- Email: [email protected]
Due-date assignment and single machine scheduling with compressible processing times
international journal of
production economics
ELSEVIER
Int. J. Production
Economics
43 (1996) 29-35
Due-date assignment and single machine scheduling with compressible processing times T.C.E. Cheng, C. O&z*, Department
of Management,
The Hong Kong Polytechnic
Received
21 February
X.D. Qi
lJniversi&, Hung Horn, Kowloon, Hong Kong
1995; accepted
11 October
1995
Abstract In this paper we consider a due-date assignment and single machine scheduling problem in which the jobs have compressible processing times. Two models are defined according to the due-date assignment methods used. The first model applies the common (constant) due-date assignment method to assign the due-dates, while in the second model the due-dates are assigned using the slack due-date assignment method. The objective is to determine the optimal sequence, the optimal due-dates and the optimal processing time compressions to minimize a total penalty function based on earliness, tardiness, due-dates and compressions. We solve the problem by formulating it as an assignment problem which is polynomially solvable. For the case that all the jobs have a common upper bound for compressions and an equal unit compression penalty, we present an O(nlogn) algorithm to obtain the optimal solution. Keywords:
Sequence;
Due-date
assignment;
Compression
1. Introduction
Recently, attention has increasingly been given to due-date assignment and scheduling problems in which the jobs have compressible processing times. The concept of compressible processing time is from the area of project planning and control, see, for example [ 11. Its motive in the field of scheduling is of the same nature, that is, the assumption of compressible processing times is justified in situations where jobs can be accomplished in shorter or longer durations by increasing or decreasing additional resources. Studies on problems with com-
* Corresponding
author.
0925-5273/96/$15.00 0 1996 Elsevier SSDI 0925-5273(95)00194-8
Science B.V. All rights reserved
pressible processing times were initiated by Vickson [2,3]. A survey on this line of scheduling research can be found in Nowicki and Zdrzalka [4]. Literature on this problem includes, among others, Van Wassenhove and Baker [6], Nowicki and Zdrzalka [4]. Daniels and Sarin [5], Zdrzalka [7] and Cheng and Janiak [8]. More recently, Panwalkar and Rajagopalan [9], Alidaee and Ahmadian [lo] and Cheng and Janiak [S] extended the problem to include the due-date aspect. Specially, Panwalkar and Rajagopalan [9] considered the common due-date assignment and single-machine scheduling problem in which the objective function is the sum of penalties based on earliness, tardiness and processing time compressions. They reduce the problem to an assignment
T.C.E. Chang et al./Int. J. Production
30
problem. Alidaee and Ahmadian [lo] extended the results of Panwalkar and Rajagopalan [9] to the parallel-machine scheduling case. Cheng and Janiak [S] further generalized the result to the case where the cost of compression is a general convex function of the amount of compression. In this paper we consider a due-date assignment and single-machine scheduling problem in which a penalty for due-dates is added to the objective function which includes the penalties for earliness, tardiness and processing time compressions. The common due-date assignment method and the slack due-date assignment method are used to assign due-dates to jobs. We solve the problem by formulating it as an assignment problem.
2. Problem formulation This paper considers a single-machine scheduling problem involving n jobs with the assigned duedates dj > 0, j = 1, 2, . . . , n. A job j has a normal processing time Esj, which can be compressed by an amount Xi > 0, 1 < j < II. Without loss of generality, we assume that the jobs are indexed in nondecreasing order of normal processing times, that is, p1 d p2 < ... < Pn. For each jobj, its compression has an upper bound Xj, 1 d j < n. Thus, if pj represents the actual processing time of job j, then pj = fjj -
x.
J’
0 < Xj d
Xj
<
pj,
1
The condition Xj < pj is necessary because if there is a job j such that Xj = pj, then it may be compressed to zero processing time, which is not reasonable in practice. In this paper we assume that all the jobs are ready at time zero and no preemption is allowed. Let Cj, Ej and Tj denote the completion time, the earliness and the tardiness of job j, respectively, then Ej = max (0, dj - Cj) and
Tj = max (0, cj - dj).
The objective function to be minimized is the total penalty based on earliness, tardiness, due-date value and compression of each job. It can be
Economics
expressed Z(fJ,
43 (1996) 29-35
as follows:
2, 2’) =
i (UEj + /ITj + ydj +fj(Xj)), j=l
(1)
where G is a job sequence, a > 0, /3 > 0 and y 3 0 are unit penalties for earliness, tardiness, and duedates, respectively. Here, d = (d,, dZ, . . . ,d,) is a vector of assigned due-dates and 2 = (x1, is a vector of job compressions. Further, x2, “. > x,) fj(xj) = bjXj> bj 3 0,
j = 1, 2, . . . 3PI,
where bj is the unit cost of compression for job j, 1 djdn. The problem is to determine the optimal sequence G*, the optimal due-date vector a* and the optimal compression vector x”* that minimizes the function z(a, d”,22). We consider the following two due-date assignment methods to assign due-dates to the jobs. (1) Common/constant (CON) due-date assignment method: All jobs are given an identical flow allowance d 2 0, that is, dj = d,
j = 1, 2, . . , n.
