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Optimal constant due-date determination and sequencing of n jobs on a single machine
259
International Journal of Production Economics, 22 ( 1991 ) 259-261 Elsevier
Technical Note
Optimal constant due-date determination and sequencing of n jobs on a single machine T.C.E. Cheng Department of Actuarial and Management Sciences, Universityof Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received July 27, 1987; accepted in revised form August 26, 1991 )
Abstract This paper considers due-date determination and sequencing of n jobs on a single-machine where each job is given a constant flow allowance. The objective is to determine the optimal value of the flow allowance and the optimal job sequence to minimize a cost function based on the flow allowance and the job earliness and tardiness values. We first propose a linear programming (LP) formulation of the problem and then derive the optimal constant flow allowance via considering the LP dual problem. We show that the optimal constant due-date value is independent of job sequence. After the theoretical treatment, a numerical example is presented for discussion.
1.
Introduction
Scheduling against due-dates has always been a popular research topic in the scheduling literature since scheduling became a field of research some thirty years ago. In the early years, scheduling research was undertaken dominantly to test the relative effectiveness of various due-date assignment rules using computer simulation. Some examples of such simulation work are: Conway [ 1 ], Eilon and Chowdhury [2], and Weeks and Fryer [ 3 ]. In recent years there has been a growing interest in taking an analytical approach to the due-date determination and scheduling problems. Many interesting results have emerged from the work of various researchers, who include Cheng [4,5], Kanet [6], Panwalker et al. [7], Seidmann et al. [8 ] and Seidmann and Smith [ 9 ], among others. 2. Problem formulation
This paper considers an n-job, single-machine problem with constant ( C O N ) due-dates. Let N be a set of n independent jobs to be processed on a single machine. It is assumed that job splitting is not allowed nor is inserted-idleness. Each job
requires ti processing time, which is fixed and known, on the machine that cannot simultaneously process more than one job. A weighting factor wi ( 0 ~ w i ~< 1, Y~7=~ wi= 1) is assigned to each job that reflects the relative importance of the job. The CON due-date assignment method is used to assign a due-date to each job. If di denote the assigned due-date of job i, Vie N, then di=s~+k, where k is the constant flow allowance and si is the time at which the machine starts processing job i. This method of assigning due-dates is widely employed in various service operations such as tax planning and photography services where due-dates are set by an appointment system. It is also commonly used in situations where due-dates are quoted based on a delayed delivery strategy - jobs are delayed until capacity becomes available - such as in an automobile repair shop or an air conditioner and furnace servicing company. Under these situations, each incoming customer is first asked to give a description of his problems. After an initial assessment, the customer will be advised of the start time and the expected duration of the service. Let rc be the set of all possible job sequences and tr be arbitrarily any one of the n! possible se-
0925-5273/91/$03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.
260 quences. Let the subscript [i] denote the job in position i in a, then Etq, Lu] and Ctq are respectively the earliness, tardiness and completion time of the ith job in a. The cost function to be minimized can be written as f ( k , a ) = ~ {ak+wul I C u l - d u l I}
(1)
i=l
maxg(x,a)= ~ tu]xu]
(8)
i=1
subject to ~ x u ] <~na
(9)
i=1
where I" I denotes the absolute value operator and a >/0 is a constant representing the due-date assignment cost per unit time. It should be noted that ifa>~ l/n, the problem has a trivial solution k*= 0, as will be shown later. Thus we make the assumption here that the condition 0~
minf(k, tr)=nak+ ~ wuj{Euj+Lt~]}
(2)
i=1
subject to dt,i +LI,I - E I , l = Ct,],
Vi~N
(3)
dtq ,Etq ,Lt/1 >t O,
Vi~N
(4)
This follows from the fact that Etq and Lul cannot be positive simultaneously at optimality. Under the CON due-date assignment method, we have dul =st~ 1 + k = C u _ l I + k
-wtq<~xt~j<~wul,
k+Ltq-Etq=tu],
Vi~N
(6)
k,Eli I ,Lul >10,
Vi ~ N
(7)
which, along with (2), completely define our LP minimization problem. It should now be clear that if a>~ 1/n, any value of k other than 0 will increase the due-date cost by nak while the tardiness cost will decrease by ~n=lwti]k. Thus there is a net increase of (na-ZT__ lwu] )k>_-0 in total cost and so k* = 0 is a trivial optimal solution. Instead of solving the LP primal directly, we consider its dual problem. Let x = ( x [ l ] , xt2 ]..... xt~ ] ) be a vector of dual variables. Since the primal consists of equality constraints, its dual must be unsymmetric and can be written as:
(10)
and xuj, Vi ~ N, are unrestricted in sign.
