Mathematical Models for a Flexible Job Shop Scheduling Problem with Machine Operator Constraints ⁎

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Mathematical Models for a Flexible Job Mathematical Models for awith Flexible Job Shop Scheduling Problem Machine Shop with OperatorProblem Constraints Shop Scheduling Scheduling Problem with Machine Machine  Operator Constraints Dominik Kress ∗∗ David Müller ∗∗

Dominik Kress David Müller Dominik Kress Kress ∗∗ David Müller Müller ∗∗ Dominik Dominik Kress ∗ David David Müller ∗ ∗ ∗ University of Siegen, Management Information Science, ∗ Kohlbettstraße 57068 Siegen, GermanyScience, ∗ of Siegen, 15, Management Information ∗ University University of Information University of Siegen, Siegen, Management Management Information Science, Science, (e-mail: {dominik.kress, david.mueller}@uni-siegen.de). Kohlbettstraße 15, 57068 Siegen, Germany Kohlbettstraße Kohlbettstraße 15, 15, 57068 57068 Siegen, Siegen, Germany Germany (e-mail: (e-mail: {dominik.kress, {dominik.kress, david.mueller}@uni-siegen.de). david.mueller}@uni-siegen.de). (e-mail: {dominik.kress, david.mueller}@uni-siegen.de). Abstract: We consider a flexible job shop scheduling problem that incorporates machine operators and aims at makespan minimization. In a detailed overview of the related literature, Abstract: Abstract: We We consider consider aa a flexible flexible job job shop shop scheduling scheduling problem problem that that incorporates incorporates machine machine Abstract: We consider flexible job shop scheduling problem that incorporates machine we reveal the fact that the research in this field is mainly concerned with (meta-)heuristic operators and aims at makespan minimization. In aa detailed overview of the related literature, operators and aims at makespan minimization. In detailed overview of the related literature, operators and aims at makespan minimization. In a detailed overview of the related literature, approaches. Only few papers consider in exact approaches. In order to promote use of exact we reveal the fact that the research this field is mainly concerned with the (meta-)heuristic we fact that the research in this mainly concerned (meta-)heuristic we reveal reveal the the fact that to thefacilitate researchthe in evaluation this field field is is concernedofwith with (meta-)heuristic approaches and in few order of mainly the In performance heuristic approaches, approaches. Only papers consider exact approaches. order to promote the use of exact approaches. Only few exact approaches. In the of approaches. Onlymathematical few papers papers consider consider exact approaches.programming In order order to to promote promote the ause useconstraint of exact exact we present two models, a mixed-integer model and approaches and in order to facilitate the evaluation of the performance of heuristic approaches, approaches and and in in order order to to facilitate facilitate the the evaluation evaluation of of the the performance performance of of heuristic heuristic approaches, approaches, approaches programming model, that are analyzed and compared with a state-of-the-art heuristic in we present two mathematical models, aa mixed-integer programming model and aa constraint we present mathematical models, we present two twotests mathematical models, a mixed-integer mixed-integer programming programming model model and and a constraint constraint computational with a standard solver. programming model, that are analyzed and compared with a state-of-the-art heuristic in programming programming model, model, that that are are analyzed analyzed and and compared compared with with aa state-of-the-art state-of-the-art heuristic heuristic in in computational computational tests tests with with aa a standard standard solver. solver. computational tests with standard solver.

