production scheduling considering outsourcing options and...

Research ArticleProduction Scheduling considering OutsourcingOptions and Carrier Costs

Byung-Cheon Choi 1 Yunhong Min2 Myoung-Ju Park 3 and Kyung Min Kim 4

1Department of Business Administration Chungnam National University 99 Daehak-ro Yuseong-gu Daejeon 34134Republic of Korea2Graduate School of Logistics Incheon National University 119 Academy-ro Yeonsu-gu Incheon 22012 Republic of Korea3Department of Industrial and Management Systems Engineering Kyung Hee University 1732 Deogyeong-daero Giheung-guYongin-si Gyeonggi-do 17104 Republic of Korea4Department of Industrial Management and Engineering Myongji University 116 MyongJi-ro Choein-gu Yongin-siGyeonggi-do 17058 Republic of Korea

Correspondence should be addressed to Kyung Min Kim kmkimmjuackr

Received 8 August 2019 Accepted 29 October 2019 Published 3 January 2020

Guest Editor Juneyoung Park

Copyright copy 2020 Byung-Cheon Choi et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We consider a single-machine scheduling problem with outsourcing options in an environment where the cost information of thedownstream is available via some information sharing technologies e due date is assigned to the position dierently from thetraditional due date Each job can be processed in-house or outsourced Note that for cost saving as many due dates as thenumber of outsourced jobs should be canceled An in-house job incurs a stepwise penalty cost for tardiness and an outsourced jobincurs an outsourcing cost us the objective is to minimize the total penalty and outsourcing cost minus the total prot fromcost savings We show that the problem is weakly NP-hard and investigate some polynomially solvable cases Due to the highcomplexity of the dynamic programming we developed heuristics and veried their performance through numerical experiments

1 Introduction

Information sharing has been known as one of the eectivetools that enable parties in a supply chain to coordinate witheach other [1 2] A number of scenarios that exploit thecapability of information sharing might be possibleaccording to the characteristics of the supply chain In thispaper we consider the problem of manufacturers who wantto schedule their processing of jobs so that the cost of acarrier as well as her own costs is simultaneously minimizedWe assume that a decision maker ie manufacturer is wellaware of the cost structure of her carrier by some infor-mation sharing technologies

Generally a manufacturer in the current business en-vironment is looking to entrust part of its production to asubcontractor and third-party logistics providers respec-tively With such outsourcing the manufacturer can deliver

goods to its consumers on time reduce operating costs andmake its organization more exible us the proper use ofoutsourcing can make a company more competitive [3 4]which provides us with motivation to consider a productionscheduling problem with an outsourcing option [5 6 7]

In this paper we consider a single-machine schedulingproblem such that a job can be processed directly by in-house resources or outsourced to a subcontractor For in-house jobs a due date is assigned depending on the jobsequence and a stepwise penalty cost for tardiness can beincurred A due date can be interpreted as the arrival time ofa particular truck operated by an independent carrier Notethat these due dates are given not for specic jobs but forspecic positions and these due dates are referred to asgeneralized due dates (GDD) [8]is reects the situation inwhich a carrierrsquos truck periodically visits a factory accordingto a planned timetable and picks up not a specic product

HindawiJournal of Advanced TransportationVolume 2020 Article ID 7069291 9 pageshttpsdoiorg10115520207069291

but one that is completed upon the truckrsquos arrival In thiscase the truckrsquos arrival time can be regarded as the due dateWhen some jobs are determined to be outsourced it isassumed that as many due dates should be canceled as thenumber of outsourced jobs In particular we assume thatany due date can be canceled is is different from thecanceling policy of Gerstl and Mosheiov [9] where the duedates are canceled in nonincreasing order

As mentioned at the outset the objective to be mini-mized of manufacturers consists of two components one fortheir own cost and the other for the cost of the carrier For agiven set of due dates the cost of the manufacturer is theoutsourcing cost e cost of the carrier on the contrary isthe difference between the waiting cost due to late com-pletion of jobs and the profit due to the canceled due datese canceling notification from the manufacturer which issufficiently earlier than the due dates enables the carrier toredirect the corresponding resources eg trucks anddrivers us we assume that the canceling of due datesincurs profit instead of cost Note that for the manufacturerall information on the carrier becomes available as a result ofsome information sharing technologies

Remark 1 In Section 3 we will show that our problem isweakly NP-hard and has a pseudopolynomial time algo-rithm Moreover in Section 4 we will introduce somepolynomially solvable cases All these results also hold forthe case where canceling due dates incurs costs instead ofprofits

e main contributions of this paper are to establish thecomputational complexities of the problem and its variantsand to design heuristics whose effectiveness was verifiedthrough numerical experiments For practitioners whoconsider our problem as an appropriate model for theirproblems polynomially solvable cases and heuristics canprovide some guidelines for tackling their own problems Inparticular manufacturers can align their productionschedules with those of carriers or other downstream firmsis practice can result in reducing the transportation costsas well as overhead costs of manufacturers [10]

e scheduling problem with GDD was introduced byHall [8] In Hall [8] and Hall et alrsquos [11] studies thecomputational complexity of cases with various perfor-mance measuresmdashsuch as maximum lateness total weightedcompletion time total weighted tardiness and the weightednumber of tardy jobsmdashis established in various machineenvironments such as a single machine parallel machinesand shops However the single-machine scheduling prob-lem that minimizes the total weighted number of tardy jobshas been proven to be NP-hard by Sriskandarajah [12] andYuan [13] and strongly NP-hard by Gao and Yuan [14]Mosheiov and Oron [15] considered the problem on parallelidentical machines to minimize the maximum tardiness andtotal tardinessey showed that the schedule ordered by theshortest processing time (SPT) performs extremely wellChoi and Park [16] considered single-machine schedulingwith GDD and identical due date intervals to minimize theweighted number of early and tardy jobs ey showed thatthe problem is strongly NP-hard and has no ρ-approximation

algorithm for any fixed value ρgt 1 Gerstl and Mosheiov [9]considered two single-machine scheduling problems withGDD and rejection options with the objective of minimizingthe total rejection cost plus the maximum tardiness or thetotal tardiness ey showed that the two machine problemsare weakly NP-hard and developed heuristics whose per-formance was verified through the numerical experiments Tothe best of our knowledge Gerstl and Mosheiov [9] were theonly ones to consider the scheduling problem with GDD andoutsourcing options

e remainder of the paper is organized as follows Section2 defines our problems In Section 3 we prove the weak NP-hardness of our problems Section 3 presents some conditionsthatmake our problems polynomially solvable In Section 4 wedevelop heuristics and conduct numerical experiments Finallyin Section 6 we present our concluding remarks