Here, the common due-date is set internally and is announced upon the arrival of jobs. It is a fixed attribute of a job. This method entirely ignores any information about the arriving jobs, jobs already in the system, future jobs, or the structure of the shop itself. It is representative of common practice where salesmen quote a uniform delivery date on all orders regardless of the order processing times. (2) Slack (SLK) due-date assignment method: Jobs are given flow allowances that reflect equal waiting times (i.e., equal slacks) k > 0, that is, dj = pj + k, j = 1, 2, . . , n. In this method, the due-dates are set internally by the scheduler as each job arrives on the basis ofjob processing times. As defined in this method, jobs are given equal slacks, which represent the practice in which customers are treated equally with regard to their waiting times. A comprehensive survey on due-date assignment methods can be found in [ 111. Therefore we have two models, namely the CON model and the SLK model, where the former corresponds to the common due-date assignment
T.C.E. Chang et aLlInt. J. Production
method and the latter corresponds to the slack due-date assignment method. The objective function can be denoted as z (a, d, 2) for the CON model and z (0, k, 2) for the SLK model, respectively. For any sequence cr, we use l-j] to denote the job in the position j of the sequence c.
3. Optimality
Economics
Lemma 4. Given 2, for any sequence, there exists an optimal value of d such that one of the jobs completes on d. Proof. The proof can be found in [12].
n(P - Ma In this section we discuss the optimality conditions for the two models. Eight lemmas are presented. The first three lemmas are applicable to both the CON model and the SLK model.
Proof. The proof follows from (l), after substituting common due-date and slack due-date for dj. As a result of Lemma 1, we assume throughout the remainder of this section.
/I > y
the early jobs
Lemma 3. Given X in an optimal sequence, the early jobs are in the longest processing time (LPT) order, and the tardy jobs are in the shortest processing time (SPT) order. Proof. The proof follows from a simple job-interchange argument. 5 are applicable
The following three lemmas are applicable SLK model only. sequence,
to the
the last
Proof. Assume that there are I early jobs in an optimal sequence D*. Suppose ptr+ il < prll. Interchanging the two jobs in positions Y and r + 1, we obtain another sequence (T’.The change in the objective function value is given by ~(a’, k, 2) - ~(a*, k, 2) = cc& - a(dc,+u
-
C;r,) + B(C;,+,,
- CI,+II) - P(C,r, - d,,,) < 0.
Therefore, the schedule is improved, which contradicts the fact that g* is an optimal sequence. Lemma 7. Given 2, for any sequence, there exists an optimal value of slack allowance k such that the last early job completes on its due-date.
Crj+ir Q dtj+ir.
Since Ctj+ 11= Ctjr + Ptj+ 11 and dtj+ 11= Ptj+ 11 + k, from the second inequality, we obtain Ctjr < k, which contradicts the first inequality.
Lemmas 4 and model only.
Proof. The proof can be found in [12].
-dir+,,)
Proof. For the CON model, the proof is trivial. For the SLK model, if in a sequence 0, the job in position j is tardy but the job in position j + 1 is early, then we have and
+ B).
Lemma 6. Given x”,in an optimal early job is the shortest job.
Lemma 1. If p < y, then d* = 0 and k* = 0.
Ctjr > dtjr
31
Lemma 5. Given x”, for any sequence, there exists an optimal due-date equal to C,,,, where r is the smallest integer greater than or equal to
conditions
Lemma 2. Given X, for any sequence, must precede the tardy jobs.
43 (1996) 29-35
Proof. Given 2 and 0, assume that for an optimal value of slack allowance k, the last early job is in position r. Suppose C,,, < dr,]. Let L = d,,, - C,,, and R = C [r+ 11- dr,+ 1,. If the slack allowance is changed to k’ such that d;,, = C,,,, then by Lemma 6, we have ~(0, k’, 1) - z (a, k, 2) = ( --ar + P(n - r) - yn)L. Similarly, if the slack allowance is changed to k” such that d[,+ 1l = C,,, 1l, then the last early job is in position r + 1 and we have
to the CON z (a, k”, 2) - z (a, k, 2) = ( -ur
+ /3(n - r) - yn) R.
32
T.C.E. Chang et al./Int. J. Production
Therefore, z (a, k’, 2) < z (6, k, 2) if ( -cxr + /? (n - r) - yn) B 0 and z(a, k”, 2) < z (a, k, 2) otherwise. The proof is complete. Lemma 8. Given 2, for any sequence, there exists an optimal value of slack allowance k equal to Ctr_lI, where r is determined by Lemma 5. Proof. Let the job in position r be the last early job
which completes on its due-date. Then, from Lemma 7, we know that the optimal slack allowance is C,,_ 1I. Therefore, for a small A > 0, if we shift the slack allowance from C,,_ rI to C,,_ 1I + A,
Economics
43 (1996) 29-35
of a job in a position with a negative position weight should be its normal processing time, and the processing time of a job in a position with a positive position weight should be its normal processing time minus the upper bound of its compression. If a position j has a zero position weight, then the optimal processing time of the job in this position may be any value between Prjl - Xtjl and Ptjl. These can be written in the notational form as follows:
~(0, C[r-l, + A, 2) - ~(a, C[r-l, + A, x”) 2 0,
if Wj < 0, if wj = 0, PC1= P;jl, P[j] X[j]> if Wj > 0, i
so
where Prjl- XfJ1d p;jld &I
P[jly
(3) and
~$1, 1
represents the optimal processing time of the job in position j. Therefore, the optimal compressions can be obtained by
(ar - fl(n - I) + yn)A 2 0,
or
Xr*il = P[jl
r 2 n (P - Y)/(@+ P). Similarly, if the slack allowance is shifted from C,,- rI to C,,_ 1I - A, we obtain r-l
-
p;Fjl,
j = 1, 2, . . . , Tl.