3. Optimal due-date It is a well-known result in LP duality theory that if k* and x * = (X~lj,X~z] ..... x~,j ) are feasible solutions to the primal and dual problems, and i f f ( k * , a ) =g(x*,tr), then k* and x* are optimal solutions to the respective problems. We shall require this result in our derivation of the optimal due-date.
Theorem I. For any job i ~ N, let s( i) = {j ~ N: tj~t,,i~j}. The optimal flow allowance is sequence-independent and is equal to one of the job processing times. That is, k* = tr, where r e N is such that
na+ ~ w i ies(r)
(5)
Substituting (5) into (3) and (4) and using tul = Ctq - Ct~- 1], we obtain
¥i~N
na+
~
ies(r)~{r}
~
i~S(r)~{r}
w,
w i - ~ wi>~O ieS(r)
Proof of Theorem 1. It follows from the assumption 0 ~
=f-wi wi
if/~s(r) i f / e S(r)
(12)
and
x*=na-
~ x*
(13)
i#r~N
Substituting (11) and (12) into (13) yields Vi ~ N and Zi~NXT~= na, SOx* is a feasible solution to g(x,tr). Now consider --Wi
261 /1
f(k*,a) =nak*+ ~ {Wlil lCti j -dl~ 1 I} i=1
=natr+ +
~, wtil{tr--t[i]}
[i]es(r)
~,
[i]eS(r)
=nat,+
Wtil{ttil--tr} E
[i]es(r)
d- E
[i]eS(r)
=nat,+
{ttil--tr}XTil
{ttil--tr}XTil tlilXTi j - t , 2 x'[il
i=1
i=1
=g(x*,tr) By the duality result of LP, it should be clear that k*=tr minimizes f ( k , a ) and since tr is independent of job sequence, we have thus completed the proof.
ear programming problem and obtained the optimal value via considering the LP dual problem. It has been shown that the optimal constant flow allowance is equal to one of the job processing times and it is independent of job sequence. A numerical example has been presented to demonstrate application of the theoretical result to determine the optimal solution.
Acknowledgements
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OPG0036424. I am thankful to two anonymous referees for their helpful comments.
References
4. An example A set of five jobs is given with t~-- l, t2=3, t3 = 5, t4 = 7, t5 = 10. The weighting factors of the jobs are w~=0.1, W E = 0 . 1 , w 3 = 0 . 2 , w4=0.4, 1 w5 = 0.2. The cost of due-date is a = 0.1 < ~. Using ( 11 ), we know that k* = tr, r ~ N and r satisfies the following conditions:
0.5+
wii~s(r)
0.5+
E
ies(r)~{r}
w,
w , - E w,> o ieS(r)
It is easy to check that job 3 satisfies the optimal condition, so k*= t3= 5 and the minimum total c o s t f ( k * , a ) = 5(0.1) + (0.1 × 4 + 0.1 ×2+0.4×2+0.2×5) =2.9.
5. Conclusion In this paper we have formulated the problem of assigning optimal constant due-dates as a lin-
Conway, R.W., 1965. Priority dispatching and job lateness in a job shop. J. Ind. Eng., 16: 228-237. Eilon, S. and Chowdhury, I.J., 1976. Due-dates in job shop scheduling, Int. J. Prod. Res., 14: 223-237. Weeks, J.K. and Fryer, J.S., 1977. A methodology for assigning minimum cost due-dates. Manage. Sci., 23: 872-881. Cheng, T.C.E., 1984. Optimal due-date determination and sequencing of n jobs on a single machine, J. Oper. Res. Soc., 35: 433-437. Cheng, T.C.E., 1986. Optimal due-date assignment in a job shop, Int. J. Prod. Res., 24:503-515. Kanet, J.J., 1981. Minimizing the average deviation of job completion times about a common due-dates, Naval Res. Logistics Q., 28:643-651. Panwalker, S.S., Smith, M.L. and Seidmann, A., 1982. Common due-date assignment to minimize total penalty for the one machine sequencing problem, Oper. Res., 30: 391399. Seidmann, A., Panwalker, S.S. and Smith, M.L., 1981. Optimal assignment of due-dates for a single processor scheduling problem. Int. J. Prod. Res, 19: 393-399. Seidmann, A. and Smith, M.L., 1981. Due-date assignment for production systems, Manage. Sci., 27:571-581.
