Parallel machine scheduling with batch delivery costs

*Correspondence address. Department of Management, TheHong Kong Polytechnic University, Kowloon, Hong Kong.

Int. J. Production Economics 68 (2000) 177}183

Parallel machine scheduling with batch delivery costs

Guoqing Wang!,",*, T.C. Edwin Cheng#

!Department of Business Administration, Jinan University, Guangzhou, People's Republic of China"Department of Management, The Hong Kong Polytechnic University, Kowloon, Hong Kong

#Ozce of the Vice President (Research and Postgraduate Studies), The Hong Kong Polytechnic University, Kowloon, Hong Kong

Received 12 May 1998; accepted 8 July 1999

Abstract

We consider a scheduling problem in which n independent and simultaneously available jobs are to be processed onm parallel machines. The jobs are delivered in batches and the delivery date of a batch is equal to the completion time ofthe last job in the batch. The delivery cost depends on the number of deliveries. The objective is to minimize the sum ofthe total #ow time and delivery cost. We "rst show that the problem is NP-complete in the ordinary sense even whenm"2, and NP-complete in the strong sense when m is arbitrary. Then we develop a dynamic programming algorithm tosolve the problem. The algorithm is pseudopolynomial when m is constant and the number of batches has a "xed upperbound. Finally, we identify two polynomially solvable cases by introducing their corresponding solutionmethods. ( 2000 Elsevier Science B.V. All rights reserved.

Keywords: Parallel machine scheduling; Batch delivery cost

1. Introduction

Batch scheduling, as combinations of sequencingand partitioning, has attracted much attention ofresearchers in recent years. Most of the results inbatch scheduling area fall into the followingthree categories: (i) item availability family schedu-ling problems, (ii) batch availability schedulingproblems, and (iii) batch processing schedulingproblems. The interested reader is referred to therecent reviews by Webster and Baker [1], Liaeeand Emmons [2], and Cheng et al. [3]

In this paper, we study a problem which falls intoa di!erent category of batch scheduling problems,

namely batch delivery problems. Batch deliveryproblems were "rst introduced by Cheng and Kahl-bacher [4]. In [4], they studied single machinebatch delivery scheduling to minimize the sum ofthe total weighted earliness and delivery costs. Theearliness of a job is de"ned as the di!erence be-tween its delivery date and completion time. Theyshowed that the problem is NP-complete in theordinary sense, and the equal weight case is poly-nomially solvable. Cheng and Gordon [5] pro-vided a dynamic programming algorithm to solvethe general problem. The algorithm is pseudo-polynomial when the number of batches has a "xedupper bound. They also provided a polynomialalgorithm to solve the common processing timecase. Cheng et al. [6] further showed that thisproblem can be formulated as a classical parallelmachine scheduling problem, thus the complexity

0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 0 5 - X

Fig. 1. Problem descriptions.

results and algorithms for the corresponding paral-lel machine scheduling problem can be easily ex-tended to the problem. Cheng et al. [7] studied thesingle machine batch delivery problem to minimizethe sum of the total weighted earliness and meanbatch delivery time.

Hermann and Lee [8] considered a single ma-chine batch delivery problem where all jobs havea common due date and the objective is to minim-ize the sum of the earliness and tardiness penaltiesand delivery costs of the tardy jobs. They provideda pseudopolynomial dynamic programming algo-rithm to solve the problem. Chen [9] showed that,when the common due date is a decision variable,the problem can be solved polynomially. Yuan [10]showed that the single machine batch deliveryproblem to minimize the sum of the total weightedearliness and tardiness and delivery costs of thetardy jobs is NP-complete in the strong sense.

