Un-restricted Common Due-Date Problem with Controllable Processing

Times

Linear Algorithm for a Given Job Sequence

Abhishek Awasthi

, J

org L

assig

and Oliver Kramer

Department of Computer Science, University of Applied Sciences Zittau/G

orlitz, G

orlitz, Germany

Department of Computing Science, Carl von Ossietzky University of Oldenburg, Oldenburg, Germany

Keywords:

Scheduling, Algorithms, Simulated Annealing, NP-Hard.

Abstract:

This paper considers the un-restricted case of the Common Due-Date (CDD) problem with controllable pro-

cessing times. The problem consists of scheduling jobs with controllable processing times on a single machine

against a common due-date to minimize the overall earliness/tardiness and the compression penalties of the

jobs. The objective of the problem is to ﬁnd the processing sequence of jobs, the optimal reduction in the

processing times of the jobs and their completion times. In this work, we ﬁrst present and prove an essential

property for the controllable processing time CDD problem for the un-restricted case along with an exact lin-

ear algorithm for optimizing a given job sequence for a single machine with a run-time complexity of O(n),

where n is the number of jobs. Henceforth, we implement our polynomial algorithm in conjunction with a

modiﬁed Simulated Annealing (SA) algorithm and Threshold Accepting (TA) to obtain the optimal/best pro-

cessing sequence while comparing the two heuristic approaches, as well. The implementation is carried out

on appended CDD benchmark instances provided in the OR-library.

1 INTRODUCTION

The Common Due-Date scheduling problem involves

sequencing and scheduling of jobs over machine(s)

against a common due-date. Each job possesses a pro-

cessing time and different penalties per unit time in

case the job is completed before or after the due-date.

For the controllable processing time case, in addition

to the CDD, the processing times of some or all the

jobs can be reduced to a certain minimum value at a

cost of some penalty per unit of reduction. This con-

trolling of the processing times can help the jobs to

reduce their earliness/tardiness penalties if the penal-

ties incurred due to the compressions are relatively

smaller than the earliness/tardiness penalties. The ob-

jective of solving the problem is to obtain the optimal

job sequence, ﬁnal processing times of the jobs and

the completion times of all the jobs to minimize the

total weighted penalty. Generally speaking, there are

two classes of common due-date problem, which have

proven to be NP-hard, namely the restrictive and the

un-restrictive CDD problem. In this work, we con-

sider the un-restrictive case of the problem, where the

common due-date is greater than or equal to the sum

of the processing times of all the jobs and each job

possesses different penalties. The CDD has already

been proven to be NP-hard, and clearly the control-

lable case is NP-hard as well (Yunqiang et al., 2013;

Biskup and Feldmann, 2001). For the controllable

processing times common due-date problem, (Biskup

and Cheng, 1999) studied it with constant penalties

for earliness/tardiness and distinct penalties for com-

pression. They also considered the penalty for the

completion time of the jobs and proved the similar-

ity of the problem to the assignment problem. (Shab-

tay and Steiner, 2007) made an extensive survey for

scheduling with controllable processing times, cov-

ering research in this ﬁeld from the last 25 years.

(Wan, 2007) studied the common due window prob-

lem with controllable processing times with constant

earliness/tardiness penalties and distinct compression

costs and discussed some properties for the optimal

solution along with a polynomial algorithm for solv-

ing the problem. (Tseng et al., 2009) studied the

general CDD problem with compressible processing

times with different due-dates and presented a heuris-

tic algorithm to minimize the total tardiness and the

compression penalties. (Nearchou, 2010) studied a

slightly different version of the problem where the

objective was to minimize the total weighted comple-

526

Awasthi A., Lässig J. and Kramer O..

Un-restricted Common Due-Date Problem with Controllable Processing Times - Linear Algorithm for a Given Job Sequence.

DOI: 10.5220/0005398205260534

In Proceedings of the 17th International Conference on Enterprise Information Systems (ICEIS-2015), pages 526-534

ISBN: 978-989-758-096-3

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

tion times and the compression costs and presented

a population based metaheuristic algorithm, consider-

ing four different heuristic approaches namely, differ-

ential evolution, particle swarm optimization, genetic

algorithms and evolution strategies. We would em-

phasize here that the CDD problem with controllable

processing times for a ﬁxed due-date has not been

studied before for the asymmetric penalties. In this

paper, we consider the single machine case for the

un-restricted CDD problem with asymmetric penal-

ties and controllable processing times with distinct

linear costs. We make a theoretical study of the prob-

lem and ﬁrst present an important property for this

problem. We then propose an O(n) exact polynomial

algorithm to optimize a given job sequence on a single

machine.

