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Balancing non-bottleneck stations using simple assembly line balancing models ⁎
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Balancing Balancing non-bottleneck non-bottleneck stations stations using using Balancing non-bottleneck stations using Balancing non-bottleneck stations using simple assembly line balancing models simple assembly line balancing models simple assembly line balancing models simple assembly line balancing models ∗ ∗ ∗,∗∗ ∗,∗∗ Lukas Lingitz ∗∗ Viola Gallina ∗∗ Csaba Kardos ∗,∗∗
∗ Viola Gallina ∗ Csaba Kardos ∗,∗∗ Lukas Lingitz Lukas Tam´ Lingitz Gallina Kardos ∗,∗∗ ∗ Viola∗∗∗ ∗ Csaba ∗∗∗ ∗,∗∗∗∗ ass Koltai Koltai Wilfried Sihn ∗,∗∗∗∗ ∗∗∗ ∗,∗∗∗∗ Lukas Tam´ Lingitz Gallina Kardos ∗,∗∗ a Wilfried Sihn ∗ Viola∗∗∗ ∗ Csaba as Koltai Wilfried Sihn ∗,∗∗∗∗ Lukas Tam´ Lingitz Viola∗∗∗ Gallina Csaba Kardos ∗,∗∗∗∗ Tam´ as Koltai ∗∗∗ Wilfried Sihn ∗,∗∗∗∗ Tam´ a s Koltai Wilfried Sihn ∗ ∗ Fraunhofer Austria Research GmbH,Theresianumgasse A-1040 ∗ Fraunhofer Austria Research GmbH,Theresianumgasse 27, 27, ∗ Austria Research GmbH,Theresianumgasse 27, A-1040 A-1040 ∗ Fraunhofer Vienna, Austria, e-mail: [email protected], Austria Research GmbH,Theresianumgasse 27, A-1040 Vienna, Austria, e-mail: [email protected], ∗ Fraunhofer Vienna, Austria, e-mail: [email protected], Fraunhofer Austria Research GmbH,Theresianumgasse 27, A-1040 [email protected], [email protected] Vienna, Austria, e-mail: [email protected], [email protected] [email protected], [email protected] ∗∗ [email protected], Vienna, Austria, e-mail: [email protected], ∗∗ Centre of Excellence Excellence in in Production Production Informatics and and Control, Control, ∗∗ [email protected], [email protected] of Informatics ∗∗ Centre of Excellence in Production Informatics andAcademy Control, of [email protected], [email protected] ∗∗ Centre Institute for Computer Science and Control, Hungarian Centre of Excellence in Production Informatics and Control, of Institute for Computer Science and Control, Hungarian Academy ∗∗ Institute for Computer Science and Control, Hungarian Academy Centre of Excellence in Production Informatics and Control, of Sciences, Kende str. 13-17, H-1111 Budapest, Hungary Institute for Computer Science and Control, Hungarian Academy of Sciences, Kende str. 13-17, H-1111 Budapest, Hungary Sciences, Kende str. 13-17, H-1111 Budapest, Hungary ∗∗∗ Institute for Computer Science and Control, Hungarian Academy of ∗∗∗ Department of Management and Business Economics, Budapest ∗∗∗ Sciences, Kende str. 13-17, H-1111 Budapest, Hungary ∗∗∗ Department of Management and Business Economics, Budapest Department of Management and Business Economics, Budapest Sciences, Kende str. 13-17, H-1111 Budapest, Hungary ∗∗∗ University of Technology Technology and Economics, Economics, Magyar tud´ osok sok krt. krt. 2., 2., Department of Management and Business Economics, University of and Magyar tud´ o ∗∗∗ University of Budapest, Technology and Economics, Magyar tud´ osokBudapest krt. 2., Department of Management ande-mail: Business Economics, Budapest H-1117, Hungary, [email protected] University of Technology and Economics, Magyar tud´ o sok krt. 2., H-1117, Budapest, Hungary, e-mail: [email protected] e-mail: [email protected] ∗∗∗∗ University of Budapest, TechnologyHungary, and Economics, Magyaroftud´ osok krt. 2., ∗∗∗∗H-1117, Vienna University of Technology, Institute Management ∗∗∗∗ Budapest, Hungary, e-mail: [email protected] ∗∗∗∗H-1117, Vienna University of Technology, Institute of Management Vienna University Hungary, of Technology, Institute of Management H-1117, Budapest, e-mail: [email protected] ∗∗∗∗ Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: Vienna University of Technology, Institute of Management Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: ∗∗∗∗ Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: Vienna University of Technology, Institute of Management [email protected] Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: [email protected] [email protected] Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: [email protected] [email protected] Abstract: One of the most important objectives in assembly line balancing is to evenly Abstract: One of the most important objectives in line balancing is to Abstract:the One of the most important objectives in assembly assembly line balancing is developed to evenly evenly distribute workload of the stations. In the literature different measures were Abstract: One of the most important objectives in assembly line balancing is developed to evenly distribute the workload of the stations. In the literature different measures were distribute the workload of the stations. In the literature different measures were developed Abstract: One of the most important objectives in assembly line balancing is to evenly for evaluating balance of an assignment. However, because of the complexity of the distribute the the workload ofquality the stations. In the literature different measures were developed for the balance quality of assignment. However, because of the complexity of the for evaluating evaluating balance of an an with assignment. However, because ofline the balancing. complexity of this the distribute the the workload ofquality thestudied stations. In exact the literature different measures were developed defined indices they are rarely rarely methods of because assembly In for evaluating the balance quality of an assignment. However, of the complexity of the defined indices they are studied with exact methods of assembly line balancing. In this defined indices they are rarely studied with exact methods of assembly line balancing. In this for evaluating the balance quality of an assignment. However, because of the complexity ofwith the paper a new balancing index is proposed by the authors and a new balancing algorithm defined they areindex rarelyis with exact methods of aaassembly line balancing. In with this paper aa indices new proposed by the authors and new algorithm paper methods new balancing balancing isstudied proposed byby the authors and new balancing balancing algorithm defined indices they areindex rarely studied with exact methods of assembly line balancing. In with this exact is introduced and illustrated means of aaand case study. paper a new balancing index is proposed by the authors a new balancing algorithm with exact methods is introduced and illustrated by means of case study. exact methods is introduced and illustrated by means of a case study. paper a new balancing index is proposed by the authors and a new balancing algorithm with Copyright 2019 IFAC exact methods and illustrated by means of a case study. ccc is introduced Copyright 2019 IFAC Copyright 2019 IFAC exact is introduced and illustrated by means of a case study. © 2019,methods IFAC c (International Copyright 2019 IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. c Copyright 2019 IFAC Keywords: Simple assembly line balancing problem (SALBP), MILP, CP, workload balancing Keywords: Keywords: Simple Simple assembly assembly line line balancing balancing problem problem (SALBP), (SALBP), MILP, MILP, CP, CP, workload workload balancing balancing Keywords: Simple assembly line balancing problem (SALBP), MILP, CP, workload balancing Keywords: Simple assembly line balancing problem (SALBP), MILP, CP, workload balancing 1. INTRODUCTION INTRODUCTION geometry of of figures figures represents represents different different product product types types of of 1. geometry 1. INTRODUCTION geometry of figures (Becker represents different product types of the same product and Scholl (2004)). One of 1. INTRODUCTION geometry of figures represents different product types of the same product (Becker and Scholl (2004)). One the same product (Becker and Scholl (2004)). One 1. INTRODUCTION geometry of figures represents different product types of the basic assembly line balancing models is the simple Assembly lines lines are are used used broadly broadly in in mass mass production production syssys- the basic same assembly product (Becker and Scholl (2004)). of line balancing models is the simple Assembly basic assembly line model-1 balancing models is which the One simple same line product (Becker and (SALBM-1) Scholl (2004)). One of Assembly lines are of used broadly instations. mass production sys- the assembly balancing minitems. They consist several work Semi-finished the basic assembly line balancing models is the simple assembly line balancing model-1 (SALBM-1) which miniAssembly lines are used broadly in mass production systems. They consist of several work stations. Semi-finished assembly line balancing model-1 (SALBM-1) which minithe basic assembly line balancing models is the simple tems. They consist of several work stations. Semi-finished Assembly lines are used broadly in mass production sysmizes the the line number of workstations workstations with given givenwhich production products pass through these stations where a of tasks assembly balancing model-1 (SALBM-1) minimizes number of with production tems. They consist of several work stations. Semi-finished products pass through these where aa set set of mizes theand number ofgiven workstations with given production line balancing model-1 (SALBM-1) which miniproducts pass through these stations stations wheremust set be of tasks tasks tems. They consist of precedence several workrelations stations. Semi-finished quantity with production time, consequently, with predetermined exe- assembly mizes the number of workstations with given production quantity and with given production time, consequently, products pass through these stations where a set of tasks with predetermined precedence relations must be exequantity with production time, consequently, theand number ofgiven workstations with given production with predetermined precedence relations must exe- mizes products pass through these stations where a set be of tasks tasks with given cycle time. The other important basic model in cuted. Assembly line balancing is the assignment of with given production time, basic consequently, with given cycle time. The other important model in with predetermined precedenceis relations must be exe- quantity cuted. Assembly line balancing the of with givenand cycle time. The other important basic modelthe in and with given production time, consequently, cuted. Assembly by lineoptimizing balancing the assignment assignment of tasks tasks with predetermined precedenceaisspecified relations must be exe- quantity assembly line balancing is SALBM-2 that minimizes to workstations objective funcwith given cycle time. The other important basic model in assembly line balancing is SALBM-2 that minimizes the cuted. Assembly line balancing is the assignment of tasks to workstations by optimizing a specified objective funcassembly line balancing is SALBM-2 that minimizes the with given cycle time. The other important basic model ina to workstations by optimizing a specified objective funccuted. Assembly line balancing is the assignment of tasks cycle time, therefore, maximizes production quantity for tion (eg. maximizing utilization of the workers or minassembly line balancing is SALBM-2 that minimizes the cycle time, therefore, maximizes production quantity for a to workstations by optimizing a specified objective function (eg. maximizing utilization of the workers or mincycle time, therefore, maximizes production quantityScholl for a assembly line balancing is SALBM-2 that minimizes the tion (eg. maximizing utilization of the workers or minto workstations by optimizing a specified objective funcgiven number of workstations (see Baybars (1986); imizing cycle time) under under certain constraints, such as given cycle time, therefore, maximizes production quantity for a number of workstations (see Baybars (1986); Scholl tion (eg. maximizing utilization of the workers or minimizing cycle time) certain constraints, such as given number of workstations (see Baybars (1986); Scholl cycle time, therefore, maximizes production quantity for a imizing cycle time) under certain constraints, such as tion (eg. maximizing utilization of availability the workersoforspecial min- and Becker (2004)). technological