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Keywords: Scheduling, Flexible job shop, Workforce constraints, Mixed-integer programming, Constraint programming Keywords: Scheduling, Scheduling, Flexible job job shop, Workforce Workforce constraints, Mixed-integer Mixed-integer programming, Keywords: Keywords: Scheduling, Flexible Flexible job shop, shop, Workforce constraints, constraints, Mixed-integer programming, programming, Constraint programming programming Constraint Constraint programming 1. INTRODUCTION (2014) introduce a variable neighborhood search. A mixedinteger programming (MIP)neighborhood model and two metaheuristic 1. INTRODUCTION (2014) 1. INTRODUCTION INTRODUCTION (2014) introduce introduce aaa variable variable neighborhood neighborhood search. search. A A mixedmixed1. introduce variable search. A mixedThe job shop scheduling problem (JSP) is a well-known (2014) approaches (simulated annealing damping integer programming programming (MIP) (MIP) model modeland andvibration two metaheuristic metaheuristic integer and two integer programming (MIP) model and two metaheuristic scheduling setting that arises in many traditional man- approaches optimization) are proposed by Yazdani et al. (2015). Zhang The job shop scheduling problem (JSP) is a well-known (simulated annealing and vibration damping The scheduling problem (JSP) is well-known (simulated annealing and damping The job job shop shop scheduling problem (JSP) et is a aal., well-known approaches (simulated annealing and vibration vibration damping ufacturing systems (see, arises e.g., Błażewicz 2007). It approaches et al. (2015) introduce a hybrid discrete particle swarm scheduling setting that in many traditional manoptimization) are proposed by Yazdani et al. (2015). Zhang scheduling setting setting that that arises arises in in many traditional traditional manman- optimization) are by Yazdani (2015). Zhang scheduling optimization)algorithm are proposed proposed by Yazdani et et al. al. (2015). Zhang assumes that each job consists of amany set of operations, each by incorporating aparticle simulated anufacturing systems (see, e.g., Błażewicz Błażewicz et al., al., 2007). 2007). It optimization et al. (2015) introduce a hybrid discrete swarm ufacturing systems (see, e.g., et It et al. (2015) introduce a hybrid discrete particle swarm ufacturing systems (see, e.g., Błażewicz et al., 2007). It et al. (2015) introduce a hybrid discrete particle swarm of which that musteach be processed onofaaspecific machine. Modnealing approach with a variable neighborhood structure. assumes job consists set of operations, each optimization algorithm algorithm by by incorporating incorporating aa simulated simulated ananassumes that each of of each assumes thatmanufacturing each job job consists consists of a a set sethowever, of operations, operations, each optimization optimization algorithm by incorporating a simulatedfruit anern flexible systems, oftentimes Zheng and Wang (2016) aneighborhood knowledge-guided of which must be processed on a specific machine. Modnealing approach with aa develop variable structure. of which must be processed on a specific machine. Modnealing approach with variable neighborhood structure. of whichmulti-purpose must be processed on athat specific machine. Mod- nealing approach with a variable neighborhood structure. feature machines allow for processing fly optimization algorithm. Penga et al. (2018) propose a ern and Wang (2016) develop knowledge-guided fruit ern flexible flexible manufacturing manufacturing systems, systems, however, however, oftentimes oftentimes Zheng and Wang develop knowledge-guided fruit ern flexible manufacturing systems, however, oftentimes Zheng and Wang (2016) (2016) develop aa Devassia knowledge-guided fruit different types of manufacturing so that opera- Zheng genetic algorithm. Vallikavungal et al. (2018) feature multi-purpose multi-purpose machines operations, that allow allow for for processing fly optimization algorithm. Peng et al. (2018) propose feature machines that processing fly optimization optimization algorithm. algorithm. Peng Peng et et al. al. (2018) (2018) propose propose a a feature machines that allow for processing fly tions aremulti-purpose typically associated to aoperations, set of eligible machines. consider a WFJSP Vallikavungal that incorporates recovery times fora different types of manufacturing so that operagenetic algorithm. algorithm. Vallikavungal Devassia et al. al. (2018) different types of manufacturing operations, so that operagenetic Devassia et (2018) different types of manufacturing operations, so that operagenetic algorithm. Vallikavungal Devassia et al. (2018) This is taken account of in the flexible job shopmachines. schedul- consider the resources. For this problem, a general variable neightions are typically associated to a set of eligible aa WFJSP that incorporates recovery times for tions are associated to a of machines. that recovery times for tionsproblem are typically typically associated to a set set of eligible eligible machines. consider search a WFJSP WFJSP that incorporates incorporates recovery times for ing (FJSP) (Brucker and Schlie, 1990). Aschedulrecent consider borhood is introduced. A further special case of the This is taken account of in the flexible job shop the resources. For this problem, a general variable neighThis is is taken taken account account of of in in the the flexible flexible job job shop shop schedulschedul- the resources. For this problem, a general variable neighThis the resources. For this problem, a general variable neighsurvey of solution approaches for the FJSP is presented WFJSP is addressed by Wu et al. (2018). The authors ing problem problem (FJSP) (FJSP) (Brucker (Brucker and and Schlie, Schlie, 1990). 1990). A A recent recent borhood ing borhood search search is is introduced. introduced. A A further further special special case case of of the the ing Chaudhry problem (FJSP) (Brucker and Schlie,a1990). A recent search is introduced. further special case of the by and Khan (2016). Usually, machine needs borhood assume workers have theA to learn, thus survey of solution approaches for the FJSP is presented WFJSP that is addressed addressed by Wu Wu etability al. (2018). (2018). Theand authors survey of solution approaches for the FJSP is presented WFJSP is by et al. The authors survey of solution approaches for the FJSP is presented WFJSP is addressed by Wu et al. (2018). The authors to be operated by some(2016). machine operator, hereafter re- assume incorporate learning intoability the WFJSP. proby Chaudhry and a machine needs that workers effects have the to learn,They and thus by and Khan Khan (2016). Usually, Usually, a needs assume that have the to thus by Chaudhry Chaudhry Khan Usually, a machine machine needs assume that workers workers have the ability ability to learn, learn, and and thus ferred to as a and worker. We(2016). will denote FJSP settings that pose a hybrid genetic algorithm which combines a genetic to be operated by some machine operator, hereafter reincorporate learning effects into the WFJSP. They to be be operated