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Optimal due-date assignment and sequencing in a single machine shop
O893-9659189 $3.00 + 0.00
Appl. Math. Lett. Vol. 2, No. 1, pp. 21-24, 1989
Copyright@ 1989 Pergamon Press plc
Printed in Great Britain 0 All rights reserved
Optimal
Due-Date
Assignment
and Sequencing
T.C.E. Department
of Actuarial
in a Single
Machine
Shop
CHENG
and Management
Sciences, Universitry
of Manitoba
Abstract. This paper considers due-date assignments and sequencing of n jobs in a singlemachine shop in which each job is given a constant waiting allowance. The objective is to End the optimal value of the waiting allowance and the optimal job sequence to minimize a cost function based on the waiting allowance and the job earliness and tardiness values. We formulate the problem as an LP and find the optimal solution via considering the LP dual problem and show that the optimal waiting allowance is independent of the job sequence.
Scheduling against due-dates has been a popular research topic in the scheduling literature. 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 (1965), Elion and Chowdhury (1976), and Weeks and Fryer (1977). M ore recently interest in analytical solution of the due-date determination and scheduling problems has been growing. Many interesting results have emerged from the work of various researchers, who include Cheng (1984, 1986), Panwalkar et al (1982), Seidmann et al (1981) and Seidmann and Smith (1981), among others PROBLEM FORMULATION
This paper considers an n-job, single machine problem in which each job is given a constant waiting allowance. Let N = { 1,2, . . . , n}be a set of rz independent jobs to be processed on a single machine. Each job requires ti amount of known and deterministic processing time on the machine which cannot, process more than one job simultaneously. A weighting factor wi (0 5 wi 5 1, Cr=‘=, wi = 1) is assigned to each job that reflects the relative importance of the job. Each job i is assigned a due-date di such that di = si + k, Vi E N, where si is the time at which the machine starts processing job i and k is a constant waiting allowance. Both job splitting and idleness between jobs are not allowed. Let II be the set of all possible job sequences and u be arbitrarily any one of the n! permutation sequences. Also let the subscript [i] denote the job in position i in cr, then E[il, L[il and C[il are respectively the earliness, tardiness and completion time of the job in position i of u. Since both assigning long due-dates and missing due-dates, be it early or tardy, will incur costs, the objective is to minimize a cost function based on the waiting allowance and individual job earliness and tardiness values as follows:
(1) i=l
where 1~1denotes the absolute value of the real number Z; a 1 0 is a constant representing the due-date assignment cost per unit time. It should be noted that if a 2 1, the problem has a trivial solution k* = 0, as will become clear later. Thus we assume that 0 5 a < 1 throughout our analysis. This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada Typeset by AM-w
21
T.C.E.
22
CHENG
The problem of minimizing f(k, u) can be converted to an LP as follows: minf(k,
u) = ok + 2
W[il{E[il+
L[i])
(2)
i=l
subject
to Vi E N
d[il + L[il - E[il = C[il
Vi E N
d[il, E[il, L[il 2 0
This follows from the fact that ElilLIil = 0 at optimality. allowance due-date assignment method, we have
d[j] = Substituting
S[i] +
(3)
(4) Under the constant
k = C[j_l] + k
(5) into (3) and (4) and using tlil = Cl, - Cli_il, Vi E N
k + L[il - E[il = t[il
Vi E N
k, E[il! L[il L 0
waiting
(5) we obtain (6)
(7)
which, along with (2), completely define the LP minimization problem. It should now be clear that if a > 1, any value of k other than 0 will increase the due-date cost by ak while the tardiness cost will decrease by Cy=‘=,wlilk. Thus there is a net increase of (o - Cy=i wlil)b = (e - 1)k 2 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 = (z’[11~[21. . . a+]) be a vector of dual variables. Since the primal consists of equality constraints, its dual must be in the asymmetric form, i.e. n
m=g(x, fl) =
Ct[ilx[i]
(8)
i=l
subject
to
2
“[i]
i
(9)
a
i=l
+qi] and zlil, Vi E N, are unrestricted
I qi] 5 W[i]
Vi E N
(IO)
in sign. OPTIMAL DUE-DATE
According
to the strong duality theory of LP, if k” and x* = (~~~1~~~1... zrnl) are
feasible solutions to the primal and dual problems, and if f(k*,a) = g(x*,c), then k” and x* are optimal solutions to the respective problems. We shall require this result in our derivation of the optimal due-date.
