Optimal single-machine sequencing and assignment of common due-dates

Computers ind. Engng Vol. 22, No. 2, pp. 115-120, 1992 Printed in Great Britain. All fights reserved

0360-8352/92 $5.00+0.00 Copyright © 1992 Pergamon Press plc

OPTIMAL SINGLE-MACHINE SEQUENCING AND ASSIGNMENT OF COMMON DUE-DATES T. C. E. CHENG Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received for publication 22 October 1991)

paper considersan n-job one-machinesequencingproblem with commondue-dates. The objective is to determinethe optimal commondue-date value and the optimaljob sequence that jointly minimize a cost functionwhich is dependent on the individualjob earliness and tardiness values. Using Kuhn-Tucker's optimalityconditionsfor constrainedconvex programmingproblems, we show that for a givenjob sequence,the optimaldue-date is a simplefunctionof the number of jobs. This result allows separation of the due-date assignmentproblem from the job sequencingproblem. A well-knowntheorem in algebra can be applied to solve the latter problem, which in turn yields the optimal solution to the overall problem. Abstrs~t--This

INTRODUCTION

The optimal due-date assignment problem has attracted considerable attention from scheduling researchers. Recently, an extensive review of scheduling research involving due-date assignment decisions has been presented by Cheng and Gupta [1]. Over the years, various methods of due-date assignment have been ploposed and studied in the scheduling literature. Examples are: Cheng [2-4], Conway [5], Eilon and Chowdhury [6], Kanet [7], Panwalkar et al. [8] and Seidmann et al. [9], among others. The popularity of due-date research is attributed to the theoretical challenge posed by the due-date determination problem itself and the paramount importance of meeting order due-dates in real-life situations. It is evident that in the current business situation where firms are constantly subject to keen competitive pressure, from both local and foreign competitors, the long-term survival and growth of a firm hinges heavily on its ability to consistently keep customer order delivery promises (or meeting order due-dates). Ragatz and Mabert [10] have given a detailed treatment of due-date management in the job shop production environment. We consider in this paper the problem of sequencing a set of independent jobs on a single machine. All jobs are assigned a common due-date. This method of common due-date assignment is widely used by firms where the salespersons are instructed to quote a uniform delivery date on all orders. The objective is to find the optimal value of the common due-date and the optimal job sequence that jointly minimize a total cost function dependent on the due-date value and the earliness and tardiness of each job. This objective function is practically relevant because both quoting long due-dates and missing the quoted due-dates will incur costs--losing customer goodwill for asking customers to wait for an unreasonably long time and contractual penalties for missing due-dates. We first formulate the problem as a constrained convex program and solve it using the Kuhn-Tucker's conditions. We show that the optimal due-date is a simple function of the number of jobs. We then apply a simple theorem in algebra to find the optimal job sequence. Cheng [11] has presented a linear programming (LP) formulation of the problem and obtained the optimal solution via considering the LP dual problem. Panwalkar et al. [8] have studied a closely related problem in which different penalty costs are incurred for assigning long due-dates as well as finishing jobs early and late. Based on the finite difference approach, Panwalkar et al. prove that the optimal due-date coincides with one of the job completion times. For the problem of finding the optimal job sequence, Kanet [7] and Panwalkar et aL [8] have presented efficient (polynomialbound) algorithms. Incidentally, both of these algorithms work on the basis of the algebraic theorem we apply in this paper to find the optimal job sequence. We remark that the general problem in which each job has a different earliness and lateness cost has been shown to be NP-hard by Hall and Posner [12]. CAm22/2--B

115 "

116

T.C.E. CHENG

PROBLEM FORMULATION Let N = { 1, 2 . . . . . n } be a set of n independent jobs to be processed on a continuously available single-machine under the common assumptions listed in Baker [13]. The machine cannot process more than one job at a time and is not allowed to have inserted idle time and job splitting. Job i requires t~ processing time on the machine which is assumed to be deterministic and known before processing begins. All jobs are available for processing at the same time and are assigned a common due-date de = k, for all i, where k I> 0 is the common due-date value. Let n be the permutation set of the n jobs and let a be arbitrarily one of the n! possible job sequences. Also let the subscript [i] denote the job in position i of a, then C[0, Etq = max{0, d t q - C[0 } and Lti] = max{0, C[t]- dr0 } denote respectively the completion time, earliness and tardiness of the job in position i of a. The problem of finding the joint optimal due-date and optimal job sequence can be formulated as a constrained convex program as follows: Minimize: f ( k , a) = ~ {E0] + Lt;l}