operated by by some some machine machine operator, operator, hereafter hereafter rere- incorporate learning learning effects effects into into the the WFJSP. WFJSP. They They proproto proexplicitly incorporate WFJSPs. however, algorithm withgenetic a variable neighborhood search. Paksi and ferred to to as as a worker. worker.workers We will willby denote FJSPNote, settings that incorporate pose aa hybrid hybrid algorithm which combines combines a genetic genetic ferred a We denote FJSP settings that pose genetic algorithm which a ferred to as a worker. We will denote FJSP settings that pose a hybrid genetic algorithm which combines a genetic that machine scheduling problems that are concerned with Ma’ruf (2016) analyze the objective of minimizing the total explicitly incorporate workers by WFJSPs. Note, however, algorithm with with aa variable variable neighborhood search. search. Paksi Paksi and and explicitly incorporate workers by WFJSPs. Note, however, explicitly incorporate workers WFJSPs. Note, however, algorithm withintroduce a variableaneighborhood neighborhood search. Paksietand two types of scheduling resources, e.g., by machines and workers, are algorithm tardiness and genetic algorithm. Kress al. that machine problems that are concerned with Ma’ruf (2016) analyze the objective of minimizing the total that machine scheduling problems that are concerned with Ma’ruf (2016) analyze the objective of minimizing the total that machine scheduling that are concerned with (2019) Ma’ruf address (2016) analyze the objective ofaccount minimizing the total sometimes also referred problems to asmachines dual-resource constrained a WFJSP that takes of sequencetwo types of resources, e.g., and workers, are tardiness and introduce a genetic algorithm. Kress et two types types of of resources, resources, e.g., e.g., machines machines and and workers, workers, are are tardiness and and introduce introduce aa genetic genetic algorithm. algorithm. Kress Kress et et al. al. two al. (DRC) systems e.g., to Xuaset dual-resource al., 2011; Treleven, 1989). tardiness dependent setupaa times. They two objectives, minsometimes also (see, referred constrained (2019) address address WFJSP thatanalyze takes account account of sequencesequencesometimes also referred to as dual-resource constrained (2019) WFJSP that takes of sometimes also referred to as dual-resource constrained (2019) address a WFJSP that takes account of sequenceimizing thesetup makespan and minimizing theobjectives, total tardiness, (DRC) systems (see, e.g., Xu et al., 2011; Treleven, 1989). dependent times. They analyze two min(DRC) systems (see, setup times. They analyze two objectives, min(DRC) systemsOverview (see, e.g., e.g., Xu Xu et et al., al., 2011; 2011; Treleven, Treleven, 1989). 1989). dependent dependent setup times.MIP They analyze two objectives, min1.1 Literature present an integrated model, and propose exact and imizing the makespan and minimizing the total tardiness, imizing the makespan and minimizing the total tardiness, imizing the makespan and minimizing the total tardiness, heuristic decomposition based solution approaches. A few 1.1 Literature Literature Overview Overview present an integrated MIP model, and propose exact 1.1 present an an integrated integrated MIP MIP model, model, and and propose propose exact exact and and 1.1 Literature Overview and Most of the research on WFJSPs is concerned with the present researchers address multiple objectives for WFJSPs. Lang heuristic decomposition decomposition based based solution solution approaches. approaches. A A few few heuristic decomposition solution approaches. A few objective of minimizing the makespan. In this streamthe of heuristic Most and Li (2011) address abased multi-objective WFJSP, where address multiple objectives for WFJSPs. Lang Most of of the the research research on on WFJSPs WFJSPs is is concerned concerned with with the the researchers researchers address multiple objectives for WFJSPs. Lang Most of the research on WFJSPs is concerned with address multiple objectives for WFJSPs. Lang publications, Xianzhou and (2011) combine a geobjective of of minimizing minimizing the Zhenhe makespan. In this this stream of researchers delivery satisfaction, process cost, energy consumption Li (2011) address aa multi-objective WFJSP, where objective the makespan. In stream of and (2011) address multi-objective WFJSP, where objective of minimizing the makespan. In this of and and Li Li (2011) address a to multi-objective WFJSP, where netic algorithm with anand immune algorithm. Lei stream and aGuo and noise pollution are be minimized. The authors publications, Xianzhou Zhenhe (2011) combine gepublications, delivery satisfaction, satisfaction, process process cost, cost, energy energy consumption consumption publications, Xianzhou Xianzhou and and Zhenhe Zhenhe (2011) (2011) combine combine aa gege- delivery delivery satisfaction, process cost, energy consumption netic algorithm with an immune algorithm. Lei and Guo  also consider uncertain processing times andThe introduce a and noise pollution are to be minimized. The authors netic algorithm with an immune algorithm. Lei and Guo This work has been supported by the European Union and and noise pollution are to be minimized. authors netic algorithm with an immune algorithm. Lei and Guo and noise pollution are to be minimized. The authors heuristic which combines simulation and a genetic algo the state North Rhine-Westphalia through the European Fund for also consider uncertain processing times and introduce a has supported the European Union and  also consider uncertain processing times and introduce a This work work has been been (EFRD). supportedIt by by the European Union and  This also consider uncertain processing times and(2013) introduce a This work has been supported by European Union and Regional Development hasthe been conducted as part rithm. Liu et al. (2011) and Zhang et al. study heuristic which combines simulation and a genetic algothe state North Rhine-Westphalia through the European Fund for heuristic which combines simulation and a genetic algothe state North Rhine-Westphalia through the European Fund for heuristic which combines simulation and a genetic algothethe state North“EKPLO: Rhine-Westphalia through the European Fundund for a bi-criteria WFJSP, where both the makespan and the of project Echtzeitnahes kollaboratives Planen Regional Development (EFRD). rithm. Regional Development (EFRD). It It has has been been conducted conducted as as part part rithm. Liu Liu et et al. al. (2011) (2011) and and Zhang Zhang et et al. al. (2013) (2013) study study Regional Development (EFRD). It has been conducted as part rithm. Liu et al. (2011) and Zhang et al. (2013) study Optimieren” (EFRE-0800463). of the project “EKPLO: Echtzeitnahes kollaboratives Planen und a bi-criteria WFJSP, where both the makespan and the the of the project “EKPLO: Echtzeitnahes kollaboratives Planen und a bi-criteria WFJSP, where both the makespan and of the project “EKPLO: Echtzeitnahes kollaboratives Planen und a bi-criteria WFJSP, where both the makespan and the Optimieren” (EFRE-0800463).