2 PROBLEM FORMULATION

In this Section, we present the mathematical notation

of the common due-date problem with the control-

lable processing times. Let n be the number of jobs

and d be the common due-date. Besides, p

and m

be the actual and minimum processing times, respec-

tively with x

being the actual reduction in the pro-

cessing time for job i. Let, E

and C

be the ear-

liness, tardiness and completion time of job i. And

,β

and γ

be the earliness, tardiness and compres-

sion penalties per time unit for any job i. Clearly, E

and T

can be expressed as E

= max{0,d −C

} and

= max{0,C

−d} for i = 1,2,...,n. Then, the ob-

jective function of the problem can be expressed as,

min

∑

i=1

·α

+ T

·β

+ x

·γ

) . (1)

−−−− −−−

t = 0

= d

Figure 1: Assume that the rth job ﬁnishes at the due-date d

in the optimal solution.

We now present some properties for both the CDD

and CDD with controllable processing times. Let the

solution value for the case when there is no compres-

sion of the processing times and the due-date lies at

the completion time of job r, as shown in Figure 1, be

Sol

, then we have

Sol

r−1

∑

i=1

∑

j=i+1

∑

i=r+1

∑

j=r+1

, (2)

where

r−1

∑

i=1

∑

j=i+1

= the total earliness for any job i and

∑

i=r+1

∑

j=r+1

= the total tardiness for any job i.

Let us assume that the reductions in the process-

ing times in the optimal schedule be x

for all i =

1,2,...,n. Then the objective function value (Sol

)

when the due-date position is at C

will be

Sol

r−1

∑

i=1

∑

j=i+1

−x

)α

∑

i=r+1

∑

j=r+1

−x

)β

∑

j=1

(3)

We ﬁrst present and prove an important property re-

garding the amount of compression of the processing

times of the jobs.

Property 1. If controlling the processing times

fetches a better solution, then the compression of the

processing times should be to their minimum value.

Proof. If the compression of the processing times

fetches a better solution, then we have Sol

≤ Sol

Using Equation (2) and (3), we obtain

r−1

∑

i=1

∑

j=i+1

∑

i=r+1

∑

j=r+1

−

∑

j=1

≥ 0 . (4)

Let us assume that instead of reducing the process-

ing times by x

, we reduce them by y

, where y

< x

∀j = 1,2,...,. Let the solution value for this case be

Sol

and x

−y

= δ

. If Sol

< Sol

, then with some

manipulation of the terms we get

r−1

∑

i=1

∑

j=i+1

∑

i=r+1

∑

j=r+1

−

∑

j=1

≤ 0 . (5)

Since x

≥0 ∀j = 1,2,...,n, Equation (4) should also

hold for any δ

> 0. However, Equation (5) is a con-

tradiction. Hence, our assumption that Sol

< Sol

wrong. This proves that the solution value only im-

proves if we reduce the processing times furthermore,

which in turn shows that the best solution value will

be obtained for maximum possible compression of the

processing times.

Let the solution value for the CDD when the (r + 1)th

job ends at the due-date be Sol

r+1

as shown in Fig-

ure 2, then

Sol

r+1

∑

i=1

r+1

∑

j=i+1

∑

i=r+2

∑

j=r+2

(6)

Un-restrictedCommonDue-DateProblemwithControllableProcessingTimes-LinearAlgorithmforaGivenJob

Sequence

527

−−−− −−−

t = 0

r+1

= d

Figure 2: Schedule with the completion time of job r + 1

lying at the due-date, C

r+1

= d.

For the optimal reductions x

in the processing times,

the solution value Sol

r+1

will be

Sol

r+1

∑

i=1

r+1

∑

j=i+1

−x

)

∑

j=1

∑

i=r+2

∑

j=r+2

−x

)

(7)

−−−− −−−

t = 0

r−1

= d

Figure 3: Schedule with the completion time of job r −1

lying at the due-date, C

r−1

= d.

