- 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)
107-l 13
Due-date assignment and single machine scheduling with compressible processing times T.C.E. Chenga, C. O&zb’*, X.D. Qib ‘Office of Vice President (Research & Postgraduate Studies) The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong bDepartment of Management, The Hong Kong Polytechnic Vniversiiy, Hung Horn, Kouloon, Hong Kong Accepted
for publication
15 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(n log n) 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, [l]. Its motive in the field of scheduling is cf the same nature, i.e., 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 compressible processing times were initiated by Vickson [2,3]. A survey on * Corresponding
author.
0925-5273/96/$15.00 Copyright SSDZO925-5273(96)00041-2
0 1996 Elsevier
this line of scheduling research can be found in [4]. Literature on this problem includes, among others, Van Wassenhove and Baker [5], Nowicki and Zdrzalka [6], Daniels and Sarin [7], Zdrzalka [8] and Cheng and Janiak [9]. h4ore recently, Panwalkar and Rajagopalan [lo], Alidaee and Ahmadian [ 1 l] and Cheng et al. [ 121 extend the problem to include the due-date aspect. Specially, Panwalkar and Rajagopalan [lo] consider 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 problem. Alidaee and Ahmadian [l l] extend the results of Panwalkar and Rajagopalan [lo] to the parallel-machine scheduling case. Cheng et al.
Science B.V. All rights reserved
108
T. C.E. Cheng et al. / Int. J. Production
[ 121 further generalize 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 duedate 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 due-dates d,BO,j = 1,2, . . . ,n. A job j has a normal processing time pi, which can be compressed by an amount Xj 2 0, 1
O
< flj, ldjdn.
The condition fj < pj is necessary because if there is a job j such that Zj = jj, 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(O, dj - Cj)
and
Tj = max(O, 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 expressed as follows: Z(O, 23-f)=
e(aEj +/?Tj + ydj + fj(Xj)),
(1)
j=l
where G is a job sequence, t( > 0, B 3 0 and y 3 0 are unit penalties for earliness, tardiness and due-dates,
Economics
43 (1996)
107-113
respectively. Here, d = (dl,dz, . . . ,d,) is a vector of assigned due-dates and x”= (x1,x2,. .,x,,) is a vector of job compressions. Further,
fj(Xj) = bjXj> bj 30, j = 1,2 ,..., n, where bj is the unit cost of compression for job j, l
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 30, i.e., 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 of job 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 duedate assignment methods can be found in [ 131. Therefore we have two models, namely the CONmodel and the SLK-model, where the former corresponds to the common due-date assignment method and the latter corresponds to the slack due-date assignment method. The objective function can be denoted as z( o, d, 2) for the CON-model and z( 0, k, 2) for the SLK-model, respectively. For any sequence 0, we use [j] to denote the job in the position j of the sequence 0.
109
T C.E. Cheng et al. / Int. J. Production Economics 43 (1996) 107-113
3. Optimality
Proof.
conditions
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.
The proof can be found in [ 141.
The following three lemmas SLK-Model only.
0
are applicable
Lemma 6. Given 2, in an optimal sequence, early job is the shortest job.
to the
the last
Lemma 1. If/I < y, then d’ = 0 and k* = 0. Proof. The proof follows from ( 1), after substituting common due-date and slack due-date for dj. 0 As a result of Lemma 1, we assume /? > y throughout the remainder of this section.
Proof. Assume that there are r early jobs in an optimal sequence o*. Suppose p[r+l] < p[r]. Interchanging the two jobs in positions r and r + 1, we obtain another sequence 0’. The change in the objective function value is given by z(a’,W
Lemma 2. Given 2, for any sequence, must precede the tardy jobs.
- z(a*,W
= a(&
- C;,.]) + WI’,+,,
the early jobs -d;,+,])
- a(d[r+tl-
-P(CI,l
- dIr1) < 0.
%+I])
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
Therefore, the schedule is improved, which contradicts the fact that cr* is an optimal sequence. 0
C[jl > dIj1
Lemma 7. Given 2, for any sequence, there exists an optimal value of slack allowance k such that the last early j’ob completes on its due-date.
and
CIj+il
Since CIi+ll = CIjl + pIj+il and d[j+il = plj+11 + k, from the second inequality, we obtain CIjl
from
Lemmas 4 and 5 are applicable only.
a
simple
job-
to the CON-model
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 [ 141.
q
Lemma 5. Given 2, for any sequence, there exists an optimal due-date equal to CL,.],where r is the smallest integer greater than or equal to n(/3 - y)/(cr + /?).