2 Notation and Problem Definition

In this section we introduce the notations used throughoutthe paper and formally define our problem

Let pj and oj be the processing time and the outsourcingcost of job j isin J ≔ 1 2 n respectively Let dg be thegth due date and qg be the profit from canceling dg forg isin D ≔ 1 2 n We assume that d1 led2 le middot middot middot ledn Letσ (h π θ) be a schedule such that the number of in-housejobs is h and

π (π(1) π(2) π(h))

θ (θ(1) θ(2) θ(h))(1)

where π(i) is the ith in-house job assigned to the θ(i)th duedate Without loss of generality assume that

θ(1)lt θ(2)lt middot middot middot lt θ(h) (2)

Let Cπ(i)(σ) be the completion time of the ith in-housejob in σ which is calculated as

Cπ(i)(σ) 1113944i

j1pπ(j) (3)

Let Vg(x) be the penalty cost function for the tardinessof the job assigned to dg which is defined as

Vg(x)

0 if x minus dg le 0

v1g if 0ltx minus dg le δ

v2g if δ ltx minus dg le 2δ

⋮vwminus 1g if (w minus 2)δ lt x minus dg le (w minus 1)δ

vwg if (w minus 1)δ lt x minus dg

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

where δ gt 0 is a given threshold Without loss of generalityassume that

0lt v1g lt v2g lt middot middot middot lt vwg for eachg isin D (5)

is stepwise tardiness penalty cost has been consideredin the field of transportation and semiconductormanufacturing [17ndash20]

2 Journal of Advanced Transportation

en our problem is to find a schedule σ to minimize

z(σ) 1113944

h

i1Vθ(i) Cπ(i)(σ)1113872 1113873 + 1113944

jisinOoj minus 1113944

gisinCqg (6)

where O and C are the sets of outsourced jobs and canceleddue dates respectively that is

O J π(1) π(2) π(h)

C D θ(1) θ(2) θ(h) (7)

Let our problem be referred to as Problem P

Proposition 1 ltere exists an optimal schedule such that thein-house jobs are sequenced by SPT order

Proof Proposition 1 holds immediately from the standardpairwise interchange scheme

Henceforth we consider only schedules satisfyingProposition 1

3 Computational Complexity

In this section we show that Problem P is weakly NP-hardFor the proof of NP-hardness we use the equal cardinalitypartition (ECP) problem which can be stated as followsgiven a setA of 2m elements a bound A isin Z+ and a size aj

for each element j isin A A can be partitioned into twodisjoint sets A1 and A2 such that

1113944jisinA1

aj 1113944jisinA2

aj A

A11113868111386811138681113868

1113868111386811138681113868 A21113868111386811138681113868

1113868111386811138681113868 m

(8)

Without loss of generality assume that for any subsetSsubeA with |S|gem + 1

1113944jisinS

aj gtA(9)

Lemma 1 Problem P is NP-hard even if

(i) lte step function for each g isin D has only two in-tervals that is

Vg(x) 0 if xledg

1 if dg lt x

⎧⎨

⎩ (10)

(ii) lte intervals between the consecutive due dates areidentical that is

dg gΔ for eachg isin D (11)

where Δgt 0 is a given threshold

Proof Given an instance of the ECP problem we canconstruct an instance of our problems as follows ere aren (2m + 1) jobs with

pj oj1113872 1113873 A3 + A2 + aj aj2A1113872 1113873 for j 1 2 2m

A3 minus mA2 minus A 1( 1113857 for j 2m + 1

⎧⎨

⎩

(12)

Let Δ A3 and

qg 0 forg 1 2 m + 1

A2 forg m + 2 m + 3 2m + 11113896 (13)

is reduction can be carried out in polynomial timeHenceforth we show that a solution to the ECP problemexists if and only if there exists a schedule σ for Problem Pwith z(σ)le 12 minus mA2

Suppose there exists a solution (A1A2) to the ECPproblem en we can construct a schedule σ (h π θ)

such that

(i) e number of in-house jobs is h m + 1(ii) π is the SPT sequence of jobs in 2m + 1 cup A1

where job 2m + 1 is the first job according toProposition 1

(iii) θ (1 2 m + 1)

Let O A2 and C m + 2 m + 3 2m + 1 Notethat for i 1 2 m + 1

Cπ(i)(σ) A3

minus mA2

minus A + 1113944i

j2pπ(j)

iA3

minus (m minus i + 1)A2

minus A + 1113944i

j2aπ(j)

(14)

Since (m minus i + 1)A2 + Agt1113936ij2aπ(j) i 1 2 m and

1113936m+1j2 aπ(j) A

Cπ(i)(σ)lt iA3

di for i 1 2 m (15)

Cπ(m+1)(σ) (m + 1)A3

dm+1 (16)

Inequalities (15) and (16) show that no tardy in-housejob exists in σ Furthermore according to the reductionscheme above

1113944

jisinO

oj 12A

1113944

jisinA2

aj 12

1113944

gisinC

qg 11139442m+1

gm+2qg mA

2

(17)

us by inequality (17)

z(σ) 1113944

jisinO

oj minus 1113944

gisinC

qg 12

minus mA2 (18)

Suppose there exists a schedule 1113954σ (1113954h 1113954π 1113954θ) for ProblemP with z(1113954σ)le 12 minus mA2 Let

1113954O J π(1) π(2) π(1113954h)1113966 1113967

1113954C D θ(1) θ(2) θ(1113954h)1113966 1113967(19)

Journal of Advanced Transportation 3

Since z(1113954σ)le 12 minus mA2

m + 2 m + 3 2m + 1 sube 1113954C (20)

and no tardy job exists in 1113954σ which implies that job 2m + 1 isthe first in-house job by Proposition 1