assembly sequence or given number of workstations (see Baybars (1986); Scholl and Becker (2004)). imizing cycle time) under certain constraints, such as technological assembly sequence or availability of special and Becker (2004)). given number of workstations (see Baybars (1986); Scholl technological assembly sequence or availability of special imizing cycle time) under certain constraints, such as equipment or workers. The problem was introduced in and Becker (2004)). technological sequence or availability of special equipment or workers. The problem was in the optimal number of workstations and the minand Becker (2004)). equipment or assembly workers. The first problem was introduced introduced in Besides technological assembly sequence or availability ofa special Besides the optimal number of and the 1955 by Salveson and was formulated as binary Besides the optimal numberequally of workstations workstations and the minminequipment or workers. The problem was introduced in 1955 by Salveson and was first formulated as a binary imal cycle time objectives, balanced workstations thetime optimal numberequally of workstations and the min1955 by Salveson and by was first formulated as a binary equipment or problem workers. The problem was introduced in Besides imal cycle objectives, balanced workstations programming Bowman (1960). imal cycle time objectives, equally balanced workstations Besides the optimal number of workstations and the min1955 by Salveson and was first formulated as a binary programming problem by Bowman (1960). mightcycle also time be important important for the management. management. Stations objectives, for equally balanced workstations might also be the Stations programming problem Bowman (1960). as a binary imal 1955 by Salveson and by was first formulated mightcycle also time be important for the management. Stations imal objectives, equally balanced workstations programming problem by Bowman (1960). The high demand for individualization of products encourmight also be important for the management. Stations programming problem by Bowman (1960). The demand for of encourThe high high demand for individualization individualization of products products encouraged the demand implementation of mixed-model mixed-model assembly lines. might also be important for the management. Stations The high for individualization of products encouraged the implementation of assembly lines. agedhigh the implementation of mixed-model assembly lines. The demand for individualization of different products encourMixed assembly lines can product aged the implementation ofmanufacture mixed-model assembly lines. Mixed assembly lines can manufacture different product Mixed assembly lines can manufacture different product aged the implementation of mixed-model assembly lines. types of the same product on the same assembly line Mixed assembly lines can manufacture different product types the product on the assembly line types of of the same same product on the same same assembly line Mixed assembly lines can manufacture different product without setup time, ensuring a flexible production envitypes of the same product on the same assembly line without setup time, ensuring a flexible production enviwithout setup time, ensuring a flexible production envitypes of the same product on the same assembly line ronment.setup Withtime, someensuring compromises, the production mixed assembly assembly without a flexible environment. With some compromises, the mixed ronment. Withproblem someensuring compromises, the production mixed assembly without setup time, atransformed flexible enviline balancing can be into a simple ronment. With some compromises, the mixed assembly line balancing problem can be transformed into a simple line balancing problem can be transformed into a simple ronment. Withbalancing some compromises, the mixed assembly assembly line problem (SALBP) (Becker and line balancing problem can be transformed into a simple assembly line balancing problem (SALBP) (Becker and assembly line Reginato balancing problem (SALBP) (Becker and line balancing problem et can be(2016)). transformed into a simple Scholl (2004); al. Figure 1. illustrates assembly line balancing problem (SALBP) (Becker and Scholl (2004); Reginato et al. (2016)). Figure 1. illustrates Scholl (2004); Reginato et al. (2016)). Figure 1. illustrates assembly line balancing problem (SALBP) (Becker and the different different types of assembly assembly lines, Figure where the the different Scholl (2004);types Reginato et al. (2016)). 1. illustrates the of lines, where different the different of assembly lines, Figure where the different Scholl (2004);types Reginato et al. (2016)). 1. illustrates the different types of assembly lines, where the different The the different types of assembly lines, where the different authors would like to acknowledge the financial support of
authors would to the support of The The authors would like like to acknowledge acknowledge the financial financial support of the Austria Research Promotion Agency and European the Austria Research Promotion Agency (FFG) (FFG) and of of support European The authors would like to acknowledge the financial of the Austria Research Promotion Agency (FFG) and of European The authors would like toStaProZell acknowledge the financial support of Commission for funding the (pr. No. 864870) and H2020 the Austria Research Promotion Agency (FFG) and of European Commission for funding the StaProZell (pr. No. 864870) and H2020 Commission for the StaProZell (pr.(FFG) No. 864870) H2020 Fig. 1. Different types of assembly lines the Austria Promotion Agency and ofand European EPIC (grant Research no.funding 739592) projects respectively. Fig. Commission for funding the StaProZell (pr. No. 864870) and H2020 EPIC (grant no. 739592) projects respectively. Fig. 1. 1. Different Different types types of of assembly assembly lines lines EPIC (grant for no.funding 739592)the projects respectively. Commission StaProZell (pr. No. 864870) and H2020 Fig. 1. Different types of assembly lines EPIC (grant no. 739592) projects respectively. Fig. 1. Different types of assembly lines EPIC (grant no. 739592) projects respectively. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2019, 2019 IFAC IFAC 1450Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 1450 Peer review© of International Federation of Automatic Copyright ©under 2019 responsibility IFAC 1450Control. Copyright © 2019 IFAC 1450 10.1016/j.ifacol.2019.11.400 Copyright © 2019 IFAC 1450
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with — near — the same workload can contribute to a stable production output, to high capacity utilization and to smooth and well controlled operation. The importance of workload smoothing in assembly lines has been studied by a few authors, however, the related research is quite scarce and all of the approaches apply heuristic procedures ˙ (Azizo˘ glu and Imat (2018)). The aim of this study is to equally balance non bottleneck stations with optimization techniques based on several versions of the SALBPs. The paper is structured as follows. In Section 2, a short overview is given regarding workload balancing in SALBP. Section 3 introduces the balancing index, a new balancing measure suggested by the authors. The algorithm that is used for balancing non-bottleneck stations in SALBP is presented in Section 4. Section 5 illustrates the application of the algorithm with the help of a case study — using two different approaches, namely binary programming and constraint programming. Conclusions and further research directions are summarized in Section 6. s 2. WORKLOAD BALANCING IN THE LITERATURE Several measures were developed for the evaluation of the solution quality of assembly line balancing problems, such as balance delay, line efficiency, line time and smoothness index (see Grzechca (2013)). The Smoothness Index (SI) or flow index, introduced in 1965 by Moodea and Young, and also discussed for example in Rekiek et al. (2002)), is one of the most frequently used measure for evaluating the balance quality of an assignment of a SALBP. It is defined as the square root of the sum of the squared differences between the maximal station time and the workstation loads of each station. Esmaeilbeigi et al. (2015) use three different linearization methods to minimize the SI in SALBP-E — where the cycle time and the number of station are the decision variables. Kucukkoc et al. (2018) and Yagmahan (2011) apply ant colony optimization for minimizing the smoothnes index in mixed assembly line balancing problem. Zacharia and Nearchou (2016) employ a multi-objective evolutionary algorithm for the solution of minimizing cycle time parallel to SI. Because of its nonlinear form and the consequent computational difficulty, it is rarely studied by researchers and all studies propose ˙ heuristic procedures (see Azizo˘ glu and Imat (2018). 3. DEFINING THE BALANCING INDEX In an assembly line, tasks are assigned to workstations. Tasks are numbered in continuously increasing order. The number i assigned to a task is called the task index. We refer to a task either by its name or by its task index. Those tasks which are not succeeded by any other tasks are called last tasks. The index set of last tasks is denoted by L. The time necessary to finish a task is denoted by ti . Workstations are also numbered in continuously increasing order. The first workstation is numbered 1 and the last workstation is numbered N. The number j assigned to a workstation is called the workstation index. Workstations are referred in the paper by the workstation index. Prior to task assignment, an assumption must be made about the possible number of stations. The number of stations used in the model is J. That is, J is the number of stations used in the mathematical model, and N is the
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actual number of stations used in the line. The assignment of tasks to workstations is expressed with the xij binary decision variable. If task i is assigned to workstation j, then xij =1, otherwise xij =0. The sum of the task times assigned to a workstation is the station time (sj ), and can be calculated as follows, sj =
I
ti xij
i=1
∀j = 1, ..., J
(1)
The classic SALBP-2 minimizes cycle time. This way the station time of the bottleneck stations will decrease and tasks will be more evenly spread. We may easily get, however, solutions in which the cycle time has the smallest possible value, but there are significant differences between the station times of the non-bottleneck stations. Operation managers try to avoid large differences between station times and strive for an evenly distributed workload for all stations. To characterize the balancing of station times a new measure of balancing is proposed. Let us assume, that the station time of all stations is listed in non-increasing order, and s[j] denotes the j -th station time in this order. Consequently, s [1] is the largest station time, which is the cycle time of the system. In case of J number of stations, s[J] is the smallest station time. The proposed Balancing Index (BI) is defined as follows, BI =
K
(s[1+k] − s[J−k] )
(2)
k=0
where K = J/2 if J is an even number, and K = (K −1)/2 if J is an uneven number. In equation 2, the differences of the largest and the smallest, the second largest and the second smallest, the third largest and the third smallest, etc. station times are summed. In the next section, an algorithm is proposed, which systematically decreases the differences in the sum of expression 2. 4. BALANCING ALGORITHM FOR NON-BOTTLENECK STATIONS The algorithm is based on the repeated solutions of some variations of SALBP-2. First, however a SALBP-1 is solved to get the minimal number of workstations necessary to perform the production task. The notations used for the description of the models are given in Table 1. The following binary linear programming formulation of SALBP-1 is used in this paper, M inN (3)
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I i=1
ti xij ≤ Tc
J
xij = 1
j=1
J j=1
j(xqj − xpj ) ≥ 0 N≥
J
jxij
j=1
(j < LJi , j > U Ji ) → xij = 0
∀j = 1, ..., J
(4)
∀i = 1, ..., I
(5)
(p, q) ∈ R
(6)
i∈L
(7)
∀j = 1, ..., J.