Proof. Given 2 and B, assume that for an optimal value of slack allowance k, the last early job is in position r. Suppose C,,l < d,,]. Let L = d[,] - Cr,] and R = C[,+II - d[,+l]. If the slack allowance is changed to k’ such that d;,] = CL,], then by Lemma 6, we have z(c, k’, 2) - Z(B, k,f)
= (-cv-
+ P(n - r) - yn)L.
Similarly,
if the slack allowance is changed to k” such that d!:+,] = C[,+il, then the last early job is in position r + 1 and we have z(o,k”,X”) - z(a,k,x”) = (ar - /?(n - r) + yn)R. Therefore,z(o,k’,2)
-r) otherwise.
The
Lemma 8. Given 2, for any sequence, there exists an optimal value of slack allowance k equal to C&l], 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
T.C.E. Cheng et al. / Int. J. Production Economics 43 (1996) 107-113
110
7, we know that the optimal slack allowance is Cl,_ 11. Therefore, for a small A > 0, if we shift the slack allowance from C[,_ll to CI,_ll + A, z(b, CL,-I] + AZ)
- z(o, C[,-l] + A$)>&
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 Pljl - &l and PIjl* These can be written in the notational form as follows:
so (ar - P(n - r) + yn)A
20,
I$] =
hjl'
if w, < 0,
61~
if
Wj
=
0,
if
Wj
>
0,
-'U]? ( P[j]
or y >n(P
- Y)l(M + 0
Similarly, if the slack allowance to C[,_il - A, we obtain ?--
is shifted from Cl,_ 11
l
where Pcjl - -f[jl d Ptjl< Prjl and pTj1> 1 dj
The proof of the lemma follows immediately.
0
(3)
j = 1,2,.
.,n.
0
(4)
4.2. SLK-model For this model, we have
4. Optimal compressions
r-1
4. I. CON-model For this model, if we substitute Cljl = c;=, p[i], d = cJ=l p[jl and xljl = Prjl - p[jl into Eq. (1) and simplify, we have
40, d,x”) =
~W
-
1) +
P
-
~~~I)PLI
A wj=
n
C (P(n-.i + l> -
-A
+Y
--~I)PL~I
+,$bjF,.
Let 1;; and $,), 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
+
+k(P(n ,=i-
b[jl)P[jl
aj+y(n+l)-bbl,
l
1 p(n-j)+y--bul,
r
1, (5)
j=r+ I
b:j]
=
Pb,’
if Gj < 0,
Pb]?
if Gj = 0,
(6)
1 &l -2~1, if Gj > 0, Let Wj
UG - 1) + yn -
=
{
D(n
-
i
+
1)
-
b[ j], b[
j],
1 djdr, rf
l
(2)
then wj, 1
where plil - $1 Q pb, < &. Using the same argument for the CON-model, we see that the optimal compressions can be obtained by it,
= plil - &,
j = 1,2,. . .,n.
(7)
Theorem 1. Given a sequence, for the CON-model and the SLK-model, the optimal compressions can 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
T.C.E. Cheng et al. / Int. J. Production Economics 43 (1996) 107-113
111
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.
in position i, and y;j = 0 otherwise, i, j = 1,2,. . . , n. Then we can formulate the sequencing problem associated with the CON-model as the following assignment problem:
Proof.
Assignment
The proof follows from the analysis above.
Problem 1
0 min
k?Cijyij i=l j=*
s.t.
2
5. Optimal sequences 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 possibIe 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. 5.1. CON-model
Yij
1,
j = 1,2 ,..., n,
1,
i = 1,2 ,..., n,
=
i=l
2
Yij =
j=l
or yij= 0,
Yij = 1
i, j = 1,2,. . . ,n.
5.2. SLK-model For this model, we can formulate the sequencing problem as the following assignment problem: Assignment
Problem 2
For i,j = 1,2 ,..., n, let a(iW;j
l)+yn-bj,
l
i=l
=
P(n-i+l)--bj,
r+l
n
if
Pj?
if
W;j
=
if
W;j
>
I
Pij =
{ pj - Xj,
W;j
<
c
s.t.
where Y is the smallest integer larger than or equal to n(P - ?)/(a + /?) as defined in Lemma 5, and P/T
j=l
yij = 1,
j = 1,2 ,..., n,
Yij =
i=
i=l
0,
2
1,
1,2 ,..., n,
j=l
0,
Yij=l
0,
OrYij=O,
i,j=
1,2 ,..., n.
where where pi - Xj
< pi. We define an n x n matrix 1
Cij
C by
^A =
W;jp;j.