(1)

i~l

k - E[,] +/..[,] = Ctq

q

(2) for all

i.

k, E[0, L[0 >I 0

(3)

In order to solve this constrained convex programming problem, we first construct its associated Lag,range, an function:

F(k, a, 2) = f (k, a) + ~ 2(t]{ Ct,1-- k + Et,1-- L[,]} i=l

= E

+ 6,]) +

i=l

k + El,]- 6,])

(4)

i=l

where 2 = (2122... An) is a vector of Lagrangean undetermined multipliers. Applying Kuhn-Tucker's conditions (see Luenberger [14]) of optimality for F(k, a, 2), we obtain, for all i in N, the following set of simultaneous non-linear equations and inequalities: ~F

t3~.iq >~ 0=~1 + 2[q >t 0,

(5)

E[,](a~[,])= O~E[,](l+ z[,])--o,

(6)

t3F aLto

Lti)(~[[,~)

t> O=M - 2tq I> O,

(7)

= O=~Lc,](1- x[,]) = o,

(8)

~--~/> 0=*.

2tO ~< O,

(9)

i~l

<,0, tgF

a2[,] o=~g,)- k + EE,J- Z,] O.

(11)

A close examination of expressions (5) through (11) leads to two observations which are presented here for later reference in the analysis of the optimal solution.

Optimal due-date assignment

117

Observation I: The Lagrangean multipliers are bounded from above and below: - 1 ~<2vl ~< 1, for all i in N. Observation 2: T h e variables Ev3 and Lt,.1 satisfy the complementary slackness condition Ev3Lt;I = 0, for all i in N, since

E[II > 0=~2[~1= - 1=-LtiI = 0

f

EVl = 0=,1 >i 2v] >I - 1=~Lvl = 0 Ltft > 0=*'2ff] = l =:'Eli I ----0

L[~]= 0=, - 1 ~< 2(0 ~< 1=~Ev] = 0

Since it is evident that k = 0 is not optimal, so k > 0. Thus, from equation (10), we can write ~ I r , j = O.

(12)

i=!

OPTIMAL C O M M O N D U E - D A T E

We now present and prove two lemmas which enable us to determine the optimal common due-date. L e m m a 1: For any given common due-date k, there exists at most one job with both its earliness and tardiness equal to zero. P r o o f o f L e m m a I: Let s ( X ) denote the size of a set X. Assume there exists a subset S of N, with s ( S ) , 1 < s ( S ) <<.n, distinct jobs, such that Et = L~ = 0, for all i in S. It follows from equation (1 1) that k = C~. Hence, C~ = Cj, for all i and j in S. This leads to contradiction because Ci and Cj are distinct for any pair of jobs i and j in S. So the converse must be true and the proof is complete. L e m m a 2: For any given job sequence a, the optimal value of the common due-date is k* = ~ CItn+ I)/21,

if n is odd (13)

L

Ct~/21+ dPtt~/2+ ij

for some 0 ~<~b ~< 1, if n is even.

P r o o f o f L e m m a 2" It is evident that the set of jobs N c a n be partitioned into 3 collectively exhaustive disjoint sets, namely, the set of early jobs E = {i ~ N : E,. > 0}, the set of tardy jobs L = {i ~ N: Lj > 0} and the set of on-time jobs T = {i e N: El = Lj = 0}. Note that T can possibly be empty. We have s ( E ) + s ( L ) + s ( T ) = n; and from L e m m a 1 we know that s ( T ) ~< 1, or equivalently, n >1s(E) + s(L) I> n - 1. Since n is either even or odd, we examine these two cases separately. Case (i): n is odd

It is clear from the above discussion that for any given s(E), s ( L ) is either equal to n - s ( E ) or n - s ( E ) - 1. Suppose s ( L ) = n - s(E), hence s ( T ) = O. Using the result of Observation 2 and equation (12), we have - s ( z ) + s(L) = 0 or /I

s(z) = ~.

04)

Since s ( E ) must be an integer and n is odd, equation (14) cannot hold and so it is impossible that s ( L ) = n - s(E). Thus s ( L ) = n - s ( E ) - 1 must hold. So s ( T ) = 1 which, from equations (6)

118

T . C . E . Cr~No

and (8), implies that there exists one job j e N such that - 1 ~<2j ~< 1. Hence from Observation 2 and equation (12), we obtain - s ( e ) + s(L) + gj = 0 or

s(e) =

(n -- 1 + 2j) 2

(15)

Since s ( E ) must be an integer, 2j = 0 for equation (15) to hold. Thus s ( E ) = s ( L ) = (n - 1)/2. It follows that the optimal common due-date should be set in such a way that will result in exactly one job on-time, (n - 1)/2 jobs early and (n - 1)/2 jobs late. Evidently, this happens only when k = k * = C[(n+ o/2].