Optimieren” (EFRE-0800463). Optimieren” 2405-8963 © (EFRE-0800463). 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 IFAC 96 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2019 IFAC 96 10.1016/j.ifacol.2019.11.144 Copyright © © 2019 2019 IFAC IFAC 96 Copyright 96

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Table 1. Overview of literature concerned with WFJSPs Publication

Objective

Math. modela

Approachb

Special characteristics

Gong et al. (2018a)

Makespan, total worker cost and green production factors Makespan, maximum workload of machines and total workload of all machines Makespan; Total tardiness

NLP

HGA

Multiple objectives

NLP

MEM

Multiple objectives

MIP

MIPS, DEC

Incorporation of sequencedependent setup times Multiple objectives, Uncertain processing times

Gong et al. (2018b) Kress et al. (2019)

-

GA

Lei and Guo (2014) Lei and Tan (2016) Liu et al. (2011) Paksi and Ma’ruf (2016) Peng et al. (2018) Vallikavungal Devassia et al. (2018)

Delivery satisfaction, process cost, energy consumption and noise pollution Makespan Makespan and total tardiness Makespan and production cost Total tardiness Makespan Makespan

MIP MIP

VNS LS HGA GA GA MIPS, VNS

Wu et al. (2018)

Makespan

NLP

HGA

Xianzhou and Zhenhe (2011) Yazdani et al. (2015) Zhang et al. (2013) Zhang et al. (2015) Zheng and Wang (2016)

Makespan Makespan Makespan and production cost Makespan Makespan

MIP NLP NLP MIP

GA MIPS, SA, VDO HPSO HPSO KF

Lang and Li (2011)

a b

Multiple objectives Multiple objectives Consideration of resource recovery constraints Consideration of learning effects for the workers Multiple objectives

The mathematical models are abbreviated as follows: MIP = Mixed-integer programming model; NLP = Non-linear programming model. The approaches are abbreviated as follows: DEC = Decomposition-based approach; KF = Knowledge-guided fruit fly optimization; (H)GA = (Hybrid) genetic algorithm; LS = Local search; MEM = Memetic algorithm; MIPS = Standard MIP solver; (H)PSO = (Hybrid) particle swarm optimization; SA = Simulated annealing; VNS = Variable neighborhood search; VDO = Vibration damping optimization

production cost are to be minimized. The former authors propose a hybrid genetic algorithm based on a Pareto approach, while a hybrid discrete particle swarm optimization algorithm is introduced by the latter authors. Lei and Tan (2016) address a WFJSP to simultaneously minimize the makespan and total tardiness. For this problem, the authors propose a local search approach with controlled deterioration. Gong et al. (2018a) consider green production indicators in a WFJSP and propose a hybrid genetic algorithm to minimize the makespan, the total worker cost and green production factors. Gong et al. (2018b) address a multi-objective WFJSP, where the makespan, the maximum workload of machines and the total workload of all machines are to be minimized. The authors propose a memetic algorithm.

for a WFJSP that aims to minimize the makespan. We evaluate the performance of both models by using CPLEX. Furthermore, we compare the performance of CPLEX on the CP model with a recently proposed heuristic approach. For an introduction to CP, we refer to Rossi et al. (2006). The remainder of this paper is organized as follows. In Section 2, we provide a formal definition of the WFJSP considered in this paper. The MIP model and the CP model are presented in Sections 3 and 4, respectively. The computational tests are subject of Section 5. A conclusion and future research directions are provided in Section 6. 2. PROBLEM DESCRIPTION The WFJSP under consideration (hereafter referred to as the WFJSP for the sake of simplicity) is composed of a set I of jobs, a set M of machines and a set W of workers. We assume that all jobs, machines and workers are available at time zero. For each job i ∈ I, we are given an ordered set of qi operations Oi = (i1 , . . . , iqi ). The ordering is such that, for any pair of operations ij , ik ∈ Oi with j < k, ik can only start to be processed after the processing of ij has finished. Each operation ij ∈ Oi , i ∈ I, must be processed without preemption on exactly one machine out of a set of eligible machines Mij ⊆ M . We define Mij ,kl := Mij ∩Mkl for all i, k ∈ I, ij ∈ Oi , kl ∈ Ok . An operation ij ∈ Oi of a job i ∈ I can only be processed on a machine, if exactly one worker of the set W is assigned to the operation for the entire processing time. We assume a heterogenous shop floor as well as workforce, so that the processing times of an operation may vary for different machines and workers. Therefore, we denote the processing time of an operation ij ∈ Oi of a job i ∈ I that is processed on an eligible machine m ∈ Mij by a worker w ∈ W by pm,w ∈ N+ . If ij some worker must not process an operation of a job on an eligible machine, the corresponding processing time is set