Let the solution value for CDD when the (r −1)th job

ends at the due-date be Sol

r−1

(Figure 3), then

Sol

r−1

r−2

∑

i=1

r−1

∑

j=i+1

∑

i=r

∑

j=r

. (8)

As earlier, for a reduction of x

in the processing

times, the solution value Sol

r−1

will be

Sol

r−1

r−2

∑

i=1

r−1

∑

j=i+1

−x

)

∑

j=1

∑

i=r

∑

j=r

−x

)

(9)

Since we know Sol

is optimal we have

Sol

≤ Sol

r+1

and (10)

Sol

≤ Sol

r−1

. (11)

Rearranging the terms in Sol

,Sol

r+1

and Sol

r−1

from

Equations (2), (6) and (8) respectively, we get

Sol

≤Sol

r+1

∑

i=r+1

≤

∑

i=1

(12)

and

Sol

≤Sol

r−1

∑

i=1

≤

∑

i=r

(13)

Equation (12) also implies that

∑

i=k+1

≤

∑

i=1

,k = r,r + 1, . . . , n −1, (14)

that is, if the sum the tardiness penalties for the jobs

r + 1 to n is less than the sum of the earliness penal-

ties for the jobs from 1 to r, then the inequality also

holds for any k ≥ r, since β

> 0 and α

> 0 for

i = 1,2,...,n.

Likewise, Equation (13) also implies that

k−1

∑

i=1

≤

∑

i=k

,k = 1, 2, . . . , r, (15)

that is, if the sum the earliness penalties for the jobs

1 to r −1 is less than the sum of the tardiness penal-

ties for the jobs from r to n, then the inequality also

holds for any k ≤ r, since β

> 0 and α

> 0 for

i = 1, 2, . . . , n. This proves that the difference of the

sum of the earliness penalties and the sum of the tar-

diness penalties changes sign before and after the op-

timal position of the due-date. The idea for Equa-

tion (14) and (15) has been adopted from (Cheng,

1989), where these properties were derived for a spe-

cial case of the CDD problem.

We now use this property of the CDD prove an es-

sential property for the un-restricted case of the CDD

with controllable processing times.

Theorem 1. If the due-date position in the optimal

schedule of un-restricted case of the CDD lies at the

completion time of some job r, then its position re-

mains unchanged for the controllable case of the un-

restricted CDD problem.

Proof. Refer to the Appendix 1.

3 THE EXACT ALGORITHM

In the previous section we proved that if the due-date

position for the general common due-date problem

lies at the completion time of a job then its position re-

mains unchanged for the controllable processing time

case as well. We now present how to utilize this prop-

erty to formulate an exact algorithm to optimize a

given job sequence for the un-restricted of the CDD

with controllable processing times.

ICEIS2015-17thInternationalConferenceonEnterpriseInformationSystems

528

Algorithm 1: Linear Algorithm to optimize any given

sequence of the un-restricted CDD with controllable

processing times.

1 C

←

∑

k=1

∀ i = 1,2, . . . , n

2 τ ← argmax

i=1,2,...,n

≤ d)

3 if (τ 6= 0) then

4 pes ←

∑

i=1

5 pls ←

∑

i=τ+1

6 if (C

≤ d) ∧(pls < pes) then

7 C

←C

−C

+ d ∀ i

8 while (τ > 0) ∧(pls < pes) do

9 pes ← pes −β

10 pls ← pls + α

11 t ← 1

12 τ ← τ −1

13 if (t = 1) then

14 C

←C

−C

τ+1

+ d ∀ i

15 σ ← argmin

i=1,2,...,n

> d)

16 pls ←

∑

i=σ

17 i ←σ

18 lShi f t ← 0

19 while (i ≤n) do

20 if (γ

≤ pls) ∧(p

> m

) then

21 dec ← p

−m

22 lShi f t ← lShi f t + dec

23 pls ← pls −β

24 C

←C

−lShi f t

25 i ← i + 1

26 σ ←σ −1

27 ple ←

∑

σ−1

i=1

28 i ←σ

29 rShi f t ← 0

30 while (i > 1) do

31 if (γ

≤ ple) ∧(p

> m

) then

32 inc ← p

−m

33 rShi f t ← rShi f t + inc

34 ple ← ple −α

i−1

35 C

i−1

←C

i−1

+ rShi f t

36 i ← i −1

37 PT

← p

38 PT

←C

−C

i−1

, i = 2,3, . . . , n

39 Compute E

, i = 1,2, 3, . . . , n

40 return

∑

i=1

·α

+ T

·β

+ γ

−PT

))