(8)
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Table 1. Summary of notations applied in the models Notation Indices: i p q j k z v Parameters: I J K N ti sj s[k] Tc T LJi UJi Sets: R L Pi Fi BS BT Decision variables: xij (z) TM ax (z) TM in
Definition = = = = = = =
index of tasks (i=1,...,I ), index of a subset of tasks, index of a subset of tasks, index of workstations (j =1,...,J ), index of the rank of station time in an order, z -th element of the sum used for calculating the balancing index, index of the elements of the balancing index,
= = = = = = = = = = =
number of tasks, number of workstations in the mathematical model, number of station time pares, actual number of workstations applied, time necessary to perform task i (task time), time necessary to perform all tasks at station j (station time), k -th station time in the ordered list of station times, cycle time of the assembly line, total available time for production, the earliest workstation which can be used by task i as a consequence of preceding tasks of task i, the latest workstation which can be used by task i as a consequence of succeeding tasks of task i,
= = = = = =
(p;q) ∈ R if task p immediate precedes task q, set of final tasks; that is, i ∈ L if task i does not precede any other tasks, index set of those tasks which must be finished before task i is started, index set of those tasks which cannot be started before task i is finished, index set of the ignored stations, index set of the ignored tasks.
= = =
0-1 decision variable; if xij =1, then task i is assigned to station j, otherwise xij =0, minimal value of the highest station time when the z-th element of the balancing index is calculated, maximal value of the smallest station time when the z-th element of the balancing index is calculated.
The objective of the model is to minimize the number of stations used in the actual system; that is, to minimize the largest index (N ) belonging to a station with task assignment. The right-hand-side of constraints (7) determines the indices of those workstations which perform last tasks. The largest such index is minimized by objective function (3). If each of these indices is smaller than or equal to N and N is minimized, then the index of the final workstation, consequently, the number of workstations, is minimized. Cycle time constraints are expressed by constraints (4), that is, the sum of task times of the assigned tasks is not allowed to exceed the cycle time. As a consequence of constraints (5), each task is assigned to one of the workstations. Constraints (6) express the precedence relations. The difference in the bracket is equal to -1, 0 or 1 for each workstation. Let us assume that task p precedes task q. Since task p must be assigned to an earlier or to the same workstation as task q, the weighted sum of these differences is always greater than or equal to 0 if the weights are the indices of the corresponding workstations. Finally, the number of variables can be reduced by constraints (8) (see Koltai and Tatay (2013); Koltai et al. (2014)). Solving the model defined by (3)-(8) the number of workstations for a given cycle time is minimized.
The right-hand side of cycle time constraints (4) is modified because the cycle time is now a variable. Consequently, (1) in constraints (4), Tc is substituted by variable TM ax . If the upper bound of the station times is minimized, then the minimum of the maximal station times is obtained. The upper bound of station times is set as follows,
In contrast to SALBP-1, SALBP-2 minimizes the cycle time for a given number of workstations (N ), that is, the objective function is as follows,
M ax(TM in ).
(1)
M in(TM ax ).
(9)
I i=1
(1)
ti xij ≤ TM ax
∀j = 1, ..., J.
(10)
The performance of each operation (5) and the precedence constraints (6) are the same as in SALBP-1. That is, SALBP-2 is determined by objective function (9) and constraints (5), (6) and (10). The model of SALBP-2 can be completed with constraints (8) applying techniques which exclude unfeasible assignments. In order to decrease the difference of the maximal and the minimal station times, the minimal station time must be maximized, while the minimum of the maximal station time, determined by SALBP-2, is satisfied. To determine (1) the maximal value of the smallest station time (TM in ) a SALBP-3 is defined. In this new problem, the objective function maximizes the smallest station time, that is, (1)
(1)
(11)
If TM in is a lower bond of the station times, and its value is maximized then the optimal value of (11) is the largest possible value of the smallest station time. Consequently, the following lower bounds are applied,
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(1)
ti xij ≥ TM in
∀j = 1, ..., J
those tasks, which are already assigned to the ignored stations, then constraints (12) are modified as follows,
(12)
I
The upper bound of the station times, determined by the SALBP-2 must also be satisfied (10). Using the above formulation, the steps of the algorithm are the following, Step 1. An SALBP-1 is solved to get the minimal number of stations (N ) necessary to perform the production plan. Step 2. An SALBP-2 is solved to get the minimal cycle time (Tc ). This cycle time is the smallest possible value of (1) the largest station time (TM ax ). Step 3. An SALBP-3 is defined which maximizes the minimum of the station times with a cycle time upper bound for all the station times. The result is the largest (1) possible value of the smallest station time (TM in ). As a consequence of Step 2 and 3 the difference between the smallest and the largest station times is minimized, that is, (1)
(1)
s[1] − s[J] = TM ax − TM in
(13)
is the smallest possible. Equation (13) provides the first element of the sum defined by equation (2). Let z denote the z -th element of the sum in equation (2). The first element (z = 1) of equation (2) was obtained by solving a SALBP-2 and a SALBP-3. To determine the subsequent elements of this sum, a modified SALBP-2 and a modified SALBP-3 must be solved consecutively. In Step 4 and Step 5 these modified models are defined and solved several times one after the other, always using the results obtained for the immediately preceding (z − 1) element. Step 4. A modified minimization SALBP-2 is solved with the following conditions: • The stations which had the minimal time when the (v) previous (z − 1) elements were calculated (TM ax , ∀v = 1, ..., Z − 1), are ignored with all the tasks assigned to them. If BS is the index set of those stations, which are ignored in the calculations, and BT is the index set of those tasks, which are already assigned to the ignored stations, then constraints (10) are modified as follows, I i=1
(1)
ti xij ≤ TM ax
∀j ∈ BS, i ∈ BT
(14)
Ignoring stations and tasks at the calculation assumes, that the ignored tasks are assigned to the corresponding ignored stations. • A lower bound for all the remaining station times is set, which is equal to the value found for the (z−1)-th element of (2), that is, I i=1
(z−1)
ti xij ≥ TM in
∀j ∈ BS, i ∈ BT
(15)
Step 5. A modified maximization SALBP-3 is solved with the following constraints: • The stations which have the maximal time in all (v) previous (z − 1) elements (TM in , ∀v = 1, ..., Z − 1) are ignored with all the tasks assigned to them. If BS is the index set of those stations, which are ignored in the calculations, and BT is the index set of
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i=1
(z)
ti xij ≥ TM in
∀j ∈ BS, i ∈ BT
(16)
Ignoring these tasks at the calculation assumes, that the ignored tasks are assigned to the corresponding ignored stations. • An upper bound for all the remaining station times is set, which is equal to the value found for the (z−1)-th element of (2), that is, I i=1
(z−1)
ti xij ≤ TM ax
∀j ∈ BS, i ∈ BT
(17)
As a consequence of Step 4 and 5 the difference between (z) (z) the z-th smallest (TM in ) and the z-th largest (TM ax ) station time is minimized, that is, (z)