Table 1 summarizes the above literature overview. 1.2 Contribution and Overview Based on the above literature overview, we conclude that mainly metaheuristic approaches have been considered for solving WFJSPs. With respect to mathematical models, some researches introduce MIP or NLP formulations. However, only very few (Yazdani et al., 2015; Kress et al., 2019; Vallikavungal Devassia et al., 2018) actually make use of their models to design exact approaches or to evaluate their heuristic approaches in computational tests. Recently, commercial constraint programming (CP) optimizers have shown to perform remarkably well for solving scheduling problems. Puget (2013), for instance, presents results of the IBM ILOG CPLEX CP Optimizer for solving the FJSP for 7 well-known instance sets from the literature. Interestingly, to the best of the authors’ knowledge, there has been no attempt to provide a CP model as a benchmark for a WFJSP. In this paper, we therefore introduce two models, a MIP model and a CP model, 97

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to infinity. Each machine and each worker can process at most one operation at a time. We denote the completion time of an operation ij ∈ Oi of job i ∈ I by Cij and the completion time of job i ∈ I by Ci . A job i ∈ I is completed if all of its operations are completed, i.e., Ci = Ciqi .



m j ,kl

:=

m,w xi j

w xi ,k j l



:=

0,



:=



vij ,kl :=

if kl is processed on m directly after ij else

1, 0,

1, 0,

 1, 

0,

if ij and kl are processed by w else

∀ ij , kl ∈ V¯ , ij = kl , w ∈ W, (3)

∀ ij , kl ∈ V¯ , ij = kl .

ti j +

w j ,kl

s.t. ti q + i

m∈Mi

qi

m,w

xiq

i

m,w

· p iq

i

≤ Cmax

∀ i ∈ I,

·

m,w pi j



m j ,kl m,w xi j

yi

w j ,kl

xi

m j ,kl

yi

with m∈Mk

j

l

∀ ij ∈ V, m ∈ Mij ,

(9)

∀ ij ∈ V¯ , kl ∈ V¯ij , m ∈ Mij ,kl ,

(10)

j

− tkl

m yi ,k j l

m,w

xi

j

B

m,w

· pi

+



j

≤ tij+1

∀ ij ∈ V¯ with j ≤ qi − 1, (11)

∀ ij ∈ V¯ with j ≤ qi − 1, m,w

xk

l

m∈Mk

j

  j



w∈W

j

≤ xi

m,w

· pi

w∈W

j

m j ,kl

yi

∀ ij , kl ∈ V¯ , ij = kl , w ∈ W,

(13)

∀ ij , kl ∈ V¯ , ij = kl ,

(14)

− tkl

≤ B · vij ,kl

≤ 2 − vij ,kl − vkl ,ij



=

(12)

−1 w j ,kl

l

m,w

xi

˜ ij ∈V kl with m∈Mij



∀ ij , kl ∈ V¯ , ij = kl , w ∈ W, m,w

xk

l

∀ kl ∈ V¯ , m ∈ Mkl ,

(15) (16)

w∈W

∈ {0, 1}

∀ ij ∈ V, kl ∈ Vij , m ∈ Mij ,kl ,

(17)

∈ {0, 1}

∀ ij ∈ V¯ , m ∈ Mij , w ∈ W,

(18)

∈ {0, 1}

∀ ij , kl ∈ V¯ , ij = kl , w ∈ W,

(19)

∀ ij , kl ∈ V¯ , ij = kl ,

(20)

vij ,kl ∈ {0, 1} +

ti j ∈ N0

∀ ij ∈ V¯ ,

+

Cmax ∈ N0 .

(21) (22)

The objective (5) minimizes the makespan, which is bounded from below by constraints (6). Constraints (7) ensure that each operation is assigned to exactly one of its eligible machines, while inequalities (8) guarantee that there is at most one operation that is processed first on each machine. Constraints (9) ensure that the variables (1) are set to their correct values with respect to the specific machine that processes an operation ij ∈ V as well as its preceding and succeeding operations (including the operation of the dummy job) on this very machine. Constraints (10) prevent an overlapping of two consecutive operations ij ∈ Oi of job i ∈ I and kl ∈ Ok of job k ∈ I processed on the same machine m ∈ Mij ,kl . The precedence relations between consecutive operations ij , ij+1 ∈ Oi , i ∈ I, are taken account of in constraints (11). Constraints (12) are redundant to constraints (11), but have shown to improve the computational performance in our tests. Constraints (13) and (14) ensure that the variables (3) and (4) are set to one when needed. Based on these variables, constraints (15) prevent overlapping of operations that are assigned to the same worker. Constraints (16) connect the sequencing variables (1) with the worker variables (2). Finally, constraints (17)–(22) define the domains of the variables.

