chosen such that either one of the positive elements in Zh is reduced to zero, or the more column or renewable resource type becomes critical. Each of these events may only occur a finite number of times, which ensures that the feasible schedule will be obtained in a finite number of iterations. For Step 1 we can apply the algorithm of [5], whose complexity is O(nm*). Let us note that the problem of finding set ED is equivalent to fmding a feasible flow in the network presented in Fig. 1. The network contains m nodes which represent machines (or rows of Zh), n + m nodes representing jobs (or columns of Zh), and p nodes representing renewable resources. There is an arc between ‘machine node’ i and a ‘job node' j iff z$ > 0, and the arc from a ‘job node’ j leads to the ‘resource node’ k iff j E Dk, The flow through each arc is limited by lower and upper capacities, L and U respectively, which have the following values: . on the arc ‘source-machine’: L = 1, U = 1, - on the arc cmachine-job’: L = 0, U = 1, _ on the arc ‘job-resource’ or ‘job-sink’: L=
1, 0,
if the job is critical, otherwise,
U= 1
9
235
9.
INFORMATION PROCESSING LETTERS
fnimbcr5
- on the arc ‘resource-sink’, L=
Bk* the number of c\itical jobs in I&,
if I?.i is critical, otherwise,
U=Bk.
Pt is asserted that for every intsgral flow in the net\qork there is a corresponding set ET) and conversely. A feasible schedule obtained by the above ’ described algorithm results in a new partition of tasks rocesgd on particular machines, although the values of x$ do not change. Thus, we have to reassign the non-renewable resources to the new task fractions while ensuring that any fraction of Trj processed in zny interval on Mi unde; the Qth allocation variant, requires the same fraction of the amount of nonrenewable resources specified in row of Sij.
4. Fhe upper bouond on the number of preemptions
Since we are studying preemptive scheduhng, pper bound on the number of preempptimal schedule is of great interest. To precise, we say that a task is preempted at time t f the job is suspended on some machine e t before its completion. the LP formulation posed in Section 2, it optimal basic feasible solution 3re than nm + q(n + m + p) positive , there will be no mox than nm + posh ve variables, wlx: L p2, p3, p5 umber of inequality constraints (2), (3) 236
16
December1979
and (5) which are satisfied with strict equality. In other words, p2 is the number of critical jobs in all time intervals, p3, the number of machines with zero idle tines in all time intervals, and ps , the number of critical renewable resource types in all time intervals. But 0 G p2 G q(m - l),O< p3 < mqandOg ps G pq. Hence in the optimal solution there will be no more than nm + 2mq + q(p - 1) strictly positive variables. If we could construct a feasible schedule without introducing additional preemptions, then the upper bound on the number of preemptions for an optimal schedule would be equal to 2mq f q(p - 1). However, the schedule construction procedure generally introduces additional preemptions. We shall now establish the upper bound on this number. Initially, it should be noted that the task aggregation made before the schedule construction for each time interval may considerably reduce the number of.’ preemptions in the resulting schedule. This follows from the fact that the number of strictly positive x$ is usually much smaller than that of X~Q.However we have to assume the worst case, where this reduction does occur. Let us consider schedule construction for time interval h. Let sh be the number of strictly positive elements in matrig Xh. Because of dummy jobs, matrix Zh will contain at most sh + m strictly positive elements. Each iteration of the schedule construetion procedure introduces at most m task fractions. We already know that after each iteration, either one of the positive elements in matrix Zh is reduced to zero, or one more column or renewable resource type becomes critical. Exactly m elements become zero in the last iteration. Thus there are at most sh + 1 iterations of the first kind, m - pi iterations of the second kind, and p - p$ iterations of the third kind (pfl and pt denote the number of critical columns in Zh and the number of critical renewable resource types, respectively, before schedule construction), hence at most sh + m - pt + p - p5 + 1 iterations in all. The number of preemptions fo,r the interval schedule is thus bounded by m(sh + m - pi + p .- p! + 1) - (sh f m), and for the final schedule by 9 m2q + pmq f C h=l
Taking
[sh(m - 1) - m(p: + p!)] .
