An MCDA-DEA approach for mixed-model assembly line balancing problem under uncertainty

Abstract

In his paper, a bi-objective balancing model for U-shaped mixed-model assembly lines to minimize total costs as well as the workload of stations, considering zoning constraints and task duplications is presented. In order the uncertain nature of the input data (i.e. monetary and time data), an interactive fuzzy possibilistic programming approach has been applied, and the bi-objective nature of the presented model is treated via an efficient multi-criteria decision analysis-DEA based approach. Finally, applicability of the presented model and its effectiveness is tested on a real industrial case study.

Keywords

Fuzzy possibilistic programming mixed-model assembly lines MCDA-DEA approach zoning constraints task duplications uncertainty

1 Introduction

Assembly lines consist of successive stations in which units continuously move through the stations in order to be assembled. Stations are defined as places where certain tasks are performed on products. Because of great competition in the market, it is a very important issue to balance such lines. Balancing assembly lines is concerned with the allocation of tasks to work stations on a production line such that defined constraints are not violated and maximum efficiency is pursued at each work station for a period of time [1, 2].

The basic problem of such lines is called the single assembly line balancing problem (SALBP) in which only one product or several products with negligible variations of setup time and task times are assembled on the line. In SALBP, task processing times are independent of their associated workstations. Moreover, none of the task processing times exceeds the cycle time. In short, the goal is to achieve a grouping of tasks that minimizes the inefficiency of the line that respects all the constraints imposed on both the tasks and the stations [3, 4].

SALBPs can be classified to four major groups;

SALBP type-1: Minimizing number of stations for the fixed and given cycle time.

SALBP type-2: Minimizing cycle time for the fixed number of stations

SALBP type-E: Maximizing the line efficiency (i.e. E = tsum/(W.C)) When neither of cycle time nor number of stations are given

SALBP type-F: Finding a feasible balance when both of the cycle time and number of stations are given [5, 6]

In modern industries with increasing competition and product variety, the importance of adequate responding to consumers, providing a mixed model assembly line (MMAL) instead of single model ones has been into a great consideration. In mixed model assembly lines, several units or products with similar production characteristics or different models of products are assembled on the same line (see Fig. 1) [7, 8].

Recently, in order to pursue the just-in-time (JIT) concept in production systems, many factories are switching their traditional straight lines to the U-shaped ones [5]. The main characteristic of a U-line that differentiates it from the straight line is that the entrance and exit of the U-line are at the same position. Configuration of a typical U-shaped assembly line with 7 tasks and 3 workstations is depicted in Fig. 2. The most remarkable and important advantage of this layout is the flexibility to increase or decrease the necessary number of workers when adapting to changes in demand. It also contains crossover workstation which includes tasks located at both front and back of the U-line (see Station #1 in Fig. 2), while a regular workstation includes tasks located at front or back of the U-line (see Stations #2 and #3 in Fig. 2) [8].

Due to dynamic nature of the respective task times and costs, it is important to consider them under uncertainty and handle them via an efficient approach.

In this paper an interactive approach to balancing a bi-objective mixed-model assembly U-line model with task duplication and zoning constraints is presented. Maximizing the line efficiency through minimizing total cost along with stations’ workload makes the proposed model to partly satisfy the JIT concept. In order to cope with epistemic uncertainty of the input data, an efficient fuzzy approach is utilized, and the behavior of the objectives are handled via an efficient MCDA-DEA approach. Finally, applicability of the model is further tested on a real manufacturing case study.

The remainder of the paper is presented as follow; in next Section (2), a comprehensive literature review on two aspects are presented. Section (3) is dedicated to problem definition and model formulation. Solution approach is addressed in Section (4). Section (5) presents the studied industrial case and sensitivity analyses; and finally, Section (6) is devoted to conclusion and future studies.

2 Literature review

In this section, current research studies are focused based on two main aspects of product variety of assembly lines (i.e., single or mixed-model), and the assembly lines’ shape (i.e., straight or U-shaped).

