A FRAMEWORK FOR MACHINE SCHEDULING PROBLEMS WITH CONTROLLABLE PROCESSING TIMES

PRODUCTION AND OPERATIONS MANAGEMENT Vol. 5. No. 4. Winter 1996

Pnnr~I in US A.

A FRAMEWORK FOR MACHINE SCHEDULING PROBLEMS WITH CONTROLLABLE

PROCESSING TIMES *

BAHRAM ALIDAEE AND GARY A.. KOCHENBERGER Department of Management and Marketing, School of Business, University of

Mississippi, University, Mississippi 38677, USA University of Colorado at Denver, College of Business,

Denver, Colorado 802 17-3364, USA

This is a study of single and parallel machine scheduling problems with controllable processing time for each job. The processing time for job j depends on the position of the job in the schedule and is a function of the number of resource units allocated to its processing. Processing time functions and processing cost functions are allowed to be nonlinear. The scheduling problems considered here have important applications in industry and include many of the existing scheduling models as special cases. For the single machine problem, the objective is minimization of total compression costs plus a scheduling measure. The scheduling measures include makespan, total flow time, total differences in completion times, total differences in waiting times, and total earliness and tardiness with a common due date for all jobs. Except when the total earliness and tardiness measure is involved, each caSe the problem is solved efficiently. Under an assumption typically satisfied in just-in-time systems, the problem with total earliness and tardiness measure is also solved efficiently. Finally, for a large class of processing time functions; parallel machine problems with total flow time and total earliness and tardiness measures are solved efficiently. In each case we reduce the problem to a transportation problem. (SINGLE AND MULTIPLE MACHINE SCHEDULING; CONTROLLABLE TIMES; APPLICATIONS OF TRANSPORTATION PROBLEM)

1. Introduction

Much of the past research in sequencing and scheduling assumed that the processing time of a job is not under managerial control. Job processing times have been treated either as parameters of the problem that are fixed and known in advance of scheduling or as random variables determined by outside forces. The scheduling objective is to determine the sequence of the jobs on machines to optimize system performance defined by one or more scheduling measures. However, in real applications scheduling problems involve resource allocation as well as sequencing of the jobs. The scheduler can accomplish a job in shorter (longer) time by increasing (decreasing) additional resources such as facilities, funds, manpower, energy, and the use of more advanced technologies such as machines with variable speeds. A simple example is the time/cost trade-off model in project management. In the time/cost model, the application of additional resources, as

* Received September 1993; revisions received January 1995 and April 1996; accepted July 1996. 391

1059-1478/96/0504/391$1.25 Copyright 0 1996, Production and Operations Management Society

392 BAHRAM ALIDAEE ANDGARY A.KOCHENBERGER

measured by their costs, reduces the time required to carry out a task. The scheduling of tooling machines with variable speeds is another example (Adiri and Yehudai 1987; Trick 1994). Tooling machines take pieces of wood, plastic, or metal, and through cutting and planing, make smaller pieces of the desired shapes and sizes. In such applications, tool wear is considerable. Performance of a machine tool is a function of the quality of its cutters. Tool wear can be decreased by running the machine at lower speeds, but this increases the time spent by the piece on the machine (Trick 1994).

In this paper, we consider a parallel machine scheduling model where the processing time of a job depends on its position in a sequence on a machine and is a function of units of resource applied to its processing. The processing cost of a job is also a function of units of resource used for its processing. The processing time and the processing cost functions are allowed to be nonlinear. The objective is to find an assignment and a sequencing of jobs on machines that minimize the total processing costs plus a second scheduling measure. For the single machine problem, the second measure includes the following: the maximum completion time (makespan), total flow time, total absolute differences in completion times ( TADC), total absolute differences in waiting times ( TADW ), and total earliness and tardiness with a common due date for all jobs. In the case of parallel machines, the second measure includes total flow time, and total earliness and tardiness with a common due date for all jobs.

Resource allocation and its impact on scheduling has been studied extensively in the project management context (see, for example, Moder and Phillips 1970; Elmaghraby 1977; Badiru 1988). The notion of controlling processing times in the shop scheduling has recently attracted much attention among the scheduling researchers. Vickson ( 1980a, 1980b) was the first to propose a single-machine scheduling model where processing times were controllable. The objectives were minimization of total cost of resource allocated to jobs (total crashing costs) plus a scheduling measure. Scheduling measures were total flow time, maximum tardiness, and total weighted completion times. Extensions of Vickson’s model in the single-machine (Daniels 1989; Tada, Ishii, and Nishida 1989; Zdrzalka 199 1; Lee 199 1; Panwalkar and Rajagopalan 1992a, 1992b) and in the mul- timachine (Grabowski and Janiak 1986; Nowicki and Zdrzalka 1988; Alidaee and Ah- madian 1993; Daniels and Mazzola 1994) systems have been studied as well.

