Compwers Opns Res. Vol. 17, No. 3, pp. 321-324, 1990 0305-0548/90 s3.00 + 0.00 Printed in Gteat Britain. All rights resewed Copyright 0 1990 Perpmon Press plc

A NOTE ON A PARTIAL SEARCH ALGORITHM FOR THE SINGLE-MACHINE OPTIMAL COMMON DUE-DATE ASSIGNMENT

AND SEQUENCING PROBLEM

T. C. E. CHENG* Department of Actuarial 8s Management Sciences, Faculty of Management, University of Manitoba,

Winnipeg, Manitoba, Canada R3T 2N2

(Received December 1988; revised June 1989)

Scope and Pmpeae-We consider in this note the problem of assigning a common due-date to a set of simultaneously available jobs and sequencing them on a single-machine. The objective is to lind an optimal combination of the due-date value and job sequence to minimize a penalty function based on the weighted sum of job earliness and tardiness values. A follow-up study to an earlier work, this note presents a new algorithm which is shown to be significantly more efiicient than the previous one for solving the problem.

Ahstrnct-This note presents a partial search algorithm to solve the single-machine common due-date assignment and sequencing problem to minimize the weighted sum of earliness and tardiness of jobs.

INTRODUCTION

This note considers the problem of assigning optimal common due-dates and sequencing n jobs on a single machine. The objective is to minimize the weighted sum of earliness and tardiness of jobs. While the general problem has been shown to be NP-hard [ 11, efficient algorithms are available for special cases of the problem. An extensive survey of research on due-date assignment and sequencing has been conducted by Cheng and Gupta [23.

So far it appears that no researcher, except perhaps Cheng [3] who has presented an O(n’2”) algorithm, has developed solution procedures to find the optimal common due-date and job sequence that jointly minimize the weighted sum of job earliness and tardiness. This note is a follow-up study of Cheng’s work and presents an improved solution algorithm for the problem.

COMMON DUE-DATE ASSIGNMENT AND SEQUENCING PROBLEM

LetN={1,2,..., n} be a set of simultaneously available jobs to be processed on a single machine under the common assumptions listed in Baker [4]. Job i requires processing time ti and has a normaliz.edweightw,(O~wi~l,~,,,,,~,= 1). Under the common due-date assignment method, job i is assigned a due-date dl = d >/ 0 which is common among the jobs. The problem is to find the joint optimal common due-date d* and job sequence o* that minimize the weighted sum of earliness and tardiness of jobs. That is

Min: f(d, a) = C Wli~IC~i]- dl lS14R

(1)

where cr is an arbitrary job sequence among the n! possible permutation sequences; [i] denotes the . .

job in positron i of Q and Cur (= C Ic,citt,,) is the completion time of job [il.

DOMINANCE PROPERTIES

We present a partial search algorithm based on the following theorems that limits the search for an optimal solution to a reduced set of feasible solutions.

*T. C. E. Cheng is Professor of Operations Management in the Faculty of Management and Adjunct Professor of Industrial Engineering in the Faculty of Engineering, University of Manitoba, Canada. He obtained his PhD in operations research from the University of Cambridge, England. His research interests are in scheduling, simulation, and inventory management. A member of the editorial advisory board of this journal, he has published over SO papers in various scholarly journals such as European Journal oj Operational Research, IIE Transactions, lnrernatkmal Journol of Production Research, Journal of the Operational Research Society and others. He is a registered professional engineer in the province of Manitoba.

321

322 T. C. E. CHENG

Theorem 1. For any 0, d* = Ct,] where r is determined by the following conditions:

c w[i] < 1/2 and l<i4r-1

l;p,iIw2~

Proof. A complete proof of this result is given in Cheng [S].

Theorem 2. In an optimal sequence, all early and on-time jobs are in the weighted longest processing time order (WLPT) and all tardy jobs are in the weighted shortest processing time order (WSPT).

Proof. This result can be established by the pairwise job interchange argument and is available in Cheng [3 J.

PARTIAL SEARCH ALGORITHM

Before presenting the partial search algorithm, we define the following terms and notation used in developing the algorithm:

6(k) = a partial sequence of size k, i.e. (Cl], 121, . . . , [k]), V[i] EN. p(k) = a feasible partial sequence of size k, i.e. ([I], [2], . . . , [k]), V[i] EN, such that

$]lW[i] 2 l[i+ l]/w[i+ 119 lGii,<k-l,forall[i]~~(k).

p’(k) = an arbitrary partial sequence of jobs not in p(k), i.e. G(n - k) containing jobs jE{iEN: i$p(k)}.

S(k) = the set of all feasible partial sequences of size k, i.e. {p(k)). F(k) = the set of all feasible full sequences generated from p(k), i.e. {G(n): G(n) =

p(k) $ p’(k) with jobs in p’(k) sequenced in WSPT}, where @ denotes the concatenation of two partial sequences.

F = the set of all feasible full sequences, i.e. u1 dkdn F(k).

(Note: To simplify notation, we write j E G(k) to denote that job j is in the partial sequence G(k)).