To optimize a given sequence for the un-restricted

case, we ﬁrst ﬁnd the optimal position of the due-

date for the uncompressed then reduce the process-

ing times of the jobs closest to the due-date mov-

ing outward. Consider Figure 1, the optimal posi-

tion of the due-date is at C

. In the next step, we

ﬁrst reduce the processing times of tardy jobs start-

ing with job r + 1. Reducing its processing time such

that C

r+1

moves closer to d will not only reduce the

tardiness of job r + 1 but of all the jobs which fol-

low, provided the penalty incurred by compressing

the processing time of job r + 1 is less than the re-

duction in the weighted tardiness penalties of the jobs

r + 1,r + 2,...,n. Thereafter, we compress job r + 2

and reduce its tardiness along with all the jobs follow-

ing it. If a compression does not offer any reduction

in the overall penalty then we move on to the next job

without compressing the current job.

We perform the same operations in the sequential

manner for the remaining jobs, starting with job r to

job 2. However, in this case the reduction in the pro-

cessing times leads the jobs to move towards right,

i.e., closer to the due-date. Notice that the reduction

in the ﬁrst job is never going to improve the penalty

since the earliness of the ﬁrst job will not be altered

by its compression but will only offer more penalty

due to compression. Algorithm 1 presents the pseudo

code for optimizing a given sequence.

4 PROOF OF OPTIMALITY

We now provide the proof of the optimality of Algo-

rithm 1 with respect to the solution value. Recall that

we consider the un-restricted case of the CDD.

Theorem 2. Algorithm 1 returns the optimal solution

value to the un-restricted case of the CDD with con-

trollable processing times for any given job sequence

with a run-time complexity of O(n).

Proof. Since there is only one way that the due-date

position may be between the completion times of two

consecutive jobs, we need to ﬁrst calculate the sum of

penalties before and after the due-date such that the

ﬁrst job starts at time zero and all the jobs follow with-

out any idle time. The schedule with t

∗

= 0 will be

optimal if the sum of the tardiness penalties is already

greater than the sum of earliness penalties. If that is

not the case, we shift all the jobs towards right, as long

as the sum of the tardiness penalties of jobs ﬁnishing

after the due-date is less than or equal to the some

of the earliness penalties of all the jobs which com-

plete before the due-date, according to Equation (14)

and (15).

Hence, we ﬁrst optimize any given sequence for

the general CDD problem and obtain the due-date po-

sition. We have already proved in Theorem 1 that

the due-date position for the general CDD and the

controllable processing times cases will be unaltered

Un-restrictedCommonDue-DateProblemwithControllableProcessingTimes-LinearAlgorithmforaGivenJob

Sequence

529

Table 1: Results obtained for single machine common due-date problem with controllable processing times. For any given

number of jobs, there are 10 different instances provided and each instance is designated a number k.

Jobs 10 20 50 100

Approach SA TA SA TA SA TA SA TA

k = 1 763 763 2576 2642 14698 14605 60156 60124

k = 2 598 598 2555 2555 11864 11890 50534 50357

k = 3 672 672 3137 3111 13840 13774 57524 57483

k = 4 757 757 2761 2761 11925 11867 60729 60837

k = 5 473 473 1949 1936 12446 12373 46217 45999

k = 6 669 669 2767 2767 12252 12194 52097 52012

k = 7 913 913 3124 3124 14848 14848 53729 53742

k = 8 497 497 1492 1492 17598 17597 68199 68044

k = 9 510 510 1774 1760 11871 11864 48789 48747

k =10 601 601 1824 1824 11856 11841 51056 51003

for the un-restricted case of the CDD. It is clear that

the optimal solution will occur only if the jobs are

brought closer to the due-date since the due-date po-

sition should not change and the best case would be

the one when all the jobs ﬁnish at the due-date, which

is impossible. Hence, we reduce the processing times

of jobs starting from the most adjacent one to the due-

date moving further away. The processing time of a

job is reduced only if the penalty incurred due to com-

pression is less than the penalty reduced by the re-

duction in the earliness (tardiness) of the jobs before

(after) it.