Adiri and Yehudai ( 1987) noted that there are situations where the service rate of a machine remains constant while a job is being processed and may be changed only upon its completion. In such environment it is possible to formulate the problem as a scheduling model where processing time of a job is dependent on the position of the job in the sequence. As an example, they studied scheduling of tooling machine. In their model, each jobj has a service requirement of tj and if scheduled in the ith position on machine k it consumes a: tj time units where a) is service factor related to machine k. The vector

k of service factors ak = (a:, . . . , a4, at, . . . , at, - - * ) is a cyclic of order q. In this form the tool is replaced after q operations. The objective was determination of assignment and sequencing of jobs on machines to minimize a scheduling measure. The scheduling measures were minimization of maximum completion time ( makespan ) , total flow time, maximum lateness, total tardiness, and number of jobs tardy.

In all previous studies two assumptions have been made: ( 1) the actual processing time of a job is a linear function of the amount of resource applied to its processing. An implicit part of this assumption [except, for the study of Adiri and Yehudai ( 1987), where the processing time of a job depends on its position in the schedule] is that the rate with which the application of a unit of resource reduces the processing time of a job is independent of both the number of units of resource allocated to the job and the time at which the resource is assigned to the job; (2) the cost of the resource allocated to a job (the processing cost of a job) is a linear function of resources applied to the job (Nowicki and Zdrzalka 1990). However, in many situations, it is more appropriate to

MACHINE SCHEDULING PROBLEMS AND PROCESSING TIMES 393

assume nonlinear functions for processing times and processing costs. For example, this would be the case when each machine can be assigned any speed, and the cost associated with each machine is a function of the machine’s speed. Ishii, Martel, Masuda, and Nishida ( 1985) noted that in real time systems that must complete a given set of tasks within a specified time window or for businesses with a predictable set of jobs that are done on a regular schedule (daily, weekly, or monthly) such assumptions are reasonable. Other examples involving nonlinear functions exist as well. For instance, the reader is referred to project scheduling work of Moder and Phillips ( 1970); Flodes and Soumis ( 1993) as well as the single machine sequencing work of Panwalkar and Rajagopalan (1992a, 1992b).

Trick ( 1994) studied a scheduling problem that has applicability in flexible machines. In his model the objectives involved the assignment of jobs to machines and the determination of optimal processing times of jobs in order to minimize ( 1) total cost of resource allocated to jobs, and (2) total cost of resource allocated to jobs plus makespan. While in the first case the total amount of resource used for processing jobs was not to exceed a global amount, i.e., a given constant, called the capacitated problem, in the second case no global capacity for use of resource was considered. In neither cases did he consider the sequencing aspects of jobs on machines. Vickson’s model and the models studied by Trick ( 1994) and Adiri and Yehudai ( 1987) are special cases of the model we put forth in this paper.

In all of these studies, the cost incurred as a result of assigning and sequencing jobs to machines and the cost incurred as a result of resource allocation are combined into a one dimensional criterion. Van Wassenhove and Baker ( 1982) introduced an approach that distinguishes between sequencing criteria and resource allocation costs. Their approach gives rise to a bicriterion formulation of the problem. The advantage of this approach is that it does not require that sequencing criteria be measurable in the same units as the resource allocation cost. Their approach constructs the trade-off curve between total amount of allocated resource and maximum tardiness. Generalizations of this model have also been studied (Daniels and Sarin 1989; Daniels 1990; Cheng, Chen, and Li 1994). Nowicki and Zdrzalka ( 1990) provided a survey of scheduling problems with controllable processing time for each job.

In this paper we present a unifying framework for a variety of problems with controllable processing times. Moreover, we demonstrate how, and under what conditions, these problems can be efficiently solved (i.e., polynomially ) by reformulating each as a transportation problem.

For reference later in the paper, we state the transportation problem (TP): Given a m X n matrix Q, choose II elements such that

(i) there is exactly one element from each column, (ii) there is at most one element from each row, (iii) the sum of elements is minimum.

In the following sections, we first consider the single machine problem. We then extend our results to the parallel machine case. Conclusions and final remarks are presented at the end.

2. Single Machine Problem

Assume there is a set of n jobs available at time zero to be scheduled on a single machine. Let u = ([l], . . . , [n]) be a sequence of the jobs where [i] is the ith job in u. Assume that the processing time of a job j (for j = 1, . . . , n) is controllable. If job j is scheduled in the ith position it consumes go( yj) units of time where yj E [ 0, Uj] is the amount of resource used for its processing. The upper bound, Uj, is the maximum resource level that can be used in processing job j. It is assumed that processing a job j in the ith

394 BAHRAM ALIDAEE AND GARY A. KOCHENBERGER

position can take as much as PO units of time (normal time), and as little as pij units of time (crash time); therefore, we have gU( 0) = Pu and gg( Uj) = pii. For a sequence (T, let rtil, Cti) and Wtil represent the processing time, completion time and waiting time of the ith job, respectively. Therefore, if job j is scheduled in the ith position and Yj units of resource are used for its processing we have mti I= ge( yj). Compression cost (processing cost) resulting from the application of yj units of resource to job j is found by a given function $( yj) . The objective is to find a sequence u of the jobs and processing times 7~ (equivalently, find cr and y = [ ytr), . . . , yml]) to minimize

n z(“9 Y) = C J;il(Y[i]) + PKta, Y)

i=l

where K( u, y) is one of the following measures:

n (l) CT[i]3 finishing all jobs as soon as possible (makespan ) .