While all prior batch delivery scheduling re-search focuses on the single machine environment,we study a parallel machine batch delivery schedul-ing problem in this paper. The problem can beformally stated as follows (see Fig. 1). There aren independent nonpreemptive jobs to be sequencedfor processing on m parallel identical machines, andpartitioned into several batches for delivery. Alljobs are available for processing at time zero. Alljobs in a batch are delivered to the customer to-gether. The batch delivery date is equal to thecompletion time of the last job in a batch. Thus the#ow time of a job is equal to the batch delivery dateon which it is delivered. The delivery cost is a non-decreasing function of the number of batch delive-

ries. The objective is to sequence and partition thejobs to minimize the sum of the total #ow time anddelivery cost.

Batch delivery is characteristic of many practicalsystems in which jobs are transported and ultimate-ly delivered in containers such as boxes or carts.For such systems, an important performance cri-terion is to minimize the work-in-process (WIP)inventories which are related to the total #ow time.As there are always some costs associated with eachdelivery, we obtain a situation which can bemodelled as the above batch delivery problem.

The rest of the paper is organized as follows. InSection 2, we introduce the notation to be used. InSection 3, we show that the problem under study isNP-complete and provide a dynamic programmingalgorithm to solve the problem optimally. In Sec-tion 4, we identify some polynomially solvablecases. Finally, we present our conclusions in the lastsection.

2. Notation

In this section, we introduce the notation to beused in the paper:

n : number of jobs;m : number of machines;J"MJ

1,2, J

nN : job set to be processed;

pi

: processing time of job Ji;

R : number of batch deliveries;a(R) : delivery cost function,

a nondecreasing functionof R;

178 G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177}183

Bl

: batch l;bl

: number of jobs in Bl;

Dl

: delivery date of Bl;

Ci

: completion time of job Ji;

Fi

: #ow tome of Ji, which is

equal to the batch deliverydate on which J

iis de-

livered;n"SB

1,2, B

RT : a schedule;

C.!9

"maxMCiN : makespan of a schedule;

G(n)"+ni/1

Fi#a(R) : total penalty of n.

Adopting the three-"eld notation introduced byGraham et al. [11], we denote our problem asPm/bd/(+F

i#a(R)).

3. NP-completeness and dynamic programming

In this section, we consider the complexity issuesof the problem. First, it is interesting to note that,when the delivery cost is negligible,Pm/bd/(+F

i#a(R)) simply reduces to the classical

parallel machine scheduling problem Pm//+Ci. It is

well known that Pm//+Ciis solved by the general-

ized shortest processing time (SPT) rule: schedulethe jobs in the order of nondecreasing processingtimes, and assign each job to the earliest availablemachine [12].

Next, we give a simple proof for the NP-com-pleteness for the problem under study. Assume thatthe batch delivery cost is so large that all jobs mustbe delivered in one batch. i.e. R"1. ThenPm/bd/(+F

i#a(R)) is equivalent to the classical

parallel machine scheduling problem Pm//C.!9

.Since Pm//C

.!9has been shown to be NP-com-

plete in the ordinary sense when m is "xed, andNP-complete in the strong sense when m is arbit-rary [13], we have the following theorem.

Theorem 1. Even when R"1, Pm/bd/(+Fi#a(R))

is NP-complete in the ordinary sense when m is xxedand m*2, and NP-complete in the strong sensewhen m is arbitrary.

The following lemma establishes several proper-ties for an optimal schedule for the problem.

Lemma 1. There exists an optimal schedulenH"SB

1,2,B

RT for Pm/bd, R);/(+F

i#a(R))

in which

(i) there is no idle time before each job;(ii) all jobs assigned to the same machine are sched-

uled in the SPT order;(iii) B

lcontains all jobs which xnish processing in the

time interval (Dl~1

,Dl], l"1,2,R.

Proof. (i) Trivial.(ii) Assume that jobs J

iand J

jare assigned to the

same machine and Jj

follows Jiimmediately such

that pi*p

jin nH. Let n@ be a schedule obtained by

swapping Ji

and Jj. It is easy to show that

G(nH)*G(n@), regardless of whether Ji

and Jj

aredelivered in the same batch or not.