C = (Cijk
cti+y(n+
where
P(n-i)+Y-bj,
C;j
=
W;jp;j.
It is noted that Cij, 1 d i, j
bij =
l)-bj,
l
Pj,
if Gij < 0,
P;‘V
if 8, = 0,
{ pj - Xj,
if Gij > 0,
~j-iTj
1,
112
T. C.E. Cheny et al. / Int. J. Production
and ylj = 1 if job j is in position otherwise, i, j = 1,2,. . .,n.
i, and yij = 0
Theorem 2. The optimal sequences for the CONModel and the SLK-Model can be obtained by solving Assignment Problem 1 and Assignment Problem 2, respectively. Proof. From the analysis above, the proof of Theorem 2 is obvious. 0
Economics
43 (1996)
107-113
Wlj+il 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 Wljl < 0 and wlj+il > 0. Interchanging the two jobs in positions j and j + 1, we obtain another sequence CJ’.The change in the value of the objective function is given by z(a’,d$)
- z(o*,d,Z)
= (c4j - 1) + yn - b)Pli+ll Since an assignment problem can be solved in 0(rz3 ) time [ 151, the optimal sequence can be found in polynomial time. Once the optimal sequence is determined, from Lemmas 5, and 8, and Theorem 1, we know that the optimal common due-date 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 O(n3).
6. A special case We now consider a special case of each model in which b, = b and XJ = X, where 0
+(cxj + yn - b)(&,.] - X) -(~0’ -(aj =
- 1) + yn - b)& f yn - b)(&+ll
- 2)
a$j] - &+1,)
< 0. The other cases can be proved in a similar manner. cl The proof of Theorem 3 is complete. In the following we present an O(n log n) algorithm for this special case. Let [x] be the smallest integer greater than or equal to x. Algorithm 1 Step 1. r’ +- n(P - y)/(cr + p). Step 2. If Y’< 0, then Y +- 0. Else Y +
[n(fi - y)/
(a + ml. Step 3. Weigh n positions by using Eq. (2) for the CON-model and Eq. (5) for the SLK-model. Step 4. Rank the position weights wj for the CONmodel and Gj for the SLK-model in descending order of magnitude such that the largest Wj(Gj) is ranked 1 and the smallest Wj(tij) 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 Eq. (7) for the SLK-model.
T.C.E.
Step 8. d* = ptII + pi;, + a;, + . . ’ + &,. STOP.
Cheng et al./Int.
J. Production
. . + pTrrl,k’ = $I1 +
Economics
43 (1996)
computational complexity of Althat step 4 can be completed in steps 3,6 and 7 can each be comHence, the overall time complexis O(n log n).
Theorem 4. Algorithm I delivers an optimal solution to the problems described in Theorem 3 and the complexity of the algorithm is O(n log n). Proof. The proof of Theorem 4 follows from the analysis above. 0
7. Conclusions
113
References [I] Elmaghraby,
To determine the gorithm 1, we note O(n log n) time and pleted in O(n) time. ity of the algorithm
107-113
[2]
[3]
[4]
[5]
[6]
[7]
[S]
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 duedates to jobs. We solved the problem by formulating it as an assignment problem. An O(n log n) algorithm is provided to obtain the optimal solution for a special case.
[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.
[15]
[9]
[lo]
[ll]
[12]
S.E., 1977. Activity Networks. Wiley, New York. Vickson, R.G., 1980. Two single-machine sequencing problems involving controllable job processing times. ABE Trans., 12: 258-262. Vickson, R.G., 1980. Choosing the job sequence and processing times to minimize total processing plus flow cost on a single machine. Oper. Res., 29: 1155-J 167. Nowicki, E. and Zdrzalka, S., 1990. A survey of results for sequencing problems with controllable processing times. Discrete Appl. Math., 26: 271-287. 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. Nowicki, E. and Zdrzalka, S., 1988. A two-machine flow shop scheduling problem with controllable job processing times, Eur. J. of Oper. Res., 34: 208-220. Daniels, R.L. and Satin, R.K., 1989. Single machine scheduling with controllable processing times and number of jobs tardy. Oper. Res., 37: 981-984. Zdaalka, 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. Automat. Control, 39: 124331246. 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. Chen, Z.L. and Li, C.L., 1996. Parallelmachine scheduling with controllable processing times, IIE Trans., 28: 177-180. Cheng, T. C.E., and Gupta, M., 1989. Survey of scheduling research involving due-date determination decisions. Eur. J. Oper. Res., 38: 156166. 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: Algotithms and Complexity. Prentice-Hall, Englewood Cliffs, N.J.
