The aim of this paper is to propose an original mathematical comparison model that, starting from the assembly system design and balancing, calculates the system total assembly time and the total energy expenditure considering the potential storage locations of the components and the related constraints at the assembly station in terms of number of storable components (Fig. 2).
The design process of an assembly system is typically based on the maximum market demand that has to be satisfied [66], which determines the maximum system throughput Qtarget and, then, the maximum number of assembly workstations (K) and of operators (Nop). In case of maximum system throughput, the two systems have the same number of workstations and of operators, and also, the assignment of the tasks to the workstations (line balancing) is the same. However, since FW and WW systems are considered to operate with a JIT approach if the throughput decreases, they have to be re-sized accordingly. In FW systems, since there is one fixed operator for each workstation, the re-sizing requires the decrease of the number of workstations as well as of the number of operators. Moreover, it is also needed to re-balance the line with the new number of workstations. On the other side, in WW systems, the number of workstations remains the same (and it is always equal to the maximum number K) while only the number of operators decreases according to the system throughput. Therefore, it is not needed to re-balance the line.
Assumptions and notations
FW and WW systems are compared in terms of flexibility, productivity, and energy expenditure; the model is based on the following assumptions, which can be easily applied to different contexts:
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1.
The systems realize one single product (simple assembly line balancing problem, SALBP)
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2.
The system has a continuous products flow (there are no buffers)
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3.
The product moves between the workstations with an automatic handling system
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4.
The considered WW system is the Rabbit Chase case (RC)
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5.
The FW system has a linear layout while the WW system has a U-shaped layout
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6.
The distance between workstations is fixed and equal for all workstations
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7.
The workstation size and the storage locations (SLs) are the ones proposed by [12]
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8.
Each task refers to one component to assemble and each component is assigned to only one task
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9.
Each task includes one single picking activity and one assembly activity
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10.
All activities times are deterministic
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11.
Failures time, waiting time, and set-up time are not considered in the problem
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12.
Workers are male operators, all with the same skill level
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13.
All workers are able to carry out all tasks in both configurations
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14.
Workers’ walking speed is constant
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15.
The effect of fatigue on tasks’ execution performance is not considered
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16.
Since the considered components’ weights are standard, these do not affect the picking time but only the picking energy expenditure [12]
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17.
The initial number of workers and workstations is the same in both configurations (FW and WW)
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18.
The changes of the system throughput are proportional to Qtarget/K and directly related to these values (i.e., the considered line configurations are related to the various maximum number of workstations)
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19.
Re-layout costs and re-balancing costs of the FW system are not considered in case of changes in productivity.
Moreover, workers’ tasks can be classified into three categories:
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Assembly activity related to the execution of the task in the workstation; both the duration and the energy expenditure of this activity are known.
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Picking activity of the component needed for the execution of the task; the time and the energy expenditure of this activity are related to the storage location (SL) of the component that varies according to the considered configuration.
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Travel activity between workstations (just in the WW system); the duration and the energy expenditure of this activity depend on the distance between the workstations and on the worker’s speed.
Table 3 presents the main variables and the notations of the model.
FW-WW balancing and comparison model
The comparison of the two systems is carried out by a mathematical model with an objective function that considers the maximum workload of the assembly line. The comparison requires the application of three subsequent steps:
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1.
Define the design of the assembly line
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2.
Balance the assembly line, minimizing the objective function under the specified constraints
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3.
Calculate the comparison parameters
Assembly line design
As first, the design of the two configurations is carried out, by defining the number of workstations (K) and of required operators (Nop). In both cases, the assembly lines are sized based on the maximum system throughput; i.e., the Qtarget of the system is calculated as the maximum market demand in the considered period.
Knowing the total assembly time (Tass) of the system, needed to assemble one finished product, it is possible to define K as:
$$ K=\left\lceil {T}_{ass}\cdotp {Q}_{\mathrm{target}}\right\rceil $$
(1)
Moreover, starting from this maximum number of stations, it is possible to define PV, which represents the production volume flexibility parameter. It takes into account how much the system is used with respect to its maximum threshold Qtarget, which is also related to the maximum number of stations K:
$$ PV=\frac{N_{op}}{K}\kern0.5em \left[\%\right] $$
(2)
Here, PV = 100% when both systems are working at their maximum rate (Nop = K for WW and Nop = KFW = K for FW). On the other side, when the system throughput is lower than Qtarget, the number of operators decreases (and, for FW, also the number of workstations decreases), and PV < 100%.