As for the run-time complexity, the ﬁrst part of

Algorithm 1 is to optimize a given sequence for the

un-restricted CDD problem to ﬁnd the optimal posi-

tion of the due-date. It can be easily observed that

the complexity for this part is of linear run-time. The

next expensive operations in terms of the run-time oc-

cur at the next two while loops and they are both of

O(n) in the worst case. The remaining steps are all

linear. Hence the overall complexity of Algorithm 1

is O(n).

5 RESULTS

Due to the unavailability of benchmark instances

for this problem as per our knowledge, we ﬁrst

present our methodology to append the benchmark

instances of the general CDD provided in the OR-

library with the extra parameters for the control-

lable processing time case (Beasley, 1990). The in-

stances provided in (Beasley, 1990) provide the pro-

cessing times, earliness/tardiness penalties and the

due-date. Hence, we append the information about

the minimum processing times and the cost of con-

trolling the processing times per unit time. We take

the minimum processing time of any job as m

∼

DU(0.6p

, p

) and γ

∼ DU(1,5), where ∼ DU (a,b)

is a discrete uniform random number between a and

b. The rest of the parameters remain the same as

in (Beasley, 1990). Our updated benchmark in-

stances can be found at www.enterprise-application-

development.org/research/benchmark-data.html.

5.1 Modiﬁed Simulated Annealing

We use a modiﬁed Simulated Annealing algorithm

to generate job sequences and Algorithm 1 to opti-

mize each sequence to its minimum penalty. Our

experiments over all the instances suggest that an

ensemble size (M) of ≈ n/10 and the maximum

number of iterations of 500n, where n is the num-

ber of jobs, work best for the provided instances

in general. The runtime for all the results is the

time after which the solutions mentioned in Table 1

are obtained on average after 10 different replica-

tions. The initial temperature (T

) is kept as twice

the standard deviation of the energy at inﬁnite tem-

perature: σ

T =∞

−hEi

T =∞

. We esti-

mate this quantity by randomly sampling the conﬁg-

uration space (Salamon et al., 2002). An exponential

schedule for cooling is adopted with a cooling rate of

1 −10

−4

. One of the modiﬁcations from the standard

SA is in the acceptance criterion. We implement two

acceptance criteria: the Metropolis acceptance prob-

ability, min{1,exp((−4E)/T )} and a constant ac-

ceptance probability of c ·10

−2

, c being a constant. A

solution is accepted with this constant probability if

it is rejected by the Metropolis criterion. This con-

cept of a constant probability is useful when the SA

is run for many iterations and the metropolis accep-

tance probability is almost zero, since the temperature

would become inﬁnitesimally small.

Another modiﬁcation is the increase in the tem-

perature after the annealing temperature becomes less

than one unit. In such a case we increase the temper-

ature to 1/10th of the initial temperature. Apart from

this, we also incorporate elitism in our modiﬁed SA.

Elitism has been successfully adopted in evolutionary

ICEIS2015-17thInternationalConferenceonEnterpriseInformationSystems

530

Table 2: Table 1 continued...

Jobs 200 500 1000

Approach SA TA SA TA SA TA

k = 1 205307 205088 1324829 1325310 5354091 5460812

k = 2 224213 224136 1425334 1426581 5138818 5259803

k = 3 215669 215528 1371225 1370031 5051903 5165823

k = 4 243170 242875 1372239 1371316 5128463 5256762

k = 5 214962 214936 1211943 1212666 5278280 5393347

k = 6 199305 199157 1189069 1189804 5073112 5185676

k = 7 210827 210815 1357730 1357399 5514919 5633111

k = 8 190068 189791 1287052 1286917 5099539 5204219

k = 9 215707 215556 1396230 1397012 5215425 5375118

k = 10 228356 228071 1264289 1264434 5187998 5292440

algorithms for solving complex optimization prob-

lems (Gen et al., 1994; Kim, 2013). L

assig and Sud-

holt made theoretical studies analysing speed-ups in

parallel evolutionary algorithms with elitism applied

to combinatorial optimization problems (L

assig and

Sudholt, 2011; L

assig and Sudholt, 2013). In (L

assig

and Hoffmann, 2009) it is shown that for a large class

of quality measures, the best possible probability dis-

tribution is a rectangular distribution over certain in-

dividuals sorted by their objective values, which can

be seen as a mild form of elitism. As for the pertur-

bation rule, we ﬁrst randomly select a certain number

of jobs in any job sequence and permute them ran-

domly to create a new sequence. The number of jobs

selected for this permutation is taken as c +b

n/10c,

where n is the number of jobs and c is a constant. This

modiﬁed SA works well for the CDD and the Air-

craft Landing Problem as well (Awasthi et al., 2013b;

Awasthi et al., 2013a).