i=l

n

t2) C c[i]9 total flow time. i=l

” n

(3) C C ICU] - C[i]l, total absolute differences in completion times ( TADC) . i-1 j-i

n n

t4) C C I wu] - w[i]17 total absolute differences in waiting times ( TADW ). i=l j=i

n

C5) C lC[i] - dl where d is a common due date for all jobs i=l

that must be found (total earliness and tardiness problem).

In the formula for Z, the value of /3 is a scale factor introduced to maintain compatibility between cost function and scheduling measure K( Q, y). In the following sections, we solve problems [problems ( 1) - (4)] efficiently for any cost function& and any processing timefunctiongU(i,j= 1,. . . , n). Then we consider a sufficient condition under which case ( 5) can be solved efficiently.

The Makespan Problem. For fixed processing times any sequence is optimal. When processing times are controllable the problem can be formulated as

n n Min z(a~ Y) = C f;i](Y[i]) + P C “[i]

i=l i=l

St: 0 I Yj I Uj forj = 1, . . . , ~1.

Given sequence u, if a job j is scheduled in the ith position and yj units of resource are used for its processing, it contributes Ho( Yj) = A( Yj) + pgo( Yj) to total cost. Since the objective function is separable, the optimal value of the amount of resource, yj*, allocated to jobj is found when Ho( yj) is minimum over [ 0, Uj] . It is easy to see that the makespan problem is solved by solving a TP where the ijth element of the n X y1 matrix Q is equal to Hij( yj*>.

LEMMA 1. For a sequence u if Hij( yi) takes minimum at one of the end points of [ 0, Uj] then job j is either processed at normal time PO or at fully compressed time pij.

ProoJ: The proof follows from the separability of Z.


COROLLARY 1. LetJ( yj) = CrjYj, and gU( yj) = PO - aijyi for yi E [0, Uj], (for i, j = 1, . . . ) n). Then there exists an optimal schedule and optimal processing times such that a job is either processed at the normal time PO or at fully compressed time pii. Furthermore, the optimal value of ~$1 for the ith job is found as follows:

I 0 Yt] =

if ‘Yljl 2 PUlil

%I if qi] < &[i]

Prooj The objective function Z can be written as

‘Cay Y) = i {PP[i] + (ali] - Pa[i])y[i]} - i=l

Now, using Lemma 1 the proof follows immediately. Example 1 in the Appendix provides an illustration of the above corollary.

The Total Flow Time Problem. The total flow time problem is formulated as follows:

Min Z(U, Y) = ifii](Y[i]) + P 5 C[i] = 5 {fiil(YIil) + P(n - i + l )TilI i=l i=l i=l

St: OlyiSUj forj = 1,. . . , n.

Given a sequence u, the optimal value of yj* for a job j is found when HU( yi) = A( J+) + p( n - i + 1 )gJ yi) is minimized over [ 0, Uj] . The problem now is to solve a TP with the ijth element of the n X n matrix Q equal to Hii( ~7).

For linear processing time and processing cost functions, results similar to those given by Lemma 1 and Corollary 1 are true for the total flow time problem. It follows from the objective function Z written as

Z(U, Y) = i { P(n - i + l)P[i] + [ali] - P(n - i + l)a[i]lY[i]} i=l

and

p( ?I - i + 1 )P[i] + (“[i] - /3( y1 - i + 1 )U[il)y[il 2 0,

that the optimal value of yfil for the ith job is found by

i

cl Yh =

if &!lil 2 /3(n - i + l)a[i]

%I if “lil < /3(n - i + l)Uljl

It is well known that the total flow time problem for fixed processing times is solved by shortest processing time. ( SPT ) order. Vickson ( 1980a) has noted that the optimal solution of the problem with controllable processing times is also in SPT order when g, = Pj - ajYj andJ=oljyj(fori,j= l,..., n) . The following two-job problem shows that this may not be true in general.

Example. Let PI1 = 5, PI2 = 100, P2, = 100, Pz2 = 2, and /3 = 1. The value of Z for schedules ( 1,2) and (2, 1 ), respectively, is 12 and 300. However, SFT order is not satisfied for the optimal sequence ( 1, 2 ) .

The TADC and TAD W Problems. Given fixed processing times, Merten and Muller ( 1972 ) considered the problem of finding a sequence to minimize completion time variance ( CTV ) given by

n

CTV(U) = (l/n) 2 (Cfil- c)2, i=l

IZ.