INFORMATION PROCESSING LETTERS
Volume 9, number 5 9
c
sh = nm + 2mq + q(r - l),
16 December 1979
the latter follows from the computational complexity of this construction, which is equal to 0(n2m3).
h=l
q
9
c h=l
P!?!=q@-
1)
and
hzl P! = pq, =
we have proved the following theorem: Theorem. An upper bound on the number of preemptions required for a cost-minimal schedule on open shop with additional resource is m2(n f 2q) + pq(m - 1) - q(2m - 1) - nm. The bound indicated by the theorem is certainly not tight, as may be seen from practical examples. It can be noted, for example that this bound was established under the assumptions that in each time interval there are m dummy jobs and that on the boundary of any two intervals there are m preemptions, which do not have to occur. If there are no dummy jobs, the bound may be reduced by mq(m - 1). To evaluate the computational effort involved in the scheduling method, we should consider the costs of solving the LP problem and constructing a feasible schedule. The former could be expressed in terms of the size of the LP problem being solved: qZg 1IS&1gij variables and nm + q(n + m + p) constraints, whereas
References [ 1] Y. Cho and S. Sahni, I&eemptive scheduling of independent jobs with release and due times on open, flow and job shops, Technical Report 78-5, Computer Science Department, University of Minnesota (1978). [ 21T. Gonzalez, A note on open shop preemptive schedules, Technical Report 214, Computer Science Department, Pennsylvania State University (1976). [ 31 T. Gonzalez and S. Sahni, Open shop scheduling to minimize finish time, J. Assoc. Comput. Mach. 23 (1976) 665-679, [4 ] R.L. Graham, E.L. Lawler, J.K. Len&a and A.1G.H. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Report BW 82/77, Mathematisch Centrum, Amsterdam (1977). [5 ] R. Stowiriski, Scheduling preemptable tasks on unrelated processors with additional resources to minimize schedule lentgh, in: G. Braccbi and P.C. Lockemann, Eds., Lecture Notes in Computer Science 65 (Springer, Berlin, 1978). 1978). [6] R. Sfowiiiski, Allocation de ressources limitdes parmi des tithes ex&uties par un ensemble de machines ind& pendantes, in: M. Pelegrin and J. Delmas, Eds., Comparison of Automatic Control and Operational Research Techniques Applied to Large Systems Analysis and Control (Pergamon Press, Oxford, 1979).
237
INFORMATION PROCESSINGLETTERS
COST-MINIMALPREEMPTIVES DATESONOPENSHOPUNDER
16 December 1979
ENTJOBSWITHRELEASEAN
Roman. SiX~WIfiSKI Institute of Control Engineering, Technical Universityof Poznari, 60965 Poznafi,Poland
Received 20 February 1979; revised version received 19 September 1979
Open shop, resource allocation, renewable and non-renewable resources, preemptive sclhedule,cost minimization
1. Introduction The open shop [3] is an ordered set {M,, ._, 14,) Of m 2 1 machines (in particular processors); n > 1 independent jobs are to be scheduled on these machines. Each job j consists of a set of tusks {Tlj, ..,, Tmj}, j = 1, .... n. Tii has to be processed on Mi in pij time units, but the-order in which the tasks are executed is immaterial (as opposed to flow and job shops). Each machine can only work on one task at a time, and a job can only be processed by one machine at a time. A review [4] of previous studies on open shop scheduling showed that all of them used the maximum completion time C,,, as an optimality criterion. In [3] it is demonstrated that preemptive and non-preemptive Cma,-optimal schedules can be obtained in linear time when m = 2. When m > 2, &,-optimal preemptive sc”sdules can still be obtained in polynomial time [2,51, whereas the problem of finding C,,,aoptimal nonpreemptive schedules becomes NP-hard [La]mThe latter problem with m = 2 and two distinct release dates is also NP-hard [4]. Recently, Cho and Sahni [ 11 investigated the problem of obtaining feasible schedules, called DD-schedules, for open shops. A DD-schedule is an assignment of tasks to machines such that the processing of every job finishes by its due date nd no job begins p:oc.:sown that the nonsing before its release date. p!:eemptive DD-scheduling problem for open shops is I!#-hard, even when m = 2, however for preemptive
case, a linear programming formulation may be obtained for m > 2 [l]. The solution to the linear program is then used in conjunction with the polynomial-in-time preemptive scheduling algorfthm of [2,3] to obtain a preemptive DD-schedule. This paper deals with an augmented open shop model where in addition to the set of machines, the set of resources is considered. It hicludes two categories of resources: renewable (or non-storable), such as facilities, personnel, input/output channels, etc., for which only total usage at every moment is constrained, and non-renewable (or storable), such as money, energy, raw .material, etc., which are consumed over the period of processing. The resources are also classified into types depending upon the functions which they perform. We shall assume the presence of p 2 1 renewable resource types G , . .. . RL, and v B 1 non-renewable resource types RY, . .. . R:. The usage of each Ri is constrained to Bk G m units at any time. Each task, besides the appropriate machine, requkes for its processing the following resources: - one unit of a.specified 1- amounts of non-rene arbitrary elements of