5.2 Threshold Accepting

Threshold Accepting is another heuristic algorithm

based on Simulated Annealing, proposed by (Dueck

and Scheuer, 1990). The basic difference from the

SA is the different acceptance rules. Unlike the stan-

dard SA where a worse solution is accepted as per the

metropolis acceptance criterion, TA instead accepts

’every new conﬁguration which is not much worse

than the old one’ (Dueck and Scheuer, 1990). We

now present the exact details of the acceptance crite-

rion adopted by us in this work.

The initial temperature (T

) is kept the same as in

Simulated Annealing. As opposed to the exponential

cooling schedule of SA, we adopt probabilistic arith-

metic cooling scheduling in TA. Let, mE

and mE

j−1

be the mean of the energy (in this optimization prob-

lem, energy is the objective function values) of all

the states in the current ( j) and the previous iteration

( j −1), respectively. Then, the temperature T

is re-

duced as

Table 3: Average run-time in seconds for all the 10 different

instances for each job over 10 iterations of the algorithm.

Jobs SA TA

10 0.173 0.490

20 1.147 0.931

50 6.017 7.768

100 13.510 21.637

200 50.973 88.822

500 171.806 221.410

1000 361.553 603.711

(

j−1

−δ, if mE

−mE

j−1

≤ prob ,

j−1

, otherwise .

(16)

In Equation (16), δ and prob are deﬁned as δ = τ ·

and prob = v ·T

√

M, where τ = c ·10

−1

and v =

c·10

−4

, c being a constant and M is the ensemble size.

The acceptance criterion for Threshold Accepting as

proposed by (Dueck and Scheuer, 1990) is the current

temperature T

. The remaining parameters such as the

ensemble size, perturbation size and the number of

iterations are kept the same for both SA and TA, so as

to exactly compare the two approaches.

In Table 1 and 2, we present our results for the

CDD with controllable processing times for the un-

restricted case, for the benchmark instances. For the

ﬁrst 10 instances with 10 jobs, SA and TA both reach

the optimal solution as it turns out by the compari-

son results with integer programming. For the small

instances with 20,50 and 100 jobs, we observed that

the Threshold Acceptance offers predominantly bet-

ter results in terms of the objective values as shown

in Table 1. TA achieves better results for 17 and

equal results for 8 instances out of 30 instances. How-

ever, for the large instances with 200,500 and 1000

jobs, the difference in the two heuristic approach was

marginal with 16 better results for SA and 14 for TA,

as shown in Table 2. Hence, as far as the quality of

the solution is concerned, Threshold Accepting and

Simulated Annealing offer almost the same quality of

results. This was not the case for the run-times of

the two approaches. In Table 3 we can see the time

Un-restrictedCommonDue-DateProblemwithControllableProcessingTimes-LinearAlgorithmforaGivenJob

Sequence

531

required by both the approaches. The run-times of

TA and SA are almost the same for small instances

till 50 jobs. However, for large instances with 100

and more jobs, Simulated Annealing performs much

faster then the Threshold Accepting. Considering, the

same quality of solution values for both, in our ex-

perience the modiﬁed Simulated Annealing algorithm

proposed in this work performs better than Threshold

Accepting for the optimization problem dealt with.

6 CONCLUSION AND FUTURE

DIRECTION

In this paper we present a novel property for the prob-

lem of scheduling against a common due-date with

controllable processing times for the un-restricted

case. We show that the due-date position in the op-

timal schedule for the un-restricted case remains the

same for both the CDD and for controllable process-

ing time cases. We then present an O(n) algorithm

for a given sequence and prove the run-time com-

plexity and its optimality with respect to the solution

value. We implement our algorithm over the bench-

mark instances provided by (Biskup and Feldmann,

2001) and appended by us for all the instances till 10

jobs. Besides, we encourage other researchers inter-

ested in this problem to test other approaches with our

benchmark instances.