(z)
s[z] − s[J−z] = TM ax − TM in (18) is the smallest possible. Step 4 and 5 can be repeated by always eliminating from the calculation the stations (z−1) and the corresponding tasks, with the minimal (TM in ) (z−1) and the maximal (TM ax ) station times found by the previously solved modified SALBP-2 and SALBP-3. The calculation stops when neither the modified SALBP-2 nor the modified SALBP-3 provides different minimum and maximum values found in the previous step and trivially at z = Int(J/2). The value of BI improves after each calculation. It can be easily proved, that as the algorithm progresses (the value of z increases) the value of BI monotonously decreases. When the value of the s[z] −s[J−z] difference decreases as a result of Step 4 and 5, tasks are reallocated among the workstations. This reallocation of tasks cannot increase any difference calculated in the preceding steps, as a consequence of the applied upper and lower bounds of the corresponding station times defined by constraints (15) and (17). Consequently, the value of BI either decreases, or does not change. In this later case, however, the algorithm stops. 5. CASE STUDY To illustrate the behaviour of the algorithm let us consider a case study from the machine industry. The problem is solved with SALBP, that was transferred from a mixed model assembly line. The applyed case was inspired by a problem at an injection molding machine assembly company — but the original data were changed. The company produces three different product types (I-III.) on the production line. The deterministic processing times (given in minutes) of each task required by the assembly operations are summarized in Table 2. Altogether we have 37 different tasks (A-DT). If a tasks is not needed to a given final product type, it is indicated with ”-” in the table. The mix of products to be assembled is presented in the last row of the table. The weighted average processing times of the tasks are calculated in the last column of Table 2 using the given volumes as weights. The equivalent precedence diagram is presented in Figure 2. In case of SALBP-1 the number of workstations must be minimized. It is assumed that the line works with a
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Table 2. Task times and the weighted average of task times (minute) Task A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA AB AC AD AE AF AG AH AI AJ DT Quantity
Product I
Product II
Product III
24 6 6 12 6 3.6 7.2 3 6 6 6 9 45 60 15 9 12 27 12 9 27 9 6 4.8 12 15 15 6 1.2 15 4.8 48
24 36 6 3 4.8 15 6 12 21 12 42 9 60 15 9 6 30 12 9 27 9 6 4.8 6 30 12 24 9 15 12 20.4 18 7.2 24
24 12 12 18 1.8 4.8 3 18 12 9 12 58 15 60 15 9 6 24 12 9 27 9 6 9.6 6 24 15 15 12 1.2 12 7.2 24
Weighted time 24 6 3 18 9 3 6 6 3 9 3 6 12 6 45 6 60 15 9 9 27 12 9 27 9 6 6 9 15 9 6 6 15 9 6 15 6
Fig. 2. The precedence relationship between the tasks of the case study, a color-map illustrates the processing time for each task.
one hour cycle time. The optimum value of the objective function of SALBP-1 (Step 1) is 8. Consequently, the minimal number of workstations required to assembly the required quantities of the three different product types is 8 — if the cycle time is one hour. The SALBP-2 (Step 2) minimizes the cycle time with a given number of workstations. If we assume, that the assembly line (1) consists of 8 stations, the optimal cycle time, TM ax is one hour. Note, that the minimal cycle time is equivalent to the highest station time. In our case, task Q with its 60 minutes processing time is the bottleneck of the assembly line and determines the cycle time of the line. SALBP-3 (Step 3) determines the maximal value of the (1) smallest station time (TM in ). The optimal value for the highest lower bound of station times is 54 minutes. That is, the 37 tasks with the given processing times and precedence relations can not be assigned to stations in a way, that the difference between the minimal (54 minutes) and the maximal (60 minutes) station times is smaller than 6 minutes. Table 3 summarizes the results of each step gained by LINGO Software (Version 17). It can be seen in Table 3, that the minimal value of the difference between the highest and smallest station times is 6 minutes
(S3: s[1] − s[8] = 6). The minimal difference between the second largest and the second smallest station times can be reduced in Step 4 and Step 5 (S4 and S5 in Table 3). In Step 4, task Q is assigned to station 5 and tasks B, E, K, P, AB, AC and AE are assigned to station 3. These tasks and stations are ignored at the calculation of the difference between the second highest and second lowest station times. The optimal value of objective function (2) of the modified SALBP-2, TM ax is 57 minutes (Step 4). After solving the modified SALBP-3 in Step 5 (the (2) maximal value of the second lowest station time, TM in is 54 minutes) the difference between the second highest and second smallest station times can not be reduced any more, the value of the BI remains 12 and the algorithm stops. Table 3. Summary of the results Iteration S1 S2 S3 S4 S5
Min sj 45 51 54 54 54
Max sj 60 60 60 60 60
s[1] − s[8] 15 9 6 6 6
s[2] − s[7] 9 6 6 3 3
BI 30 24 18 12 12
In order to further validate the algorithm, the problem of the case study was modelled and solved using discrete optimization. The integer processing times allowed formulating a Constraint Programming (CP) representation of the model. Examples can be found in the literature for using CP to solve SALBP-1 and SALBP-2 (Topaloglu et al. (2012)), thus the complete description of the model is not given here. Using the effective and expressive constraints and objective modeling capabilities of CP the effects of the restrictions made in Step 4 and Step 5 were inspected. The CP
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model combines Step 4 and Step 5 into one step with less restrictions on the task assignment. According to (15), (14), (16), (17), in the z = α iteration of the algorithm the tasks assigned to a station in a previous z < α iteration are ignored and they remain unchanged. However, in the CP model by applying a sorting global constraint (Older et al. (1995)) to the array of S = {sj } station times, the resulting s[z] − s[J−z] station time differences from each z = α iteration are added as a constraint for the subsequent z = α + 1 iteration. The calculated station times are displayed in Fig. 3, with the objective function being s[z] − s[J−z] for each iteration and with assigning every task in every iteration. It can be seen that the optimal solution for station times after the last step is equal with the solution obtained by the subsequent solution of the modified SALBPs proposed in Section 4.
Fig. 3. Station times after each iteration of the algorithm.
6. CONCLUSION AND OUTLOOK In this paper a new balancing index is introduced, which measures the distribution of workstation workloads. This balancing index is monotonous decreasing as the distribution of workload improves, consequently, it can be a good indicator in an optimization algorithm. The authors proposed a balancing algorithm for equally balance the workload of non-bottleneck stations with binary and constraint programming. Finally, the application of the proposed balancing index and algorithm were demonstrated with a case study, where a mixed model assembly line problem was simplified to a SALBP. Further research should be undertaken to investigate the feasibility of the solution of the original problem gained by SALBP. Instead of ignoring stations and tasks in step 4 and 5 of the algorithm, formulating the minimal upper and maximal lower bounds as constraints is a topic of our further research as well. ACKNOWLEDGEMENTS The authors would like to acknowledge the financial support of the Austria Research Promotion Agency (FFG) and of European Commission for funding the StaProZell (pr. No. 864870) and H2020 EPIC (grant no. 739592) projects respectively.