(4)
4.2. SLK Model
For this model, we have r-l ~(0, k, 2) = C (Mj+ ~(n + 1) - btjl)Ptjl
The proof of the lemma follows immediately.
j=l
4. Optimal compressions + f: 4. I. CON model
(P(n-A + Y -
For this model, if we substitute, CIj, = xi= 1pli], d = xi= lp[j] and Xtjl = Prjl - Prjl into Eq. (1) and simplify, we have
~(0, d, -f) = f: (~0’ - 1) + yn - b,j])Prj]
Glj + Y(n + 1) * wj= i fi(n-j)+y-btjl, P[jl>
jz$+ l(B(n -j
+ l) -
&jl)P[jJ + i
i&l =
bjPi.
j=l
Let w,= I
bjlTj.
j= 1
Let $j and p^&, 1 d j < n, denote the position weight of position j and the optimal processing time of the job in position j, respectively, then
j=l
+
b[jl)P[jl + i
j=l
P;jl, i Prj] - Xrj],
b[j],
1
if ~j < 0, if 9j = 0, if &j > 0,
(5)
05)
Using where&j1 - -$jl< p;jl< FIjl. cc(j-l)+yn--tjl, i P(n -j + l) -
b[j],
1 dj
(2)
then wj, 1 d j < n, represents the position weight of position j in the sequence (T.Since x7= lbjpj is a constant, for any sequence, the optimal processing time
the same argument for the CON model, we see that the optimal compressions can be obtained by a:jj, = P[jr - $r*j]F j=
1, 2, . . . ,n.
(7)
Theorem 1. Given a sequence, for the CON model and the SLK model, the optimal compressions can
T.C.E. Chang et al./Int. J. Production
be determined as follows: the compression of the job in a negative-weight position is zero; the compression of the job in a positive-weight position is its upper bound; if the position weight of a position is zero, then the compression of the job in this position can be any value between its lower bound and its upper bound. Proof. The proof follows from the analysis above.
Economics
43 (1996) 29-35
33
position i of a sequence. Furthermore, let Yij = 1 if job j is in position i, and Yij = 0 otherwise, i, j = 1, 27 ... 3n. Then we can formulate the sequencing problem associated with the CON model as the following assignment problem: Assignment
min i
i=lj=l
f:
Problem 1 CijYij
s.t.
5. Optimal sequences
i$lYij = 19 j = 192, ... $4 Now we discuss the optimal sequences for the
two models. In view of the analysis in the previous sections that the optimal due-dates and compressions can be computed for any given sequence, the problem reduces to a pure sequencing problem. In order to obtain the optimal sequence, we formulate the problem as an assignment problem. This is based on the observation that the problem is to form an optimal sequence by assigning n jobs to n possible positions with the condition that each job can only be assigned to one position and each position is to take exactly one of the jobs.
i=l,2, Yij=l
or
. . ..?I. i,j==l,2,
Yij=O,
. . . . ?l.
5.2. SLK model
For this model, we can formulate the sequencing problem as the following assignment problem: Assignment Problem 2
min i f c^ijYij i=lj=i
5.1. CON model For i, j = 1, 2, . . . , n, let E(i-l)+yn--j,
w,,= IJ
i
s.t.
1
/?(n-i+l)-bj,
where r is the smallest integer larger than or equal to n(fl - y)/(cr + /?) as defined in Lemma 5, and
i$lYij =1, j=l,2,
. . . . n,
j$i Yij = 1, i=l,2,
. . . . n,
Yij = 1 Pj7 Pij =
Pj,
i-fpj
-
Xj,
if if if
Wij < Wij = Wij >
0, 0, 0,
(3)
where pj - Xj < pJ < pj. We define an n x n matrix C by C = (Cij), where cij =
wijpij.
It is noted that cij, 1 < i,j < n, is the minimum cost resulting from an optimal combination of the position weight and compression if job j is assigned to
or
Yij = 0,
i, j = 1, 2, . . . , It.
where, A A A Cij = Wijpijs A
wij =
~i + JJ(~ +
/I(n -
1) - bj,
i) + Y - bj,
if 9ij < 0, if ~ij = 0, if tiij > 0,
1 < i < r - 1, r < i < n.
(3)
34
T.C.E. Chang et al./Int. J. Production
and yij = 1 if job j is in position otherwise, i, j = 1, 2, . . . , n.
i, and
yij = 0
Theorem 2. The optimal sequences for the CON model and the SLK model can be obtained by solving Assignment Problem 1 and Assignment Problem 2, respectively. Proof. From the analysis Theorem 2 is obvious.
above,
the
proof
of
Economics
43 (1996) 29-35
jobs in positions j and j + 1 may be early or tardy jobs, and w~jl and W~j+ i1 may be positive, negative or zero. For brevity, we only show a case for the CON model in which the two jobs in positions j and j + 1 are all early jobs, and Wtjl < 0 and Wtj+ il > 0. Interchanging the two jobs in positions j and j + 1, we obtain another sequence G’. The change in the value of the objective function is given by z(cr’, d, 2) - ~(a*, d, 2)
Since an assignment problem can be solved in O(n3) time [13], the optimal sequence can be found in polynomial time. Once the optimal sequence is determined, from Lemma 5, Lemma 8 and Theorem 1, we know that the optimal common duedate for the CON model, the optimal allowance for the SLK model and the optimal processing time compressions for both models can be obtained. Since the dominant step in this solution procedure is the assignment algorithm used to obtain the optimal sequence, the overall computational complexity of the problem is 0(n3).