(2)

(4)

4. CONSTRAINT PROGRAMMING MODEL

(5)

 

kl ∈Vi

≤ tij+1

m∈Mi

xi

j

m,w

xi



−

≤ 1−

min j

m∈Mi

Let B be a large positive number. Then, a MIP model for WFJSP is as follows. min Cmax

m,w xi j

m∈Mi

pm,min ij

∀ ij ∈ V¯ , m ∈ Mij , w ∈ W,

m l ,ij

 



(8)

=0

tij + p i

∀ ij ∈ V, kl ∈ Vij , m ∈ Mij ,kl , (1)

if the processing of kl starts before the processing of ij finishes else



tij +

if ij is processed by w on m else

(7)

∀ m ∈ M, yk

w∈W

Now, for all operations ij ∈ V¯ , we define a continuous variable tij ∈ N+ 0 that represents the point in time at which operation ij is started to be processed, a continuous variable Cmax ∈ N+ 0 that represents the makespan, as well as the following binary variables:

yi

≤1



tij +

We furthermore define pm,min := minw∈W pm,w for all i ∈ ij ij

 1,

∀ kl ∈ V¯ ,

=1

˜ kl ∈V ij with m∈Mkl

Kress et al. (2019) propose a MIP model for a WFJSP with sequence-dependent setup times based on modelling approaches for the vehicle routing problem. We adjust this model to the simplified setting considered in this paper. To do so, we define a dummy job 0 with exactly one operation 01 and M01 = M . We set M01 ,ij = Mij ,01 = Mij for all i ∈ I and ij ∈ Oi . Moreover, as in Kress et al. (2019), we define the following sets:  • V := i∈I Oi ∪ {01 } • V¯ := V \ {01 } • Vij := V \ {ik |k ≤ j} for all ij ∈ V • V˜ij := V \ {ik |k ≥ j} for all ij ∈ V • V¯ij := Vij \ {01 } for all ij ∈ V := minm∈Mij

m 1 ,ij

y0

¯ ij ∈V

3. MIXED-INTEGER PROGRAMMING MODEL

I, ij ∈ Oi and m ∈ Mij , and set for all i ∈ I, ij ∈ Oi .

m j ,kl

yi

˜ ij ∈V kl m∈Mij ,kl

The problem is to allocate the operations to eligible machines and workers and to determine corresponding feasible sequences of the operations, such that the makespan Cmax := maxi∈I Ci is minimized. Based on the results of Lenstra and Rinnooy Kan (1979), it can easily be seen that this problem is strongly NP-hard.

pmin ij



The IBM ILOG CPLEX CP Optimizer provides a CP engine which enables the modelling and solving of scheduling problems. In order to cover temporal dimensions, the optimizer provides interval variables and sequence variables.

(6)

w∈W

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Table 3. Random testbed

The former variables are used to model the start and the end of the processing of the operations, while the latter variables are used to represent the sequencing decisions, i.e., orderings of interval variables. Moreover, the optimizer provides several special constraint types, which we assume the reader to be familiar with for the sake of brevity. Details are given in Laborie et al. (2018), as well as in the online documentation (IBM, 2016a,b) that includes detailed examples that make use of IBM’s Optimization Programming Language (OPL). For WFJSP, we define interval and sequencing variables as illustrated in Table 2.

Definition

IOP {ij }

Interval variable for each operation, i.e., for all i ∈ I, ij ∈ Oi Interval variable for each processing mode, i.e., each eligible combination of an operation ij ∈ Oi of job i ∈ I, a machine, a worker and a (finite) processing time Sequence variable for each machine m ¯ ∈ M; related to all interval variables IM O{ij ,m,w,p} with m = m ¯ Sequence variable for each worker w ¯ ∈ W; related to all interval variables IM O{ij ,m,w,p} with w = w ¯

IM O{ij ,m,w,p}

Sm ¯

Sw¯

Using the structures and notation provided by IBM’s CP Optimizer, a compact formulation of the objective and the special constraint types for WFJSP is as follows. min max(endOf (IOP {iq i∈I

i

} ))

.

(23)

. ∀ i ∈ I, j ≤ qi − 1,

(24)

s.t. endBef oreStart(IOP {ij } , IOP {ij+1 } )

alternative(IOP {ij } , all IM O{ij ,m,w,p} ). ∀ i ∈ I, ij ∈ Oi , noOverlap(Sm ) noOverlap(Sw )

. ∀ m ∈ M, . ∀ w ∈ W.