Claim 1 1113954C m + 2 m + 3 2m + 1

Proof By relation (20) 1113954hlem + 1 us Claim 1 holds onlyby proving 1113954h m + 1 Suppose that 1113954hlem en |1113954O|gem + 1holds and by assumption (9) we have

z(1113954σ)ge 1113944

jisin1113954O

oj minus 1113944

gisin1113954C

qg 12A

1113944

jisin1113954O

aj minus mA2 gt

12

minus mA2

(21)

is is a contradiction

Claim 2 1113936jisin1113954O

oj 12

Proof Since z(1113954σ)le 12 minus mA2 and 1113954C m + 2 m + 3

2m + 1

1113944

jisin1113954O

oj le12 (22)

Suppose that 1113936jisin1113954O

oj lt 12 en

1113944

m+1

j2a1113954π(j)gtA (23)

By inequality (23) we have

C1113954π(m+1)(1113954σ) 1113944

m+1

j1p1113954π(j)

(m + 1)A3

minus A + 1113944m+1

j2a1113954π(j)gtdm+1

(24)

and job 1113954π(m + 1) becomes tardy is is a contradictionLet 1113954A1 1113954O and 1113954A2 1 2 2m 1113954Oen 1113954A1 and 1113954A2

become the solution to the ECP problemHenceforth for simplicity assume that

p1 lep2 le middot middot middot lepn (25)

Lemma 2 Problem P can be solved in pseudopolynomialtime

Proof We reduce Problem P to the shortest path (SP)problem Let N(0 0 0) and N(n + 1 n + 1 middot) be the sourceand sink nodes respectively Let N(a b C) be the nodeindicating

(i) Which jobs in 1 2 a have to be outsourced(ii) Which due dates in d1 d2 db1113864 1113865 have to be

canceled(iii) e total processing time of in-house jobs in

1 2 a is C

If 0le aprime lt ale n and 0le bprime lt ble n then let N(aprime bprime C minus

pa) be connected to N(a b C) with length

Vb(C) + 1113944aminus 1

japrime+1

oj minus 1113944bminus 1

gbprime+1

qg (26)

If 0le ale n and 0le ble n then let N(a b C) be connectedto the sink node with length

1113944

n

ja+1oj minus 1113944

n

gb+1qg (27)

e objective is to find the SP from the source to the sinknode In the reduced SP problem the number of nodes isobserved to be O(n21113936

nj1pj) and each node has O(n2)

outgoing arcs Since the graph of the reduced SP problem isacyclic the reduced SP problem can be solved inO(n41113936

nj1pj) by the algorithm of Ahuja et al [21]

Theorem 1 Problem P is weakly NP-hard

Proof eorem 1 holds immediately from Lemmas 1 and2

4 Polynomially Solvable Cases

In this section we introduce three polynomially solvablecases

Theorem 2 Problem P is polynomially solvable if the pro-cessing times are identical that is pj p j 1 2 n

Proof We will prove eorem 2 by reducing Problem P tothe SP problem such that the total number of nodes isbounded by n Let N(0 0 0) and N(n + 1 n + 1) be thesource and sink nodes respectively Let N(a b h) be thenode indicating



1 2 a is hp

If 0le aprime lt ale n and 0le bprime lt ble n then let N(aprime bprime h minus 1)

be connected to N(a b h) with length

Vb(hp) + 1113944aminus 1

japrime+1


gbprime+1

qg (28)

If 0le ale n and 0le ble n then let N(a b h) be connectedto the sink node with length

1113944

n


n

gb+1qg (29)

e objective is to find the SP from the source to the sinknode In the reduced SP problem the number of nodes isobserved to be O(n3) and each node has O(n2) outgoingarcs respectively Since the graph of the reduced SP problem


is acyclic the reduced SP problem can be solved in O(n5) bythe algorithm of Ahuja et al [21]

Theorem 3 Problem P is polynomially solvable if

(i) lte intervals between the consecutive due dates areidentical that is


where Δ is a given threshold

(ii) pj leΔ for each j isin J

Proof In an optimal schedule the total penalty cost fortardiness is observed to be zero by sequencing the in-housejobs in SPT order Based on this observation we will proveeorem 3 by reducing Problem P to the weighted matchingproblem

Construct a graph G (LcupRE) such that

(i) L a(l) | l 1 2 n andR b(r) | r 1 2 n

(ii) For each edge (i j) i isinL and j isinR

For l 1 2 n and r 1 2 n the weight of edge(a(l) b(r)) is calculated as

max 0 qr minus ol( 11138571113864 1113865 (31)

e objective is to find the set of edges Elowast whose totalweight is maximized Note that if edge (a(l) b(r)) isin Elowastthen in an optimal schedule of Problem P job l is out-sourced and due date dr is canceled e reduced weightedmatching problem can be solved in O(n3) by the Hungarianmethod

Theorem 4 Given π (π(1) π(h)) it is polynomiallysolvable to determine θ (θ(1) θ(h)) to minimize z(σ)

for Problem P

Proof We will prove eorem 4 by reducing Problem P tothe SP problem such that the number of nodes is bounded byn Let Cπ(i) be the completion time of job π(i) Let N(0 0)

and N(h + 1 n + 1) be the source and sink nodes respec-tively Let N(i b) be the node indicating

(i) Which due dates in d1 d2 db1113864 1113865 have to becanceled

(ii) In-house job π(i) is assigned to db

If 0lt ile h and 0le bprime lt ble n then let N(i minus 1 bprime) beconnected to N(i b) with length

Vb Cπ(i)1113872 1113873 minus 1113944bminus 1

gbprime+1

qg (32)

If b n and 0le ilt h then let N(i b) be connected to thesink node with length infin If i h then let N(i b) beconnected to the sink node with length

minus 1113944n

gb+1qg (33)

e objective is to find the SP from the source to the sinknode In the reduced SP problem the number of nodes isobserved to be O(n2) and each node has O(n) outgoing arcsrespectively Since the graph of the reduced SP problem isacyclic the reduced SP problem can be solved in O(n3) bythe algorithm of Ahuja et al [21]