6. A special case We now consider a special case of each model in which bj=b and Xj=X, where O
= (~0’ - 1) + in - b)PIj+ 11 + (olj + yn - b)(&
- 2) - (LY(~- 1)
+ JJ~ - b)ptj, f (aj + yn - b)(Ptj+ 11- X) =
cc(P[j]-P[j+l])
< 0.
The other cases can be proved in a similar manner. The proof of Theorem 3 is complete. In the following we present an 0 (n log n) algorithm for this special case. Let r x 1 be the smallest integer greater than or equal to x. Algorithm 1 step
1.
I’ +- n (p -
y)/(cc
+ p).
Step 2. If I’ < 0, then r + 0. Else
-r 4~ - ~)ib+ 8) 1.
r Step 3. Weigh n positions by using equation (2) for the CON model and equation (5) for the SLK Model. Step 4. Rank the position weights Wj for the CON model and Gj for the SLK model in descending order of magnitude such that the largest Wj(9j) is ranked 1 and the smallest Wj(Gj) is ranked n. Break ties arbitrarily. Step 5. Find the optimal sequence by matching the position weights in descending order with the jobs in ascending order of their normal processing times. Step 6. Calculate the optimal processing times by using Eq. (3) for the CON model and Eq. (6) for the SLK-Model. Step 7. Determine the optimal compressions by using Eq. (4) for the CON model and (7) for the SLK model.
T.C.E. Chang et al./Int. J. Production
Step 8. d* = $I, + p& + ... + p&, k* = p^Fll+ &+& + ... + p^r*,_ll.STOP.
Economics
Theorem 4. Algorithm 1 delivers an optimal solution to the problems described in Theorem 3 and the complexity of the algorithm is 0 (n log n). Proof. The proof analysis above.
of Theorem
4 follows
[Z]
[3]
[S]
[S]
from the
7. Conclusions In this paper we consider the n-job, single-machine scheduling problem in which processing times are compressible. The objective is to jointly determine the optimal due-dates, the optimal sequence and the optimal processing time compressions to minimize the total penalty based on job earliness, tardiness, due-dates and processing time compressions. The common due-date assignment method and the slack due-date assignment method are used to assign due-dates to jobs. We solved the problem by formulating it as an assignment problem. An 0 (nlogn) algorithm is provided to obtain the optimal solution for a special case.
[6]
[7]
[S]
[9]
[lo]
[l l]
[12]
[13]
Acknowledgements [14]
This research was supported in part by The Hong Kong Polytechnic University under grant number 0340-864-A3-230. We are thankful to a referee for his/her constructive comments.
35
References [l]
To determine the computational complexity of Algorithm 1, we note that Step 4 can be completed in O(nlogn) time and Steps 3,6 and 7 can each be completed in O(n) time. Hence, the overall time complexity of the algorithm is 0 (n log n).
43 (1996) 29-35
[15]
Elmaghraby, SE., 1977. Activity Networks. Wiley, New York. Vickson, R.G., 1980a. Two single-machine sequencing problems involving controllable job processing times. AIIE Trans., 12: 258-262. Vickson, R.G., 1980b. Choosing the job sequence and processing times to minimize total processing plus flow cost on a single machine. Oper. Res., 29:1155-l 167. Nowicki, E. and Zdrzalka, S., 1988. A two-machine flow shop scheduling problem with controllable job processing times. Eur. J. Oper. Res., 34: 208-220. Van Wassenhove, L.N. and Baker, K.R., 1982. A bicriterion approach to time/cost trade-offs in sequencing. Eur. J. Oper. Res., 11: 48-52. Daniels, R.L. and Sarin, R.K., 1989. Single machine scheduling with controllable processing times and number of jobs tardy. Oper. Res., 37: 981-984. Zdrzalka, S., 1991. Scheduling jobs on a single machine with release dates, delivery times and controllable processing times: worst-case analysis. Oper. Res. Lett., 10: 519-524. Cheng, T.C.E. and Janiak, A., 1994. Resource optimal control in some single-machine scheduling problems. IEEE Trans. Autom. Control, 39: 1243-1246. Panwalkar, S.S. and Rajagopalan, R., 1992. A single machine sequencing problem with controllable processing times. Eur. J. Oper. Res., 59: 298-302. Alidaee, B. and Ahmadian, A., 1993. Two parallel machine sequencing problems involving controllable job processing times. Eur. J. Oper. Res., 70: 335-341. Cheng, T.C.E. and Gupta, M., 1989. Survey of scheduling research involving due-date determination decisions. Eur. J. Oper. Res., 38: 156-166. Panwalkar, S.S., Smith, M.L. and Seidmann, A., 1982. Common due-date assignment to minimize total penalty for the one machine scheduling problem. Oper Res., 30: 391-399. Papadimitrous, C.H. and Stieglitz, K., 1982. Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ. Nowicki, E. and Zdrzalka, S., 1990. A survey of results for sequencing problems with controllable processing times. Discrete Appl. Math., 26: 271-287. Cheng, T.C.E., Chen, Z.L. and Li, C.L., 1996. Parallelmachine scheduling with controllable processing times, IIE Trans., 28: 177-180.