396 BAHRAM ALIDAEE AND GARY A.KOCHENBERGER

Merten and Muller have noted that in an organization of computer files in systems with large data sets it is desirable to provide uniform response time to users. In such cases the objective is to determine the arrangement that minimizes variation of access time to different records in the file. It has been shown by Kanet ( 198 la) that the CTV problem is equivalent to the problem of finding a sequence u to minimize

n n TSDC(u) = c c (CD] - c[i])'

i=l j-i

called total squared differences in completion times ( TSDC). Because of the difficulty involved in solving the CTV problem [ Kubiak ( 1993) has recently proved that the CTV problem is NP-complete] , and based on the equivalency between CTV and TSDC, Kanet ( 198 la) proposed the TADC problem as a heuristic for the CTV problem. The TADC problem is to find a sequence u to minimize

n n

Following Kanet’s formulation for fixed processing times, the TADC problem with controllable processing times can be formulated as

Min z(a~Y)= l5Ai](Y[i])+Bi:(i- l)(n-i+ ljr[i] i=l i=l

St: OlyiSUj forj = 1,. . . , ~1.

Given a sequence u the optimal value of ~7 for a jobj is found when the function HU( x) = A( yi) + /3( i - I)( n - i + 1 )gJ vj) is minimum over [ 0, Uj] . Now, the problem can be solved by solving a TP when the 0th element of the y1 X II matrix Q is equal to HU( J$’ ). It follows from the objective function Z written as

z(U, .Y) = I5 { P(n - i + l)p[i] + [“ii] - P(i - l>(n - i + l>a[i]lY[i]} i=l

that for linear processing times and processing costs the optimal value of yri) for the ith job is found by

I 0 Yri] =

if Lytil 2 /3(i - l)(n - i + l)CZ[i]

%I if Ct!til < p(i - l)(?Z - i + l)U[i] *

A similar argument holds true for the TADW problem. In the TADW problem we have

TADW(U) = 5 5 ) wb, - Wrill = i i(n - i)"li] ix, j=i i=l

from which we have Hi/(Yj) =A(&) + Pi(n - i)gij(yj), for i, j = 1,. . . , n. The Total Earliness and Tardiness Problem. The problem is formulated as follows:

Min Z(u, Y, 4 = 5 hil(Yril) + P i IC[il - dl i=l i=l

St: OlyjlUj forj= l,...,n.

If~=crjyj,andg,j=Pj-Ujyj(fori,j=l,..., n) the problem has been solved efficiently by Panwalkar and Rajagopalan ( 1992). For the general case, however, the problem is not solved efficiently.


The total earliness and tardiness problem has been introduced by Kanet ( 198 lb); Panwalkar, Smith, and Seidmann ( 1982) for fixed processing times. Such a performance measure is known to be nonregular as opposed to a regular measure of performance that is a nondecreasing function of the jobs’ completion times (Baker 1974). This is in line with the just-in-time (JIT) concept, in which we would like the jobs to be completed neither too early nor too late. In this situation we would like jobs be competed as close as possible to the desired due date d, preferably completed (or started) exactly at the time d. Lemma 2 given below is a well-known result (Kanet 198 1 b; Panwalkar, Smith, and Seidmann 1982) providing a basis for our further analysis of the total earliness and tardiness problem.

LEMMA 2. Consider the problem with fixed processing times. Given a sequence c there exists an optimal value of d that coincides with a completion time of a job in the position r, and the objective function 2 can be written as

z(a, 4 = (O)PI,I + (1 V’t21 +. . . + (r - 1 U=I,I

+ (n - r)P[r+l] + * * * + 24,-l] + ( 1 )P[n].

Furthermore, an optimal sequence can be found by assigning the largest job to the position with the smallest penalty (position 1)) the second largest job in a position with the next smallest penalty (position 2 or n), and so on. Finally, the optimal due date for the sequence will coincide with the completion time of the set of nontardy jobs and the schedule starts at some time R, where the optimal value of due date is d = R + z1 ;=I Plil. The optimal sequence thus obtained is V-shaped, i.e., jobs scheduled before the smallest job in a nonincreasing order of processing times and the rest of jobs in a nondecreasing order of processing times.

LEMMA 3. Consider the problem for controllable processing times. Given a sequence u and processing times ?r, an optimal due date d will coincide with the completion time of some job in a position r. Furthermore, the objective function Z can be written as

Z(u, Y, 4 = (~~II(YIII) + P(Obqt,) + (J;21(~[21) + PC 1 h23) + - - -

+ (fir](Y[rl) + P(r - 1 h[rl) + (.fi~+ll(Y[r+ll) + P(n - ~h[r+ll)

+* - - + (fin-Il(Y[n-11) + #42>7qn-II) + (Afi,l(Y[fll) + P( 1 )*m1).