production economics
ELSEVIER
Int. J. Production
Economics
43 (1996)
107-l 13
Due-date assignment and single machine scheduling with compressible processing times T.C.E. Chenga, C. O&zb’*, X.D. Qib ‘Office of Vice President (Research & Postgraduate Studies) The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong bDepartment of Management, The Hong Kong Polytechnic Vniversiiy, Hung Horn, Kouloon, Hong Kong Accepted
for publication
15 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(n log n) 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, [l]. Its motive in the field of scheduling is cf the same nature, i.e., 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 compressible processing times were initiated by Vickson [2,3]. A survey on * Corresponding
author.
0925-5273/96/$15.00 Copyright SSDZO925-5273(96)00041-2
0 1996 Elsevier
this line of scheduling research can be found in [4]. Literature on this problem includes, among others, Van Wassenhove and Baker [5], Nowicki and Zdrzalka [6], Daniels and Sarin [7], Zdrzalka [8] and Cheng and Janiak [9]. h4ore recently, Panwalkar and Rajagopalan [lo], Alidaee and Ahmadian [ 1 l] and Cheng et al. [ 121 extend the problem to include the due-date aspect. Specially, Panwalkar and Rajagopalan [lo] consider 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 problem. Alidaee and Ahmadian [l l] extend the results of Panwalkar and Rajagopalan [lo] to the parallel-machine scheduling case. Cheng et al.
Science B.V. All rights reserved
108
T. C.E. Cheng et al. / Int. J. Production
[ 121 further generalize 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 duedate 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 due-dates d,BO,j = 1,2, . . . ,n. A job j has a normal processing time pi, which can be compressed by an amount Xj 2 0, 1
O
< flj, ldjdn.
The condition fj < pj is necessary because if there is a job j such that Zj = jj, 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(O, dj - Cj)
and
Tj = max(O, 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 expressed as follows: Z(O, 23-f)=
e(aEj +/?Tj + ydj + fj(Xj)),
(1)
j=l
where G is a job sequence, t( > 0, B 3 0 and y 3 0 are unit penalties for earliness, tardiness and due-dates,
Economics
43 (1996)
107-113
respectively. Here, d = (dl,dz, . . . ,d,) is a vector of assigned due-dates and x”= (x1,x2,. .,x,,) is a vector of job compressions. Further,
fj(Xj) = bjXj> bj 30, j = 1,2 ,..., n, where bj is the unit cost of compression for job j, l
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 30, i.e., 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 of job 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 duedate assignment methods can be found in [ 131. Therefore we have two models, namely the CONmodel and the SLK-model, where the former corresponds to the common due-date assignment method and the latter corresponds to the slack due-date assignment method. The objective function can be denoted as z( o, d, 2) for the CON-model and z( 0, k, 2) for the SLK-model, respectively. For any sequence 0, we use [j] to denote the job in the position j of the sequence 0.
109
T C.E. Cheng et al. / Int. J. Production Economics 43 (1996) 107-113
3. Optimality
Proof.
conditions
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.
The proof can be found in [ 141.
The following three lemmas SLK-Model only.
0
are applicable
Lemma 6. Given 2, in an optimal sequence, early job is the shortest job.
to the
the last
Lemma 1. If/I < y, then d’ = 0 and k* = 0. Proof. The proof follows from ( 1), after substituting common due-date and slack due-date for dj. 0 As a result of Lemma 1, we assume /? > y throughout the remainder of this section.
Proof. Assume that there are r early jobs in an optimal sequence o*. Suppose p[r+l] < p[r]. Interchanging the two jobs in positions r and r + 1, we obtain another sequence 0’. The change in the objective function value is given by z(a’,W
Lemma 2. Given 2, for any sequence, must precede the tardy jobs.