production economics
ELSEVIER
Int. J. Production
Economics
43 (1996) 29-35
Due-date assignment and single machine scheduling with compressible processing times T.C.E. Cheng, C. O&z*, Department
of Management,
The Hong Kong Polytechnic
Received
21 February
X.D. Qi
lJniversi&, Hung Horn, Kowloon, Hong Kong
1995; accepted
11 October
1995
Abstract In this paper we consider a due-date assignment and single machine scheduling problem in which the jobs have compressible processing times. Two models are defined according to the due-date assignment methods used. The first model applies the common (constant) due-date assignment method to assign the due-dates, while in the second model the due-dates are assigned using the slack due-date assignment method. The objective is to determine the optimal sequence, the optimal due-dates and the optimal processing time compressions to minimize a total penalty function based on earliness, tardiness, due-dates and compressions. We solve the problem by formulating it as an assignment problem which is polynomially solvable. For the case that all the jobs have a common upper bound for compressions and an equal unit compression penalty, we present an O(nlogn) algorithm to obtain the optimal solution. Keywords:
Sequence;
Due-date
assignment;
Compression
1. Introduction
Recently, attention has increasingly been given to due-date assignment and scheduling problems in which the jobs have compressible processing times. The concept of compressible processing time is from the area of project planning and control, see, for example [ 11. Its motive in the field of scheduling is of the same nature, that is, the assumption of compressible processing times is justified in situations where jobs can be accomplished in shorter or longer durations by increasing or decreasing additional resources. Studies on problems with com-
* Corresponding
author.
0925-5273/96/$15.00 0 1996 Elsevier SSDI 0925-5273(95)00194-8
Science B.V. All rights reserved
pressible processing times were initiated by Vickson [2,3]. A survey on this line of scheduling research can be found in Nowicki and Zdrzalka [4]. Literature on this problem includes, among others, Van Wassenhove and Baker [6], Nowicki and Zdrzalka [4]. Daniels and Sarin [5], Zdrzalka [7] and Cheng and Janiak [8]. More recently, Panwalkar and Rajagopalan [9], Alidaee and Ahmadian [lo] and Cheng and Janiak [S] extended the problem to include the due-date aspect. Specially, Panwalkar and Rajagopalan [9] considered the common due-date assignment and single-machine scheduling problem in which the objective function is the sum of penalties based on earliness, tardiness and processing time compressions. They reduce the problem to an assignment
T.C.E. Chang et al./Int. J. Production
30
problem. Alidaee and Ahmadian [lo] extended the results of Panwalkar and Rajagopalan [9] to the parallel-machine scheduling case. Cheng and Janiak [S] further generalized the result to the case where the cost of compression is a general convex function of the amount of compression. In this paper we consider a due-date assignment and single-machine scheduling problem in which a penalty for due-dates is added to the objective function which includes the penalties for earliness, tardiness and processing time compressions. The common due-date assignment method and the slack due-date assignment method are used to assign due-dates to jobs. We solve the problem by formulating it as an assignment problem.
2. Problem formulation This paper considers a single-machine scheduling problem involving n jobs with the assigned duedates dj > 0, j = 1, 2, . . . , n. A job j has a normal processing time Esj, which can be compressed by an amount Xi > 0, 1 < j < II. Without loss of generality, we assume that the jobs are indexed in nondecreasing order of normal processing times, that is, p1 d p2 < ... < Pn. For each jobj, its compression has an upper bound Xj, 1 d j < n. Thus, if pj represents the actual processing time of job j, then pj = fjj -
x.
J’
0 < Xj d
Xj
<
pj,
1
The condition Xj < pj is necessary because if there is a job j such that Xj = pj, then it may be compressed to zero processing time, which is not reasonable in practice. In this paper we assume that all the jobs are ready at time zero and no preemption is allowed. Let Cj, Ej and Tj denote the completion time, the earliness and the tardiness of job j, respectively, then Ej = max (0, dj - Cj) and
Tj = max (0, cj - dj).
The objective function to be minimized is the total penalty based on earliness, tardiness, due-date value and compression of each job. It can be
Economics
expressed Z(fJ,
43 (1996) 29-35
as follows:
2, 2’) =
i (UEj + /ITj + ydj +fj(Xj)), j=l
(1)
where G is a job sequence, a > 0, /3 > 0 and y 3 0 are unit penalties for earliness, tardiness, and duedates, respectively. Here, d = (d,, dZ, . . . ,d,) is a vector of assigned due-dates and 2 = (x1, is a vector of job compressions. Further, x2, “. > x,) fj(xj) = bjXj> bj 3 0,
j = 1, 2, . . . 3PI,
where bj is the unit cost of compression for job j, 1 djdn. The problem is to determine the optimal sequence G*, the optimal due-date vector a* and the optimal compression vector x”* that minimizes the function z(a, d”,22). We consider the following two due-date assignment methods to assign due-dates to the jobs. (1) Common/constant (CON) due-date assignment method: All jobs are given an identical flow allowance d 2 0, that is, dj = d,
j = 1, 2, . . , n.