The optimal due date is d = R + C := 1 r[i] where the schedule starts at some time R. ProoJ For given processing times the value of 2 F-1 fiil(vli)) is constant. Now, the

desired result follows from Lemma 2. It follows from Z that an early job in the ith position will add

&j(fi) =“$(J$) + P(i i lkij(yj), for some yj E [0, Uj]

to the total cost, and a tardy job in the ith position will add

&j(fi) =J(yJ + P(n - i + lkij(yj), for some yi E [0, Uj]

to the total cost. Furthermore, the optimal value of ~7 for an early job occurs at a minimum of HeJ vi) and for a tardy job it occurs at a minimum of H,o( vj) over [ 0, Uj] . Now, define a (2n) X 12 matrix Q as

where Qe and Qt are two y1 X n matrices with the ijth element equal to H& ~2) and H& J$), respectively. In the following we consider an assumption under which the


problem of earliness and tardiness will be solved efficiently. The assumption is based on the objective function Z as presented in Lemma 2. Consider two early positions i and k where k > i (i.e., k is a position closer than the ith position to the due date d). If a job j is scheduled in the ith position it will add (i - 1 )Pj to Z while the same job if scheduled in the kth position will add (k - 1 )Pj to Z. Similarly, for two tardy positions i and k where k < i (i.e., k is a position closer than the ith position to due date d), if a job j is scheduled in the ith position it will add (~1 + i - 1 )Pj to Z, and if it is scheduled in the kth position it will add (n + k - 1 )Pj to Z. Hence, in both cases job j is adding more to Z if scheduled in a position closer to the due date d. This can be interpreted to mean that jobs are competing to be scheduled closer to the due date. From an overall system’s standpoint we are willing to accept extra costs to schedule a job closer to the desired due date. In the following this idea is used as a general assumption under which the earliness and tardiness problem can be solved efficiently.

ASSUMPTION. Assume that if a jobj is early and scheduled in the ith position it adds less to total cost Z compared to the case when the same job is early and scheduled in the kth position where k > i. Likewise, assume jobj is tardy and scheduled in the ith position it adds less to the total cost Z compared to the case when the same job is tardy and scheduled in the kth position where k < i. Alidaee and Ahmadian ( 1993) showed that the above assumption holds true for the earliness and tardiness problem with controllable processing times when A( v) = ajy, and go = Pj - ajy for i, j = 1, . . . , IZ . Now consider the TP (with solution conditions i, ii, and iii given at the end of section 1) with Q defined as above. A subset of elements of Q that satisfies (i-ii) is denoted by G. If it also satisfies (iii), it is denoted by G*.

PROPOSITION 1. Let G* be an optimal solution of the transportation problem. If the ijth element of Qe belongs to G*, then for each k < i there exists some column r such that the krth element of Qe belongs to G *. If the ijth element of Qr belongs to G*, then for each k < i there exists some column r such that the krth element of Qt belongs to G*.

Proof: Assume that the ijth element of Q,, belongs to G*. Suppose for some k < i, no element of row k belongs to G*. Let q be the first integer larger than k such that the qrth element of Qe belongs to G* for some column r. It follows from the above assumption that for any integer s and k < s I q we have

which violates the optimality of G*. The same argument can be given for Qt.

PROPOSITION 2. G* is an optimal solution to the total earliness and tardiness problem. Proof: It follows from Proposition 1 that an optimal solution to the transportation

problem is a feasible solution to the total earliness and tardiness problem. The optimality of G* follows from the fact that the set of feasible solutions to the total earliness and tardiness problem is contained in the set of feasible solutions to the transportation problem (for which G * is optimal ) .

In Lemma 4 we show that for a class of processing time functions the stated assumption is satisfied; hence, the earliness and tardiness problem can be solved by using Propositions 1 and 2.

LEMMA 4. Let gg = gj( Yj), Pu = Pi, and pv = pj, for i, j = 1, . . . , n, then the total earliness and tardiness problem can be solved by using Propositions 1 and 2.

Proof It suffices to show that the above assumption is satisfied. Since functions& and gj can take only positive or zero values, then the assumption follows from the fact that

mifuj { A( yj) + Pkgjt pi) 1 5 oyy”u, {.6( yi) + @isi( vj) 19 for k > i. / J


The class of processing time functions considered in Lemma 4 includes those that have been studied previously, except the one studied by Adiri and Yehudai ( 1987). In Lemma 2 it was indicated that for fixed processing times an optimal solution to the problem is V-shaped. Panwalkar and Rajagopalan ( 1992) proved that the V-shaped property is also satisfied whenJ( y) = ajy, and go = Pj - ajy for i, j = 1, . . . , n. The following four-job problem shows that in general the V-shaped property may not be satisfied.

Example. Let the processing times be as follows:

P,, = 1 P*2 = 10 P33 = 15 P44 = 2 PO = al

for all other i, and j, and 8 = 1.

Clearly, the optimal sequence is u = ( 1, 2, 3, 4). By Lemma 3 there are five possible cases of due dates to be considered. Four of the cases have due dates set at the start of each job and one case where the due date is set at completion of fourth job. The value of 2 for the case when the due date is set at the start ofjob 3 or 4 is (0)PIll + P12] + 2P13] + PL4]. This value is smaller than the value of 2 for other possible due dates. Clearly this schedule does not satisfy the V-shaped property, as the processing times are ?r = ( 1, 10, 152).