2.1 Product variety of assembly lines

Balancing single assembly lines has widely been studied among papers;

Klein and Scholl [9] considered maximization of the production rate for simple assembly line balancing with fixed number of stations using branch and bound procedure. Lapierre et al. [10] proposed a new tabu search algorithm for solving Single assembly line type-1. The efficiency of the model is further tested on a real industrial case. Nicosia et al. [11] presented a dynamic programming algorithm for assigning operations to an ordered sequence of non-identical workstations, observing precedence relationships and cycle time restrictions in a single assembly line balancing problem. Scholl et al. [12] studied single assembly balancing problem by considering sequence-dependent task times to formulate several versions of a mixed-integer program. Rekiek et al. [13] focused on a comprehensive survey of existing methods the line-balancing and resource planning of single assembly lines.

In the field of mixed-model assembly lines, Yagmahan [14] studied a mixed-model assembly line balancing type-1 problem with multi-objective ant colony optimization algorithm in order to minimize the balance delay, the smoothness index between and within stations for a given cycle time. Simaria and Vilarinho [15] presented a mathematical programming model and an iterative genetic algorithm-based procedure for the mixed-model assembly line balancing problem with consideration of parallel workstations and objective of maximization of production rate of line as the objective function. Niu and Moreno [16] studied the Japan (Toyota) pattern in MMAL environment in straight line with minimization of the idle time and stoppage using genetic algorithm. Haq et al. [17] used a hybrid genetic algorithm approach to mixed-model assembly line-balancing type-1 problem with combined precedence diagram consideration. Akpina & Bayhan [18] proposed a hybrid genetic algorithm to solve mixed-model assembly line balancing problem with: minimization the number of workstations, maximization the workload smoothness between workstations, and the workload smoothness within workstations.

A common assumption on an MMAL balancing problem is assigning common tasks of multiple models to different stations. Assigning such tasks to different stations is called task duplication which was proposed by Bukchin and Rabinowitch [19]. A branch-and-bound based heuristic is then developed for solving problems. Kazemi et al. [20] also considered task duplication in MMAL balancing problem for U-shaped lines using two-stage genetic algorithm.

2.2 Assembly lines’ shape

The straight line, in terms of not to be considered as U-lines, have been widely studies in various aspects and terms (i.e., straight, parallel, two-sided line, etc.), and in what follows, some recent studies are discussed.

Kucukkoc and Zhang [21] presented both balancing and sequencing of mixed-model parallel two-sided assembly lines and solving it via an agent-based ACO approach. Akpinar and Baykasoğlu[22, 23] presented a mixed integer linear programming model for a mixed-model assembly line balancing problem considering setup times. The problem is then been solved utilizing a multiple colony hybrid bees algorithm for the larger sizes.

With consideration of huge studies under the term of mixed-model assembly lines, only a few papers have dealt with the U-shaped balancing problem of MMALs. The U-shaped assembly systems have been first studied by Sparling and Miltenburg [24] with consideration of the weighted average of task times for each task in different models. Rabbani et al. [5] developed a new approach to balance a mixed model U-shaped production system without consideration of task sequence. Maximizing the line efficiency with minimizing total number of crossover stations were considered simultaneously. The genetic algorithm is then used for solving the problem in large scale. Kara et al. [25] presented a multi-objective model to both balancing and sequencing of U-shaped MMALs in order to minimize the absolute deviations of workloads across workstations, part usage rate, and setup costs. To solve the presented model in the larger size, a simulated annealing approach is utilized. Hwang and Katayama [26] focused on balancing of an MMAL problem with minimization of number of workstations and the variation of workload, simultaneously. Validity of the proposed model is tested on a real case study. Özcan et al. [27] presented an integrated balancing and sequencing problem for a U-shaped MMAL problem considering stochastic task times. A genetic algorithm was then utilized to solve the model for large instances.