Algorithm

Step I:

Step 2:

Step 3:

Step 4:

Step 5:

Initialization. Let k = 0 be the size of an initial feasible partial sequence p(O) E S(0). Define job CO] as an initial job sequenced in position 0 with weight wtol = 0 and processing time tiol = 6 (an arbitrary positive number). Let the sets of feasible partial sequences and feasible full sequences of all sizes be empty, i.e. S(k), F(k) = a,1 < k < n. For each and every feasible partial sequence of size k, i.e. p(k)ES(k), generate a new feasible partial sequence of size k + 1, if possible, i.e. p(k + 1) = p(k) @ G(l) where j c&(l) for some j E {iEN: i$p(k)} such that t,/wi 6 tIkI/wP1. If the sum of the weights of jobs in the newly generated partial sequence is less than l/2, let the sequence be an element of the set of feasible partial sequences of size k + 1, i.e. if Ciepu+ 1) wi K l/2, p(k + 1) + S(k + 1). Otherwise, construct a com- plementary partial sequence by arranging the remaining jobs in WSPT order and then concatenate the complementary sequence to the original sequence to form a feasible full sequence, i.e. construct p’(k + 1) which is in WSPT, p(k + 1) @ p’(k + 1) + F(k + 1). If either (i) the set of feasible partial sequences of size k + 1 is empty, i.e. S(k + 1) = 0, or (ii) the sequence size k is greater than or equal to n - 1, go to Step 5. Otherwise, increment the sequence size by one, i.e. k = k + 1, and go to Step 2. Search for the optimal solution from the set of all feasible full sequences, i.e. find

CT* = CT of f*(d, a) and d* =d of f*(d, 6) where F =

This algorithm systematically applies the results of Theorems 1 and 2, which are necessary conditions for optimality, to screen the set of feasible solutions and reduce the full set to a smaller size. The essence of the algorithm in Steps 2 and 3 which are direct applications of Theorems 2 and 1 respectively.

Partial search algorithm 323

AN EXAMPLE

To illustrate the operation of the algorithm, consider thr! 4-job example shown in Table 1. Applying the algorithm to this given job set yields the results shown in Table 2. It is interesting to note that the partial search algorithm substantially reduces the full feasible solution set from 4! = 24 to 5 sequences from which the optimal solution is easily determined.

TIME COMPLEXITY OF THE ALGORITHM

At each stage k, F(k) is the set of feasible full sequences generated for comparison in the final step of the algorithm. Since the maximum number of elements in F(k) cannot exceed the number of elements in S(k), an upper bound on (F(k)( is IS(k where (X( denotes the cardinality of the set X. It is noted that an upper bound on IS(k)1 is C(n, k), a binomial coefficient representing the total number of different ways to form a sub-sequence of size k. Also, it is clear that if (F(k)1 = IS(k)(, S(k + 1) = 0 and the algorithm will terminate. Therefore an upper bound on the maximum number of feasible full sequences generated by the algorithm, IFI, is given by

IFI < max kC(n, k) lQk6n

= rcwh b/m (2)

where [x] denotes the largest integer not exceeding x. Letting n + co and applying Stirling formula to approximate n!, we obtain from (2)

lim IFI < lim {f122”-1/7r)}1/2. (3) “-gr “‘co

Since the final step of the algorithm is a search of the feasible set F of size (FI to find the optimal solution, the search time is a linear function of IFI. It follows from (3) that the time complexity of the algorithm, T(n), for some constant c, is given by

T(n)= lim clF( n-m

< lim ~(n2~“- l/n)} l/2 n-m

= O(n”22”).

Table 1. Data of the example problem

Job i I 2 3 4

f,

A,

i.1 4 8 10 w 0.2 0.2 05

10 20 40 20

Table 2. Solution of the example problem

Step I: Initialization. Set k = 0; BE S(0); wIo) = 0; ttO, =6>0andS(k),F(k)=a, l<k<4.

Steps 2. 3 and 4: k - 0, P(1) = (11 (2), (3), (4). S(1) = {(lb (2). (3% F(l)= ((4,l. 2,3)};

k= 1, Pm - (29 1). c&4)* (3,l). (3,2), (3,4). S(2) = i(2.1). (3.1). (3.2,). F(2) = ((2.4 t. 3). (3.4.1,2)};

k = 2, p(3) - (3.2, l), (3.2.4). S(3) = 0,

Srep 5:

F(3)= {(3,2,1.4). (3,2.4,1));

F = {(4,1.2* 3). (2,4,1,3), (3.4.1.2). (3.2.1.4). (3.2.4.1,). f’ (d, a) - 3.1; d+ = 18 and u* = (3.4.1.2).

324 T. C. E. CHENG

While this partial search algorithm is not polynomial-bound, it represents a significant improvement in computational eficiency of the order of O(n4) over that presented in Cheng [3] whi.ch is of O(n22”) time complexity and provides a systematic method of solution for the single-machine common due-date assignment and sequencing problem.

Acknowledyemenrs-This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OPGO036424. The author is thankful to an anonymous referee for his/her constructive comments.

REFERENCES

1. N. G. Hall and M. E. Posner, Weighted deviation of completion times about a common due date. Working Paper, College of Administrative Sciences, Ohio State Umversity (1989).

2. T. C. E. Cheng and M. C. Gupta, Survey of scheduling research involving due-date determination decisions. Eur. J. Opl Res. 38, 156-166 (1989).

3. T. C. E. Cheng, An algorithm for the CON due-date determination and sequencing problem. Computers Opns Res. 14, 537-542 (1987).

4. K. R. Baker, Introduction to Sequencing nnd Scheduling. Wiley, New York (1974). 5. T. C. E. Cheng, On optimal common due-date determination. IMA J. Math. Mymt 1, 39-43 (1986).