Set |I| |M | |W | qi

|Mij | |Wm | Set |I| |M | |W | qi

R1 R2 R3 R4 R5 R6 R7 R8

[1, 2] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2] [1, 2]

3 3 5 5 5 5 7 7

2 2 2 2 3 3 2 2

2 2 2 2 3 3 2 2

[2, 4] [4, 8] [2, 4] [4, 8] [2, 4] [4, 8] [2, 4] [4, 8]

2 2 2 2 2 3 2 2

R9 R10 R11 R12 R13 R14 R15 R16

7 7 10 10 10 20 20 20

3 3 5 5 5 10 10 10

3 3 5 5 5 10 10 10

[2, 4] [4, 8] [4, 8] [2, 10] [5, 10] [5, 10] [5, 15] [10, 15]

|Mij | |Wm | [1, 2] [1, 2] [1, 3] [1, 3] [2, 3] [1, 3] [2, 3] [2, 3]

2 3 3 3 3 5 5 5

features 10 instances with the number of jobs |I|, machines |M |, workers |W |, as well as the number of workers that can operate each machine (denoted by |Wm |, where Wm refers to the actual set of workers that is determined randomly) being fixed. The testbed was generated randomly. For each instance, the number of operations qi , i ∈ I, and the number of eligible machines |Mij | for operations ij ∈ Oi , i ∈ I, were drawn from uniform distributions over the intervals given in Table 3. The process of generating the processing times of the operations was as follows (cf. Kress et al., 2019). First, auxiliary integer parameters pij were drawn from a uniform distribution over [10, 100] for all i ∈ I and ij ∈ Oi . Based on these parameters, we generated varying processing times over the corresponding eligible machines m ∈ Mij by drawing auxiliary integer parameters pm ij from uniform distributions over the interval [0.9·pij , 1.1·pij ]. Finally, we incorporated dependencies on workers w ∈ W by drawing integer parameters pm,w ij , m ∈ Mw ∩ Mij , from uniform distributions over [0.9 · m pm ij , 1.1 · pij ]. Here, Mw defines the set of machines that can be operated by worker w ∈ W . It can easily be constructed based on the sets Wm , m ∈ M .

Table 2. Variables for the CP model Variables

97

In order to evaluate our results, we use a lower bound on the makespan introduced by Lei and Guo (2014), which we simplify to take account of the facts that all jobs are available at time zero and that |M | ≤ |I| and |W | ≤ |I| for all considered instances. For a given instance of WFJSP, the bound is defined as follows:          P P . , , pmin LB := max max  ij i∈I |M | |W | ij ∈Oi   Here, P := i∈I ij ∈Oi pmin ij . Note that, for the sake of brevity, we do not explicitly state the concrete instance in the definition of the bound.

(25) (26) (27)

The objective function (23) represents the minimization of the makespan. Constraints (24) capture the precedence constraints among the operations of the jobs. Constraints (25) guarantee that an eligible processing mode is chosen for each operation. Constraints (26) and (27) ensure that each machine and each worker processes at most one operation at a time. 5. COMPUTATIONAL STUDY

Let Inst be a given instance of WFJSP and denote by model (Inst) the (not necessarily optimal) makespan reCmax turned by CPLEX for the CP or the MIP model (model ∈ {CP, M IP }) within the time limit. Then, we define the quality ratio

In order to compare the performance of CPLEX on the MIP model and the CP model, we performed computaR CoreTM i7-4770 CPU, tional tests on a PC with an Intel running at 3.4 GHz, with 16 GB of RAM under a 64-bit version of Windows 8. The models were implemented in Java using Eclipse (Eclipse IDE for Java Developers (Oxygen 4.7)), where OPL was applied as a modelling language for the CP model. We used the MIP and CP solvers of IBM ILOG CPLEX in version 12.7 and the Java Runtime Environment (JRE) in version 1.8.0_191. If not stated otherwise, the time limit for each call of the CPLEX solvers was set to 3,600 seconds.

model Cmax (Inst) − LB LB as a measure for the quality of the corresponding solution.

Qmodel (Inst) := 100 ·

Table 4 presents the computational results for the instances of our random testbed. For each instance set, the table presents information about the average lower bound (column LBavg ), the number of test instances for which a feasible and optimal solution was obtained within the given time limit (columns “feas.” and “opt.”), the average quality ratios (columns “Qmodel avg ”, model ∈ {CP, M IP })

Our random testbed is composed of 16 instances sets, denoted by R1–R16 and summarized in Table 3. Each set 99

2019 IFAC MIM 98 Berlin, Germany, August 28-30, 2019

Dominik Kress et al. / IFAC PapersOnLine 52-13 (2019) 94–99

and the average runtimes (columns “tavg ”). Entries “tl” denote cases in which the time limit was reached for all instances of the set. Table 4. Performance of MIP and CP models on random testbed MIP

CP

Set

IP LBavg feas. opt. QM avg

tavg [s] feas. opt. QCP avg

tavg [s]

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

204.2 453.2 344.7 712.2 233.5 473.4 485.1 1029.1 323.4 668.2 563.6 570.6 664.1 669.5 861.8 1075

0.21 484.23 1415.75 tl 21.32 3308.13 tl tl 3264.19 tl tl tl -

0.07 2.77 5.05 2837.9 0.68 1348.27 1328.93 tl 366.9 tl tl 3260.93 tl tl tl tl

10 10 10 10 10 10 10 3 10 8 4 1 0 0 0 0

10 9 7 0 10 1 0 0 1 0 0 0 0 0 0 0

9.48 7.57 3.84 6.26 21.26 11.42 3.65 11.32 14.19 23.72 55.42 40.63 -

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

10 10 10 3 10 8 7 0 9 0 0 1 0 0 0 0

9.48 7.57 3.74 2.72 21.26 7.53 1.93 2 13.05 6.48 11.01 12.72 11.06 18.55 20.5 21.28