This paper presents a bi-objective model for balancing of the mixed-model U-lines (MMUL) considering task duplication and zoning constraint. Maximizing the line efficiency through minimizing total costs and workload of stations makes the proposed model to partly satisfy the JIT concept. In order to elevate the efficiency of the proposed model in real world application, some input data was considered under uncertainty, and an efficient interactive approach is utilized to handle such issue. To cope with the bi-objective nature of the model, an ɛ-constraint method is used, and further a common weighted MCDA-DEA approach is applied to reach the unique global solution.

3 Problem definition and mathematical formulation

Since the considered assembly line is U-shaped, the concepts of phantom precedence diagram and combined precedence graph is utilized. In straight lines, a task can only be processed if all of its predecessors are already assigned. However, in U-shaped lines a task can be assigned to a station if either all of its predecessors or its successors are already assigned [5]. A simple precedence diagram added to its phantom diagram is depicted in Fig. 3.

The proposed model is based on following assumptions:

The considered assembly line is U-shaped.

Since the current problem only considers balancing problem, no predetermined model sequence is given (see [5]).

Zoning constraints are existed, which means some tasks should be assigned to same station while some others should not assigned to same ones.

Labor traveling time is only considered for crossover stations.

No parallel workstations or tasks are considered.

Product models with similar production characteristics are produced on the same U-line.

Performance times of tasks common between models are not necessarily identical for different models.

The model we cope with in this study in MMULBP-Type 1 (in which the cycle time is known and the primary balance goal is minimizing number of stations.)

Parameter, decision variables and formulated model are presented as follows;

It should be noted that uncertain parameters are differentiated from the deterministic ones using a tilde on.

Indices

index of tasks (i = 1, …, I)

index of models (j = 1, …, J)

index of stations (k = 1, …, K_max)

Parameters

${\tilde{t}}_{ij}$ processing time of task i for product model j

λ an extra coefficient for crossover workstations

C cycle time of the line

$\tilde{δ}$ an extra time added to the workload of crossover workstations considering operator’s travel times

$S_{j}^{'}$ set of precedence relationships for the combined (original) precedence diagram

$(v, w) \in S_{j}^{'}$ a precedence relationship; assembly task v must be done before task w

$\tilde{SC}$ fixed cost of opening a station

${\tilde{TC}}_{i}$ cost of assigning task i to any station

ZP set of tasks with positive zoning restrictions

ZN set of tasks with negative zoning restrictions

Decision variables

z_k 1; if station k is utilized and regular, 0; otherwise.

s_k 1; if station k is utilized and crossover, 0; otherwise.

α_ik (β_ik) 1; if task i of any models is assigned to station k from original (phantom) diagram, 0; otherwise.

u_jk utilization of product model j on workstation k

a_j average utilization of model j for total workstations

x_ijk (y_ijk) 1; if task i of model j is assigned to station k from the original (phantom) diagram, 0; otherwise.

o_k (g_k) 1; if any task is assigned to station k from the original (phantom) diagram, 0; otherwise.

Model formulation $\begin{matrix} \underset{1}{minW} = \tilde{SC} [\sum_{k}^{k_{\max}} (z_{k} + (1 + λ) s_{k})] \\ + \sum_{i}^{I} \tilde{{TC}_{i}} \sum_{k}^{k_{\max}} (α_{ik} + β_{ik}) \end{matrix}$ (1) $\begin{matrix} \underset{2}{minW} = \frac{1}{J} \times \\ \sum_{j} \sqrt{(\frac{1}{\sum_{k}^{k_{\max}} (z_{k} + s_{k})} \sum_{k}^{k_{\max}} {(u_{jk} - a_{j})}^{2})} \end{matrix}$ (2)

$\begin{matrix} s . t . \\ \sum_{k}^{k_{\max}} (x_{ijk} + y_{ijk}) = 1, \forall i, j, \end{matrix}$ (3)