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REFERENCES ˙ Azizo˘glu, M., Imat, S., 2018. Workload smoothing in simple assembly line balancing. Computers & Operations Research 89, 51–57. Baybars, I., 1986. A survey of exact algorithms for the simple assembly line balancing problem. Management Science 32, 909–932. Becker, C., Scholl, A., 2004. A survey on problems and methods in generalized assembly line balancing. European Journal of Operational Research 168 (3), 694–715. Bowman, E. H., 1960. Assembly-line balancing by linear programming. Massachusetts Institute of Technology, 385–389. Esmaeilbeigi, R., Naderi, B., Charkhgard, P., 2015. The type e simple assembly line balancing problem: A mixed integer linear programming formulation. Computers & Operations Research 64, 168–177. Grzechca, W., 2013. Idle times analysis in two-sided assembly line balancing problem. IFAC Proceedings Volumes 46 (9), 1720–1725. Koltai, T., Kall´o, N., Tatay, V., 2014. Application of the results of simple assembly line balancing models in practice: the case of a bicycle manufacturer. Internation Jounal of Computer Integrated Manufacturing 27 (9), 887–898. Koltai, T., Tatay, V., 2013. Formulation of workforce skill constraints in assembly line balancing models. Optimization and engineering 14 (4), 529–545. Kucukkoc, I., Li, Z., Karaoglan, A. D., Zhang, D. Z., 2018. Balancing of mixed-model two-sided assembly lines with underground workstations: A mathematical model and ant colony optimization algorithm. International Journal of Production Economics 205, 228–243. Older, W., Swinkels, G., Emden, M., 1995. Getting to the real problem: experience with bnr prolog. In: in OR, Practical Application of Prolog, PAP95. Citeseer, pp. 1–14. Reginato, G., Anzanello, M. J., Kahmann, A., Schmidt, L., 2016. Balanceamento de linha de montagem mista em cen´arios com distintos mix de produtos. Gest˜ ao & Produ¸c˜ao 23 (2), 294–307. Rekiek, B., Delchambre, A., Dolgui, A., Bratcu, A., 2002. Assembly line design: A survey. IFAC Proceedings Volumes 35 (1), 155–166. Scholl, A., Becker, C., 2004. State-of-the-art exact and heuristic solution procedures for simple assembly line balancing. European Journal of Operational Research 168 (3), 666–693. Topaloglu, S., Salum, L., Supciller, A. A., 2012. Rule-based modeling and constraint programming based solution of the assembly line balancing problem. Expert Systems with Applications 39 (3), 3484–3493. Yagmahan, B., 2011. Mixed-model assembly line balancing using a multi-objective ant colony optimization approach. Expert Systems with Applications 38 (10), 12453–12461. Zacharia, P. T., Nearchou, A. C., 2016. A population-based algorithm for the bi-objective assembly line worker assignment and balancing problem. Engineering Applications of Artificial Intelligence 49, 1–9.
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Balancing Balancing non-bottleneck non-bottleneck stations stations using using Balancing non-bottleneck stations using Balancing non-bottleneck stations using simple assembly line balancing models simple assembly line balancing models simple assembly line balancing models simple assembly line balancing models ∗ ∗ ∗,∗∗ ∗,∗∗ Lukas Lingitz ∗∗ Viola Gallina ∗∗ Csaba Kardos ∗,∗∗
∗ Viola Gallina ∗ Csaba Kardos ∗,∗∗ Lukas Lingitz Lukas Tam´ Lingitz Gallina Kardos ∗,∗∗ ∗ Viola∗∗∗ ∗ Csaba ∗∗∗ ∗,∗∗∗∗ ass Koltai Koltai Wilfried Sihn ∗,∗∗∗∗ ∗∗∗ ∗,∗∗∗∗ Lukas Tam´ Lingitz Gallina Kardos ∗,∗∗ a Wilfried Sihn ∗ Viola∗∗∗ ∗ Csaba as Koltai Wilfried Sihn ∗,∗∗∗∗ Lukas Tam´ Lingitz Viola∗∗∗ Gallina Csaba Kardos ∗,∗∗∗∗ Tam´ as Koltai ∗∗∗ Wilfried Sihn ∗,∗∗∗∗ Tam´ a s Koltai Wilfried Sihn ∗ ∗ Fraunhofer Austria Research GmbH,Theresianumgasse A-1040 ∗ Fraunhofer Austria Research GmbH,Theresianumgasse 27, 27, ∗ Austria Research GmbH,Theresianumgasse 27, A-1040 A-1040 ∗ Fraunhofer Vienna, Austria, e-mail: [email protected], Austria Research GmbH,Theresianumgasse 27, A-1040 Vienna, Austria, e-mail: [email protected], ∗ Fraunhofer Vienna, Austria, e-mail: [email protected], Fraunhofer Austria Research GmbH,Theresianumgasse 27, A-1040 [email protected], [email protected] Vienna, Austria, e-mail: [email protected], [email protected] [email protected], [email protected] ∗∗ [email protected], Vienna, Austria, e-mail: [email protected], ∗∗ Centre of Excellence Excellence in in Production Production Informatics and and Control, Control, ∗∗ [email protected], [email protected] of Informatics ∗∗ Centre of Excellence in Production Informatics andAcademy Control, of [email protected], [email protected] ∗∗ Centre Institute for Computer Science and Control, Hungarian Centre of Excellence in Production Informatics and Control, of Institute for Computer Science and Control, Hungarian Academy ∗∗ Institute for Computer Science and Control, Hungarian Academy Centre of Excellence in Production Informatics and Control, of Sciences, Kende str. 13-17, H-1111 Budapest, Hungary Institute for Computer Science and Control, Hungarian Academy of Sciences, Kende str. 13-17, H-1111 Budapest, Hungary Sciences, Kende str. 13-17, H-1111 Budapest, Hungary ∗∗∗ Institute for Computer Science and Control, Hungarian Academy of ∗∗∗ Department of Management and Business Economics, Budapest ∗∗∗ Sciences, Kende str. 13-17, H-1111 Budapest, Hungary ∗∗∗ Department of Management and Business Economics, Budapest Department of Management and Business Economics, Budapest Sciences, Kende str. 13-17, H-1111 Budapest, Hungary ∗∗∗ University of Technology Technology and Economics, Economics, Magyar tud´ osok sok krt. krt. 2., 2., Department of Management and Business Economics, University of and Magyar tud´ o ∗∗∗ University of Budapest, Technology and Economics, Magyar tud´ osokBudapest krt. 2., Department of Management ande-mail: Business Economics, Budapest H-1117, Hungary, [email protected] University of Technology and Economics, Magyar tud´ o sok krt. 2., H-1117, Budapest, Hungary, e-mail: [email protected] e-mail: [email protected] ∗∗∗∗ University of Budapest, TechnologyHungary, and Economics, Magyaroftud´ osok krt. 2., ∗∗∗∗H-1117, Vienna University of Technology, Institute Management ∗∗∗∗ Budapest, Hungary, e-mail: [email protected] ∗∗∗∗H-1117, Vienna University of Technology, Institute of Management Vienna University Hungary, of Technology, Institute of Management H-1117, Budapest, e-mail: [email protected] ∗∗∗∗ Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: Vienna University of Technology, Institute of Management Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: ∗∗∗∗ Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: Vienna University of Technology, Institute of Management [email protected] Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: [email protected] [email protected] Science, Theresianumgasse 27, A-1040 Wien, Austria, e-mail: [email protected] [email protected] Abstract: One of the most important objectives in assembly line balancing is to evenly Abstract: One of the most important objectives in line balancing is to Abstract:the One of the most important objectives in assembly assembly line balancing is developed to evenly evenly distribute workload of the stations. In the literature different measures were Abstract: One of the most important objectives in assembly line balancing is developed to evenly distribute the workload of the stations. In the literature different measures were distribute the workload of the stations. In the literature different measures were developed Abstract: One of the most important objectives in assembly line balancing is to evenly for evaluating balance of an assignment. However, because of the complexity of the distribute the the workload ofquality the stations. In the literature different measures were developed for the balance quality of assignment. However, because of the complexity of the for evaluating evaluating balance of an an with assignment. However, because ofline the balancing. complexity of this the distribute the the workload ofquality thestudied stations. In exact the literature different measures were developed defined indices they are rarely rarely methods of because assembly In for evaluating the balance quality of an assignment. However, of the complexity of the defined indices they are studied with exact methods of assembly line balancing. In this defined indices they are rarely studied with exact methods of assembly line balancing. In this for evaluating the balance quality of an assignment. However, because of the complexity ofwith the paper a new balancing index is proposed by the authors and a new balancing algorithm defined they areindex rarelyis with exact methods of aaassembly line balancing. In with this paper aa indices new proposed by the authors and new algorithm paper methods new balancing balancing isstudied proposed byby the authors and new balancing balancing algorithm defined indices they areindex rarely studied with exact methods of assembly line balancing. In with this exact is introduced and illustrated means of aaand case study. paper a new balancing index is proposed by the authors a new balancing algorithm with exact methods is introduced and illustrated by means of case study. exact methods is introduced and illustrated by means of a case study. paper a new balancing index is proposed by the authors and a new balancing algorithm with Copyright 2019 IFAC exact methods and illustrated by means of a case study. ccc is introduced Copyright 2019 IFAC Copyright 2019 IFAC exact is introduced and illustrated by means of a case study. © 2019,methods IFAC c (International Copyright 2019 IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. c Copyright 2019 IFAC Keywords: Simple assembly line balancing problem (SALBP), MILP, CP, workload balancing Keywords: Keywords: Simple Simple assembly assembly line line balancing balancing problem problem (SALBP), (SALBP), MILP, MILP, CP, CP, workload workload balancing balancing Keywords: Simple assembly line balancing problem (SALBP), MILP, CP, workload balancing Keywords: Simple assembly line balancing problem (SALBP), MILP, CP, workload balancing 1. INTRODUCTION INTRODUCTION geometry of of figures figures represents represents different different product product types types of of 1. geometry 1. INTRODUCTION geometry of figures (Becker represents different product types of the