For the sake of comparability, we set the time limit for the CP solver to the average (as the authors initiate multiple runs of their algorithm) runtime of the KF approach as stated in Zheng and Wang (2016) for each instance. Moreover, we deactivated the parallel processing mode in the CPLEX CP Optimizer. The computational results are presented in Table 5. For each instance, the table presents Table 5. Performance of CP model for literature instances

It can be seen that the CP model clearly outperforms the MIP model in its ability to determine feasible solutions. In fact, it returned a feasible solution for all considered instances. Moreover, CP is the clear winner with respect to determining optimal solutions, average quality ratios and runtimes for the small instances in the sets R1–R10. For the sets of large instances (R11–R16), only one instance could be solved to optimality by using the CP model. In order to be able to analyze the influence of a varying staffing level (SL), defined as the ratio of the number of workers and the number of machines, on the performance of the CP solver for large instances, we ran additional tests. Here, we used modified instance sets R11–R16, where |W | was decreased before generating the instances to achieve staffing levels of 60% and 80%. The corresponding results are illustrated in Fig. 1. We observe that the average CP (SL 100%) CP (SL 80%) CP (SL 60%)

QCP avg

20

instances from the literature, MK1–M10 (Brandimarte, 1993) and DP1–DP12 (Dauzère-Pérès and Paulli, 1997), that have been adapted to include worker information by Lei and Guo (2014) by providing the sets W and Mw . Unfortunately, the generation of the processing times is not clearly described in Lei and Guo (2014), so that (based on the information given by Lei and Guo, 2014) we propose to draw the processing times pm,w ij , m ∈ Mw ∩ Mij , from ¯m a uniform distribution over the interval [¯ pm ij + δij ] for ij , p m all w ∈ W , i ∈ I and ij ∈ Oi . Here, p¯ij is the processing time stated for the literature instances, and δij is drawn from a uniform distribution over the interval [2, 8].

Inst.

KF LB QCP QKF avg tavg [s] Inst.

LB

KF QCP QKF avg tavg [s]

MK1 MK2 MK3 MK4 MK5 MK6 MK7 MK8 MK9 MK10

66 65 182 80 295 78 213 488 443 289

2881 2881 2881 2862 2832 2799 2843 2835 2824 2840 2786 2723

6.77 1.8 1.7 5.45 2.72 3.93 4.82 1.83 2.51 5.53 4.45 5.77

19.7 20 39.01 35 6.78 62.82 12.68 28.48 20.54 20.42

8.2 7.69 41 26.28 9.21 36.9 13.38 19.03 28.16 37.1

3.14 3.23 13.44 5.28 15.11 11.23 10.47 59.22 52.86 49.43

DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8 DP9 DP10 DP11 DP12

11.18 9.94 10.65 13.11 11.43 10.19 27.01 25.22 27.32 26.87 31.21 30.09

35.23 35.33 35.64 35.78 35.12 34.44 52.31 52.12 52.35 52.56 52.98 53.15

information about the lower bound (column LB), the quality ratio resulting from the CP solver (column “QCP ”) and the average quality ratio (column “QKF avg ”) as well as KF the average runtime (column “tavg ”) of the KF approach as stated by Zheng and Wang (2016) for the corresponding similar instance. The results indicate that the CP solver tends to outperform KF for most instances. This effect is particularly pronounced for the DP instances. 6. CONCLUSION AND FUTURE RESEARCH

10

16 R

15 R

14 R

13 R

12 R

R

11

0

Fig. 1. Staffing level impact (CP model) quality ratios decrease for smaller staffing levels. The above results indicate that heuristic approaches will need to be evaluated and compared against CP approaches. As an example, we will now analyze one of the most recent metaheuristic approaches, i.e., the knowledgeguided fruit fly optimization algorithm (denoted by KF) by Zheng and Wang (2016), that the authors find to be “more effective than the existing algorithms.” The study of Zheng and Wang (2016) is based on two sets of FJSP benchmark 100

In this paper, we have addressed a flexible job shop scheduling problem that aims to minimize the makespan and takes account of machine operators with differing skills. We have provided an overview of the related research and have presented a MIP model and a CP model that we have then compared by using the standard solvers provided by CPLEX. We found that the CP solver clearly outperforms the MIP solver for the considered modelling approaches. The CP solver tends to provide high quality solutions within reasonable time. It was especially interesting to see that it also tends to outperform a state-ofthe-art metaheuristic approach. For future research, one will therefore have to provide (meta-)heuristics and exact approaches that prove to be competitive when compared with the use of standard CP solvers. Moreover, detailed benchmark sets will need to be published so that fair comparisons become possible.

2019 IFAC MIM Berlin, Germany, August 28-30, 2019

Dominik Kress et al. / IFAC PapersOnLine 52-13 (2019) 94–99

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