${\begin{matrix} \sum_{k}^{k_{\max}} k (x_{v jk} - x_{w jk}) \leq 0, \forall (v, w) \\ \sum_{k}^{k_{\max}} k (y_{w jk} - y_{v jk}) \leq 0, \in S_{j}^{'}, \end{matrix}$ (4)

$\begin{matrix} \sum_{i}^{I} {\tilde{t}}_{ij} (x_{ijk} + y_{ijk}) \leq \\ {Cz}_{k} + (C - \tilde{δ}) s_{k}, \forall j, k, \end{matrix}$ (5) $α_{ik} \geq \frac{1}{m} \sum_{j = 1}^{J} x_{ijk}, \forall i, j,$ (6) $β_{ik} \geq \frac{1}{m} \sum_{j = 1}^{J} y_{ijk}, \forall i, k,$ (7) ${\begin{matrix} \sum_{i}^{I} α_{ik} - n \times o_{k} \geq 1 - n, \\ \sum_{i}^{I} α_{ik} - n \times o_{k} \leq 0, \\ \sum_{i}^{I} β_{ik} - n \times g_{k} \geq 1 - n, \\ \sum_{i}^{I} β_{ik} - n \times g_{k} \leq 0, \\ o_{k} + g_{k} - 2 s_{k} - z_{k} = 0, \end{matrix} \forall i, k,$ (8) $u_{jk} = \frac{1}{C} \sum_{i}^{I} {\tilde{t}}_{ij} (x_{ijk} + y_{ijk}), \forall j, k,$ (9) $a_{j} = \frac{1}{\sum_{k}^{k_{\max}} (z_{k} - s_{k})} \sum_{k}^{k_{\max}} u_{jk}, \forall j,$ (10)

$\begin{matrix} \sum_{k}^{k_{\max}} (x_{ijk} + y_{ijk}) - \forall (ij, qp) \\ \sum_{k}^{k_{\max}} (x_{qpk} + y_{qpk}) = 0, \in ZP, \end{matrix}$ (11)

$\begin{matrix} (x_{ijk} + y_{ijk}) + \forall (ij, qp) \\ (x_{qpk} + y_{qpk}) \leq 1, \in ZN, k, \end{matrix}$ (12)

$\begin{matrix} x_{ijk}, y_{ijk}, α_{ik}, β_{ik}, \\ z_{k}, s_{k}, o_{k}, g_{k} \in {0, 1}, \forall i, j, k, \end{matrix}$ (13) $u_{jk}, a_{j} \geq 0, \forall j, k .$ (14)

The objective function (1) consists of two parts. The first part minimizes total costs of opening workstations, and the second one minimizes task duplication costs. The objective function (2) which is the modified version of the objective function proposed by Hwang and Katayama [26], minimizes total variation of workload.

Constraint set (3) assures that each task must be assigned to a single station regarding a task can be assigned to a station only once using the original or phantom diagram. Set of constraints (4) shows precedence relationships between tasks. Constraint set (5) guarantees that sum of operational tasks on each station and movement time (in case of crossover stations) cannot exceed the cycle time. Set of constraints (6) checks whether task i of any model is assigned to workstation k from the original diagram, while constraint (7) checks whether task i of any model is assigned to workstation k from the phantom diagram. Defining whether a station is regular or crossover, which is the modified version of the model proposed by Kara and Tekin [28], is considered in constraint (8). Equations (9) calculate the utilization of product model j on workstation k. Average utilization of model j for total stations is computed by Equations (10). Constraint set (11) shows positive zoning tasks, which should be assigned to the same stations; while constraint set (12) shows negative zoning tasks, which should not be assigned to the same workstations. Finally constraints (13) to (16) define the decision variables.

Because of unavailability or incompleteness of data in the real world situations, some of the parameters embedded in such problem (i.e. task time, extra time for crossover stations and monetary data) have an imprecise nature.