International Journal of Production Economics, 22 ( 1991 ) 259-261 Elsevier
Technical Note
Optimal constant due-date determination and sequencing of n jobs on a single machine T.C.E. Cheng Department of Actuarial and Management Sciences, Universityof Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received July 27, 1987; accepted in revised form August 26, 1991 )
Abstract This paper considers due-date determination and sequencing of n jobs on a single-machine where each job is given a constant flow allowance. The objective is to determine the optimal value of the flow allowance and the optimal job sequence to minimize a cost function based on the flow allowance and the job earliness and tardiness values. We first propose a linear programming (LP) formulation of the problem and then derive the optimal constant flow allowance via considering the LP dual problem. We show that the optimal constant due-date value is independent of job sequence. After the theoretical treatment, a numerical example is presented for discussion.
1.
Introduction
Scheduling against due-dates has always been a popular research topic in the scheduling literature since scheduling became a field of research some thirty years ago. In the early years, scheduling research was undertaken dominantly to test the relative effectiveness of various due-date assignment rules using computer simulation. Some examples of such simulation work are: Conway [ 1 ], Eilon and Chowdhury [2], and Weeks and Fryer [ 3 ]. In recent years there has been a growing interest in taking an analytical approach to the due-date determination and scheduling problems. Many interesting results have emerged from the work of various researchers, who include Cheng [4,5], Kanet [6], Panwalker et al. [7], Seidmann et al. [8 ] and Seidmann and Smith [ 9 ], among others. 2. Problem formulation
This paper considers an n-job, single-machine problem with constant ( C O N ) due-dates. Let N be a set of n independent jobs to be processed on a single machine. It is assumed that job splitting is not allowed nor is inserted-idleness. Each job
requires ti processing time, which is fixed and known, on the machine that cannot simultaneously process more than one job. A weighting factor wi ( 0 ~ w i ~< 1, Y~7=~ wi= 1) is assigned to each job that reflects the relative importance of the job. The CON due-date assignment method is used to assign a due-date to each job. If di denote the assigned due-date of job i, Vie N, then di=s~+k, where k is the constant flow allowance and si is the time at which the machine starts processing job i. This method of assigning due-dates is widely employed in various service operations such as tax planning and photography services where due-dates are set by an appointment system. It is also commonly used in situations where due-dates are quoted based on a delayed delivery strategy - jobs are delayed until capacity becomes available - such as in an automobile repair shop or an air conditioner and furnace servicing company. Under these situations, each incoming customer is first asked to give a description of his problems. After an initial assessment, the customer will be advised of the start time and the expected duration of the service. Let rc be the set of all possible job sequences and tr be arbitrarily any one of the n! possible se-
0925-5273/91/$03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.
260 quences. Let the subscript [i] denote the job in position i in a, then Etq, Lu] and Ctq are respectively the earliness, tardiness and completion time of the ith job in a. The cost function to be minimized can be written as f ( k , a ) = ~ {ak+wul I C u l - d u l I}
(1)
i=l
maxg(x,a)= ~ tu]xu]
(8)
i=1
subject to ~ x u ] <~na
(9)
i=1
where I" I denotes the absolute value operator and a >/0 is a constant representing the due-date assignment cost per unit time. It should be noted that ifa>~ l/n, the problem has a trivial solution k*= 0, as will be shown later. Thus we make the assumption here that the condition 0~
minf(k, tr)=nak+ ~ wuj{Euj+Lt~]}
(2)
i=1
subject to dt,i +LI,I - E I , l = Ct,],
Vi~N
(3)
dtq ,Etq ,Lt/1 >t O,
Vi~N
(4)
This follows from the fact that Etq and Lul cannot be positive simultaneously at optimality. Under the CON due-date assignment method, we have dul =st~ 1 + k = C u _ l I + k
-wtq<~xt~j<~wul,
k+Ltq-Etq=tu],
Vi~N
(6)
k,Eli I ,Lul >10,
Vi ~ N
(7)
which, along with (2), completely define our LP minimization problem. It should now be clear that if a>~ 1/n, any value of k other than 0 will increase the due-date cost by nak while the tardiness cost will decrease by ~n=lwti]k. Thus there is a net increase of (na-ZT__ lwu] )k>_-0 in total cost and so k* = 0 is a trivial optimal solution. Instead of solving the LP primal directly, we consider its dual problem. Let x = ( x [ l ] , xt2 ]..... xt~ ] ) be a vector of dual variables. Since the primal consists of equality constraints, its dual must be unsymmetric and can be written as:
(10)
and xuj, Vi ~ N, are unrestricted in sign.