Here, the common due-date is set internally and is announced upon the arrival of jobs. It is a fixed attribute of a job. This method entirely ignores any information about the arriving jobs, jobs already in the system, future jobs, or the structure of the shop itself. It is representative of common practice where salesmen quote a uniform delivery date on all orders regardless of the order processing times. (2) Slack (SLK) due-date assignment method: Jobs are given flow allowances that reflect equal waiting times (i.e., equal slacks) k > 0, that is, dj = pj + k, j = 1, 2, . . , n. In this method, the due-dates are set internally by the scheduler as each job arrives on the basis ofjob processing times. As defined in this method, jobs are given equal slacks, which represent the practice in which customers are treated equally with regard to their waiting times. A comprehensive survey on due-date assignment methods can be found in [ 111. Therefore we have two models, namely the CON model and the SLK model, where the former corresponds to the common due-date assignment
T.C.E. Chang et aLlInt. J. Production
method and the latter corresponds to the slack due-date assignment method. The objective function can be denoted as z (a, d, 2) for the CON model and z (0, k, 2) for the SLK model, respectively. For any sequence cr, we use l-j] to denote the job in the position j of the sequence c.
3. Optimality
Economics
Lemma 4. Given 2, for any sequence, there exists an optimal value of d such that one of the jobs completes on d. Proof. The proof can be found in [12].
n(P - Ma In this section we discuss the optimality conditions for the two models. Eight lemmas are presented. The first three lemmas are applicable to both the CON model and the SLK model.
Proof. The proof follows from (l), after substituting common due-date and slack due-date for dj. As a result of Lemma 1, we assume throughout the remainder of this section.
/I > y
the early jobs
Lemma 3. Given X in an optimal sequence, the early jobs are in the longest processing time (LPT) order, and the tardy jobs are in the shortest processing time (SPT) order. Proof. The proof follows from a simple job-interchange argument. 5 are applicable
The following three lemmas are applicable SLK model only. sequence,
to the
the last
Proof. Assume that there are I early jobs in an optimal sequence D*. Suppose ptr+ il < prll. Interchanging the two jobs in positions Y and r + 1, we obtain another sequence (T’.The change in the objective function value is given by ~(a’, k, 2) - ~(a*, k, 2) = cc& - a(dc,+u
-
C;r,) + B(C;,+,,
- CI,+II) - P(C,r, - d,,,) < 0.
Therefore, the schedule is improved, which contradicts the fact that g* is an optimal sequence. Lemma 7. Given 2, for any sequence, there exists an optimal value of slack allowance k such that the last early job completes on its due-date.
Crj+ir Q dtj+ir.
Since Ctj+ 11= Ctjr + Ptj+ 11 and dtj+ 11= Ptj+ 11 + k, from the second inequality, we obtain Ctjr < k, which contradicts the first inequality.
Lemmas 4 and model only.
Proof. The proof can be found in [12].
-dir+,,)
Proof. For the CON model, the proof is trivial. For the SLK model, if in a sequence 0, the job in position j is tardy but the job in position j + 1 is early, then we have and
+ B).
Lemma 6. Given x”,in an optimal early job is the shortest job.
Lemma 1. If p < y, then d* = 0 and k* = 0.
Ctjr > dtjr
31
Lemma 5. Given x”, for any sequence, there exists an optimal due-date equal to C,,,, where r is the smallest integer greater than or equal to
conditions
Lemma 2. Given X, for any sequence, must precede the tardy jobs.
43 (1996) 29-35
Proof. Given 2 and 0, assume that for an optimal value of slack allowance k, the last early job is in position r. Suppose C,,, < dr,]. Let L = d,,, - C,,, and R = C [r+ 11- dr,+ 1,. If the slack allowance is changed to k’ such that d;,, = C,,,, then by Lemma 6, we have ~(0, k’, 1) - z (a, k, 2) = ( --ar + P(n - r) - yn)L. Similarly, if the slack allowance is changed to k” such that d[,+ 1l = C,,, 1l, then the last early job is in position r + 1 and we have
to the CON z (a, k”, 2) - z (a, k, 2) = ( -ur
+ /3(n - r) - yn) R.
32
T.C.E. Chang et al./Int. J. Production
Therefore, z (a, k’, 2) < z (6, k, 2) if ( -cxr + /? (n - r) - yn) B 0 and z(a, k”, 2) < z (a, k, 2) otherwise. The proof is complete. Lemma 8. Given 2, for any sequence, there exists an optimal value of slack allowance k equal to Ctr_lI, where r is determined by Lemma 5. Proof. Let the job in position r be the last early job
which completes on its due-date. Then, from Lemma 7, we know that the optimal slack allowance is C,,_ 1I. Therefore, for a small A > 0, if we shift the slack allowance from C,,_ rI to C,,_ 1I + A,
Economics
43 (1996) 29-35
of a job in a position with a negative position weight should be its normal processing time, and the processing time of a job in a position with a positive position weight should be its normal processing time minus the upper bound of its compression. If a position j has a zero position weight, then the optimal processing time of the job in this position may be any value between Prjl - Xtjl and Ptjl. These can be written in the notational form as follows:
~(0, C[r-l, + A, 2) - ~(a, C[r-l, + A, x”) 2 0,
if Wj < 0, if wj = 0, PC1= P;jl, P[j] X[j]> if Wj > 0, i
so
where Prjl- XfJ1d p;jld &I
P[jly
(3) and
~$1, 1
represents the optimal processing time of the job in position j. Therefore, the optimal compressions can be obtained by
(ar - fl(n - I) + yn)A 2 0,
or
Xr*il = P[jl
r 2 n (P - Y)/(@+ P). Similarly, if the slack allowance is shifted from C,,- rI to C,,_ 1I - A, we obtain r-l
-
p;Fjl,
j = 1, 2, . . . , Tl.