Assembly line balancing
Two binary variables are defined as:
$$ {x}_{jk}=\left\{\begin{array}{cc}1& \mathrm{if}\ \mathrm{task}\ \mathrm{j}\ \mathrm{is}\ \mathrm{assigned}\ \mathrm{to}\ \mathrm{workstation}\ k\kern1.5em \\ {}0&\ \mathrm{if}\ \mathrm{task}\ \mathrm{j}\ \mathrm{is}\ \mathrm{not}\ \mathrm{assigned}\ \mathrm{to}\ \mathrm{workstation}\ k\end{array}\right. $$
(3)
$$ {z}_{jkw}=\left\{\begin{array}{cc}1& \mathrm{if}\ \mathrm{task}\ \mathrm{j}\ \mathrm{is}\ \mathrm{assigned}\ \mathrm{to}\kern0.5em \mathrm{workstation}\ k\ \mathrm{in}\ SL\ w\kern1.25em \\ {}0&\ \mathrm{if}\ \mathrm{task}\ \mathrm{j}\ \mathrm{is}\ \mathrm{not}\ \mathrm{assigned}\ \mathrm{workstation}\ k\ \mathrm{in}\ SL\ w\ \end{array}\right. $$
(4)
The assembly line balancing problem is solved by an objective function that aims to minimize the maximum workload of each workstation WLk. It is defined as:
$$ \theta =\min\ \left({\max}_k{WL}_k\right)\ \left[s/\mathrm{piece}\right]\kern0.75em $$
(5)
where WLk is the workload of workstation k that includes the assembly and the picking activities. It is defined as:
$$ {WL}_k={AT}_k+{PT}_k={\sum}_{j=1}^n\left({x}_{jk}\cdotp {t}_j\right)+{\sum}_{j=1}^n{\sum}_{w=0}^W\left(\ {z}_{jk w}\cdotp {t}_{p,w}\right)\ \left[\mathrm{s}/\mathrm{piece}\right]\kern3.75em $$
(6)
where ATk indicates the total assembly time in the workstation k, while PTk indicates the total picking time in workstation k. In particular, tj is the assembly time of task j; tp, w is the picking time from the SL w.
In order to consider the energy expenditure of both configurations, the energy expenditure of workstation k Ek is defined as:
$$ {E}_k={E}_{pos}\cdotp {WL}_k+{\sum}_{j=1}^n\left({AE}_{jk}+{PE}_{jk}\right)={E}_{pos}\cdotp {WL}_k+{\sum}_{j=1}^n\left({x}_{jk}\cdotp {e}_j\right)+{\sum}_{j=1}^n{\sum}_{w=0}^W\left(\ {z}_{jk w}\cdotp {e}_{p, jw}\right)\kern0.5em \left[\mathrm{kcal}/\mathrm{piece}\right] $$
(7)
where AEjk is the total energy expenditure of the assembly activity of task j in the workstation k, while PEj indicates the total energy expenditure of the picking activity of task j related to workstation k. In particular, Epos is the standard energy expenditure for the maintenance of the standing posture and it is calculated using Garg et al.’s formula [67], as:
$$ {E}_{pos}=\frac{0.024\cdotp BW}{60}\kern0.5em \left[\mathrm{kcal}/\mathrm{s}\right] $$
(8)
where BW is the worker’s body weight. The notations ej and ep, jw indicate the energy expenditure for the assembly activity of task j and the energy expenditure for the picking activity of task j from SL w, respectively. They are also calculated deriving an energy expenditure per piece from the Garg et al.’s formulas [67], depending on the components’ weight Lj, the operators walking speed, and the characteristics of the various picking and assembly activities [47, 63].