(iii) Let us number all jobs in the order of theircompletion time in nH. Assume that the batches arenumbered in accordance with the numbers of theirlast jobs. It is clear that, without loss of generality,we can assume D

1(2(D

R. Let J

jbe the "rst

job in BR. If there are any jobs between J

jand J

n(the last job in B

R) which are assigned to other

batches, then there must be at least one batch, sayBl, such that C

j)D

l(D

R. Let n@ be a schedule

obtained by simply assigning Jjto B

l. It is obvious

that G(nH)*G(n@), a contradiction. Following thesame argument with the jobs in batch B

R~1and so

on, we can show that there exists an optimal sched-ule in which all batches consist of a number of jobswhich "nish processing contiguously. Since all jobswhich processing at D

l, l"1,2, R, can all be as-

signed to Bl

in any optimal schedule, we haveshown that B

lcontains all jobs which "nish pro-

cessing in the time interval (Dl~1

, Dl]. h

Based on Lemma 1, we can develop a dynamicprogramming algorithm to solve the problem. Let; be an upper bound for the number of batchdeliveries, and P"+n

i/1pi. The algorithm is for-

mally described as follows.Algorithm PMBD-1:

(a) Renumber the jobs in the SPT order, i.e.p1)p

22)p

n.

(b) De"ne HR(j, t

1,2, t

m,D

1,2,D

R) as the min-

imum total #ow time if we have scheduled jobsJ1

up to Jjsuch that the total processing time

G. Wang, T.C.E. Cheng / Int. J. Production Economics 68 (2000) 177}183 179

of the jobs assigned to machine u istu, u"1,2,m, and the delivery date is D

lfor

batch Bl, l"1,2, R.

(c) Recursive relations: For j"0,2, n, tu"0,2,

P , u " 1 , 2 , m , Dl"D

l~1# 1 , 2 , P ,

l"1,2, R, D0"0, and R"1,2,;,

HR( j, t

1,2, t

m, D

1,2, D

R)" min

1xuxm

MXuN, (1)

where

Xu"H

R( j!1, t

1,2, t

u!p

j,2, t

m,D

1,2, D

R)

#Fj; (2)

Fj"MD

lDD

l~1(t

u)D

l, D

0"0,

l"1,2, RN. (3)

(d) Initial conditions: For each tk"0,2,P,

u " 1 , 2 , m , Dl" D

l~1# 1 , 2 , P ,

l"1,2, R, D0"0, and R"1,2,;,

HR( j, t

1,2, t

m, D

l,2, D

R)

"G0 if j"0, t

1"t

2"2"t

m"0,

R otherwise.

(e) Optimal solution: GH"minMHR(n, t

1,2, t

m,

D1,2, D

R)#a(R)N over all t

u"0,2,P,

u"1,2, m, Dl"D

l~1#1,2, P, l"1,2,

R, D0"0, and R"1,2,;.

Lemma 2. Algorithm PMBD-1 solves the problemPm/bd, R);/+(F

i#a(R)) in O(nm;2PmÙ~1) time.

Proof. Due to Lemma 1, there exists an optimalschedule with jobs assigned to each machine in theSPT order, and each job in assigned to the "rstbatch after the completion of the job. If J

jis as-

signed to machine u, then Cj"t

u, and if

Dl~1

(tu)D

l, then J

jis assigned to B

l, and so

Fj"D

l. This justi"es Eqs. (2) and (3). Since

HR(j, t

1,2, t

m, D

1,2, D

R) is determined by the

minimum assignment by de"nition, we have justi-"ed the validity of the recursive relations. So thealgorithm PMBD-1 solves the problemPm/bd, R);/(+F

i#a(R)).