resource types. The tasks of one job require the same renewable resource type. In tirder to express the amounts of 233
INFORMATION PROCESSING L!ETTERS
Volume 9, number 5
non-renewable resources which may be required by Tij- let US introduce tht:(gij X V) matrix Sii = [Fii**-’ FijvJ, where the Qthrow of Sij, [rijla **rijva],is composed of feasible numbers of units of RE, k = 1, . . . . v, which may take part together in the performance of Tiiq Q= 1, . ..) gij; i = 1, . .. . m; j = l,* .... n. The Qth row of matrix Sij determines the Qthvariantfor the allocation of non-renewable resources to Tii- Associated with each task is the vector pijn], where pijn, Q= 1, .... , represents the total processing time required to mpfete task Ttj, if the task is processed exclusively by machine Mi with the required renewable resource unit and the amount of non-renewable resources cified in variant Q. In addition to task times, job j o has associated with it a releasedate aj, at which j becomes available for processing, and a due date 4; {or deadiine) by which j must be completed. The processing of each task may be arbitrarily interrupted and restarted later with no time penalty (preemption), possibly with another variant of non-renewable resource allocation. The fractions of a task require the same fractions of the amount of non-renewable resources allotted to the whole task, that is, each task consumes non-renewable resources at the constant rate rijkP/pije, k = 1, . . . . V, for any variant Q. AS has been stated, for each task Tij there are known gij 2 1 different variants of non-renewable resource allocation with corresponding task processing times. It is assumed that the allocation of a larger amount of non-renewable resources leads to a shorter time being required for task execution, The problem we wish to consider is that of finding a feasible schedule for which the consumption (or rather its cost) of non-renewable resources is;minimized. In the following we shal show how the linear ,programming (LP) formulation of [l ] can be extended to handle the above problem. We shall also outline a new polynomial-in-time algorithm which should be used in conjunction with the LP solution to obtain a feasible schedule. The LP formulation of the considered scheduling problem provides a means for esta&ishing the upper bound on the m.rmber of preemptions required for an optimal schedule. The complexity of remains open, since up to now there is lynomial-bounded algorithm for the neral LP problem (even though experience indicates e LP algoritlm is usualiy very practical), nor problem been shown to be NP-hard.
16 December 1979
2. LP formulation of the cost-minimal scheduling problem
l
234
Let us introduce the set of cost coefficients {cl, .,.., c,) associated with the consumption of one unit of non-renewable resources Ry, .... Rt. Let bl < bz < a* be the ordered collection of all distinct
vmgI%
minimize K =
ckrijk*x$Z
k=ii=l
j=l
h=l
Q=l
PijQ
(1)
subject to m
Bij
cc i=l
Q=l
x& G Ih,
j = 1, . . .. n, h = 1, . . . . q
(2)
X~CG Ih,
i= l,..., m, h= l,.,,., q,
(3)
gij kc j=l P=l
q
gij
C h=l
C XiQ/pijQ = Q=l
m
gij
cc CX&d&,, i=l jEDk P=l
1,
i = 1, .... m, j - 1, .... n,
k=l,...,
p, h=l,...,
(4)
q, (5)
h
Xijh
20, 0, i =
ifaj < bh anddj > bh+l, otherwise.
(6)
In (1) K represents the cost of non-renewable resources consumed during the processing of all jobs; let us note that in particular, K may be treated as the cost of processing. Inequality (2) requires that no job be scheduled for more than Ih time units in any interval. Inequality (3)‘requires that the amount of processing assigned to any machine be no more than the interval length. Equality (4) requires that each job be completed. Inequality (5) requires that :he constraint on the availability of renewable resources is to be observed in any interval. The constraints on
Volume 9, number 5
INFORMATIONPROCESSING LETTERS
x$~ ensure that no job is assigned to a machine either before its release date or after its due date. It is evident that the values of K and xfin for the feasible schedule constitute a feasible solution to the LP problem (l)-(6). We hold that the converse is also true, that is, for any feasible solution to the LP problem, there is a feasible schedule with the same values of X~Qand K. In the next section we outline an algorithm for finding such a feasible schedule in polynomial time. It is based on the algorithm of [S,6] which has been applied in conjunction with the LB solution for the problem of preemptive scheduling on unrelated parallel machines under resource constraints.