In this field, Kara et al. [29] applied binary fuzzy goal programming using Chang [30] approach to some models for balancing straight and U-shaped assembly lines with fuzzy goals. Mahdavi et al. [31] proposed a fuzzy multi-objective linear programming, in which they considered minimization of total utility work, total setup cost and variation of total production rate by using a two-phase linear programming approach. Hop [32] presented the mixed-model line balancing problem with fuzzy processing time with an improvement heuristic approach. The problem is then formulated as an MIP model and the heuristic method is then being developed using a flexible exchange sequence procedure for assigning tasks to workstations.

4 Crisp counterpart formulation

To cope with the uncertainty of such situation, an interactive and efficient approach of Jimenez et al. [33], which is based on the definition of the “expected value” and the “expected interval” has been applied. Noteworthy, possibility distribution of the imprecise parameters is determined to be triangular (seeFig. 4).

Regarding the afore-mentioned descriptions, the equivalent auxiliary crisp model can be formulated as below.

$\begin{matrix} \underset{1}{minW} = (\frac{SC (p) + 2 SC (m) + SC (o)}{4}) \times \\ [\sum_{k}^{k_{\max}} (z_{k} + (1 + β) s_{k})] + \\ \sum_{i}^{I} (\frac{{TC}_{i} (p) + 2 {TC}_{i} (m) + {TC}_{i} (o)}{4}) \times \\ \sum_{k}^{k_{\max}} (α_{ik} + β_{ik}) \end{matrix}$ (15) $\begin{matrix} s . t . \\ \sum_{i}^{I} [\begin{matrix} φ (\frac{t_{ij} (m) + t_{ij} (o)}{2}) + \\ (1 - φ) (\frac{t_{ij} (p) + t_{ij} (m)}{2}) \end{matrix}] \\ (x_{ijk} + y_{ijk}) + \forall j, k, \\ [\begin{matrix} φ (\frac{δ (m) + δ (o)}{2}) + \\ (1 - φ) (\frac{δ (p) + δ (m)}{2}) \end{matrix}] s_{k} \leq C, \end{matrix}$ (16) $\begin{matrix} u_{jk} \geq \frac{1}{C} \times \\ \sum_{i}^{I} [\begin{matrix} \frac{φ}{2} (\frac{t_{ij} (m) + t_{ij} (o)}{2}) + \\ (1 - \frac{φ}{2}) (\frac{t_{ij} (p) + t_{ij} (m)}{2}) \end{matrix}] \\ (x_{ijk} + y_{ijk}), \forall j, k, \end{matrix}$ (17) $\begin{matrix} u_{jk} \leq \frac{1}{C} \times \\ \sum_{i}^{I} [\begin{matrix} (1 - \frac{φ}{2}) (\frac{t_{ij} (m) + t_{ij} (o)}{2}) + \\ \frac{φ}{2} (\frac{t_{ij} (p) + t_{ij} (m)}{2}) \end{matrix}] \\ (x_{ijk} + y_{ijk}), \forall j, k, \end{matrix}$ (18) $φ \in {0, 1},$ (19)

Constraints (2)-(4), (6)-(8), (10)-(14).

where φ indicates minimum confidence level of satisfying the uncertain constraints. If the decision maker establishes a high level of confidence level for a solution, the feasible solution set shrinks and as a result, the objective function value become worse. Consequently, the decision maker should set atradeoff between the highest optimality of the objective function and the degree of satisfaction of constraints [33].

5 Solution approach

5.1 Calculation of cycle time

In order to efficiently estimate the cycle time due to the concept of bottleneck measure, Bukchin [34] presented a stochastic procedure of calculating the expected value cycle time, where the number of stations are known (for further information, please refer to [34]). However, this section represents a heuristic approach to determine an interval for the cycle time calculation, where the number of stations are unknown. This solution provides a valuable managerial insight to optimally determine the cycle time within an interval.