same product and Scholl (2004)). One of 1. INTRODUCTION geometry of figures represents different product types of the same product (Becker and Scholl (2004)). One the same product (Becker and Scholl (2004)). One 1. INTRODUCTION geometry of figures represents different product types of the basic assembly line balancing models is the simple Assembly lines lines are are used used broadly broadly in in mass mass production production syssys- the basic same assembly product (Becker and Scholl (2004)). of line balancing models is the simple Assembly basic assembly line model-1 balancing models is which the One simple same line product (Becker and (SALBM-1) Scholl (2004)). One of Assembly lines are of used broadly instations. mass production sys- the assembly balancing minitems. They consist several work Semi-finished the basic assembly line balancing models is the simple assembly line balancing model-1 (SALBM-1) which miniAssembly lines are used broadly in mass production systems. They consist of several work stations. Semi-finished assembly line balancing model-1 (SALBM-1) which minithe basic assembly line balancing models is the simple tems. They consist of several work stations. Semi-finished Assembly lines are used broadly in mass production sysmizes the the line number of workstations workstations with given givenwhich production products pass through these stations where a of tasks assembly balancing model-1 (SALBM-1) minimizes number of with production tems. They consist of several work stations. Semi-finished products pass through these where aa set set of mizes theand number ofgiven workstations with given production line balancing model-1 (SALBM-1) which miniproducts pass through these stations stations wheremust set be of tasks tasks tems. They consist of precedence several workrelations stations. Semi-finished quantity with production time, consequently, with predetermined exe- assembly mizes the number of workstations with given production quantity and with given production time, consequently, products pass through these stations where a set of tasks with predetermined precedence relations must be exequantity with production time, consequently, theand number ofgiven workstations with given production with predetermined precedence relations must exe- mizes products pass through these stations where a set be of tasks tasks with given cycle time. The other important basic model in cuted. Assembly line balancing is the assignment of with given production time, basic consequently, with given cycle time. The other important model in with predetermined precedenceis relations must be exe- quantity cuted. Assembly line balancing the of with givenand cycle time. The other important basic modelthe in and with given production time, consequently, cuted. Assembly by lineoptimizing balancing the assignment assignment of tasks tasks with predetermined precedenceaisspecified relations must be exe- quantity assembly line balancing is SALBM-2 that minimizes to workstations objective funcwith given cycle time. The other important basic model in assembly line balancing is SALBM-2 that minimizes the cuted. Assembly line balancing is the assignment of tasks to workstations by optimizing a specified objective funcassembly line balancing is SALBM-2 that minimizes the with given cycle time. The other important basic model ina to workstations by optimizing a specified objective funccuted. Assembly line balancing is the assignment of tasks cycle time, therefore, maximizes production quantity for tion (eg. maximizing utilization of the workers or minassembly line balancing is SALBM-2 that minimizes the cycle time, therefore, maximizes production quantity for a to workstations by optimizing a specified objective function (eg. maximizing utilization of the workers or mincycle time, therefore, maximizes production quantityScholl for a assembly line balancing is SALBM-2 that minimizes the tion (eg. maximizing utilization of the workers or minto workstations by optimizing a specified objective funcgiven number of workstations (see Baybars (1986); imizing cycle time) under under certain constraints, such as given cycle time, therefore, maximizes production quantity for a number of workstations (see Baybars (1986); Scholl tion (eg. maximizing utilization of the workers or minimizing cycle time) certain constraints, such as given number of workstations (see Baybars (1986); Scholl cycle time, therefore, maximizes production quantity for a imizing cycle time) under certain constraints, such as tion (eg. maximizing utilization of availability the workersoforspecial min- and Becker (2004)). technological assembly sequence or given number of workstations (see Baybars (1986); Scholl and Becker (2004)). imizing cycle time) under certain constraints, such as technological assembly sequence or availability of special and Becker (2004)). given number of workstations (see Baybars (1986); Scholl technological assembly sequence or availability of special imizing cycle time) under certain constraints, such as equipment or workers. The problem was introduced in and Becker (2004)). technological sequence or availability of special equipment or workers. The problem was in the optimal number of workstations and the minand Becker (2004)). equipment or assembly workers. The first problem was introduced introduced in Besides technological assembly sequence or availability ofa special Besides the optimal number of and the 1955 by Salveson and was formulated as binary Besides the optimal numberequally of workstations workstations and the minminequipment or workers. The problem was introduced in 1955 by Salveson and was first formulated as a binary imal cycle time objectives, balanced workstations thetime optimal numberequally of workstations and the min1955 by Salveson and by was first formulated as a binary equipment or problem workers. The problem was introduced in Besides imal cycle objectives, balanced workstations programming Bowman (1960). imal cycle time objectives, equally balanced workstations Besides the optimal number of workstations and the min1955 by Salveson and was first formulated as a binary programming problem by Bowman (1960). mightcycle also time be important important for the management. management. Stations objectives, for equally balanced workstations might also be the Stations programming problem Bowman (1960). as a binary imal 1955 by Salveson and by was first formulated mightcycle also time be important for the management. Stations imal objectives, equally balanced workstations programming problem by Bowman (1960). The high demand for individualization of products encourmight also be important for the management. Stations programming problem by Bowman (1960). The demand for of encourThe high high demand for individualization individualization of products products encouraged the demand implementation of mixed-model mixed-model assembly lines. might also be important for the management. Stations The high for individualization of products encouraged the implementation of assembly lines. agedhigh the implementation of mixed-model assembly lines. The demand for individualization of different products encourMixed assembly lines can product aged the implementation ofmanufacture mixed-model assembly lines. Mixed assembly lines can manufacture different product Mixed assembly lines can manufacture different product aged the implementation of mixed-model assembly lines. types of the same product on the same assembly line Mixed assembly lines can manufacture different product types the product on the assembly line types of of the same same product on the same same assembly line Mixed assembly lines can manufacture different product without setup time, ensuring a flexible production envitypes of the same product on the same assembly line without setup time, ensuring a flexible production enviwithout setup time, ensuring a flexible production envitypes of the same product on the same assembly line ronment.setup Withtime, someensuring compromises, the production mixed assembly assembly without a flexible environment. With some compromises, the mixed ronment. Withproblem someensuring compromises, the production mixed assembly without setup time, atransformed flexible enviline balancing can be into a simple ronment. With some compromises, the mixed assembly line balancing problem can be transformed into a simple line balancing problem can be transformed into a simple ronment. Withbalancing some compromises, the mixed assembly assembly line problem (SALBP) (Becker and line balancing problem can be transformed into a simple assembly line balancing problem (SALBP) (Becker and assembly line Reginato balancing problem (SALBP) (Becker and line balancing problem et can be(2016)). transformed into a simple Scholl (2004); al. Figure 1. illustrates assembly line balancing problem (SALBP) (Becker and Scholl (2004); Reginato et al. (2016)). Figure 1. illustrates Scholl (2004); Reginato et al. (2016)). Figure 1. illustrates assembly line balancing problem (SALBP) (Becker and the different different types of assembly assembly lines, Figure where the the different Scholl (2004);types Reginato et al. (2016)). 1. illustrates the of lines, where different the different of assembly lines, Figure where the different Scholl (2004);types Reginato et al. (2016)). 1. illustrates the different types of assembly lines, where the different The the different types of assembly lines, where the different authors would like to acknowledge the financial support of