- z(a*,W
= a(&
- C;,.]) + WI’,+,,
the early jobs -d;,+,])
- a(d[r+tl-
-P(CI,l
- dIr1) < 0.
%+I])
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
Therefore, the schedule is improved, which contradicts the fact that cr* is an optimal sequence. 0
C[jl > dIj1
Lemma 7. Given 2, for any sequence, there exists an optimal value of slack allowance k such that the last early j’ob completes on its due-date.
and
CIj+il
Since CIi+ll = CIjl + pIj+il and d[j+il = plj+11 + k, from the second inequality, we obtain CIjl
from
Lemmas 4 and 5 are applicable only.
a
simple
job-
to the CON-model
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 [ 141.
q
Lemma 5. Given 2, for any sequence, there exists an optimal due-date equal to CL,.],where r is the smallest integer greater than or equal to n(/3 - y)/(cr + /?).
Proof. Given 2 and B, assume that for an optimal value of slack allowance k, the last early job is in position r. Suppose C,,l < d,,]. Let L = d[,] - Cr,] and R = C[,+II - d[,+l]. If the slack allowance is changed to k’ such that d;,] = CL,], then by Lemma 6, we have z(c, k’, 2) - Z(B, k,f)
= (-cv-
+ P(n - r) - yn)L.
Similarly,
if the slack allowance is changed to k” such that d!:+,] = C[,+il, then the last early job is in position r + 1 and we have z(o,k”,X”) - z(a,k,x”) = (ar - /?(n - r) + yn)R. Therefore,z(o,k’,2)
-r) otherwise.
The
Lemma 8. Given 2, for any sequence, there exists an optimal value of slack allowance k equal to C&l], 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
T.C.E. Cheng et al. / Int. J. Production Economics 43 (1996) 107-113
110
7, we know that the optimal slack allowance is Cl,_ 11. Therefore, for a small A > 0, if we shift the slack allowance from C[,_ll to CI,_ll + A, z(b, CL,-I] + AZ)
- z(o, C[,-l] + A$)>&
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 Pljl - &l and PIjl* These can be written in the notational form as follows:
so (ar - P(n - r) + yn)A
20,
I$] =
hjl'
if w, < 0,
61~
if
Wj
=
0,
if
Wj
>
0,
-'U]? ( P[j]
or y >n(P
- Y)l(M + 0
Similarly, if the slack allowance to C[,_il - A, we obtain ?--
is shifted from Cl,_ 11
l
where Pcjl - -f[jl d Ptjl< Prjl and pTj1> 1 dj
The proof of the lemma follows immediately.
0
(3)
j = 1,2,.
.,n.
0
(4)
4.2. SLK-model For this model, we have
4. Optimal compressions
r-1
4. I. CON-model For this model, if we substitute Cljl = c;=, p[i], d = cJ=l p[jl and xljl = Prjl - p[jl into Eq. (1) and simplify, we have
40, d,x”) =
~W
-
1) +
P
-
~~~I)PLI
A wj=
n
C (P(n-.i + l> -
-A
+Y
--~I)PL~I
+,$bjF,.
Let 1;; and $,), 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
+
+k(P(n ,=i-
b[jl)P[jl
aj+y(n+l)-bbl,
l
1 p(n-j)+y--bul,
r
1, (5)
j=r+ I
b:j]
=
Pb,’
if Gj < 0,
Pb]?
if Gj = 0,
(6)
1 &l -2~1, if Gj > 0, Let Wj
UG - 1) + yn -
=
{
D(n
-
i
+
1)
-
b[ j], b[
j],
1 djdr, rf
l
(2)
then wj, 1
where plil - $1 Q pb, < &. Using the same argument for the CON-model, we see that the optimal compressions can be obtained by it,
= plil - &,
j = 1,2,. . .,n.
(7)
Theorem 1. Given a sequence, for the CON-model and the SLK-model, the optimal compressions can 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
T.C.E. Cheng et al. / Int. J. Production Economics 43 (1996) 107-113
111
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.
in position i, and y;j = 0 otherwise, i, j = 1,2,. . . , n. Then we can formulate the sequencing problem associated with the CON-model as the following assignment problem:
Proof.
Assignment
The proof follows from the analysis above.
Problem 1
0 min
k?Cijyij i=l j=*
s.t.