5.1.1 Notations

operation time of model j (j′) on station k (k′)

$T_{k}^{'}$ station time of station k′ when model j is served by station k

τ_j The probability of model j to be the next model entering the line*

β _jk ${\begin{matrix} 1 if t_{jk} \geq t_{j^{'} k^{'}}, \\ 0 otherwise \end{matrix}$

* τ_{j} = \frac{The quantity of production of modelj}{The quantity of total production}

5.1.2 Estimation of the cycle time

There are various cycle times for each MMAL during certain periods of time. This section calculates which model in each station has the largest time, which can then be considered as the cycle time of the line.

The probability of t_jk to be the cycle time for each line, or in other words the station k to be the bottleneck, is:

$\begin{matrix} Pr (model j is assembled in stationk) \\ \Pr (t_{jk} \geq max_{k^{'} \neq k} (T_{k^{'}})), \end{matrix}$ (20)

With considering independency between stations, we would have: $\begin{matrix} Pr (t_{jk} \geq max_{k^{'} \neq k} (T_{k^{'}})) = \\ \prod_{k^{'}}^{K_{\max}^{'} \in [K_{lower}^{'}, K_{upper}^{'}]} Pr (t_{jk} \geq T_{k^{'}}), \end{matrix}$ (21)

Lemma 5.1.2.1.The lower bound of number of workstations equals to the numbers of tasks belonging to the negative zone restriction (NNZ).

Proof. If there are at least two tasks with negative zone restriction, they should not be assigned to the same site. Consequently, there should be a separate station for each that equals to the number of corresponding tasks. □

Lemma 5.1.2.2. The upper bound of number of workstations equals to the maximum number of tasks minus tasks belonging to the positive zone restriction (NPZ) plus 1.

Proof. If there are at least two tasks with positive zone restriction, they should be assigned to the same site. As a result, they will be considered as a consolidated task, one of which should be ignored calculating total number of tasks. □

and $Pr (t_{jk} \geq T_{k^{'}}) = \sum_{j} τ_{j} β_{jk},$ (22)

The expected cycle time can be calculated as follows; $\begin{matrix} E (C) = \\ \sum_{j} \sum_{k}^{K_{\max} \in [NNZ, Jmax - NPZ + 1]} \\ Pr (t_{jk} is the largest time in station) t_{jk}, \end{matrix}$ (23)

Since the number of stations and their arrangement is not given, we assume that the service time for model j is identical for different stations. Hence, we would have; $t_{jk} \equiv t_{j} = \sum_{i} {\tilde{t}}_{ij} = \sum_{i} (\frac{t_{ij} (p) + 2 t_{ij} (m) + t_{ij} (o)}{4}),$ (24)

Then, Equation (23) can be transformed into two equations below; $\begin{matrix} E (C_{\min}) = \\ \sum_{i, j} \frac{Pr (t_{j} is the largest time in station) (\frac{t_{ij} (p) + 2 t_{ij} (m) + t_{ij} (o)}{4})}{J_{\max} - NPZ + 1}, \end{matrix}$ (25)

and $\begin{matrix} E (C_{\max}) = \\ \sum_{i, j} \frac{Pr (t_{j} is the largest time in station) (\frac{t_{ij} (p) + 2 t_{ij} (m) + t_{ij} (o)}{4})}{NNZ}, \end{matrix}$ (26)

where $E (C_{\min}) \leq C \leq E (C_{\max})$ (27)

5.2 ɛ-constraint method

In this part, the well-known ɛ-constraint method in order to better depict the controversial nature of the objective functions and also providing a good approximation of Pareto frontier for the DMs is provided.

This method is categorized as a posteriori or generation multi-objective method. In fact, using this method a good approximation of Pareto optimal solutions could be achieved which facilitates the process of decision making when facing bi-objective problems. For this purpose, first the model is solved via each objective function separately in order to find two extreme efficient points of the Pareto frontier. Then, shifting one of the objective functions to the constraint set and relaxing the value of right hand side (ɛ parameter) step-by-step, other Pareto optimal solutions could be achieved. It is clear that by implying small changes in right hand side value, much more Pareto optimal solutions will be achieved (if exist) which in turn takes more time and vice versa.