16 December 1979
_
exactly Bk jobs belonging to Dk if Ri is critical, and no more than Bk , otherwise. We shall use the set ED to construct a partial schedule with length 6 > 0. The construction of a feasible schedule for any particular time interval proceeds via a series of partial schedules in the following way [S] : Step 1. Find the set ED. Step 2, Calculate the partial schedule length 6: zmin 9
6 = L
Ih -
Zm,,,
if Ih - zmin > zmax, otherwise,
(7)
where Zmin =
min
[z$],
h
ZijEED
3. Construction of a feasible schedule First, let us aggregate all the fractions resulting from the solution of LP problem (l)-(6) of the same task processed in interval h under different nonrenewable resource allocation variants. The processing time of the aggregated task is xi = #, X$Q.We will construct a feasible schedule for each time interval and the aggregated tasks, and finally we will reassign the appropriate amounts of non-renewable resources to the fractions of tasks resulting from the schedule construction procedure. Let Xh denote the m X m matrix of nonnegative elements which are the optimal values of xh. Column j (job j) of matrix Xh will be called ctitic~2 if e&x8 = Ih. Similarly, resource RL will be called criticalif it is fully utilized in the interval h, that is if EC
xb = BkIh.
i=l jE&
Next, let Yh be the m X m diagonal matrix of nonnegative machine idle times: yt = Ih - x& 1 x8, i= 1, . .. . m. The columns of Yh will represent dummy jobs which do not require additional resources. We shall denote by Zh the m X (n + m) matrix composed of matrices Xh and Yh in the following way: Zh = [Xh 1Yh]. Finally, let us introduce the set ED composed of m positive elements of matrix Zh, with exactly one element of the set in each row and no more than one element in any column. Specifically, the elements of ED should represent : _ each critical job,
zmax =
max
h Z ij$ ED
Step 3. Decrease It, and all elements of ED by 6. if Ih = 0 than stop, otherwise go to Step 1. It can be seen that set ED is constructed such that at the end of each iteration, the elements of Xh fulfil conditions (2), (3), (5) for the present value of Ih. Let us also note that for each set ED, S is chosen such that either one of the positive elements in Zh is reduced to zero, or the more column or renewable resource type becomes critical. Each of these events may only occur a finite number of times, which ensures that the feasible schedule will be obtained in a finite number of iterations. For Step 1 we can apply the algorithm of [5], whose complexity is O(nm*). Let us note that the problem of finding set ED is equivalent to fmding a feasible flow in the network presented in Fig. 1. The network contains m nodes which represent machines (or rows of Zh), n + m nodes representing jobs (or columns of Zh), and p nodes representing renewable resources. There is an arc between ‘machine node’ i and a ‘job node' j iff z$ > 0, and the arc from a ‘job node’ j leads to the ‘resource node’ k iff j E Dk, The flow through each arc is limited by lower and upper capacities, L and U respectively, which have the following values: . on the arc ‘source-machine’: L = 1, U = 1, - on the arc cmachine-job’: L = 0, U = 1, _ on the arc ‘job-resource’ or ‘job-sink’: L=
1, 0,
if the job is critical, otherwise,
U= 1
9
235
9.
INFORMATION PROCESSING LETTERS
fnimbcr5
- on the arc ‘resource-sink’, L=
Bk* the number of c\itical jobs in I&,
if I?.i is critical, otherwise,
U=Bk.
Pt is asserted that for every intsgral flow in the net\qork there is a corresponding set ET) and conversely. A feasible schedule obtained by the above ’ described algorithm results in a new partition of tasks rocesgd on particular machines, although the values of x$ do not change. Thus, we have to reassign the non-renewable resources to the new task fractions while ensuring that any fraction of Trj processed in zny interval on Mi unde; the Qth allocation variant, requires the same fraction of the amount of nonrenewable resources specified in row of Sij.