The constraints of the model are:
$$ {\sum}_{k=1}^K{x}_{jk}\le 1\kern0.5em \forall j=1\dots n\kern0.5em $$
(9)
$$ {\sum}_{k=1}^K{\sum}_{w=0}^W{z}_{jkw}\le 1\kern0.5em \forall j=1\dots n $$
(10)
$$ {\sum}_{k\epsilon \left[{E}_j,{L}_j\right]}k\cdotp {x}_{ik}\le {\sum}_{k\in \left[{E}_l,{L}_l\right]}k\cdotp {x}_{jk}\kern0.75em \forall \left(i,j\right)\in A $$
(11)
$$ {\sum}_{j=1}^n{z}_{jkw}\le {C}_{wk\kern0.75em }\forall w=0\dots W,\forall k=1\dots K $$
(12)
$$ {z}_{jk w}\le {x}_{jk}\kern1.5em \forall j=1\dots n,\forall k=1\dots K,\forall w=1\dots W\kern0.5em $$
(13)
$$ {\sum}_{w=0}^W{\sum}_{j=1}^n\ \left[\left({z}_{jk w}\cdotp {t}_{p,w}\right)+\left({x}_{jk}\cdotp {t}_j\right)\right]\le \max WLk $$
(14)
$$ \frac{E_k}{WL_k}\cdotp 60\le 4.29\kern0.5em \forall k=1\dots K $$
(15)
$$ {WL}_k>0\kern2.25em \forall k=1\dots K $$
(16)
$$ {x}_{jk}\in \left\{0,1\right\}\kern1.25em \forall j=1\dots n,\forall k=1\dots K $$
(17)
$$ {z}_{jkw}\in \left\{0,1\right\}\kern1.75em \forall j=1\dots n,\forall k=1\dots K,\forall w=1\dots W $$
(18)
Constraints (9) and (10) assure that each task j is assigned only to one workstation k and that each component of task j is assigned just to one SL w. Constraints (11) are for precedence relations, with A set of arcs in the precedence diagram. Equation (12) is for the capacity constraint of each workstation k: the number of components stored in each SL w has to be equal or lower of Cwk, which is the maximum capacity of SL w in workstation k. Constraint (13) assures a proper assignment of the tasks to the available storage locations, while Eq. (14) limits the cycle time of the assembly line; i.e., the total assembly and picking time for every task j must be minor of the final cycle time. Equations (15) and (16) refer to the energy expenditure of each station, which has to be lower than the limit suggested by [68] to avoid the need for rest allowances. Constraints (17) and (18) set the domains of the decision variables.
Evaluation and comparison of the systems configurations
In order to evaluate FW and WW systems, some parameters are defined. First, it is considered the productivity of the system as:
$$ Q=\frac{3600}{\max \left({WL}_k\right)}\kern1em \left[\mathrm{pieces}/h\right] $$
(19)
Then, the total workload for the two configurations Ttot, FW and Ttot, WW are calculated with:
$$ {T}_{tot, FW}={\sum}_{k=1}^K{WL}_k\kern1.25em \left[s/\mathrm{piece}\right]\kern4.75em $$
(20)
$$ {T}_{tot, WW}={\sum}_{k=1}^K{WL}_k+K\cdotp \frac{D}{v}\kern1.75em \left[s/\mathrm{piece}\right]\kern3.75em $$
(21)
and compared with:
$$ \Delta T=\frac{T_{tot, WW}-{T}_{tot, FW}}{T_{tot, WW}}\kern1em \left[\%\right] $$
(22)
Then, if ∆T > 0, the FW configuration is preferable from a workload point of view; otherwise, if ∆T < 0, the WW configuration is the best one. A similar formulation is defined also for the total energy expenditures per finished product:
$$ {E}_{tot, FW}={\sum}_{k=1}^K{E}_k\kern2.25em \left[ kcal/\mathrm{piece}\right] $$
(23)
$$ {E}_{tot, WW}={\sum}_{k=1}^K{E}_k+K\cdotp {e}_t\kern2.25em \left[ kcal/\mathrm{piece}\right] $$
(24)
$$ \Delta E=\frac{E_{tot, WW}-{E}_{tot, FW}}{E_{tot, WW}}\kern0.75em \left[\%\right] $$
(25)
If ∆E > 0, the FW configuration is preferable from an energy expenditure point of view; otherwise, if ∆E < 0, the WW configuration is the best one.