(4)
4.2. SLK Model
For this model, we have r-l ~(0, k, 2) = C (Mj+ ~(n + 1) - btjl)Ptjl
The proof of the lemma follows immediately.
j=l
4. Optimal compressions + f: 4. I. CON model
(P(n-A + Y -
For this model, if we substitute, CIj, = xi= 1pli], d = xi= lp[j] and Xtjl = Prjl - Prjl into Eq. (1) and simplify, we have
~(0, d, -f) = f: (~0’ - 1) + yn - b,j])Prj]
Glj + Y(n + 1) * wj= i fi(n-j)+y-btjl, P[jl>
jz$+ l(B(n -j
+ l) -
&jl)P[jJ + i
i&l =
bjPi.
j=l
Let w,= I
bjlTj.
j= 1
Let $j and p^&, 1 d j < n, denote the position weight of position j and the optimal processing time of the job in position j, respectively, then
j=l
+
b[jl)P[jl + i
j=l
P;jl, i Prj] - Xrj],
b[j],
1
if ~j < 0, if 9j = 0, if &j > 0,
(5)
05)
Using where&j1 - -$jl< p;jl< FIjl. cc(j-l)+yn--tjl, i P(n -j + l) -
b[j],
1 dj
(2)
then wj, 1 d j < n, represents the position weight of position j in the sequence (T.Since x7= lbjpj is a constant, for any sequence, the optimal processing time
the same argument for the CON model, we see that the optimal compressions can be obtained by a:jj, = P[jr - $r*j]F j=
1, 2, . . . ,n.
(7)
Theorem 1. Given a sequence, for the CON model and the SLK model, the optimal compressions can
T.C.E. Chang et al./Int. J. Production
be determined as follows: the compression of the job in a negative-weight position is zero; the compression of the job in a positive-weight position is its upper bound; if the position weight of a position is zero, then the compression of the job in this position can be any value between its lower bound and its upper bound. Proof. The proof follows from the analysis above.
Economics
43 (1996) 29-35
33
position i of a sequence. Furthermore, let Yij = 1 if job j is in position i, and Yij = 0 otherwise, i, j = 1, 27 ... 3n. Then we can formulate the sequencing problem associated with the CON model as the following assignment problem: Assignment
min i
i=lj=l
f:
Problem 1 CijYij
s.t.
5. Optimal sequences
i$lYij = 19 j = 192, ... $4 Now we discuss the optimal sequences for the
two models. In view of the analysis in the previous sections that the optimal due-dates and compressions can be computed for any given sequence, the problem reduces to a pure sequencing problem. In order to obtain the optimal sequence, we formulate the problem as an assignment problem. This is based on the observation that the problem is to form an optimal sequence by assigning n jobs to n possible positions with the condition that each job can only be assigned to one position and each position is to take exactly one of the jobs.
i=l,2, Yij=l
or
. . ..?I. i,j==l,2,
Yij=O,
. . . . ?l.
5.2. SLK model
For this model, we can formulate the sequencing problem as the following assignment problem: Assignment Problem 2
min i f c^ijYij i=lj=i
5.1. CON model For i, j = 1, 2, . . . , n, let E(i-l)+yn--j,
w,,= IJ
i
s.t.
1
/?(n-i+l)-bj,
where r is the smallest integer larger than or equal to n(fl - y)/(cr + /?) as defined in Lemma 5, and
i$lYij =1, j=l,2,
. . . . n,
j$i Yij = 1, i=l,2,
. . . . n,
Yij = 1 Pj7 Pij =
Pj,
i-fpj
-
Xj,
if if if
Wij < Wij = Wij >
0, 0, 0,
(3)
where pj - Xj < pJ < pj. We define an n x n matrix C by C = (Cij), where cij =
wijpij.
It is noted that cij, 1 < i,j < n, is the minimum cost resulting from an optimal combination of the position weight and compression if job j is assigned to
or
Yij = 0,
i, j = 1, 2, . . . , It.
where, A A A Cij = Wijpijs A
wij =
~i + JJ(~ +
/I(n -
1) - bj,
i) + Y - bj,
if 9ij < 0, if ~ij = 0, if tiij > 0,
1 < i < r - 1, r < i < n.
(3)
34
T.C.E. Chang et al./Int. J. Production
and yij = 1 if job j is in position otherwise, i, j = 1, 2, . . . , n.
i, and
yij = 0
Theorem 2. The optimal sequences for the CON model and the SLK model can be obtained by solving Assignment Problem 1 and Assignment Problem 2, respectively. Proof. From the analysis Theorem 2 is obvious.
above,
the
proof
of
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43 (1996) 29-35
jobs in positions j and j + 1 may be early or tardy jobs, and w~jl and W~j+ i1 may be positive, negative or zero. For brevity, we only show a case for the CON model in which the two jobs in positions j and j + 1 are all early jobs, and Wtjl < 0 and Wtj+ il > 0. Interchanging the two jobs in positions j and j + 1, we obtain another sequence G’. The change in the value of the objective function is given by z(cr’, d, 2) - ~(a*, d, 2)
Since an assignment problem can be solved in O(n3) time [13], the optimal sequence can be found in polynomial time. Once the optimal sequence is determined, from Lemma 5, Lemma 8 and Theorem 1, we know that the optimal common duedate for the CON model, the optimal allowance for the SLK model and the optimal processing time compressions for both models can be obtained. Since the dominant step in this solution procedure is the assignment algorithm used to obtain the optimal sequence, the overall computational complexity of the problem is 0(n3).