The time complexity of the algorithm can beestablished as follows. Since only m!1 of thevalues t

1,2, t

mare independent, the number of

di!erent states of the recursive relations is at most

nPmÙ~1 for R"1,2,;. For each state, the right-hand side of Eq. (1) can be calculated in O(m;)time. Thus, the overall computational complexityof Algorithm PMBD-1 is O(nm;2PmÙ~1). h

Lemma 2 implies that the problem Pm/bd,R);/(+F

i#a(R)) is not strongly NP-complete

for any constant m and ;. But it is not clearwhether Pm/bd,R);/(+F

i#a(R)) is strongly

NP-complete or pseudopolynomically solvable fora constant m and an arbitrary ;.

4. Polynomially solvable cases

In this section, we "rst consider a special casewhere the job assignment is predetermined. It isevident that the problem reduces to an optimalbatching problem in this case. This special casecharacterizes the practical scenario where each ma-chine is dedicated to a special group of jobs. Ac-cording to Lemma 1, we can provide a backwarddynamic programming algorithm to solve the opti-mal batching problem as follows.

Algorithm PMBD-2:

(a) Schedule the jobs on each machine in the SPTorder, and then renumber all the jobs in ac-cordance with the job completion times.

(b) De"ne HR(j, l) as the minimum total comple-

tion time of the jobs Jj,2, J

nwhen they are

assigned to the delivery batches Bl,2, B

R.

(c) Recursive relations: For R"1,2, n,l"R,2, 1, and j"n,2, l,

HR( j, l)" min

j:kxn`1

MHR(k, l#1)#(k!j)C

k~1N. (4)

(d) Initial conditions: For each j"1,2, nl"1,2, R, and R"1,2, n,

HR(j, l)"G

0 if j"n#1, and l"R#1;

R otherwise.

(e) Optimal solution:G(nH)" min

1xRxn

MHR(1, 1)#a(R)N.

While the optimality of the algorithm can be easilyjusti"ed, it is also not di$cult to see that the timecomplexity of the algorithm is O(n4).


Fig. 2. Example 1.

Now we present a numerical example for thespecial case to demonstrate the optimality of thealgorithm.

Example 1. Consider the instance with J"MJ

1,2, J

4N, m"2, and a(R)"7R. Assume that

J1

and J4

are assigned on M1,J

2and J

3are

assigned on M3, and all jobs are sequenced in the

SPT order on each machine (as shown in Fig. 2).Now using algorithm PMBD-2 to solve the in-stance, we have the following results:

When R"1, we have

H1(4, 1)"H

1(5, 2)#C

4"12,

H1(3, 1)"H

1(5, 2)#2C

4"24,

H1(2, 1)"H

1(5, 2)#3C

4"36,

H1(1, 1)"H

1(5, 2)#4C

4"48,

and so

G(nH1)"7R#H

1(1, 1)"55.

When R"2, we have

H2(4, 2)"H

2(5, 3)#C

4"12,

H2(3, 2)"H

2(5, 3)#2C

4"24,

H2(2, 2)"H

2(5, 3)#3C

4"36,

H2(1, 1)"minMH

2(4, 2)#3C

3,

H2(3, 2)#2C

2, H

2(2, 2)#C

3N"32,

and so

G(nH2)"7R#H

1(1, 1)"46.

When R"3, we have

H3(4, 3)"H

3(5, 4)#C

4"12,

H3(3, 3)"H

3(5, 4)#2C

4"24,

H3(2, 2)"minMH

3(4, 3)#2C

3,

H2(3, 3)#C

2N"28,

H3(1, 1)"H

3(2, 2)#C

1"31.

Thus

G(nH3)"7R#H

3(1, 1)"52.

When R"4, we have

H4(4, 4)"H

4(5, 5)#C

4"12,

H4(3, 3)"H

4(4, 4)#C

3"20,

H4(2, 2)"H

4(3, 3)#C

4"24,

H4(1, 1)"H

4(2, 2)#C

1"27,

and so

G(nH4)"7R#H

4(1, 1)"55.

Hence, we obtained an optimal schedule nH"SB

1, B

2T with B

1"MJ

1, J

2N and B

2"MJ

3,J

4N.