3. Optimal due-date It is a well-known result in LP duality theory that if k* and x * = (X~lj,X~z] ..... x~,j ) are feasible solutions to the primal and dual problems, and i f f ( k * , a ) =g(x*,tr), then k* and x* are optimal solutions to the respective problems. We shall require this result in our derivation of the optimal due-date.
Theorem I. For any job i ~ N, let s( i) = {j ~ N: tj
na+ ~ w i ies(r)
(5)
Substituting (5) into (3) and (4) and using tul = Ctq - Ct~- 1], we obtain
¥i~N
na+
~
ies(r)~{r}
~
i~S(r)~{r}
w,
w i - ~ wi>~O ieS(r)
Proof of Theorem 1. It follows from the assumption 0 ~
=f-wi wi
if/~s(r) i f / e S(r)
(12)
and
x*=na-
~ x*
(13)
i#r~N
Substituting (11) and (12) into (13) yields Vi ~ N and Zi~NXT~= na, SOx* is a feasible solution to g(x,tr). Now consider --Wi
261 /1
f(k*,a) =nak*+ ~ {Wlil lCti j -dl~ 1 I} i=1
=natr+ +
~, wtil{tr--t[i]}
[i]es(r)
~,
[i]eS(r)
=nat,+
Wtil{ttil--tr} E
[i]es(r)
d- E
[i]eS(r)
=nat,+
{ttil--tr}XTil
{ttil--tr}XTil tlilXTi j - t , 2 x'[il
i=1
i=1
=g(x*,tr) By the duality result of LP, it should be clear that k*=tr minimizes f ( k , a ) and since tr is independent of job sequence, we have thus completed the proof.
ear programming problem and obtained the optimal value via considering the LP dual problem. It has been shown that the optimal constant flow allowance is equal to one of the job processing times and it is independent of job sequence. A numerical example has been presented to demonstrate application of the theoretical result to determine the optimal solution.
Acknowledgements
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OPG0036424. I am thankful to two anonymous referees for their helpful comments.
References
4. An example A set of five jobs is given with t~-- l, t2=3, t3 = 5, t4 = 7, t5 = 10. The weighting factors of the jobs are w~=0.1, W E = 0 . 1 , w 3 = 0 . 2 , w4=0.4, 1 w5 = 0.2. The cost of due-date is a = 0.1 < ~. Using ( 11 ), we know that k* = tr, r ~ N and r satisfies the following conditions:
0.5+
wii~s(r)
0.5+
E
ies(r)~{r}
w,
w , - E w,> o ieS(r)
It is easy to check that job 3 satisfies the optimal condition, so k*= t3= 5 and the minimum total c o s t f ( k * , a ) = 5(0.1) + (0.1 × 4 + 0.1 ×2+0.4×2+0.2×5) =2.9.
5. Conclusion In this paper we have formulated the problem of assigning optimal constant due-dates as a lin-
Conway, R.W., 1965. Priority dispatching and job lateness in a job shop. J. Ind. Eng., 16: 228-237. Eilon, S. and Chowdhury, I.J., 1976. Due-dates in job shop scheduling, Int. J. Prod. Res., 14: 223-237. Weeks, J.K. and Fryer, J.S., 1977. A methodology for assigning minimum cost due-dates. Manage. Sci., 23: 872-881. Cheng, T.C.E., 1984. Optimal due-date determination and sequencing of n jobs on a single machine, J. Oper. Res. Soc., 35: 433-437. Cheng, T.C.E., 1986. Optimal due-date assignment in a job shop, Int. J. Prod. Res., 24:503-515. Kanet, J.J., 1981. Minimizing the average deviation of job completion times about a common due-dates, Naval Res. Logistics Q., 28:643-651. Panwalker, S.S., Smith, M.L. and Seidmann, A., 1982. Common due-date assignment to minimize total penalty for the one machine sequencing problem, Oper. Res., 30: 391399. Seidmann, A., Panwalker, S.S. and Smith, M.L., 1981. Optimal assignment of due-dates for a single processor scheduling problem. Int. J. Prod. Res, 19: 393-399. Seidmann, A. and Smith, M.L., 1981. Due-date assignment for production systems, Manage. Sci., 27:571-581.