2
5. Optimal sequences 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 possibIe 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. 5.1. CON-model
Yij
1,
j = 1,2 ,..., n,
1,
i = 1,2 ,..., n,
=
i=l
2
Yij =
j=l
or yij= 0,
Yij = 1
i, j = 1,2,. . . ,n.
5.2. SLK-model For this model, we can formulate the sequencing problem as the following assignment problem: Assignment
Problem 2
For i,j = 1,2 ,..., n, let a(iW;j
l)+yn-bj,
l
i=l
=
P(n-i+l)--bj,
r+l
n
if
Pj?
if
W;j
=
if
W;j
>
I
Pij =
{ pj - Xj,
W;j
<
c
s.t.
where Y is the smallest integer larger than or equal to n(P - ?)/(a + /?) as defined in Lemma 5, and P/T
j=l
yij = 1,
j = 1,2 ,..., n,
Yij =
i=
i=l
0,
2
1,
1,2 ,..., n,
j=l
0,
Yij=l
0,
OrYij=O,
i,j=
1,2 ,..., n.
where where pi - Xj
< pi. We define an n x n matrix 1
Cij
C by
^A =
W;jp;j.
C = (Cijk
cti+y(n+
where
P(n-i)+Y-bj,
C;j
=
W;jp;j.
It is noted that Cij, 1 d i, j
bij =
l)-bj,
l
Pj,
if Gij < 0,
P;‘V
if 8, = 0,
{ pj - Xj,
if Gij > 0,
~j-iTj
1,
112
T. C.E. Cheny et al. / Int. J. Production
and ylj = 1 if job j is in position otherwise, i, j = 1,2,. . .,n.
i, and yij = 0
Theorem 2. The optimal sequences for the CONModel and the SLK-Model can be obtained by solving Assignment Problem 1 and Assignment Problem 2, respectively. Proof. From the analysis above, the proof of Theorem 2 is obvious. 0
Economics
43 (1996)
107-113
Wlj+il 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 Wljl < 0 and wlj+il > 0. Interchanging the two jobs in positions j and j + 1, we obtain another sequence CJ’.The change in the value of the objective function is given by z(a’,d$)
- z(o*,d,Z)
= (c4j - 1) + yn - b)Pli+ll Since an assignment problem can be solved in 0(rz3 ) time [ 151, the optimal sequence can be found in polynomial time. Once the optimal sequence is determined, from Lemmas 5, and 8, and Theorem 1, we know that the optimal common due-date 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 O(n3).
6. A special case We now consider a special case of each model in which b, = b and XJ = X, where 0
+(cxj + yn - b)(&,.] - X) -(~0’ -(aj =
- 1) + yn - b)& f yn - b)(&+ll
- 2)
a$j] - &+1,)
< 0. The other cases can be proved in a similar manner. cl The proof of Theorem 3 is complete. In the following we present an O(n log n) algorithm for this special case. Let [x] be the smallest integer greater than or equal to x. Algorithm 1 Step 1. r’ +- n(P - y)/(cr + p). Step 2. If Y’< 0, then Y +- 0. Else Y +
[n(fi - y)/
(a + ml. Step 3. Weigh n positions by using Eq. (2) for the CON-model and Eq. (5) for the SLK-model. Step 4. Rank the position weights wj for the CONmodel and Gj for the SLK-model in descending order of magnitude such that the largest Wj(Gj) is ranked 1 and the smallest Wj(tij) 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 Eq. (7) for the SLK-model.
T.C.E.
Step 8. d* = ptII + pi;, + a;, + . . ’ + &,. STOP.
Cheng et al./Int.
J. Production
. . + pTrrl,k’ = $I1 +
Economics
43 (1996)
computational complexity of Althat step 4 can be completed in steps 3,6 and 7 can each be comHence, the overall time complexis O(n log n).
Theorem 4. Algorithm I delivers an optimal solution to the problems described in Theorem 3 and the complexity of the algorithm is O(n log n). Proof. The proof of Theorem 4 follows from the analysis above. 0
7. Conclusions
113
References [I] Elmaghraby,
To determine the gorithm 1, we note O(n log n) time and pleted in O(n) time. ity of the algorithm
107-113
[2]
[3]
[4]
[5]
[6]
[7]
[S]
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 duedates to jobs. We solved the problem by formulating it as an assignment problem. An O(n log n) algorithm is provided to obtain the optimal solution for a special case.
[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.
[15]
[9]
[lo]
[ll]
[12]
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