In Proposition 2 we established only that an optimal solution to the TP is an optimal solution to the scheduling problem. If assignment G satisfies only (i-ii), this may not be a feasible solution to the scheduling problem. To show this, consider the case when y1 is even. The assignment defined as follows satisfies (i-ii) but is not a schedule. Choose element (1, l), (3, 3), (5, 5),- * * of Q, and (2, 2), (4, 4); . - ofQ,. This means we process job 1 in the first position and the second position is not filled and then the third position is filled with job 3, and so on. At the other end we process job 2 in the last position, n, and then the position n - 1 is not filled while position n - 2 is filled with job 4, and so forth. Clearly this is impossible, since two jobs j and k are scheduled on the ith and the (i + 2)th positions then there must be a job to be scheduled in the (i + 1)th position. The same argument is true for the tardy jobs. Without the stated assumption there is no guarantee that such a situation will not occur. In fact, even for the linear case given as J = &jYj and gij( yj) = PO - aijyj, for i, j = 1, . . . , n, there is no guarantee that an assignment G will be a feasible schedule to the earliness and tardiness problem. Example 2 in the Appendix provides an illustration of the earliness and tardiness problem with the above assumption satisfied.

3. Parallel Machine Problem

Assume there is a set of n jobs (j = 1, . . . , n) available at time zero to be processed bymmachines(k= l,..., m). Each job can be processed by any one of the machines and a machine can process only one job at a time. Related to the it jobs and m machines the following notation is defined:

Pi, the normal time of job j if scheduled in the ith position on machine k, pi, the crash time of job j if scheduled in the ith position on machine k, uJ, the maximum amount of resource that can be used to process job j by machine k, yj” E [0, u,“] , amount of resource used for processing job j by machine k, &, processing time of job j if scheduled in the ith position on machine k, gi(yj”), processing time function of job j if scheduled in the ith position on machine k, fT( y!), cost function of job j if scheduled in the ith position on machine k, u=(ul,..., Q”‘), schedule of jobs on m machines, rJk = (111, * * * 9 bkl), a schedule of nk jobs on machine k,


7r = (R’, . . . ) ?r “) , vector of processing times, 7rk = (a;,], . . .) ?rf,,$, vector of processing times of 12 k jobs scheduled on machine k, Cfil, completion time of ith job on machine k The objective is to find a sequence u of the jobs and processing times 7r (equivalently,

find u and ytll, . . . , ytnkl for k = 1, . . . , riz) to minimize

Z(a9 Y) = 5 ifi ffi]CY:i]J + PKta, VI, k=l i=l

where K( 6, y) is one of the following measures: m nk

(l) C z1 Cfi], total flow time. k=l i=l

m nk

t2) C C ICfi]-dl, where d is a common due date to all jobs k=l i=l

that must be found, (total earliness and tardiness problem).

Again, the value of p is a scale factor introduced to maintain compatibility between cost function and scheduling measure K( u, y). When J = 0 and g$ = a: tj with tj and a: being constant numbers (for i, j = 1, . . . , n) we have the model studied by Adiri and Yehudai ( 1987). In their model no compression costs were involved. They gave an 0( n”’ log n) algorithm for the total flow time problem.

In the rest of the paper we consider a class of processing time functions in which for each job j we have

&Yj”> = k$<Yj”> where Pj” 5 gj”( yf ) I pi” for some value of yj” E [ 0, ~$1.

This class of functions includes those studied by Vickson, Van Wassenhove and Baker, and Trick. In the following we show that for such processing time functions the flow time problem and the total earliness and tardiness problem can be solved efficiently.

The Total Flow Time Problem. The problem can be formulated as follows:

Min i=l I

St: 0 I yj” I I.47 fork= l,..., m,andj= l,..., nk.

The problem for a machine k and a subset of rr k jobs is to find a schedule uk and processing times 7rk to minimize

nk nk Zk( uk, Yk) = Z f:i](Y:i]) + P C Cfi] = $ {f fil(Yfil) + P(nk - i + l >rt] > *

i=l i=l i=l

Now, we have a single machine nk job problem that can be solved by a transportation problem. It follows from Z k that, independent of the number of jobs n k, for a machine k if a job j is scheduled in the ith position from the end it contributes H$(yf) = fj”( yj”) + pig!< yj”) to total cost.

LEMMA 5. For a machine k, a job j contributes more to the total cost if scheduled in the ith position compared to the case when it is scheduled in the wth position with w < i.

ProojI Since functions fr and g$ can take only positive or zero values, then the desired result follows from


Now, define an mn X n matrix Q by

where Q’ (for i = 1, . . . , n) is an m X n matrix and the kjth element of Q’ is equal to E&y,* ‘), representing contribution of job j to the total cost if scheduled in the ith position from the end on machine k. Function Hi( yf ) takes on minimum at yy k E [ 0, $1. Now, for the TP with Q defined as above a subset of elements of Q that satisfies (i- ii) is denoted by G. If it also satisfies (iii), it is denoted by G*.