We now consider the special case with identicalprocessing times, Pm/bd, p

i"p/(+F

i#a(R)). Let

n0"vn/mw , g"n

0m!n. Let n

ube the number of

jobs processed on machine u under a speci"c sched-ule. Then we have the following lemma.

Lemma 3. There exists an optimal schedule forPm/bd, p

i"p/(+F

i#a(R)) in which n

0!1)

nu)n

0,u"1,2, m.

Proof. Suppose there exists an optimal schedulenH in which the condition is not satis"ed. Accord-ing to Lemma 1, we can assume that there is noinserted idle time in nH. Then there must be a pairof machines u and v such that n

u*n

v#2 and the

last job on machine u is also the last job of the lastbatch delivery. It is clear that moving the last jobon machine u to the last position on machine v willnot increase the total penalty. Repeating this pro-cess, we can obtain a desired optimal schedule. h

Let pHR"SB

1,2, B

RT be an optimal schedule

with R batch deliveries for the problem Pm/bd,pi"p/(+F

i#a(R)). Let bM

l"b

l/m, l"1,2, R!1.

We have

Lemma 4. There exists an optimal schedule withR batch deliveries for Pm/bd, p

i"p/(+F

i#a(R)) in

which bMl, l"1,2,R!1, is integral.


Proof. According to Lemma 1, we can assume thatin nH

Rthere is no inserted idle time and B

lcontains

all jobs which "nish processing in the time interval(D

l~1, D

l], l"1,2, R!1. It is evident that

(Dl!D

l~1)/p is integral, l"1,2,R. Since

bl"(D

l!D

l~1)m/p, l"1,2,R!1, bM

lis also in-

tegral. h

Now assume that nHR

satis"ed Lemmas 3 and 4.Let bM

l"vb

l/mw . We have

Lemma 5. There exists an optimal schedule withR batch deliveries for Pm/bd, p

i"p/(+F

i#a(R)) in

which DbMk!bM

lD)1 for any pair of k and l, where

k"1,2, R, l"1,2,R.

Proof. We "rst show that changing the sequence ofBl, l"1,2,R!1, will not cause any increase in

the total penalty. It is not di$cult to see that thesequencing of batch deliveries B

l, l"1,2, R!1,

is equivalent to the classical single machine totalweighted #ow time scheduling problem, denoted as1//+w

lC

l, with w

l"bM

lm and p

l"bM

lp. It is well

known that the total weighted #ow time is mini-mized by sequencing the jobs in the weighted shor-test processing time (WSPT) order [14]. Sincep1/w

1"2"p

R~1/w

R~1"p/m, B

i, l"1,2, R

!1, can be sequenced in an arbitrary order.Now, we can prove the lemma by showing

DbMl~1

!bMlD)1, l"2,2,R. (5)

We assume that bMl'1, l"2,2,R. By de"ni-

tion, we have

Dl~1

bl~1

#Dlbl)(D

l~1!p)(b

l~1!m)

# (Dl#p)(b

l#m), (6)

Dl~1

bl~1

#Dlbl)(D

l~1#p)(b

l~1#m)

# (Dl!p)(b

l!m). (7)

Then we can easily obtain the desired results.Now, assume that there are some batches such

that bMl"1. Note that b

lmay be less than m in this

case. Since it is trivial when bMl~1

"1 (or bMl`1

"1),we suppose bM

l~1'1 (or bM

l`1'1). From (6) (or

(7)), we can easily show that bMl~1

)2 or (bMl`1

)2),and thus (5) holds again.

This completes the proof. h

Based on these results, we can easily construct anoptimal schedule with R batch deliveriesnHR"SB

1,2,B

RT such that

bMl"bM

0!1, l"1,2, h,

bMl"bM

0, l"h#1,2, R,

where bM0"vn

0/Rw and h"bM

0R!n

0. The asso-

ciated total penalty can be calculated as

G(nHR)"a(R)#

h+l/1

lmp(bM0!1)2

#h(bM0!1)p ) bM

0(R!h)m

#

R~h+l/1

lmpbM 20!gn

0p )

"a(R)#(RbM0!h#bM

0)(RbM

0!h)mp/2

!h(bM0!1)mp/2!gn

0p.

