optimal assignment of nop due-dates and sequencing in a single machine shop

ZOR - Methods and Models of Operations Research (1992) 36:333- 341

Optimal Assignment of NOP Due-Dates and Sequencing in a Single Machine Shop

B. Alidaee 1

Abstract: We consider the problem of optimal assignment of NOP due-dates to n jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.

Key words: Single machine, due date determination.

1 Introduction

Let N = {1, 2 . . . . . n} be a set of n independent jobs to be processing on a single machine. Associated with each job i are Pi and di as processing time and due- date respectively. We assume that the due-dates are determined on the basic of number of operations to be performed on the job (NOP). Thus di = ri + kMi for i = 1 . . . . . n, where r i, and Mi are ready time and number of operations to be performed on job i respectively, and k is a constant waiting allowance common among all jobs. We also assume that all jobs will be ready at the same time (rl = r2 = . . . = rn). Both job splitting and idleness between jobs are not allowed.

L e t / / b e the set of all n! permutations of n jobs and tr = ([1] . . . . . In]) to be an arbitrary job sequence where [i] is the ith job in sequence tr, then Cti I and T[i ] = ( C [ i ] - d [ i ] ) + are completion time and tardiness of the job in position i. The objective is to find an optimal job ordering tr* and an optimal waiting allowance k* that jointly minimize the penalty function given by

1 Bahram Alidaee, Mathematics and Physical Sciences Department, West Texas State University, Canyon, TX 79016.

0340-9422/92/4/333-341 $2.50 �9 1992 Physica-Verlag, Heidelberg

334 B. Alidaee

Z(cr, k ) - a k + m a x T[i ] (1) l<-.i<n

where a _> 0 is the cost per unit time of waiting time allowance. The significance of assigning accurate due dates to jobs in a production

system is well recognized by researchers [2, 4, 7]. There has been a variety of decision rules suggested to assign due dates to jobs in a job shop production system. Some of the well studied due date decision rules are as follows [4].

(1) SLK: jobs are given flow allowances that reflect equal waiting times or equal slacks (di = r i + P i + k , i = 1 . . . . . n ) (2) TWK: due dates are based on total work content (di = r i+kpi , i = 1 . . . . , n ) (3) NOP: due dates are determined on the basis of number of operations to be performed on the job (d/= ri+ kMi, i = 1 . . . . . n ) (4) CON: all jobs are given exactly the same flow allowance ( d i = r i + k , i = 1 . . . . . n).

For the penalty function given in (1) if the rule is SLK due date, an analytical solution procedure is presented by Cheng [3] and Alidaee [1]. In this paper we consider NOP due dates for the objective function (1), and will present an efficient algorithm to solve the problem. All results in this paper are easily applicable to TWK due date also.

The rest of the paper proceeds as follows. In Section 2 we give some theoretical results and rules for solutions of the problem. In Section 3 an efficient algorithm is presented.

2 Theoretical Results

Since r 1 = r2 = , . . . . r,, we may let ri = 0 v i e N , then the penalty function (1) will be the same as,

Z(tr, k ) = a k + max {max (O, C[i ] - kM[il) } (2) l <.i<_n

The following lemma gives an optimal ordering for any fixed value of k.

Lemma." Given a value of k, the earliest due date (EDD) order minimizes the penalty function Z ( a , k ) .

Optimal Assignment of NOP Due-Dates and Sequencing in a Single Machine Shop 335

Proof." For each job i let f (Ci ) = a k + (C i -d i ) + and apply the famous theorem of Lawler [5], then the desired result follows immediately.

From the above lemma it is clear that we only need to consider EDD orders for optimal solution of the problem. If the due date rule is NOP, the EDD order will be the same as an ordering where M1 - M 2 - . . . . . _</I//,. From now on we let the jobs be numbered such that M1 <-M2 <- . . . . . <_M n. Now the problem is reduced to finding an optimal value of k* such that the penalty function given in Equation (3) is minimized.