PROPOSITION 3. Suppose G* is an optimal solution of the transportation problem. If the kjth element of Q’ belongs to G* then for each w < i there exists some column r such that krth element of Q” belongs to G*.

PROPOSITION 4. G* is an optimal solution to the total flow time problem. Proofs of Propositions 3 and 4 are similar to proofs for Propositions 1 and 2, respectively.

The Total Earliness and Tardiness Problem. The problem is formulated as follows:

Min f P 2 ICfiI - dl i=l I

St: 0 5 yj” I uj” for k = 1,. . . , m, and j = 1, . . . , nk.

Now, the problem for a machine k and a subset of nk jobs is to find a schedule uk, processing times ?yk and a common due date dk to minimize

Zk( uk 2 Yk, dk) = 5 {ffil(Yfil) + BICll - dkl}. i=l

This is a single machine n k job problem that can be solved by a transportation problem. It follows from Lemma 3 that there exists a job in the rkth position that must be completed at due date dk = C $ i 7r 11. Furthermore, we have

zk(uk,yk,dk)=(ft(lI(Yt(l~)+B(O)~~,I)+(ffiI(~I)+P(1)~~~~)

+. . . + (.fbk~ CY~~I) + P( rk - lb+])+ (f$+*l(g++l])

+B(nk-rk)&+lI)+* ’ ‘+(f:nk-,](Y:nk-,])+16(2)1Tt(nk-,])

+u-~iJq(Y~iiq)+~w~~nq).

It follows from Z k that, independent of the number of jobs n k, for a machine k if a job j is early and scheduled in the ith position it contributes H$(yjk) = fj”( yj”) + p( i - l)gt(yf) to total cost. Similarly if a job j is tardy and scheduled in the ith position from the end its contribution to total cost is H$(JJjk) = fjk(y;) + 8( i)gjk( yj”). Now, define a 2mn X n matrix Q by


Qt

Q; Q= .

The k jth element of the IZ X n matrix QL (for i = 1, . . . , n) is equal to H$( yzk), where y$” makes this function minimum over [0, ~71. Similarly, the kjth element of n X II matrix Qf (for i = 1, . . . , n) is equal to H$j(y$ k), where yt k makes this function minimum over [ 0, ~11.

PROPOSITION 5. Let G* be an optimal solution to the transportation problem where Q is defined as above. If the kjth element of Qi belongs to G* then for each w < i there exists some column Y such that the krth element of Q: belongs to G*. If the kjth element of Qf belongs to G*, then for each w > i there exists some column r such that the krth element of QY belongs to G*.

PROPOSITION 6. G* is an optimal solution to the total earliness and tardiness problem. Propositions 5 and 6 provide a sequence u and processing times a to the earliness and tardiness problem. Then the optimal due date can be found by d* = maxl&sm { dk } , where dk = 9lkl rfil, and the first job on machine k starts at time Rk = d* - dk, for k = 1 . . 2 m.

P&ofs for Propositions 5 and 6 are similar to the proofs for Propositions 1 and 2, respectively.

4. Conclusion and Remarks

In this paper we considered single and parallel machine scheduling problems with controllable processing times for each job. The processing time of a job j was assumed to depend on the position of the job in the schedule and is a function of units of resource applied for its processing. For the single machine problem, the objective was minimization of total compression costs plus a scheduling measure. The scheduling measures included makespan, total flow time, total differences in completion times, total differences in waiting times, and total earliness and tardiness with a common due date for all jobs. Except for the total earliness and tardiness measure, all variations were solved efficiently. Under an assumption that is typically satisfied in a JIT environment, the problem with total earliness and tardiness measure was also solved efficiently. For a large class of processing time functions, the parallel machine problem with total flow time, and total earliness and tardiness measures was solved efficiently. In all cases the problems were reduced to a transportation problem which is known to be solved in 0(n3) time (Pa- padimitriou and Stieglitz 1982 ).

We have restricted the use of resource yj for a job j up to an upper bound Uj. It is reasonable to consider problems under an assumption that the total resource level used by all jobs must not exceed a global amount. With these restrictions, given a sequence u, since the objective function 2 is separable, then the problem of allocating resources to jobs can be solved efficiently by available algorithms [refer to Ibaraki and Katoh ( 1988); for special cases refer to Vidal ( 1987)]. Under the global restriction, the problem of allocating jobs and sequencing them on a single machine may not, however, be easy to solve.


In the case of parallel machines, some of the problems are known to be N&hard [refer to Adiri and Yehudai ( 1987) and Trick ( 1994)]. For such difficult problems, some work (on particular cases) is starting to appear in the literature. For example, Daniels ( 1989) formulated the multiproject scheduling problem as a single machine scheduling problem to minimize maximum tardiness under a global resource constraint. Moreover, based on the apparent urgency ( AP) algorithm of Vepsalainen and Morton ( 1987) for the weighted tardiness problem, Lawrence and Morton ( 1993) gave good heuristic algorithms for multiproject resource constraint scheduling problems to minimize the weighted tardiness costs when jobs have fixed processing times. It is the subject of further studies to apply similar heuristic algorithms to the case when processing times are controllable.’