6. A special case We now consider a special case of each model in which bj=b and Xj=X, where O
= (~0’ - 1) + in - b)PIj+ 11 + (olj + yn - b)(&
- 2) - (LY(~- 1)
+ JJ~ - b)ptj, f (aj + yn - b)(Ptj+ 11- X) =
cc(P[j]-P[j+l])
< 0.
The other cases can be proved in a similar manner. The proof of Theorem 3 is complete. In the following we present an 0 (n log n) algorithm for this special case. Let r x 1 be the smallest integer greater than or equal to x. Algorithm 1 step
1.
I’ +- n (p -
y)/(cc
+ p).
Step 2. If I’ < 0, then r + 0. Else
-r 4~ - ~)ib+ 8) 1.
r Step 3. Weigh n positions by using equation (2) for the CON model and equation (5) for the SLK Model. Step 4. Rank the position weights Wj for the CON model and Gj for the SLK model in descending order of magnitude such that the largest Wj(9j) is ranked 1 and the smallest Wj(Gj) is ranked n. Break ties arbitrarily. Step 5. Find the optimal sequence by matching the position weights in descending order with the jobs in ascending order of their normal processing times. Step 6. Calculate the optimal processing times by using Eq. (3) for the CON model and Eq. (6) for the SLK-Model. Step 7. Determine the optimal compressions by using Eq. (4) for the CON model and (7) for the SLK model.
T.C.E. Chang et al./Int. J. Production
Step 8. d* = $I, + p& + ... + p&, k* = p^Fll+ &+& + ... + p^r*,_ll.STOP.
Economics
Theorem 4. Algorithm 1 delivers an optimal solution to the problems described in Theorem 3 and the complexity of the algorithm is 0 (n log n). Proof. The proof analysis above.
of Theorem
4 follows
[Z]
[3]
[S]
[S]
from the
7. Conclusions In this paper we consider the n-job, single-machine scheduling problem in which processing times are compressible. The objective is to jointly determine the optimal due-dates, the optimal sequence and the optimal processing time compressions to minimize the total penalty based on job earliness, tardiness, due-dates and processing time compressions. The common due-date assignment method and the slack due-date assignment method are used to assign due-dates to jobs. We solved the problem by formulating it as an assignment problem. An 0 (nlogn) algorithm is provided to obtain the optimal solution for a special case.
[6]
[7]
[S]
[9]
[lo]
[l l]
[12]
[13]
Acknowledgements [14]
This research was supported in part by The Hong Kong Polytechnic University under grant number 0340-864-A3-230. We are thankful to a referee for his/her constructive comments.
35
References [l]
To determine the computational complexity of Algorithm 1, we note that Step 4 can be completed in O(nlogn) time and Steps 3,6 and 7 can each be completed in O(n) time. Hence, the overall time complexity of the algorithm is 0 (n log n).
43 (1996) 29-35
[15]
Elmaghraby, SE., 1977. Activity Networks. Wiley, New York. Vickson, R.G., 1980a. Two single-machine sequencing problems involving controllable job processing times. AIIE Trans., 12: 258-262. Vickson, R.G., 1980b. Choosing the job sequence and processing times to minimize total processing plus flow cost on a single machine. Oper. Res., 29:1155-l 167. Nowicki, E. and Zdrzalka, S., 1988. A two-machine flow shop scheduling problem with controllable job processing times. Eur. J. Oper. Res., 34: 208-220. Van Wassenhove, L.N. and Baker, K.R., 1982. A bicriterion approach to time/cost trade-offs in sequencing. Eur. J. Oper. Res., 11: 48-52. Daniels, R.L. and Sarin, R.K., 1989. Single machine scheduling with controllable processing times and number of jobs tardy. Oper. Res., 37: 981-984. Zdrzalka, S., 1991. Scheduling jobs on a single machine with release dates, delivery times and controllable processing times: worst-case analysis. Oper. Res. Lett., 10: 519-524. Cheng, T.C.E. and Janiak, A., 1994. Resource optimal control in some single-machine scheduling problems. IEEE Trans. Autom. Control, 39: 1243-1246. Panwalkar, S.S. and Rajagopalan, R., 1992. A single machine sequencing problem with controllable processing times. Eur. J. Oper. Res., 59: 298-302. Alidaee, B. and Ahmadian, A., 1993. Two parallel machine sequencing problems involving controllable job processing times. Eur. J. Oper. Res., 70: 335-341. Cheng, T.C.E. and Gupta, M., 1989. Survey of scheduling research involving due-date determination decisions. Eur. J. Oper. Res., 38: 156-166. Panwalkar, S.S., Smith, M.L. and Seidmann, A., 1982. Common due-date assignment to minimize total penalty for the one machine scheduling problem. Oper Res., 30: 391-399. Papadimitrous, C.H. and Stieglitz, K., 1982. Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ. Nowicki, E. and Zdrzalka, S., 1990. A survey of results for sequencing problems with controllable processing times. Discrete Appl. Math., 26: 271-287. Cheng, T.C.E., Chen, Z.L. and Li, C.L., 1996. Parallelmachine scheduling with controllable processing times, IIE Trans., 28: 177-180.