Now, we can construct a simple algorithm tosolve the problem as follows.

Algorithm PMBD-3:1. n

0:"vn/mw ; g :"n

0m!n; RH :"0; GH :"R;

2. for R"1 to n0

dobegin

bM0

:"vn0/Rw ; h :"bM

0R!n

0;

G(nHR) :"a(R)#(RbM

0!h#bM

0)

(RbM0!h)mp/2!h(bM

0!1)mp/2!gn

0p;

if GH'G(nHR) then RH :"R; GH :"G(nH

R);

endend

It is clear that Algorithm PMBD-3 solves the prob-lem Pm/bd, p

i"p/(+F

i#a(R)) in O(n/m) time. It

should be pointed out that, the algorithm, althoughe$cient, is not actually polynomial when each batchdelivery has an equal delivery cost c, i.e. a(R)"cR.

The following numerical example demonstratesthe optimality of the algorithm.

Example 2. Consider the instance with J"MJ1,

2, J7N, m"2, a(R)"7R, p

i"p"2, i"1,2, 7.

From Lemma 3, we know that there exists anoptimal schedule in which all jobs are sequenced asshown in Fig. 3. It is clear that n

0"4, g"1. Using

algorithm PMBD-3 to solve the instance, we havethe following results.

When R"1, we have bM0"4 and h"0, and so

G(nH1)"7R#28p"63.


Fig. 3. Example 2.


G(nH2)"7R#20p"54.


G(nH3)"7R#18p"57.


G(nH4)"7R#16p"60.

Now we can obtain an optimal schedulenH"SB

1, B

2T with B

1"MJ

1, J

2,J

3,J

4,N and

B2"MJ

5, J

6, J

7N.

5. Conclusions

In this paper, we have studied the parallel ma-chine scheduling with batch delivery costs. We haveshown that the problem to minimize the sum of thetotal #ow time and delivery cost is NP-complete inthe strong sense. We have then provided a dynamicprogramming algorithm to solve the problem. Thealgorithm is pseudopolynomial when the numberof machines is constant and the number of batcheshas a "xed upper bound. We also have providedtwo polynomial time algorithms to solve the specialcases where the job assignment is given or the jobprocessing times are equal.

There are a number of issues which are of interestfor further research. First, it is interesting to investi-gate the open problem posed in the paper, i.e.whether it is pseudopolynomially solvable or strong-ly NP-complete when the number of machines isconstant and the number of batches is arbitrary. It isalso interesting to investigate polynomial time algo-rithms for the special case where the job processingtimes are equal and the batch delivery cost functionis linear. Another interesting issue is to develope!ective heuristics to solve the general problem, andit is evident that a viable strategy is to combine thelist scheduling procedure for the classical parallelmachine scheduling problems [15] with the optimalbatching algorithm proposed in this paper.

Acknowledgements

This research was supported in part by TheHong Kong Polytechnic University under grantnumber 350/239.

References

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[3] T.C.E. Cheng, J.N.D. Gupta, G. Wang, A review of #ow-shop scheduling research with setup times, Production andOperations Management (1999), to appear.

[4] T.C.E. Cheng, H.G. Kahlbacher, Scheduling with deliveryand earliness penalties, Asia-Paci"c Journal of Opera-tional Research 10 (1993) 145}152.

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[10] J. Yuan, A note on the complexity of single-machine sched-uling with a common due date, earliness}tardiness, andbatch delivery costs, European Journal of OperationalResearch 94 (1996) 203}205.

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[15] R.L. Graham, Bounds for certain multiprocessinganomalies, Bell System Technical Journal 45 (1996)1563}1583.


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Parallel machine scheduling with batch delivery costs