Optimal
Due-Date
Assignment
and Sequencing
23
THEOREM 1. ForanyjobiENlets(i)={jEN: tj
c
a+
c
wj -
iEs(r)
c
a+
wico
iCS(r)U{~l wj -
iCs(r)uCr}
c
Wi >
(11)
0
S(r)
Proof of Theorem 1: Since 0 5 a < 1, there exists a job satisfied. Let k* = t, and define the following dual variables: xf =
-Wi
{ W
r E N such that
(11) is
if i E s(r)
(12)
if i E S(r)
and
(13) Substituting (11) and (12) into (13) yields -Wi < X: 5 Wi. Since -Wi Cyzl xP = a, x* is a feasible solution to g(x, u). Now consider
<
Xp
5
Wi,
Vi E
N and
f(k*, g) = uL” + k{w[i]
ICiil - d[d]II
i=l
at, + C
=
W[i]{tr -t[i]I
+
MEs(r) = Ut,
C
C
W[i]{t[i]
-
trI
[ilES(r)
{t[i] - tv}Xs] + C
It[i] - trYI”c]
[ilES(r)
[iEs(r) kt[ilxs]
=
i=l =
$7(x*, a)
f(k, u). Since t, is independent
By the duality result of LP, k” = t, minimizes sequence, the proof is complete.
of job
REFERENCES 1. T.C.E. Cheng, Optimal Due-Date Determination and Sequencing of n jobs on a single machine, Journal of the Operational Research Society 35 (1984), 433-437. 2. T.C.E. Cheng, Optimal Due-Date Assignment in a Job Shop, International Journal of Production Research 24 (1986), 503-515. 3. R.W. Conway, Priority Dispatching and Job Lateness in a Job Shop, Journal of Industrial Engineering 16 (1965), 228-237. 4. S. Elion, I.J. Chowdhury, Due-Dates Research 14 (1976), 223-237,
in Job
Shop
Scheduling, International
Journal of Production
Common Due-Date Assignment to Minimize Total Sequencing Problem, Operations research 30 (1982), 391-399. S.S. Panwalkar, M.L. Smith, Optimal Assignment of Due-Dates fos a Single Proces-
5. S.S. Panwalkar, M.L. Smith, A. Seidmann, Penalty
for
the One
Machine
6. A. Seidmann, 4OT Scheduling Problem,
International Journal of Production Research 19 (1981), 393-399.
24
T.C.E.
CHENG
7. A. Seidmann, M.L. Smith, Due-Date Arsignment for Production 27 (1981), 571-581. 8. J.K. Weeks, J.S. Fkyer, A Methodology for Assigning Minimum Science 23 (1977), 872-881.
Department of Actuarial and Management Winnipeg, Manitoba, Canada R3T 2N2
Sciences, Univetitry
Systems, Cost
of Manitoba
Management
Due-Date+
Science
Management
Appl. Math. Lett. Vol. 2, No. 1, pp. 21-24, 1989
Copyright@ 1989 Pergamon Press plc
Printed in Great Britain 0 All rights reserved
Optimal
Due-Date
Assignment
and Sequencing
T.C.E. Department
of Actuarial
in a Single
Machine
Shop
CHENG
and Management
Sciences, Universitry
of Manitoba
Abstract. This paper considers due-date assignments and sequencing of n jobs in a singlemachine shop in which each job is given a constant waiting allowance. The objective is to End the optimal value of the waiting allowance and the optimal job sequence to minimize a cost function based on the waiting allowance and the job earliness and tardiness values. We formulate the problem as an LP and find the optimal solution via considering the LP dual problem and show that the optimal waiting allowance is independent of the job sequence.
Scheduling against due-dates has been a popular research topic in the scheduling literature. 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 (1965), Elion and Chowdhury (1976), and Weeks and Fryer (1977). M ore recently interest in analytical solution of the due-date determination and scheduling problems has been growing. Many interesting results have emerged from the work of various researchers, who include Cheng (1984, 1986), Panwalkar et al (1982), Seidmann et al (1981) and Seidmann and Smith (1981), among others PROBLEM FORMULATION
This paper considers an n-job, single machine problem in which each job is given a constant waiting allowance. Let N = { 1,2, . . . , n}be a set of rz independent jobs to be processed on a single machine. Each job requires ti amount of known and deterministic processing time on the machine which cannot, process more than one job simultaneously. A weighting factor wi (0 5 wi 5 1, Cr=‘=, wi = 1) is assigned to each job that reflects the relative importance of the job. Each job i is assigned a due-date di such that di = si + k, Vi E N, where si is the time at which the machine starts processing job i and k is a constant waiting allowance. Both job splitting and idleness between jobs are not allowed. Let II be the set of all possible job sequences and u be arbitrarily any one of the n! permutation sequences. Also let the subscript [i] denote the job in position i in cr, then E[il, L[il and C[il are respectively the earliness, tardiness and completion time of the job in position i of u. Since both assigning long due-dates and missing due-dates, be it early or tardy, will incur costs, the objective is to minimize a cost function based on the waiting allowance and individual job earliness and tardiness values as follows:
(1) i=l
where 1~1denotes the absolute value of the real number Z; a 1 0 is a constant representing the due-date assignment cost per unit time. It should be noted that if a 2 1, the problem has a trivial solution k* = 0, as will become clear later. Thus we assume that 0 5 a < 1 throughout our analysis. This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada Typeset by AM-w
21
T.C.E.