5 Heuristics

In this section due to the high complexity of the dynamicprogramming in Lemma 2 we present three heuristics Weconduct numerical experiments to evaluate each heuristicHeuristic H1 is based on the heuristic of Gerstl andMosheiov[9] First Heuristic H1 prioritizes the jobs to be outsourcedand the due dates to be canceled in advance and constructsnew schedules by deleting jobs and due dates in order ofpriority from the initial schedule en Heuristic H1 selectsthe best schedule among the constructed schedules edetailed description of Heuristic H1 is as follows

51 Heuristic H1

Step 1 Obtain two sequences τ1 and τ2 by sequencingthe jobs in the nonincreasing orders of (oj minus pj) andojpj respectively and a sequence ς by sequencing thedue dates in the nonincreasing orders of dgqgStep 2 Set

πi0 (1 2 n)

θi0 (1 2 n)

for i 1 2

(34)

Step 3 Construct

S1 n minus k πik θik1113872 111387311138681113868111386811138681113868 i 1 2 k 1 2 n1113882 1113883 (35)

where πik and θik are the subsequences constructedfrom πikminus 1 and θikminus 1 by deleting job τi(k) and ς(k)respectivelyStep 4 Return σlowast such that

z σlowast( 1113857 min z(σ) | σ isin S1 cup n π10 θ101113872 11138731113966 11139671113966 1113967 (36)

Heuristic H1 can be improved by the property ineorem 4 Instead of θik in Step 3 of Heuristic H1 HeuristicH2 assigns due dates according to the algorithm in the proofof eorem 4 e detailed procedure is as follows

52 Heuristic H2

Step 1 Obtain two sequences τ1 and τ2 by sequencingthe jobs in the nonincreasing orders of (oj minus pj) andojpj respectively


Step 2 Set

πi0 (1 2 n) for i 1 2 (37)

Step 3 Construct

S2 πik θik1113872 111387311138681113868111386811138681113868 i 1 2 k 1 2 n1113882 1113883 (38)

where πik is the subsequence from πikminus 1 by deleting jobτi(k) and θik is the sequence obtained by applying theapproach in eorem 4 when π πikStep 4 Return σlowast such that

z σlowast( 1113857 min z((σ) | σ isin S2 cup n π10 θ101113872 11138731113966 11139671113966 1113967 (39)

Unlike Heuristics H1 and H2 Heuristic H3 constructsnew schedules by optimally deleting jobs and due dates fromthe initial schedule based oneorem 4 en Heuristic H3selects the best schedule among the constructed schedulese detailed description of Heuristic H3 is as follows

53 Heuristic H3

Step 1 Set σlowast (n π θ) and h n where

π (1 2 n)

θ (1 2 n)(40)

Step 2 Construct

S3 h πj θg11138731113872 111387311138681113868111386811138681113868 j isin J g isin D1113882 1113883 (41)

where πj and θg are the subsequences constructed fromπ and θ by deleting job j and due date g respectivelyStep 3 Find σlowastjg (h πlowastjg θlowastjg) such that

z σlowastjg1113872 1113873 min z(σ) | σ isin S31113864 1113865 (42)

Step 4 Set h h minus 1 π πlowastjg θ θlowastjg J J j1113864 1113865 andD D g1113864 1113865Step 5 If z(σlowastjg)lt z(σlowast) then set σlowast σlowastjgStep 6 If J empty then return σlowast and STOP otherwisego to Step 2

6 Experimental Results

To evaluate the performance of Heuristics H1ndashH3 weconducted numerical experiments with randomly generatedinstances in various settings We implemented the heuristicswith Python language To implement the pseudopolynomialtime algorithm introduced in Lemma 2 the Python Net-workX package was used All our experiments were per-formed on a laptop computer with 32GB of RAM and a400GHz processor

61 Instances All instances were categorized with respect tothe number of jobs (n) and the densities of the generalizeddue dates (α) as defined by Gerstl and Mosheiov [9] Morespecifically we considered n 20 40 60 and 80 jobs ejob processing times and outsourcing costs were randomlygenerated in the intervals [1 40] and [1 20] respectivelye generalized due dates were generated uniformly fromthe interval [1 dn] where dn α1113936

nj1pj We considered the

cases α 02 04 06 08 and 10 e profits for cancelingdue dates were randomly generated in the interval [1 40]Finally for the tardiness penalty cost Vg(x) introduced inSection 1 vig qg times βi β 13 15 17 20 and δ is thestandard deviation of the job processing timesWe generated30 instances for each (n α β)

62 Results To evaluate the quality of the solutions obtainedby Heuristics H1ndashH3 we compare these three solutions withthe optimal solution which is obtained by the pseudopo-lynomial time algorithm in Lemma 2 For instances withnge 40 however the optimal solutions are not available dueto lack of computer memory

Table 1 shows the objective values of the schedules byapplying Heuristics H1ndashH3 and the optimal algorithm forthe instances under the setting (n α β) (20 08 15) H3outperforms the other heuristics in all instances and inparticular finds optimal solutions for 80( 2430) ofinstances us Heuristic H3 is the best of the threeheuristics

Table 2 compares the computation times of the threeheuristics with the optimal algorithm Heuristic H1 is thefastest since it simply sorts the jobs and due datesaccording to predefined measures Heuristic H3 rankssecond though it requires repetitive enumerationsOn the contrary although Heuristic H2 is similar toHeuristic H1 it exhibits longer computation times thanHeuristic H1 since the SP problem needs to be solved as asubroutine

According to the results in Tables 1 and 2 Heuristic H3could be the most competitive alternative among the optimalalgorithm and heuristics Indeed a similar conclusion can bedrawn for (n 20 α 08) setting with different β values

e effect of the densities of generalized due dates(controlled by α in our experiments) is also reported inTable 3 Interestingly Heuristic H3 finds the optimal so-lutions for all 30 instances when α 10

In Table 4 we investigate the effects of β Note thatHeuristic H1 does not consider β in its implementation andthus finds the same solution regardless of β On the con-trary Heuristics H2 and H3 consider β and thus find dif-ferent solutions for different β values