’ The authors thank the three referees whose comments helped greatly improve the presentation ofthis paper. We also thank Susan Blake-Caldwell a student of English at West Texas A&M University for helping in presentation of the paper.

Appendix

Example 1. This example provides an illustration of a IO-job and one machine makespan problem. Let gr,(fi)=Pj-uiifi,andJ(yj)=ajfifori,j=l ,..., 10, and let normal time P, crash time p, value of (Y and the matrix A = [ ati] be as follows:

Pj: II 6 13 22 18 11 10 24 16 3 Pj: 2333 111 3 3 2

CYjZ 2313 231332

4 2 3 3 4 1 4 2 2 4

4 I 2 3 3 12242

I 4 3 3 3 13144

3 3 3 4 3 2 2 4 2 3

A=4 3 3 4 1 3 4 3 4 4 3 2 1 3 3 44221

1 2 1 4 3 1 3 1 4 1 3 2 3 1 3 4 1 4 1 3

11422 14 144

3 3 2 1 4 3 3 2 1 4

Since pj I rrj 5 Pj, then we must have the values of uj for j = 1, . . , n as follows:

uj: 2.25 .I5 2.5 4.75 4.25 2.5 2.25 5.25 3.25 .25.

We let the value of factor fl = 1. Calculating matrix Q we have

6.5 6 8 22 9.5 1 I 3.25* 24 16 2.5

6.5* 6 10.25 22 13.75 11 7.75 24 12.75 3

11 5.25* 8 22 13.75 11 5.5 24 12.75 2.5

8.75 6 8 17.25* 13.75 11 7.75 18.75 16 2.75

Q = 6.5 6 8 17.25 18 11 3.25 24 12.75 2.5* 8.75 6 13 22 13.75 s.5* 3.25 24 16 3

11 6 13 17.25 13.75 11 5.5 24 12.75* 3

8.75 6 8 22 13.75 8.5 10 18.75* 16 2.75

11 6 5.5* 22 18 11 3.25 24 12.75 2.5

8.75 6 10.25 22 9.5* 11 5.5 24 16 2.5

Indicated values by an asterisk in the matrix represent optimal solution that was given using LINDO. Hence the sequence picked by this solution and the actual processing times are as follows:

j: 7, 1,2,4, 10, 6, 9, 8, 3, 5

Xj: 1, 2, 3, 3, 2, 1, 3, 3, 3, 1


All jobs will finish in 22 units of time (makespan). Example 2. This example provides an illustration of an eight-job and one machine total earliness and

tardiness problem. Letgu(y,)=Pj/(fi+ l)andJ(-(yi)=ajyiforj= l,..., 8, where normal time and P, crash time p and the

value of (Y are as follows:

Pjt 7 4 7 13 10 6 6 14 Pi: 2333 1 1 1 3 ffj: 231 3 2 3 1 3

Since pj s 3 s Pi then the value of Uj for j = 1, . . . , n is as follows:

Uj: :::i955y

We let the value of factor /3 = 1. Calculating matrix Q we have. Calculating matrices Q. and Q, to form the matrix Q we have

0 0 0 0 0 0 0 0*

5.48 3.92 4.33 9.48* 6.94 5.48 3.89 9.96 8.58 7 7.33 14.66 10.65* 9 5.92 15.33

Qe=

10.96 10* 10.33 18.63 13.5 11.69 7.48 19.44 13 13 13.33 21.98 15.88 13.97 8.8* 22.92 15 16 16.33 25 18 15.97 9.95 26 17 19 19.33 28 19.9 17.48 11 29 19 22 22.33 31 21.66 19.44 13 32

21 25 25.33 34 23.29 19 22 22.33 31 21.66 17 19 19.33 28 19.9

Q1 = :: :: ::::: 2::8 li.888 10.96* 10 10.33 18.63 13.5 8.58 7 7.33* 14.66 10.65 5.48 3.92 4.33 9.48 6.94

21 14 35 19.44 13 32 17.48 I1 29 15.97 9.95 26 13.97 8.8 22.92 11.69 7.48 19.44

9 5.92 15.33 5.48* 3.89 9.96

Indicated values by an asterisk in the matrix represent optimal solution that was given using LINDO. Hence the sequence picked by the solution and the actual processing times are as follows:

j: 8, 4, 5, 2, 7, 1, 3 6 3: 14, 6.25, 3.16, 3, 1.22, 2.16, 3, 4.26

Jobs 8, 4, 5, 2, 7 are early and the rest of jobs are tardy. Optima1 values for due date and objective function, respectively, are 27.63 and 62.7. The schedule starts at time zero.

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Documents

A FRAMEWORK FOR MACHINE SCHEDULING PROBLEMS WITH CONTROLLABLE PROCESSING TIMES