22
CHENG
The problem of minimizing f(k, u) can be converted to an LP as follows: minf(k,
u) = ok + 2
W[il{E[il+
L[i])
(2)
i=l
subject
to Vi E N
d[il + L[il - E[il = C[il
Vi E N
d[il, E[il, L[il 2 0
This follows from the fact that ElilLIil = 0 at optimality. allowance due-date assignment method, we have
d[j] = Substituting
S[i] +
(3)
(4) Under the constant
k = C[j_l] + k
(5) into (3) and (4) and using tlil = Cl, - Cli_il, Vi E N
k + L[il - E[il = t[il
Vi E N
k, E[il! L[il L 0
waiting
(5) we obtain (6)
(7)
which, along with (2), completely define the LP minimization problem. It should now be clear that if a > 1, any value of k other than 0 will increase the due-date cost by ak while the tardiness cost will decrease by Cy=‘=,wlilk. Thus there is a net increase of (o - Cy=i wlil)b = (e - 1)k 2 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 = (z’[11~[21. . . a+]) be a vector of dual variables. Since the primal consists of equality constraints, its dual must be in the asymmetric form, i.e. n
m=g(x, fl) =
Ct[ilx[i]
(8)
i=l
subject
to
2
“[i]
i
(9)
a
i=l
+qi] and zlil, Vi E N, are unrestricted
I qi] 5 W[i]
Vi E N
(IO)
in sign. OPTIMAL DUE-DATE
According
to the strong duality theory of LP, if k” and x* = (~~~1~~~1... zrnl) are
feasible solutions to the primal and dual problems, and if f(k*,a) = g(x*,c), then k” and x* are optimal solutions to the respective problems. We shall require this result in our derivation of the optimal due-date.
Optimal
Due-Date
Assignment
and Sequencing
23
THEOREM 1. ForanyjobiENlets(i)={jEN: tj
c
a+
c
wj -
iEs(r)
c
a+
wico
iCS(r)U{~l wj -
iCs(r)uCr}
c
Wi >
(11)
0
S(r)
Proof of Theorem 1: Since 0 5 a < 1, there exists a job satisfied. Let k* = t, and define the following dual variables: xf =
-Wi
{ W
r E N such that
(11) is
if i E s(r)
(12)
if i E S(r)
and
(13) Substituting (11) and (12) into (13) yields -Wi < X: 5 Wi. Since -Wi Cyzl xP = a, x* is a feasible solution to g(x, u). Now consider
<
Xp
5
Wi,
Vi E
N and
f(k*, g) = uL” + k{w[i]
ICiil - d[d]II
i=l
at, + C
=
W[i]{tr -t[i]I
+
MEs(r) = Ut,
C
C
W[i]{t[i]
-
trI
[ilES(r)
{t[i] - tv}Xs] + C
It[i] - trYI”c]
[ilES(r)
[iEs(r) kt[ilxs]
=
i=l =
$7(x*, a)
f(k, u). Since t, is independent
By the duality result of LP, k” = t, minimizes sequence, the proof is complete.
of job
REFERENCES 1. T.C.E. Cheng, Optimal Due-Date Determination and Sequencing of n jobs on a single machine, Journal of the Operational Research Society 35 (1984), 433-437. 2. T.C.E. Cheng, Optimal Due-Date Assignment in a Job Shop, International Journal of Production Research 24 (1986), 503-515. 3. R.W. Conway, Priority Dispatching and Job Lateness in a Job Shop, Journal of Industrial Engineering 16 (1965), 228-237. 4. S. Elion, I.J. Chowdhury, Due-Dates Research 14 (1976), 223-237,
in Job
Shop
Scheduling, International
Journal of Production
Common Due-Date Assignment to Minimize Total Sequencing Problem, Operations research 30 (1982), 391-399. S.S. Panwalkar, M.L. Smith, Optimal Assignment of Due-Dates fos a Single Proces-
5. S.S. Panwalkar, M.L. Smith, A. Seidmann, Penalty
for
the One
Machine
6. A. Seidmann, 4OT Scheduling Problem,
International Journal of Production Research 19 (1981), 393-399.
24
T.C.E.
CHENG
7. A. Seidmann, M.L. Smith, Due-Date Arsignment for Production 27 (1981), 571-581. 8. J.K. Weeks, J.S. Fkyer, A Methodology for Assigning Minimum Science 23 (1977), 872-881.
Department of Actuarial and Management Winnipeg, Manitoba, Canada R3T 2N2
Sciences, Univetitry
Systems, Cost
of Manitoba
Management
Due-Date+
Science
Management