Table 5 demonstrates the average objective values foreach setting of n and α We found that the performances areconsistent with the previous results

Finally we report the increases in computation timeswith respect to the number of jobs Table 6 shows the av-erages and standard deviations of the computation times forthe instances with (α β) (08 15) Heuristic H1 findssolutions fastest with little variation since it simply sorts


jobs On the contrary the computation times of HeuristicH2 show steep increases as the number of jobs increasessince it uses an SP path subroutinee computation time ofHeuristic H3 lies between those of H1 and H2

7 Concluding Remarks

Information sharing among parties in a supply chain pro-vides an opportunity for members of the supply chain to

Table 2 Average and standard deviation of computation times for instances with (n α β) (20 08 15)

OPT H1 H2 H3Avg times 475589 s 0002 s 0723 s 0066 sStd 69201 s 0001 s 0284 s 0085 s

Table 3 Average objective values for instances with (n β) (20 15) and different αrsquos

α OPT H1 H2 H302 minus 68104 minus 5967 minus 63175 minus 6647104 minus 83550 minus 5900 minus 73117 minus 8057506 minus 89683 minus 16067 minus 801 minus 8726708 minus 93233 minus 23100 minus 82867 minus 9213310 minus 92500 minus 22933 minus 80400 minus 92500

Table 4 Average objective values for instances with (n α) (20 02) and different βs

β OPT H1 H2 H313 minus 68956 minus 5967 minus 63696 minus 6771615 minus 68104 minus 5967 minus 63175 minus 6647117 minus 65744 minus 5967 minus 62949 minus 6459120 minus 67133 minus 5967 minus 62800 minus 63433

Table 1 Objective values for instances with (n α β) (20 08 15)

No OPT H1 H2 H31 minus 90 minus 22 minus 76 minus 882 minus 61 minus 1 minus 56 minus 613 minus 71 minus 13 minus 54 minus 714 minus 81 +12 minus 71 minus 815 minus 97 minus 15 minus 95 minus 976 minus 127 minus 46 minus 118 minus 1277 minus 65 minus 2 minus 48 minus 658 minus 93 minus 19 minus 86 minus 939 minus 76 minus 16 minus 71 minus 7410 minus 85 minus 20 minus 70 minus 8511 minus 140 minus 74 minus 125 minus 14012 minus 104 minus 54 minus 93 minus 9513 minus 98 minus 38 minus 88 minus 9814 minus 118 minus 31 minus 90 minus 11815 minus 66 minus 2 minus 59 minus 6616 minus 81 minus 12 minus 72 minus 8117 minus 96 minus 35 minus 87 minus 9618 minus 92 minus 24 minus 81 minus 8819 minus 118 minus 31 minus 90 minus 11820 minus 66 minus 2 minus 59 minus 6621 minus 101 minus 7 minus 91 minus 10022 minus 109 minus 55 minus 98 minus 10923 minus 99 minus 38 minus 96 minus 9924 minus 73 minus 20 minus 62 minus 6725 minus 82 minus 47 minus 72 minus 8226 minus 94 minus 7 minus 75 minus 9427 minus 103 minus 23 minus 100 minus 10328 minus 101 minus 11 minus 89 minus 10129 minus 109 minus 16 minus 100 minus 10930 minus 96 minus 19 minus 90 minus 96


coordinate with each other In this paper we consider analignment problem of production and transportationschedules that can be modeled as a single-machine sched-uling problem with generalized due dates and outsourcingoptions Tardiness and outsourcing costs occur because ofin-house and outsourced jobs respectively and cost savingsarise from canceled due dates e objective is to minimizethe total outsourcing and canceling costs minus the totalbenefit from cost savings We showed that the problem isNP-hard even if each step function has only two intervalsand we developed a pseudopolynomial time algorithm Dueto the high complexity of the pseudopolynomial time al-gorithm however we developed three heuristics and verifiedtheir performance by conducting numerical experiments invarious settings ese findings can be used to design andimplement production schedules considering the alignmentof production and transportation to downstream firms

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the 2019 Research Fund ofMyongji University

References

[1] H L Lee and S Whang ldquoInformation sharing in a supplychainrdquo International Journal of Manufacturing Technologyand Management vol 1 no 1 pp 79ndash93 2000

[2] H Zhou Jr and W C Benton ldquoSupply chain practice andinformation sharingrdquo Journal of Operations Managementvol 25 no 6 pp 1348ndash1365 2007

[3] G P Cachon and P T Harker ldquoCompetition and outsourcingwith scale economiesrdquo Management Science vol 48 no 10pp 1314ndash1333 2002

[4] MWebster C Alder and A P Muhlemann ldquoSubcontractingwithin the supply chain for electronics assembly manufac-turerdquo International Journal of Operations amp ProductionManagement vol 17 no 9 pp 827ndash841 1997

[5] M Haoues M Dahane and N K Mouss ldquoOutsourcingoptimization in two-echelon supply chain network underintegrated production-maintenance constraintsrdquo Journal ofIntelligent Manufacturing vol 30 no 2 pp 701ndash725 2019

[6] B Liao Q Song J Pei S Yang and P M Pardalos ldquoParallel-machine group scheduling with inclusive processing set re-strictions outsourcing option and serial-batching under theeffect of step-deteriorationrdquo Journal of Global Optimization2018

[7] H Safarzadeh and F Kianfar ldquoJob shop scheduling with theoption of jobs outsourcingrdquo International Journal of Pro-duction Research vol 57 no 10 pp 3255ndash3272 2019

[8] N G Hall ldquoScheduling problems with generalized due datesrdquoIIE Transactions vol 18 no 2 pp 220ndash222 1986

[9] E Gerstl and G Mosheiov ldquoSingle machine schedulingproblems with generalised due-dates and job-rejectionrdquo In-ternational Journal of Production Research vol 55 no 11pp 3164ndash3172 2017

[10] Y C E Lee C K Chan A Langevin and H W J LeeldquoIntegrated inventory-transportation model by synchronizingdelivery and production cyclesrdquo Transportation Research PartE Logistics and Transportation Review vol 91 pp 68ndash892016