ACKNOWLEDGEMENTS

The research project was promoted and funded by the

European Union and the Free State of Saxony, Ger-

many.

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APPENDIX 1

Proof of Theorem 1

Proof. Since there is only one way that the due-date

position may be between the completion times of two

consecutive jobs (i.e., when the ﬁrst job starts at time

t = 0), we need to ﬁrst calculate the sum of penal-

ties before and after the due-date such that the ﬁrst

job starts at time zero and all the jobs follow without

any idle time. Thereafter, we shift all the jobs towards

right as long as the sum of the tardiness of jobs ﬁnish-

ing after the due-date is less than or equal to the some

of the earliness penalties of all the jobs which com-

plete before the due-date. Rearranging the terms of

Sol

from Equation (3), we have

Sol

r−1

∑

i=1

∑

j=i+1

−x

)α

∑

j=1

∑

i=r+1

∑

j=r+1

−x

)β

Sol

r−1

∑

i=1

∑

j=i+1

−

r−1

∑

i=1

∑

j=i+1

−

∑

i=r+2

∑

j=r+2

∑

j=1

∑

i=r+1

r+1

−x

r+1

)β

∑

i=r+2

∑

j=r+2

(17)

With some manipulations of terms, Sol

can be also

written as

Sol

r−2

∑

i=1

r−1

∑

j=i+1

−

r−2

∑

i=1

r−1

∑

j=i+1

r−1

∑

i=1

−x

)α

∑

j=1

∑

i=r+1

∑

j=r+1

−

∑

i=r+1

∑

j=r+1

(18)

Similarly, the terms of Sol

r+1

in Equation (7) can re-

arranged such that

Sol

r+1

∑

i=1

r+1

∑

j=i+1

−x

)α

∑

i=r+2

∑

j=r+2

−x

)β

∑

j=1

r−1

∑

i=1

∑

j=i+1

−

r−1

∑

i=1

∑

j=i+1

∑

i=r+2

∑

j=r+2

∑

i=1

r+1

−x

r+1

)α

−

∑

i=r+2

∑

j=r+2

∑

j=1

(19)

Likewise Sol

r−1

in Equation (9) can also be expressed

Sol

r−1

r−2

∑

i=1

r−1

∑

j=i+1

−x

)α

∑

i=r

∑

j=r

−x

)β

∑

j=1

r−2

∑

i=1

r−1

∑

j=i+1

−

r−2

∑

i=1

r−1

∑

j=i+1

∑

i=r+1

∑

j=r+1

∑

i=r

−x

)β

−

∑

i=r+1

∑

j=r+1

∑

j=1

(20)

Now we prove by contradiction that the position of

the due-date will not change even after the optimal

reduction in the processing times. Let us assume that

Sol

is not optimal, then we have

Sol

> Sol

r+1

and (21)

Sol

> Sol

r−1

. (22)

Substituting values of Sol

from Equation (17) and

Sol

r+1

from Equation (19) in Equation (21), we have

Sol

>Sol

r+1

∑

i=r+1

r+1

−x

r+1

)β

∑

i=1

r+1

−x

r+1

)α

and

r+1

−x

r+1

)

∑

i=r+1

−

∑

i=1

> 0 . (23)

Now, using Equation (12), we obtain

r+1

< x

r+1

. (24)

Likewise, substituting the values of Sol

from Equa-

tion (18) and Sol

r−1

from Equation (20) in Equa-

tion (22), we get

−x

)

r−1

∑

i=1

−

∑

i=r

> 0 . (25)

Equation (13) then fetches

< x

. (26)

Equation (24) and (26) show that if the optimal so-

lution for the uncompressed case occurs such when

the due-date position is at C

for some r, then for the

Un-restrictedCommonDue-DateProblemwithControllableProcessingTimes-LinearAlgorithmforaGivenJob

Sequence

533

compressed case, the position of the due-date will re-

main ﬁxed at C

as well, since a change in the position

of the due-date will require a compression in the pro-

cessing time which is more than the actual processing

time itself.

ICEIS2015-17thInternationalConferenceonEnterpriseInformationSystems

534