$\begin{matrix} \underset{1}{minW} \\ s . t . \\ W_{2} \leq ɛ, \\ X \in F (X) . \end{matrix}$ (28)

5.3 MCDA-DEA approach

The DEA approach computes a measure of the associate efficiency of each DMU. This is done by comparing each DMU to all of the remaining DMUs. The problem of evaluation of each DMU is formulated in form of a linear programming model. Evaluation of the performance of n different DMUs includes the solution of n different linear programming problems [35, 36].

One of the well-known DEA models, namely the CCR model, is proposed by Charnes et al. [37]. In this model, there are n DMUs and associate numerical data for each of the m inputs and s outputs for all DMUs. To obtain values for the input weights (vi; i = 1, ... ,m) and the output weights (ur; r = 1, ... ,s) variables, a fractional mathematical programming model should be solved. The mentioned model is as follows (see Model (29)): $\begin{matrix} max h (u, v) = \sum_{r} u_{r} y_{{rj}_{0}} / \sum_{i} v_{i} x_{{ij}_{0}} \\ s . t . \\ \sum_{r} u_{r} y_{rj} / \sum_{i} v_{i} x_{ij} \leq 1, \forall j, \\ u_{r}, v_{j} \geq 0, \forall r, j . \end{matrix}$ (29)

Index j₀ denotes the DMU being evaluated. The objective function of the model is to maximize the ratio of output to input of the DMU under evaluation by computing the appropriate weights v_i and u_r [36]. The constraint of the aforementioned model guarantees that this ratio will not be greater than 1 for each DMU. This implies that the value of objective function will be in [0-1]; the latter value infers that the DMU under evaluation is efficient [35, 36].

Zhou et al. [38] developed a mathematical programming model based on CCR model considering composite indicators (CIs). A CI is a mathematical aggregation of a set of individual indicators including multi-dimensional concepts so that usually have no same units of measurement [36, 39]. Hatefi and Torabi [40] extended the model developed by Zhou et al. [38]. Their model applied common weight approach to construct the CIs so that owns more discriminating power than the best weights that were utilized by Zhou et al. [38] and Azadeh et al. [36]. Since the developed model of this paper copes with CIs (cost sub-indicator; as objective function #1, and inconsistency sub-indicator; as objective function #2), it uses the common weighted MCDA-DEA method developed by Hatefi and Torabi [40]. The associate model of this method is as follows (see Model (30)): $\begin{matrix} min M \\ s . t . \\ M \geq d_{i}, \forall i, \\ \sum_{j} w_{j} I_{ij} + d_{i} = 1, \forall i, \\ w_{j} \geq ɛ_{j}, d_{i} \geq 0, \forall i, j . \end{matrix}$ (30)

In the above model, the decision variables are weights (w_j) and deviations (d_i), and I_ij addresses the value of sub-indicator j regarding to DMU_i. The DMU_i with greatest efficiency score (1- d_i) will be the best solution among the all DMUs [36, 40].

6 Computational experiments

In this section, the effectiveness of the model and the proposed solution approach are implemented on a real industrial case study. In Section 6.1. some preliminary information on the presented case study is provided, and Section 6.2. is dedicated to some sensitivity analyses and managerial insights. It should be noted that all mathematical models are coded and solved by GAMS v.22.9 with CPLEX solver in a computer with core i5 and 4GB of RAM.

6.1 Industrial case study

The presented case study contains a manufacturing company developing for two different models of boiler (i.e. oil and gas boiler). The assembly process of each product is almost the same and both models consist of 16 tasks each, with different precedence diagram. The precedence diagram for each model along with their combined model is presented in Fig. 5. According to some safety policy of the considered company, the cooling task (Task #13) and punching process (Task #16) should not assigned at the same station, while since only one operator serves both tasks #4 and #15, they should be assigned to the same station. Task times of each model are presented in Table 1. It should be noted that due to experts’ knowledge and historical data, task times are considered with perturbation±5 seconds.