[11] N G Hall S P Sethi and C Sriskandarajah ldquoOn thecomplexity of generalized due date scheduling problemsrdquoEuropean Journal of Operational Research vol 51 no 1pp 100ndash109 1991

[12] C Srikandarajah ldquoA note on the generalized due datesscheduling problemrdquo Naval Research Logistics vol 37 no 4pp 587ndash597 1990

[13] J J Yuan ldquoe NP-hardness of the single machine commondue date weighted tardiness problemrdquo Systems Science andMathematical Science vol 5 no 4 pp 328ndash333 1992

[14] Y Gao and J Yuan ldquoUnary NP-hardness of minimizing totalweighted tardiness with generalized due datesrdquo OperationsResearch Letters vol 44 no 1 pp 92ndash95 2016

Table 5 Average objective values for instances with β 15 and different n and α values

αn 40 n 60 n 80

H1 H2 H3 H1 H2 H3 H1 H2 H302 minus 7200 minus 134375 minus 139500 +16400 minus 191440 minus 200026 minus 4183 minus 253879 minus 28125804 minus 20283 minus 157508 minus 168700 minus 26867 minus 238733 minus 253233 minus 55900 minus 335833 minus 36126706 minus 15467 minus 153633 minus 168733 minus 30267 minus 237233 minus 267333 minus 53750 minus 320200 minus 36450008 minus 36100 minus 170033 minus 194733 minus 49633 minus 241667 minus 283733 minus 74500 minus 337800 minus 38243310 minus 35133 minus 160033 minus 185033 minus 46667 minus 252433 minus 290833 minus 57033 minus 320700 minus 373533

Table 6 Averages and standard deviations of the computation times for instances with (α β) (08 15) and different n values

nH1 H2 H3

Avg Std Avg Std Avg Std20 0002 s 0001 s 0723 s 0284 s 0066 s 0085 s40 0006 s 0002 s 11468 s 0792 s 1014 s 0110 s60 0011 s 0002 s 54759 s 2740 s 5664 s 0341 s80 0016 s 0000 s 110108 s 3014 s 13913 s 1109 s


[15] G Mosheiov and D Oron ldquoA note on the SPT heuristic forsolving scheduling problems with generalized due datesrdquoComputers amp Operations Research vol 31 no 5 pp 645ndash6552004

[16] B-C Choi and M-J Park ldquoJust-in-time scheduling withgeneralized due dates and identical due date intervalsrdquo Asia-Pacific Journal of Operational Research vol 35 no 6 ArticleID 1850046 2018

[17] S N Chaurasia S Sundar and A Singh ldquoHybrid meta-heuristic approaches for the single machine total stepwisetardiness problem with release datesrdquo Operational Researchvol 17 no 1 pp 275ndash295 2017

[18] J Curry and B Peters ldquoRescheduling parallel machines withstepwise increasing tardiness and machine assignment sta-bility objectivesrdquo International Journal of Production Re-search vol 43 no 15 pp 3231ndash3246 2005

[19] B Detienne S Dauzere-Peres and C Yugma ldquoSchedulingjobs on parallel machines to minimize a regular step total costfunctionrdquo Journal of Scheduling vol 14 no 6 pp 523ndash5382011

[20] C-T Tseng and K-H Chen ldquoAn electromagnetism-likemechanism for the single machine total stepwise tardinessproblem with release datesrdquo Engineering Optimizationvol 45 no 12 pp 1431ndash1448 2013

[21] R K Ahuja K Mehlhorn J Orlin and R E Tarjan ldquoFasteralgorithms for the shortest path problemrdquo Journal of theACM vol 37 no 2 pp 213ndash223 1990


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but one that is completed upon the truckrsquos arrival In thiscase the truckrsquos arrival time can be regarded as the due dateWhen some jobs are determined to be outsourced it isassumed that as many due dates should be canceled as thenumber of outsourced jobs In particular we assume thatany due date can be canceled is is different from thecanceling policy of Gerstl and Mosheiov [9] where the duedates are canceled in nonincreasing order

As mentioned at the outset the objective to be mini-mized of manufacturers consists of two components one fortheir own cost and the other for the cost of the carrier For agiven set of due dates the cost of the manufacturer is theoutsourcing cost e cost of the carrier on the contrary isthe difference between the waiting cost due to late com-pletion of jobs and the profit due to the canceled due datese canceling notification from the manufacturer which issufficiently earlier than the due dates enables the carrier toredirect the corresponding resources eg trucks anddrivers us we assume that the canceling of due datesincurs profit instead of cost Note that for the manufacturerall information on the carrier becomes available as a result ofsome information sharing technologies

Remark 1 In Section 3 we will show that our problem isweakly NP-hard and has a pseudopolynomial time algo-rithm Moreover in Section 4 we will introduce somepolynomially solvable cases All these results also hold forthe case where canceling due dates incurs costs instead ofprofits

e main contributions of this paper are to establish thecomputational complexities of the problem and its variantsand to design heuristics whose effectiveness was verifiedthrough numerical experiments For practitioners whoconsider our problem as an appropriate model for theirproblems polynomially solvable cases and heuristics canprovide some guidelines for tackling their own problems Inparticular manufacturers can align their productionschedules with those of carriers or other downstream firmsis practice can result in reducing the transportation costsas well as overhead costs of manufacturers [10]

e scheduling problem with GDD was introduced byHall [8] In Hall [8] and Hall et alrsquos [11] studies thecomputational complexity of cases with various perfor-mance measuresmdashsuch as maximum lateness total weightedcompletion time total weighted tardiness and the weightednumber of tardy jobsmdashis established in various machineenvironments such as a single machine parallel machinesand shops However the single-machine scheduling prob-lem that minimizes the total weighted number of tardy jobshas been proven to be NP-hard by Sriskandarajah [12] andYuan [13] and strongly NP-hard by Gao and Yuan [14]Mosheiov and Oron [15] considered the problem on parallelidentical machines to minimize the maximum tardiness andtotal tardinessey showed that the schedule ordered by theshortest processing time (SPT) performs extremely wellChoi and Park [16] considered single-machine schedulingwith GDD and identical due date intervals to minimize theweighted number of early and tardy jobs ey showed thatthe problem is strongly NP-hard and has no ρ-approximation

algorithm for any fixed value ρgt 1 Gerstl and Mosheiov [9]considered two single-machine scheduling problems withGDD and rejection options with the objective of minimizingthe total rejection cost plus the maximum tardiness or thetotal tardiness ey showed that the two machine problemsare weakly NP-hard and developed heuristics whose per-formance was verified through the numerical experiments Tothe best of our knowledge Gerstl and Mosheiov [9] were theonly ones to consider the scheduling problem with GDD andoutsourcing options

e remainder of the paper is organized as follows Section2 defines our problems In Section 3 we prove the weak NP-hardness of our problems Section 3 presents some conditionsthatmake our problems polynomially solvable In Section 4 wedevelop heuristics and conduct numerical experiments Finallyin Section 6 we present our concluding remarks