authors would to the support of The The authors would like like to acknowledge acknowledge the financial financial support of the Austria Research Promotion Agency and European the Austria Research Promotion Agency (FFG) (FFG) and of of support European The authors would like to acknowledge the financial of the Austria Research Promotion Agency (FFG) and of European The authors would like toStaProZell acknowledge the financial support of Commission for funding the (pr. No. 864870) and H2020 the Austria Research Promotion Agency (FFG) and of European Commission for funding the StaProZell (pr. No. 864870) and H2020 Commission for the StaProZell (pr.(FFG) No. 864870) H2020 Fig. 1. Different types of assembly lines the Austria Promotion Agency and ofand European EPIC (grant Research no.funding 739592) projects respectively. Fig. Commission for funding the StaProZell (pr. No. 864870) and H2020 EPIC (grant no. 739592) projects respectively. Fig. 1. 1. Different Different types types of of assembly assembly lines lines EPIC (grant for no.funding 739592)the projects respectively. Commission StaProZell (pr. No. 864870) and H2020 Fig. 1. Different types of assembly lines EPIC (grant no. 739592) projects respectively. Fig. 1. Different types of assembly lines EPIC (grant no. 739592) projects respectively. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2019, 2019 IFAC IFAC 1450Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 1450 Peer review© of International Federation of Automatic Copyright ©under 2019 responsibility IFAC 1450Control. Copyright © 2019 IFAC 1450 10.1016/j.ifacol.2019.11.400 Copyright © 2019 IFAC 1450
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with — near — the same workload can contribute to a stable production output, to high capacity utilization and to smooth and well controlled operation. The importance of workload smoothing in assembly lines has been studied by a few authors, however, the related research is quite scarce and all of the approaches apply heuristic procedures ˙ (Azizo˘ glu and Imat (2018)). The aim of this study is to equally balance non bottleneck stations with optimization techniques based on several versions of the SALBPs. The paper is structured as follows. In Section 2, a short overview is given regarding workload balancing in SALBP. Section 3 introduces the balancing index, a new balancing measure suggested by the authors. The algorithm that is used for balancing non-bottleneck stations in SALBP is presented in Section 4. Section 5 illustrates the application of the algorithm with the help of a case study — using two different approaches, namely binary programming and constraint programming. Conclusions and further research directions are summarized in Section 6. s 2. WORKLOAD BALANCING IN THE LITERATURE Several measures were developed for the evaluation of the solution quality of assembly line balancing problems, such as balance delay, line efficiency, line time and smoothness index (see Grzechca (2013)). The Smoothness Index (SI) or flow index, introduced in 1965 by Moodea and Young, and also discussed for example in Rekiek et al. (2002)), is one of the most frequently used measure for evaluating the balance quality of an assignment of a SALBP. It is defined as the square root of the sum of the squared differences between the maximal station time and the workstation loads of each station. Esmaeilbeigi et al. (2015) use three different linearization methods to minimize the SI in SALBP-E — where the cycle time and the number of station are the decision variables. Kucukkoc et al. (2018) and Yagmahan (2011) apply ant colony optimization for minimizing the smoothnes index in mixed assembly line balancing problem. Zacharia and Nearchou (2016) employ a multi-objective evolutionary algorithm for the solution of minimizing cycle time parallel to SI. Because of its nonlinear form and the consequent computational difficulty, it is rarely studied by researchers and all studies propose ˙ heuristic procedures (see Azizo˘ glu and Imat (2018). 3. DEFINING THE BALANCING INDEX In an assembly line, tasks are assigned to workstations. Tasks are numbered in continuously increasing order. The number i assigned to a task is called the task index. We refer to a task either by its name or by its task index. Those tasks which are not succeeded by any other tasks are called last tasks. The index set of last tasks is denoted by L. The time necessary to finish a task is denoted by ti . Workstations are also numbered in continuously increasing order. The first workstation is numbered 1 and the last workstation is numbered N. The number j assigned to a workstation is called the workstation index. Workstations are referred in the paper by the workstation index. Prior to task assignment, an assumption must be made about the possible number of stations. The number of stations used in the model is J. That is, J is the number of stations used in the mathematical model, and N is the
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actual number of stations used in the line. The assignment of tasks to workstations is expressed with the xij binary decision variable. If task i is assigned to workstation j, then xij =1, otherwise xij =0. The sum of the task times assigned to a workstation is the station time (sj ), and can be calculated as follows, sj =
I
ti xij
i=1
∀j = 1, ..., J
(1)
The classic SALBP-2 minimizes cycle time. This way the station time of the bottleneck stations will decrease and tasks will be more evenly spread. We may easily get, however, solutions in which the cycle time has the smallest possible value, but there are significant differences between the station times of the non-bottleneck stations. Operation managers try to avoid large differences between station times and strive for an evenly distributed workload for all stations. To characterize the balancing of station times a new measure of balancing is proposed. Let us assume, that the station time of all stations is listed in non-increasing order, and s[j] denotes the j -th station time in this order. Consequently, s [1] is the largest station time, which is the cycle time of the system. In case of J number of stations, s[J] is the smallest station time. The proposed Balancing Index (BI) is defined as follows, BI =
K
(s[1+k] − s[J−k] )
(2)
k=0
where K = J/2 if J is an even number, and K = (K −1)/2 if J is an uneven number. In equation 2, the differences of the largest and the smallest, the second largest and the second smallest, the third largest and the third smallest, etc. station times are summed. In the next section, an algorithm is proposed, which systematically decreases the differences in the sum of expression 2. 4. BALANCING ALGORITHM FOR NON-BOTTLENECK STATIONS The algorithm is based on the repeated solutions of some variations of SALBP-2. First, however a SALBP-1 is solved to get the minimal number of workstations necessary to perform the production task. The notations used for the description of the models are given in Table 1. The following binary linear programming formulation of SALBP-1 is used in this paper, M inN (3)
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I i=1
ti xij ≤ Tc
J
xij = 1
j=1
J j=1
j(xqj − xpj ) ≥ 0 N≥
J
jxij
j=1
(j < LJi , j > U Ji ) → xij = 0
∀j = 1, ..., J
(4)
∀i = 1, ..., I
(5)
(p, q) ∈ R
(6)
i∈L
(7)
∀j = 1, ..., J.
(8)
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Table 1. Summary of notations applied in the models Notation Indices: i p q j k z v Parameters: I J K N ti sj s[k] Tc T LJi UJi Sets: R L Pi Fi BS BT Decision variables: xij (z) TM ax (z) TM in
Definition = = = = = = =
index of tasks (i=1,...,I ), index of a subset of tasks, index of a subset of tasks, index of workstations (j =1,...,J ), index of the rank of station time in an order, z -th element of the sum used for calculating the balancing index, index of the elements of the balancing index,
= = = = = = = = = = =
number of tasks, number of workstations in the mathematical model, number of station time pares, actual number of workstations applied, time necessary to perform task i (task time), time necessary to perform all tasks at station j (station time), k -th station time in the ordered list of station times, cycle time of the assembly line, total available time for production, the earliest workstation which can be used by task i as a consequence of preceding tasks of task i, the latest workstation which can be used by task i as a consequence of succeeding tasks of task i,
= = = = = =
(p;q) ∈ R if task p immediate precedes task q, set of final tasks; that is, i ∈ L if task i does not precede any other tasks, index set of those tasks which must be finished before task i is started, index set of those tasks which cannot be started before task i is finished, index set of the ignored stations, index set of the ignored tasks.
= = =
0-1 decision variable; if xij =1, then task i is assigned to station j, otherwise xij =0, minimal value of the highest station time when the z-th element of the balancing index is calculated, maximal value of the smallest station time when the z-th element of the balancing index is calculated.
The objective of the model is to minimize the number of stations used in the actual system; that is, to minimize the largest index (N ) belonging to a station with task assignment. The right-hand-side of constraints (7) determines the indices of those workstations which perform last tasks. The largest such index is minimized by objective function (3). If each of these indices is smaller than or equal to N and N is minimized, then the index of the final workstation, consequently, the number of workstations, is minimized. Cycle time constraints are expressed by constraints (4), that is, the sum of task times of the assigned tasks is not allowed to exceed the cycle time. As a consequence of constraints (5), each task is assigned to one of the workstations. Constraints (6) express the precedence relations. The difference in the bracket is equal to -1, 0 or 1 for each workstation. Let us assume that task p precedes task q. Since task p must be assigned to an earlier or to the same workstation as task q, the weighted sum of these differences is always greater than or equal to 0 if the weights are the indices of the corresponding workstations. Finally, the number of variables can be reduced by constraints (8) (see Koltai and Tatay (2013); Koltai et al. (2014)). Solving the model defined by (3)-(8) the number of workstations for a given cycle time is minimized.
The right-hand side of cycle time constraints (4) is modified because the cycle time is now a variable. Consequently, (1) in constraints (4), Tc is substituted by variable TM ax . If the upper bound of the station times is minimized, then the minimum of the maximal station times is obtained. The upper bound of station times is set as follows,
In contrast to SALBP-1, SALBP-2 minimizes the cycle time for a given number of workstations (N ), that is, the objective function is as follows,
M ax(TM in ).
(1)
M in(TM ax ).
(9)
I i=1
(1)
ti xij ≤ TM ax
∀j = 1, ..., J.