Z(EDD, k) = a k + max {max(O, C i - k M i ) } l<-i<-n

(3)

Since for different values of k the value of max {max (O, C / - kMi) } might l<-i<-n

occur at a different job i e N , then minimizing penalty function (3) is the same as solving a multiple objective piecewise linear programming problem with single variable k, and the objective is minimization of vector u(k) = (ul(k) . . . . . un(k)) where we have,

ui(k) = ak + max (0, C i - kMi) , i = I, 2 . . . . . n (4)

Definition 1: Let S1 = { i e N : a =Mi) , S 2 = { i e N : O<_a<Mi} and S 3 = { i e N : a > Mi}.

Clearly S i i = 1,2,3 give a partition of N and S i • Sj = 0 for i * j . Consider function ui(k) for a job i. The value of ui(k) is equal to ak if

k>_Ci/M i and its value depends on the size of the value of a and M i for k < C/ /M i as given in the following

for all i eS1

for all i eS2

for all i e S 3

I a k if k>_Ci/M i

C i if k < Ci /M i

I a k if k > Ci /M i

- (M i - a )k+ Ci if k < CJMi

I ak i fk> Ci/Mi

(a - Mi )k + Ci if k < Ci/Mi

Consider Figure 1 which represents graph of ui(k) for different jobs. For a fixed value of k, let say k -- t, the value of Z(EDD, t), can be found on the heavy lines on the graph. The problem of minimizing Z(EDD, k) geometrically means: as th~ vertical line k = t moves along k-axis we want to find those values of k where the value of Z(EDD, k) on the heavy line is smallest.

336 B. Alidaee

Z ~k

I k = t

Fig. 1. Maximum value of Z as k = t moves along k-axis

I, k

Let a _> O takes all possible values as shown in ( 1 ) - ( 7 ) in the following

(1) M I = M 2 = , . . . . M n - ~ = M n = a

(a) a < M ~ <-M2<- . . . . . <-Mn-~ <-Mn

(3) M I < M a < . . . . , < M n _ l < M n < a

(4) M1 = M 2 = , . . . . = M k = a < M k + l < - . . . . . <-Mn

(5) M I <_M2 < _ . . . . . < _ M k < a = M k + 1 = , . . . . = M n

(6) M I < M 2 < . . . . . < M k < a < M k + l < . . . . . < M n

($1 * 0, $2 = 0, $3 = O)

(s~ =o, s2,o, s3 =9)

(s~ =o, s2=o, s~,o)

( S ~ , O, $ 2 , 0 , $ 3 = 0 )

( s l , o, s2=0, s3,0)

(sl =o, s2.o, s3.o)

(7) M I <_M2 < _ . . . . . < _ M k < a = M k + 1 = , . . . . = M m < M m + l < - . . . . . < - M n

(S~ . 0 , $ 2 . 0 , $ 3 . 0 )

In the proposi t ion (1) - (7) we consider case (1) - (7) above, respectively and solve the problem.

P r o o f o f proposi t ions ( 1 ) - (3) are clear f rom Figure 2(a), (b) and (c) respectively.


/ l

I I

L Ci/M i

Fig. 2a. a = M i, i= 1 . . . . . n

Ci/(M i- ~)

Fig. 2c. a > M i, i= 1 . . . . . n

a k

~ k

Z ek

I I 1

C• i

Fig. 2b. a < M i, i= 1 . . . . . n

~ a k

Z=

l !

C2/M •

Ci/M i

-k

aCi/M i

, - k

Propos i t ion 1: Let a = Mi for i = 1 . . . . . n then, any value of k * < max C J a = l <_i<_n

Cn/a is opt imal so lu t ion and Z ( E D D , k * ) = Cn.