2 Notation and Problem Definition

In this section we introduce the notations used throughoutthe paper and formally define our problem

Let pj and oj be the processing time and the outsourcingcost of job j isin J ≔ 1 2 n respectively Let dg be thegth due date and qg be the profit from canceling dg forg isin D ≔ 1 2 n We assume that d1 led2 le middot middot middot ledn Letσ (h π θ) be a schedule such that the number of in-housejobs is h and

π (π(1) π(2) π(h))

θ (θ(1) θ(2) θ(h))(1)

where π(i) is the ith in-house job assigned to the θ(i)th duedate Without loss of generality assume that

θ(1)lt θ(2)lt middot middot middot lt θ(h) (2)

Let Cπ(i)(σ) be the completion time of the ith in-housejob in σ which is calculated as

Cπ(i)(σ) 1113944i

j1pπ(j) (3)

Let Vg(x) be the penalty cost function for the tardinessof the job assigned to dg which is defined as

Vg(x)

0 if x minus dg le 0

v1g if 0ltx minus dg le δ

v2g if δ ltx minus dg le 2δ

⋮vwminus 1g if (w minus 2)δ lt x minus dg le (w minus 1)δ

vwg if (w minus 1)δ lt x minus dg

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

where δ gt 0 is a given threshold Without loss of generalityassume that

0lt v1g lt v2g lt middot middot middot lt vwg for eachg isin D (5)

is stepwise tardiness penalty cost has been consideredin the field of transportation and semiconductormanufacturing [17ndash20]



z(σ) 1113944

h

i1Vθ(i) Cπ(i)(σ)1113872 1113873 + 1113944


gisinCqg (6)


O J π(1) π(2) π(h)

C D θ(1) θ(2) θ(h) (7)








1113944jisinA1

aj 1113944jisinA2

aj A

A11113868111386811138681113868

1113868111386811138681113868 A21113868111386811138681113868

1113868111386811138681113868 m

(8)


1113944jisinS

aj gtA(9)



Vg(x) 0 if xledg

1 if dg lt x

⎧⎨

⎩ (10)







⎧⎨

⎩

(12)

Let Δ A3 and

qg 0 forg 1 2 m + 1

A2 forg m + 2 m + 3 2m + 11113896 (13)



such that



(iii) θ (1 2 m + 1)


Cπ(i)(σ) A3

minus mA2

minus A + 1113944i

j2pπ(j)

iA3


minus A + 1113944i

j2aπ(j)

(14)


1113936m+1j2 aπ(j) A

Cπ(i)(σ)lt iA3

di for i 1 2 m (15)

Cπ(m+1)(σ) (m + 1)A3

dm+1 (16)


1113944

jisinO

oj 12A

1113944

jisinA2

aj 12

1113944

gisinC

qg 11139442m+1

gm+2qg mA

2

(17)


z(σ) 1113944

jisinO

oj minus 1113944

gisinC

qg 12

minus mA2 (18)


1113954O J π(1) π(2) π(1113954h)1113966 1113967

1113954C D θ(1) θ(2) θ(1113954h)1113966 1113967(19)



m + 2 m + 3 2m + 1 sube 1113954C (20)


Claim 1 1113954C m + 2 m + 3 2m + 1


z(1113954σ)ge 1113944

jisin1113954O

oj minus 1113944

gisin1113954C

qg 12A

1113944

jisin1113954O

aj minus mA2 gt

12

minus mA2

(21)


Claim 2 1113936jisin1113954O

oj 12


2m + 1

1113944

jisin1113954O

oj le12 (22)


oj lt 12 en

1113944

m+1

j2a1113954π(j)gtA (23)


C1113954π(m+1)(1113954σ) 1113944

m+1

j1p1113954π(j)

(m + 1)A3

minus A + 1113944m+1


(24)








1 2 a is C



Vb(C) + 1113944aminus 1

japrime+1


gbprime+1

qg (26)


1113944

n


n

gb+1qg (27)













1 2 a is hp




japrime+1


gbprime+1

qg (28)


1113944

n


n

gb+1qg (29)











(i) L a(l) | l 1 2 n andR b(r) | r 1 2 n



max 0 qr minus ol( 11138571113864 1113865 (31)



for Problem P







gbprime+1

qg (32)


minus 1113944n

gb+1qg (33)


5 Heuristics


51 Heuristic H1


πi0 (1 2 n)

θi0 (1 2 n)

for i 1 2

(34)

Step 3 Construct





52 Heuristic H2



Step 2 Set

πi0 (1 2 n) for i 1 2 (37)

Step 3 Construct

S2 πik θik1113872 111387311138681113868111386811138681113868 i 1 2 k 1 2 n1113882 1113883 (38)




53 Heuristic H3


π (1 2 n)

θ (1 2 n)(40)

Step 2 Construct
































Data Availability




Acknowledgments


References
















αn 40 n 60 n 80



nH1 H2 H3













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



z(σ) 1113944

h

i1Vθ(i) Cπ(i)(σ)1113872 1113873 + 1113944


gisinCqg (6)


O J π(1) π(2) π(h)

C D θ(1) θ(2) θ(h) (7)








1113944jisinA1

aj 1113944jisinA2

aj A

A11113868111386811138681113868

1113868111386811138681113868 A21113868111386811138681113868

1113868111386811138681113868 m

(8)


1113944jisinS

aj gtA(9)