Thus the optimal due-date is unique when n is odd. Case (iO: n is even

For any s ( E ) , as in Case (i) above, s ( L ) is either equal to n - s ( E ) or n - s ( E ) 1. Suppose s ( L ) = n - s ( E ) - 1, so that s ( T ) = 1 and there exists one j o b j e N such that - 1 ~<2j ~< 1. Using the result of Observation 2 and equation (12), we write - s ( E ) + s(Z ) + ,~j = 0 or

s(e) =

(n - 1 + 2j) 2

(16)

Since s ( E ) must be an integer, equation (16) holds only when 2j = 1 or - 1. If 2j = 1, s ( E ) = n/2 and s ( L ) --- (n/2) - 1. If ~j = - 1, s ( E ) = (n/2) - 1 and s ( L ) = n/2. Next suppose s ( L ) = n - s ( E ) , so that s ( T ) = O. From Observation 2 and equation (12), we obtain - s ( E ) + s(L) = 0 or n

s(Z) = s(L) = 5"

(17)

Therefore the optimal common due-date should be set in such a way that will give rise to one of the following three possible situations: (i) (n/2) - 1 jobs early and n/2 jobs late, (ii) n/2 early and (n/2) - 1 jobs late, and (iii) n/2 jobs early and late. This is achieved by having k = k * = Ct,/21 + (~t[a/2+ I1,

for an arbitrary 0 ~<~b ~< 1. Thus there is an infinite number of optimal due-dates when n is even. Finally, it is noted that a common due-date value k* has been derived which satisfies the Kuhn-Tucker's conditions for optimality. Since, for any given a, f ( k , a) is convex in k and the set of constraints (2) and (3) is linear, the Kuhn-Tucker's conditions are both necessary and sufficient. So k* as expressed in equation (13) is a global minimum of the cost function f ( k , a). This completes the proof. O P T I M A L JOB S E Q U E N C E

Using the resutls of L e m m a 2, we can find the optimal due-date value which is a simple function of n. To facilitate the derivation of the optimal job sequence, we define a new variable r as follows: f4-in_21,

if n i s o d d (18)

r

if n is even.

Optimal due-date assignment

119

Table 2. A list of all possible optimal

solutions to the example problem S~lUOn~ No.

a*

k*

1 2 3

%5--4-2-1-3-6 7--5-4-1-2-3-6 %5-3-2-1--4-6

43 36 41

4 5 6 7 8

7-5-3-1-2-4--6 7--6--4-2-1-3-5 7--6--4-1-2-3-5 7--6--3-2-1--4-5 7--6-3-1-2-4-5

39 47 45 45 43

Table I. One possible assignment of jobs according to H a r d y et al.'s theorem Position i

I

Positional index tlff [i]

3

4

5

6

7

0 I 2 21 15 8 7 6 4

2

3 I I

3 3 2

2 6 3

! II 5

For any job sequence a, let k* be the optimal common due-date as determined from equation (13). Then the cost function can be written as

f

~{Ctrl-Ctq}+i =~r + l

i=1

{Ctq-Cl,l},

if n i s o d d

f ( k * , a)

~{Ct,l+dptt,+,l-Ctq}+ ~ {Ct,)-Ct,l-dPtt,+,l}, if

i:1

niseven.

(19)

i:r+l

i

Substituting C[~1= ~ t[jI into equation (19) and simplifying yields j=l

f(k*,a)= iE(i-1)ttq+ =1

.L

~

(n+l-i)ttq.

(20)

i--r+l

The optimal job sequence a* f o r f ( k * , o') as expressed in equation (20) can be found using a simple theorem in algebra. The theorem states that the sum of pairwise products of two sequences of real numbers is minimized by arranging one sequence in increasing order and the other sequence in decreasing order (see Hardy [15]). In equation (20), the coefficient of each tt~] can be viewed as a positional index for position i in a job sequence. To obtain the optimal job sequence, we assign jobs to positions in such a way that the job processing times match the positional indices according to the theorem.