Propos i t ion 2: Let O < - a < M 1 then, k* = max C i / M i is opt imal so lu t ion of the p rob lem and Z ( E D D , k*) = a k * . 1 <_i<_n

Propos i t ion 3: Let a > M n then , k* = min C i / ( M i - a ) is opt imal solut ion of the p rob lem and Z ( E D D , k*) = 0 l<_i<_n

Def in i t ion 2: Define a set of jobs {s, l, j } such that ,

Cs = m a x Ci , a C t ~ M r = m a x a C i / M i and a C j / M i = m a x a C i / M i i e S 1 i e S 2 i e S 3

Propos i t ion 4: Let a = M i for i = 1 . . . . . k and a < M i for i = k + 1 . . . . . n then,

338 B. Alidaee

If Cs<_aCt/Mt then, k* = Ct/M t is optimal solution and Z(EDD, k*) = aCt /M 1 I f Cs> aCt~Mr then, k* e [max (Cs -C i ) / (a -Mi ) , Cs/a] is optimal solution and Z(EDD, k*) = Cs. iEs2

Proof." Consider the Figure 1. Since the set $3 is empty then all functions ui(k), i = l . . . . . n belong to the jobs in the sets S~ or $2. It is easy to see that if Cs<_ aCt/Mr then the heavy lines stay on the graph of the job l e $2 and therefore k* = Ct/M t minimizes the function Z and Z(EDD, k*) = aCt /M t. If Cs> aC/MI then we have a situation that the heavy lines stay on graph of the job s e S~ up to some point and then continues to stay on graph of a job i e $2. Since the intersection of graph of a job i e $2 and the job s e $1 occur at k = (Cs-Ci) / ( a - M / ) then it is obvious that any k * e [max (Cs -C i ) / (a -Mi ) , Cs/a] is op-

i e S 2

timal solution and Z(EDD, k*) = C s. This completes the proof. Using Figure 1, similar proofs may be given for propositions (5) - (6) . We

omit the proofs here.

Proposition 5: Let a > Mi for i = 1,2 . . . . . k and a = M i for i = k + 1 . . . . , n then, If Cs>_aCJMj then any value k*<_Cs/Ms is optimal solution of the problem and Z(EDD, k*) = C s If Cs< a CJMj then any value k* _< min (Cs- C i ) / ( a - Mi) is optimal solution of the problem and Z(EDD, k * ) = Cs ies3

Proposition 6: Let a>Mi for i = 1,2 . . . . . k and a<M~ for i = k + 1 . . . . . n then, If aCt/Mt>aCy/Mj then k* = Ct/M l is optimal solution of the problem and Z(EDD, k*) = aCl /M t I f aCt /Mt<aC/M] then k* = max (Ci -Crn) / (Mi -gm) is optimal solution

ie S 2, rueS 3

of the problem and Z(EDD, k*) = ( a - M y ) k * + Cy, where y is a job in S 3 such that the maximum k* is obtained.

Proposition 7: Let a > M i for i= 1 . . . . . k, a = M i for i = k + l . . . . ,m and a < M i for i = m + 1 . . . . . n then, the optimal solution to the problem can be found by the following rules.

(1) If aCj/Mj<_max{C s, aC/Mt} then

(a) k* = Ct/Mt and Z(EDD, k*) = a C / M l for Cs<_aCl/Mt

(b) k*e [max (C s - Ci)/(a -Mi) , Cs/a] and Z(EDD, k*) = C s i eS 2

for Cs> aCi /M t


(2) If aCj/Mj>max[Cs, aCJMt} then

(a) k* = max (Ci-Cm)/(Mi-Mm), Z(EDD, k*) = (a-My)k* +Cy ieS2,m~S 3

for U s <_ aCt~Mr

where y is a job in S a such that the maximum k* is obtained

(b) k* = max IieS2,maxmeS3 (Ci-Cm)/(Mi-Mm), max(Cs-Ci)/(a-Mi) 1 , i e s 3 and

Z (EDD, k*) = (a - My)k* + Cy for C s > aCt/M t

where y is a job in $3 such that the maximum k* is obtained.

Proof." If aCJMj<max{Cs, aCt~Mr}, it is obvious that graph of any job ieS3 does not have any role in finding optimal value of k*, therefore by applying the rules in Proposition (4) we can find an optimal solution of the problem. Now suppose aCj/Mj > max [Cs, aCl/Mt}. If Cs < aCt~Mr then any job i e $1 does not have any role in finding optimal value of k*, therefore by applying the rule in second part of proposition (6) we can find optimal value of k*. If C~> aCt/Mr (i.e., aCj/Mj> Cs>aCJMt) the heavy lines in Figure 1 stays on graph of some job m ~ $3 up to some point then it continues to stay on graph of some job i e $2 or the job s e $1. Since intersection of graph of a job m ~ $3 and graph of a job i e $2 occur at k * = (Ci-Cm)/(Mi-Mm), and intersection of graph of a job m ~ $3 and graph of the job s ~ $1 occur at k* = (C~- Ci)/(a -Mi) then the desired result follows immediately.