6.2 Sensitivity analyses

With aid of Section 5.1., the interval of the cycle time is calculated as 70≤C≤525, and since no task time exceed the minimum value for C, it can be considered as the line’s cycle time. Setting the parameters as Tables 2 and 3, the Pareto front of the proposed ɛ-constraint solution approach have been evaluated under different uncertainty levels (i.e. φ ∈ [0.1, 0.9]). The non-dominant solutions are depicted in Fig. 6.

As can be seen, with increasing the uncertainty level, both objective functions increase and move to a higher level (in terms of values). Since the scale of the first objective function is quite large, Fig. 6 does not show any significant changes between levels. Nevertheless, for better clarification, Table 4 specifically shows the values of OFV₁ under different uncertainty levels.

Figure 7 shows total number of stations (i.e. both regular and crossover) under different uncertainty levels. As the results show, with increasing the minimum satisfaction level of uncertain constraint sets, total number of workstation pursue a non-decreasing behavior. More precisely, for φ= 0.5, we see an increasing value in number of regular stations, while such increase occurs for φ= 0.6 for the crossover ones. Consequently, φ= {0.5,0.6} imposes a high infrastructural change, which should be determined based on preferences of top managers in strategic decision making team.

As mentioned earlier, in this paper, a common weighted MCDA-DEA method developed by Hatefi and Torabi [40], is used to determine the most preferred solution in the obtained Pareto frontiers. The non-dominated solutions are assumed as DMUs for selecting the most preferred solution [36]. Based on decision maker’s preference, we assume that a low level of risk is allowable, so it is considered the Pareto optimal solutions related to the uncertainty level 0.9. The objective functions are assumed as sub-indicators that their values (I_ij) are provided in Table 5. Deviation values (d_i) for each DMU are addressed in Table 6. Efficiency scores would be gained easily by subtracting deviation values from 1 (1 - d_i). Therefore, the final ranking would be as 8 > 7 > 10 > 1 > 9 > 8 > 7 > 6 > 5 > 2 > 4 > 3. About the other uncertainty levels, the ranking of the Pareto optimal solutions similarly can be provided and utilized.

The most preferred solution for the problem related to the uncertainty level 0.9 obtained by the tenth solution is that the value for its objective function #1 is 43449270.55 and the value for the objective function #2 is 0.186202. Based on the results, the common weighted MCDA-DEA method chooses the preferred Pareto frontier solution minimizing total variation of workload that has an important role in a production unit.

7 Conclusion and future research

With recent intense competitiveness and the necessity of product variety, the importance of mixed-model assembly lines have been highlighted. To partially satisfy the just-in-time concept, many companies are changing their line from the traditional straight lines to the U-shaped ones. In this regard, in this paper an efficient solution approach to a bi-objective model for mixed-model U-shaped assembly lines with minimization of the total costs along with the workload of stations as the objective functions are presented. To cope with the uncertain nature of the input data, an interactive fuzzy method is utilized, and further the bi-objective nature of the presented model is handled using an efficient MCDA-DEA based approach. The model is then implemented on a real industrial case study.

Handling the uncertain nature of the input data using some other efficient approaches (e.g., robust optimization), considering other measures in the DEA model, and solving the model on larger scales via some novel and efficient (meta-) heuristic algorithms, are the authors’ main focus for further researches in this field.

Footnotes

Acknowledgments

The authors would like to thank the Editor-in-Chief (Professor R. Langari) of the “Journal of Intelligent and Fuzzy Systems” and anonymous referees for their helpful comments, which greatly improved the presentation of this paper. Additionally, the second author would like to acknowledge the partially financial support of the University of Tehran for this research under Grant No. 8106043/1/26.

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