etaL

AN EXAMPLE

A set of seven jobs is given with tl = 1, t 2 = 3 , h = 6 , t4=8, t s = 11, t6= 15, t7 = 21. Using the results of 1, r = 4 since there are n = 7 jobs. Thus the optimal due-date value is k* = CE41 which can easily be found once the optimal sequence a* is determined. We construct Table 1 and apply the pairwise matching theorem to find a*. It is clear that the optimal job sequences a* corresponding to the job assignment as shown in Table 1 is 7 - 6 - 4 - 1 - 2 - 3 - 5 which yields an optimal due-date value k * = Ct43= 21 + 15 + 8 + 1 = 4 5 and the resulting minimum cost is a*) = 24 + 9 + 1 + 3 + 9 + 20 = 66. It is interesting to note that, except for the first position, the positional indices are symmetric about k*, so there are alternate optimal solutions. Bagchi [16] have shown that the number of alternate optimal solutions is 2'-~ if n is odd and 2' if n is even. Thus, for our problem, there are a total of 8 optimal solutions (Table 2), each of which yields the same minimum cost.

Lemma

f(k*,

et al.

CONCLUSIONS AND DISCUSSIONS

We have considered the common due-date assignment and sequencing problem. We have formulated the problem as a constrained convex program and derived the optimal solution using the Kuhn-Tucker's conditions. We have shown that the optimal value of the common due-date is a simple function of the number of jobs. This result allows separation of the due-date assignment problem from the job sequencing problem, whose optimal solution can be easily obtained using a simple theorem in algebra. A similar optimal result for the due-date value has been obtained by Panwalkar [8] using the finite difference approach, which is mathematically simpler than our constrained convex

et al.

120

T.C.E. Cm~o

programming approach. However, when the number of jobs is even, the finite difference method can only show that k* is either equal to CI,/21 or Cl¢~/2)+ ,l. In such a case, there is in fact an infinite number of optimal due-dates, as stated in/.,emma 2, which can be any value between the completion times of the [n/2]th and [(n/2) + 1]th jobs. This latter optimal result is readily available from the Kuhn-Tucker conditions. As for the problem of finding the optimal job sequence, Kanet [7] and Panwalkar et al. [8] have presented efficient algorithms both of which, we have noted, are implicitly based on the theorem of Hardy et al. [15]. Acknowledgements--This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OPG0036424. We are thankful to two anonymous referees for their constructive comments on an earlier version of this paper. REFERENCES 1. T. C. E. Cheng and M. C. Gupta. Survey of scheduling research involving due-date determination decisions. Fur. J. Opl Res. 38, 156-166 (1989). 2. T. C. E. Cheng. Optimal due-date determination and sequencing of n jobs on a single machine. J. Opl Res. Soc. 35, 433-437 (1984). 3. T. C. E. Cheng. Optimal total-work-content-power due-date determination and sequencing. Computers Math. Applic. 14, 579-582 (1987). 4. T. C. E. Cheng. Optimal assignment of slack due-date and sequencing of jobs with random processing times on a single machine. Fur. J. Opl Res. 51, 348-353 (1991). 5. R. W. Conway. Priority dispatching and job lateness in a job shop. J. lndustr. Engng 16, 228-237 (1965). 6. S. Eilon and I. G. Chowdhury. Due-date in job shop scheduling. Int. J. Prodn Res. 14, 223-238 (1976). 7. J. J. Kanet. Minimizing the average deviation of job completion times about a common due-date. Naval Res. Log. Q. 28, 634-651 (1981). 8. S. S. Panwalkar, M. L. Smith and A. Seidmann. Common due-date assignment to minimize total penalty for the one machine scheduling problem. O/ms Res. 30, 391-399 (1982). 9. A. Seidmann, S. S. Panwalkar and M. L. Smith. Optimal assignment of due-dates for a single processor scheduling problem. Int. J. Prodn ICes. 19, 393-399 (1981). 10. G. L. Ragatz and V. A. Mabert. A framework for the study of due-date management in job shops. Int. J. Prodn Res. 22, 685-695 (1984). 11. T. C. E. Cheng. A duality approach to optimal due-date determination. Engng Opt. 9, 127-130 (1985). 12. 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 University, (1989). 13. K. R. Baker. Introduction to Sequencing and Scheduling. Wiley, New York (1974). 14. D. G. Luenberger. Linear and Non-Linear Programming. Addison-Wesley, Reading. (1984). 15. G. H. Hardy, J. E. Littlewood and G. Polya. Inequalities. Cambridge University Press, Cambridge (1988). 16. U. Bagchi, R. S. Sullivan and Y. L. Chang. Minimizing mean absolute deviation of completion times about a common due-date. Naval Res. Log. Q. 33, 227-240 (1986).