3 Algorithm for Optimal Solution of the Problem

Consider again the Figure 1 and the fact that we are only interested in k>_ 0. It is clear that the optimal value of k has an upper bound /~= max {Ci/Mi}. It

1<-i<_n

should also be clear that all jobs in $1 are irrelevant except the job s, and all jobs i ~ $3 with Ci/Mi<_ Cs/M s or Ci/Mi<_max Cp/Mp are irrelevant.

PeS 2

For a given value k ~ [0,/~], compute Z(EDD, k). Exactly one of the following is true.

(i) k = O, Z occurs on a job in S 3 and k is optimal,

340 B. Alidaee

(ii) Z occurs on job s or intersection of a job from Si and a job from Sp, p * i and k is optimal

(iii) Z only occurs on a job in $2 and optimal solution is to the right of k

(iv) Z only occurs on a job in $3 and optimal solution is to the left of k.

The following algorithm is based on (i)- (iv) given above and the results of earlier properties. For O<k<_l~, the algorithm checks the maximum values of Z(EDD, k) and find the optimal solution. The algorithm can easily be im- plemented in O(n) time [Preparata/Shamos: pp 279-289].

Algorithm:

Step 1: (Initialization) From the set of jobs keep the jobs S 2, the job s and the jobs from $3 where C i / M i > Cs/Ms or Ci/Mi>max C p / M p and number

P~S 2 these jobs in an non-decreasing order of Ms.

Step 2: For k = 0, compute Z(EDD, k), If Z occurs on a job in $3 then k = 0 is the optimal solution If Z occurs on the job s then [O, kR] is optimal where

kR = min limisn (Cs-Ci)/(a-Mi) , Cs/Ms 1 �9

Otherwise name the job where Z has occurred to be m and go to Step 3.

Step 3: Let kz = min{kl,kE, k3} where,

k 1 = ( C s - C m ) / ( a - M m )

k2 = min (Cm - Ci)/(Mm - Mi) ieS 3

k3= min (Cm-Ci)/(g,n-Mi) ieS 3 i:#m

If kL = kl then [kL, kR] is optimal where

kR = min (min (Cs- Ci)/(a -M.A, Cs/Ms) ~ ieS 3

If kL = k2 then k L is the optimal solution


If kL = k3, rename the job m to be the job where k3 has occurred and start over Step 3.

Acknowledgement: The author would like to thank the anonymous referee, whose comments and suggestions on earlier versions of this paper greatly improved the presentation of the paper. I also would like to thank Killgore Research Center at West Texas State University for the partial support of this research.

References

1. Alidaee B (1991) Optimal Assignment of Slack Due-Dates and Sequencing in a Single-Machine Shop, Appl Math Lett 4 :9 -11

2. Baker KR, Scudder GD (1990) Sequencing with Earliness and Tardiness Penalties: A Review. Operat Res 38:22-36

3. Cheng TCE (1989) Optimal Assignment of Slack Due-Dates and Sequencing in a Single-Machine Shop, Appl Math Lett Vol 2 No 4:333- 335

4. Cheng TCE, Gupta MC (1989) Survay of Scheduling Research Involving Due-date Determination Decisions. Eur J Operat Res 38:156- 166

5. Lawler EL (1973) Optimal Sequencing of a Single Machine Subject to Precedence Constraints, Management Sci 19:544-546

6. Preparrata PP, Shamos IS (1985) Computational Geometry. Springer Verlag, New York 7. Smith ML, Dudek RA, Blair EL (1986) In: Stecke KE, Suri R (Eds) Flexible Manufacturing

Systems. Characteristics of US flexible manufacturing system. - A survey. Elsevier, Amsterdam

Received April 1990 Revised version received June 1991

optimal assignment of nop due-dates and sequencing in a single machine shop

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