(10)
The performance of each operation (5) and the precedence constraints (6) are the same as in SALBP-1. That is, SALBP-2 is determined by objective function (9) and constraints (5), (6) and (10). The model of SALBP-2 can be completed with constraints (8) applying techniques which exclude unfeasible assignments. In order to decrease the difference of the maximal and the minimal station times, the minimal station time must be maximized, while the minimum of the maximal station time, determined by SALBP-2, is satisfied. To determine (1) the maximal value of the smallest station time (TM in ) a SALBP-3 is defined. In this new problem, the objective function maximizes the smallest station time, that is, (1)
(1)
(11)
If TM in is a lower bond of the station times, and its value is maximized then the optimal value of (11) is the largest possible value of the smallest station time. Consequently, the following lower bounds are applied,
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(1)
ti xij ≥ TM in
∀j = 1, ..., J
those tasks, which are already assigned to the ignored stations, then constraints (12) are modified as follows,
(12)
I
The upper bound of the station times, determined by the SALBP-2 must also be satisfied (10). Using the above formulation, the steps of the algorithm are the following, Step 1. An SALBP-1 is solved to get the minimal number of stations (N ) necessary to perform the production plan. Step 2. An SALBP-2 is solved to get the minimal cycle time (Tc ). This cycle time is the smallest possible value of (1) the largest station time (TM ax ). Step 3. An SALBP-3 is defined which maximizes the minimum of the station times with a cycle time upper bound for all the station times. The result is the largest (1) possible value of the smallest station time (TM in ). As a consequence of Step 2 and 3 the difference between the smallest and the largest station times is minimized, that is, (1)
(1)
s[1] − s[J] = TM ax − TM in
(13)
is the smallest possible. Equation (13) provides the first element of the sum defined by equation (2). Let z denote the z -th element of the sum in equation (2). The first element (z = 1) of equation (2) was obtained by solving a SALBP-2 and a SALBP-3. To determine the subsequent elements of this sum, a modified SALBP-2 and a modified SALBP-3 must be solved consecutively. In Step 4 and Step 5 these modified models are defined and solved several times one after the other, always using the results obtained for the immediately preceding (z − 1) element. Step 4. A modified minimization SALBP-2 is solved with the following conditions: • The stations which had the minimal time when the (v) previous (z − 1) elements were calculated (TM ax , ∀v = 1, ..., Z − 1), are ignored with all the tasks assigned to them. If BS is the index set of those stations, which are ignored in the calculations, and BT is the index set of those tasks, which are already assigned to the ignored stations, then constraints (10) are modified as follows, I i=1
(1)
ti xij ≤ TM ax
∀j ∈ BS, i ∈ BT
(14)
Ignoring stations and tasks at the calculation assumes, that the ignored tasks are assigned to the corresponding ignored stations. • A lower bound for all the remaining station times is set, which is equal to the value found for the (z−1)-th element of (2), that is, I i=1
(z−1)
ti xij ≥ TM in
∀j ∈ BS, i ∈ BT
(15)
Step 5. A modified maximization SALBP-3 is solved with the following constraints: • The stations which have the maximal time in all (v) previous (z − 1) elements (TM in , ∀v = 1, ..., Z − 1) are ignored with all the tasks assigned to them. If BS is the index set of those stations, which are ignored in the calculations, and BT is the index set of
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i=1
(z)
ti xij ≥ TM in
∀j ∈ BS, i ∈ BT
(16)
Ignoring these tasks at the calculation assumes, that the ignored tasks are assigned to the corresponding ignored stations. • An upper bound for all the remaining station times is set, which is equal to the value found for the (z−1)-th element of (2), that is, I i=1
(z−1)
ti xij ≤ TM ax
∀j ∈ BS, i ∈ BT
(17)
As a consequence of Step 4 and 5 the difference between (z) (z) the z-th smallest (TM in ) and the z-th largest (TM ax ) station time is minimized, that is, (z)
(z)
s[z] − s[J−z] = TM ax − TM in (18) is the smallest possible. Step 4 and 5 can be repeated by always eliminating from the calculation the stations (z−1) and the corresponding tasks, with the minimal (TM in ) (z−1) and the maximal (TM ax ) station times found by the previously solved modified SALBP-2 and SALBP-3. The calculation stops when neither the modified SALBP-2 nor the modified SALBP-3 provides different minimum and maximum values found in the previous step and trivially at z = Int(J/2). The value of BI improves after each calculation. It can be easily proved, that as the algorithm progresses (the value of z increases) the value of BI monotonously decreases. When the value of the s[z] −s[J−z] difference decreases as a result of Step 4 and 5, tasks are reallocated among the workstations. This reallocation of tasks cannot increase any difference calculated in the preceding steps, as a consequence of the applied upper and lower bounds of the corresponding station times defined by constraints (15) and (17). Consequently, the value of BI either decreases, or does not change. In this later case, however, the algorithm stops. 5. CASE STUDY To illustrate the behaviour of the algorithm let us consider a case study from the machine industry. The problem is solved with SALBP, that was transferred from a mixed model assembly line. The applyed case was inspired by a problem at an injection molding machine assembly company — but the original data were changed. The company produces three different product types (I-III.) on the production line. The deterministic processing times (given in minutes) of each task required by the assembly operations are summarized in Table 2. Altogether we have 37 different tasks (A-DT). If a tasks is not needed to a given final product type, it is indicated with ”-” in the table. The mix of products to be assembled is presented in the last row of the table. The weighted average processing times of the tasks are calculated in the last column of Table 2 using the given volumes as weights. The equivalent precedence diagram is presented in Figure 2. In case of SALBP-1 the number of workstations must be minimized. It is assumed that the line works with a
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Table 2. Task times and the weighted average of task times (minute) Task A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA AB AC AD AE AF AG AH AI AJ DT Quantity
Product I
Product II
Product III
24 6 6 12 6 3.6 7.2 3 6 6 6 9 45 60 15 9 12 27 12 9 27 9 6 4.8 12 15 15 6 1.2 15 4.8 48
24 36 6 3 4.8 15 6 12 21 12 42 9 60 15 9 6 30 12 9 27 9 6 4.8 6 30 12 24 9 15 12 20.4 18 7.2 24
24 12 12 18 1.8 4.8 3 18 12 9 12 58 15 60 15 9 6 24 12 9 27 9 6 9.6 6 24 15 15 12 1.2 12 7.2 24
Weighted time 24 6 3 18 9 3 6 6 3 9 3 6 12 6 45 6 60 15 9 9 27 12 9 27 9 6 6 9 15 9 6 6 15 9 6 15 6
Fig. 2. The precedence relationship between the tasks of the case study, a color-map illustrates the processing time for each task.
one hour cycle time. The optimum value of the objective function of SALBP-1 (Step 1) is 8. Consequently, the minimal number of workstations required to assembly the required quantities of the three different product types is 8 — if the cycle time is one hour. The SALBP-2 (Step 2) minimizes the cycle time with a given number of workstations. If we assume, that the assembly line (1) consists of 8 stations, the optimal cycle time, TM ax is one hour. Note, that the minimal cycle time is equivalent to the highest station time. In our case, task Q with its 60 minutes processing time is the bottleneck of the assembly line and determines the cycle time of the line. SALBP-3 (Step 3) determines the maximal value of the (1) smallest station time (TM in ). The optimal value for the highest lower bound of station times is 54 minutes. That is, the 37 tasks with the given processing times and precedence relations can not be assigned to stations in a way, that the difference between the minimal (54 minutes) and the maximal (60 minutes) station times is smaller than 6 minutes. Table 3 summarizes the results of each step gained by LINGO Software (Version 17). It can be seen in Table 3, that the minimal value of the difference between the highest and smallest station times is 6 minutes
(S3: s[1] − s[8] = 6). The minimal difference between the second largest and the second smallest station times can be reduced in Step 4 and Step 5 (S4 and S5 in Table 3). In Step 4, task Q is assigned to station 5 and tasks B, E, K, P, AB, AC and AE are assigned to station 3. These tasks and stations are ignored at the calculation of the difference between the second highest and second lowest station times. The optimal value of objective function (2) of the modified SALBP-2, TM ax is 57 minutes (Step 4). After solving the modified SALBP-3 in Step 5 (the (2) maximal value of the second lowest station time, TM in is 54 minutes) the difference between the second highest and second smallest station times can not be reduced any more, the value of the BI remains 12 and the algorithm stops. Table 3. Summary of the results Iteration S1 S2 S3 S4 S5
Min sj 45 51 54 54 54
Max sj 60 60 60 60 60
s[1] − s[8] 15 9 6 6 6
s[2] − s[7] 9 6 6 3 3
BI 30 24 18 12 12
In order to further validate the algorithm, the problem of the case study was modelled and solved using discrete optimization. The integer processing times allowed formulating a Constraint Programming (CP) representation of the model. Examples can be found in the literature for using CP to solve SALBP-1 and SALBP-2 (Topaloglu et al. (2012)), thus the complete description of the model is not given here. Using the effective and expressive constraints and objective modeling capabilities of CP the effects of the restrictions made in Step 4 and Step 5 were inspected. The CP
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model combines Step 4 and Step 5 into one step with less restrictions on the task assignment. According to (15), (14), (16), (17), in the z = α iteration of the algorithm the tasks assigned to a station in a previous z < α iteration are ignored and they remain unchanged. However, in the CP model by applying a sorting global constraint (Older et al. (1995)) to the array of S = {sj } station times, the resulting s[z] − s[J−z] station time differences from each z = α iteration are added as a constraint for the subsequent z = α + 1 iteration. The calculated station times are displayed in Fig. 3, with the objective function being s[z] − s[J−z] for each iteration and with assigning every task in every iteration. It can be seen that the optimal solution for station times after the last step is equal with the solution obtained by the subsequent solution of the modified SALBPs proposed in Section 4.
Fig. 3. Station times after each iteration of the algorithm.
6. CONCLUSION AND OUTLOOK In this paper a new balancing index is introduced, which measures the distribution of workstation workloads. This balancing index is monotonous decreasing as the distribution of workload improves, consequently, it can be a good indicator in an optimization algorithm. The authors proposed a balancing algorithm for equally balance the workload of non-bottleneck stations with binary and constraint programming. Finally, the application of the proposed balancing index and algorithm were demonstrated with a case study, where a mixed model assembly line problem was simplified to a SALBP. Further research should be undertaken to investigate the feasibility of the solution of the original problem gained by SALBP. Instead of ignoring stations and tasks in step 4 and 5 of the algorithm, formulating the minimal upper and maximal lower bounds as constraints is a topic of our further research as well. ACKNOWLEDGEMENTS The authors would like to acknowledge the financial support of the Austria Research Promotion Agency (FFG) and of European Commission for funding the StaProZell (pr. No. 864870) and H2020 EPIC (grant no. 739592) projects respectively.
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