4. Fhe upper bouond on the number of preemptions
Since we are studying preemptive scheduhng, pper bound on the number of preempptimal schedule is of great interest. To precise, we say that a task is preempted at time t f the job is suspended on some machine e t before its completion. the LP formulation posed in Section 2, it optimal basic feasible solution 3re than nm + q(n + m + p) positive , there will be no mox than nm + posh ve variables, wlx: L p2, p3, p5 umber of inequality constraints (2), (3) 236
16
December1979
and (5) which are satisfied with strict equality. In other words, p2 is the number of critical jobs in all time intervals, p3, the number of machines with zero idle tines in all time intervals, and ps , the number of critical renewable resource types in all time intervals. But 0 G p2 G q(m - l),O< p3 < mqandOg ps G pq. Hence in the optimal solution there will be no more than nm + 2mq + q(p - 1) strictly positive variables. If we could construct a feasible schedule without introducing additional preemptions, then the upper bound on the number of preemptions for an optimal schedule would be equal to 2mq f q(p - 1). However, the schedule construction procedure generally introduces additional preemptions. We shall now establish the upper bound on this number. Initially, it should be noted that the task aggregation made before the schedule construction for each time interval may considerably reduce the number of.’ preemptions in the resulting schedule. This follows from the fact that the number of strictly positive x$ is usually much smaller than that of X~Q.However we have to assume the worst case, where this reduction does occur. Let us consider schedule construction for time interval h. Let sh be the number of strictly positive elements in matrig Xh. Because of dummy jobs, matrix Zh will contain at most sh + m strictly positive elements. Each iteration of the schedule construetion procedure introduces at most m task fractions. We already know that after each iteration, either one of the positive elements in matrix Zh is reduced to zero, or one more column or renewable resource type becomes critical. Exactly m elements become zero in the last iteration. Thus there are at most sh + 1 iterations of the first kind, m - pi iterations of the second kind, and p - p$ iterations of the third kind (pfl and pt denote the number of critical columns in Zh and the number of critical renewable resource types, respectively, before schedule construction), hence at most sh + m - pt + p - p5 + 1 iterations in all. The number of preemptions fo,r the interval schedule is thus bounded by m(sh + m - pi + p .- p! + 1) - (sh f m), and for the final schedule by 9 m2q + pmq f C h=l
Taking
[sh(m - 1) - m(p: + p!)] .
INFORMATION PROCESSING LETTERS
Volume 9, number 5 9
c
sh = nm + 2mq + q(r - l),
16 December 1979
the latter follows from the computational complexity of this construction, which is equal to 0(n2m3).
h=l
q
9
c h=l
P!?!=q@-
1)
and
hzl P! = pq, =
we have proved the following theorem: Theorem. An upper bound on the number of preemptions required for a cost-minimal schedule on open shop with additional resource is m2(n f 2q) + pq(m - 1) - q(2m - 1) - nm. The bound indicated by the theorem is certainly not tight, as may be seen from practical examples. It can be noted, for example that this bound was established under the assumptions that in each time interval there are m dummy jobs and that on the boundary of any two intervals there are m preemptions, which do not have to occur. If there are no dummy jobs, the bound may be reduced by mq(m - 1). To evaluate the computational effort involved in the scheduling method, we should consider the costs of solving the LP problem and constructing a feasible schedule. The former could be expressed in terms of the size of the LP problem being solved: qZg 1IS&1gij variables and nm + q(n + m + p) constraints, whereas
References [ 1] Y. Cho and S. Sahni, I&eemptive scheduling of independent jobs with release and due times on open, flow and job shops, Technical Report 78-5, Computer Science Department, University of Minnesota (1978). [ 21T. Gonzalez, A note on open shop preemptive schedules, Technical Report 214, Computer Science Department, Pennsylvania State University (1976). [ 31 T. Gonzalez and S. Sahni, Open shop scheduling to minimize finish time, J. Assoc. Comput. Mach. 23 (1976) 665-679, [4 ] R.L. Graham, E.L. Lawler, J.K. Len&a and A.1G.H. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Report BW 82/77, Mathematisch Centrum, Amsterdam (1977). [5 ] R. Stowiriski, Scheduling preemptable tasks on unrelated processors with additional resources to minimize schedule lentgh, in: G. Braccbi and P.C. Lockemann, Eds., Lecture Notes in Computer Science 65 (Springer, Berlin, 1978). 1978). [6] R. Sfowiiiski, Allocation de ressources limitdes parmi des tithes ex&uties par un ensemble de machines ind& pendantes, in: M. Pelegrin and J. Delmas, Eds., Comparison of Automatic Control and Operational Research Techniques Applied to Large Systems Analysis and Control (Pergamon Press, Oxford, 1979).
237