Vg(x) 0 if xledg

1 if dg lt x

⎧⎨

⎩ (10)







⎧⎨

⎩

(12)

Let Δ A3 and

qg 0 forg 1 2 m + 1

A2 forg m + 2 m + 3 2m + 11113896 (13)



such that



(iii) θ (1 2 m + 1)


Cπ(i)(σ) A3

minus mA2

minus A + 1113944i

j2pπ(j)

iA3


minus A + 1113944i

j2aπ(j)

(14)


1113936m+1j2 aπ(j) A

Cπ(i)(σ)lt iA3

di for i 1 2 m (15)

Cπ(m+1)(σ) (m + 1)A3

dm+1 (16)


1113944

jisinO

oj 12A

1113944

jisinA2

aj 12

1113944

gisinC

qg 11139442m+1

gm+2qg mA

2

(17)


z(σ) 1113944

jisinO

oj minus 1113944

gisinC

qg 12

minus mA2 (18)


1113954O J π(1) π(2) π(1113954h)1113966 1113967

1113954C D θ(1) θ(2) θ(1113954h)1113966 1113967(19)



m + 2 m + 3 2m + 1 sube 1113954C (20)


Claim 1 1113954C m + 2 m + 3 2m + 1


z(1113954σ)ge 1113944

jisin1113954O

oj minus 1113944

gisin1113954C

qg 12A

1113944

jisin1113954O

aj minus mA2 gt

12

minus mA2

(21)


Claim 2 1113936jisin1113954O

oj 12


2m + 1

1113944

jisin1113954O

oj le12 (22)


oj lt 12 en

1113944

m+1

j2a1113954π(j)gtA (23)


C1113954π(m+1)(1113954σ) 1113944

m+1

j1p1113954π(j)

(m + 1)A3

minus A + 1113944m+1


(24)








1 2 a is C



Vb(C) + 1113944aminus 1

japrime+1


gbprime+1

qg (26)


1113944

n


n

gb+1qg (27)













1 2 a is hp




japrime+1


gbprime+1

qg (28)


1113944

n


n

gb+1qg (29)











(i) L a(l) | l 1 2 n andR b(r) | r 1 2 n



max 0 qr minus ol( 11138571113864 1113865 (31)



for Problem P







gbprime+1

qg (32)


minus 1113944n

gb+1qg (33)


5 Heuristics


51 Heuristic H1


πi0 (1 2 n)

θi0 (1 2 n)

for i 1 2

(34)

Step 3 Construct





52 Heuristic H2



Step 2 Set

πi0 (1 2 n) for i 1 2 (37)

Step 3 Construct

S2 πik θik1113872 111387311138681113868111386811138681113868 i 1 2 k 1 2 n1113882 1113883 (38)




53 Heuristic H3


π (1 2 n)

θ (1 2 n)(40)

Step 2 Construct
































Data Availability




Acknowledgments


References
















αn 40 n 60 n 80



nH1 H2 H3













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



m + 2 m + 3 2m + 1 sube 1113954C (20)


Claim 1 1113954C m + 2 m + 3 2m + 1


z(1113954σ)ge 1113944

jisin1113954O

oj minus 1113944

gisin1113954C

qg 12A

1113944

jisin1113954O

aj minus mA2 gt

12

minus mA2

(21)


Claim 2 1113936jisin1113954O

oj 12


2m + 1

1113944

jisin1113954O

oj le12 (22)


oj lt 12 en

1113944

m+1

j2a1113954π(j)gtA (23)


C1113954π(m+1)(1113954σ) 1113944

m+1

j1p1113954π(j)

(m + 1)A3

minus A + 1113944m+1


(24)








1 2 a is C



Vb(C) + 1113944aminus 1

japrime+1


gbprime+1

qg (26)


1113944

n


n

gb+1qg (27)













1 2 a is hp




japrime+1


gbprime+1

qg (28)


1113944

n


n

gb+1qg (29)











(i) L a(l) | l 1 2 n andR b(r) | r 1 2 n



max 0 qr minus ol( 11138571113864 1113865 (31)



for Problem P







gbprime+1

qg (32)


minus 1113944n

gb+1qg (33)


5 Heuristics


51 Heuristic H1


πi0 (1 2 n)

θi0 (1 2 n)

for i 1 2

(34)

Step 3 Construct





52 Heuristic H2



Step 2 Set

πi0 (1 2 n) for i 1 2 (37)

Step 3 Construct

S2 πik θik1113872 111387311138681113868111386811138681113868 i 1 2 k 1 2 n1113882 1113883 (38)




53 Heuristic H3


π (1 2 n)

θ (1 2 n)(40)

Step 2 Construct
































Data Availability




Acknowledgments


References
















αn 40 n 60 n 80



nH1 H2 H3













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










(i) L a(l) | l 1 2 n andR b(r) | r 1 2 n



max 0 qr minus ol( 11138571113864 1113865 (31)



for Problem P







gbprime+1

qg (32)


minus 1113944n

gb+1qg (33)


5 Heuristics


51 Heuristic H1


πi0 (1 2 n)

θi0 (1 2 n)

for i 1 2

(34)

Step 3 Construct





52 Heuristic H2



Step 2 Set

πi0 (1 2 n) for i 1 2 (37)

Step 3 Construct

S2 πik θik1113872 111387311138681113868111386811138681113868 i 1 2 k 1 2 n1113882 1113883 (38)




53 Heuristic H3


π (1 2 n)

θ (1 2 n)(40)

Step 2 Construct
































Data Availability




Acknowledgments


References
















αn 40 n 60 n 80



nH1 H2 H3













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Step 2 Set

πi0 (1 2 n) for i 1 2 (37)

Step 3 Construct

S2 πik θik1113872 111387311138681113868111386811138681113868 i 1 2 k 1 2 n1113882 1113883 (38)




53 Heuristic H3


π (1 2 n)

θ (1 2 n)(40)

Step 2 Construct
































Data Availability




Acknowledgments


References
















αn 40 n 60 n 80



nH1 H2 H3













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia















Data Availability




Acknowledgments


References
















αn 40 n 60 n 80



nH1 H2 H3













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


production scheduling considering outsourcing options and...

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