In cases where the balance problem of an assembly line with the aim to distribute the work loads among the stations as equal as possible, the concept of entropy function can be used. In this paper, a typical assembly line balancing problem with different objective functions such as entropybased objective function plus two more objective functions like equipment purchasing cost and worker timedependent wage is formulated. The nonlinear entropybased objective function is approximated as a linear function using the bounded variable method of linear programming. A new hybrid fuzzy programming approach is proposed to solve the proposed multiobjective formulation efficiently. The extensive computational experiments on some test problems proves the efficiency of the proposed solution approach comparing to the available approaches of the literature.
In recent competitive industrial environment, a manufacturer should be able to produce qualitative products with ontime delivery to the customers. So, designing a manufacturing environment including production department, machines’ layout, etc. is an important issue to reach the goals like better quality and ontime delivery. An effective way to have such design is to establish and balance a production (assembly) line. An assembly line consists of some tasks to be performed in a given order for producing the final product. The order of tasks is determined according to their precedence graph which defines the relationships among the tasks. The line is balanced when the tasks are assigned to some stations in order to optimize a given criterion (or a set of criteria). The stations are usually connected with a conveyor and the parts and semiproducts are moved among the stations on the conveyor to be completed at the end of the line. In a balanced line each station consists of one or more tasks to be operated by usually one (in some cases more than one) worker in a given common time for all stations called cycle time of the line. The cycle time forces the line to send out a product from its last station in each cycle time. The order of stations and assigning the tasks to them must be determined in a way that would respect the precedence relationships of the tasks (precedence graph). The usual criteria used in an optimization problem of an assembly line balancing can be cycle time minimization, number of stations minimization, equipment purchasing cost minimization, workerrelated cost minimization, etc. As a line balancing problem, one or more than one of these criteria may be considered for an assembly line. As an instance given by Fig.
A graphical illustration of input (a) and output (b and c) for a simple assembly line balancing problem.
Assembly lines are classified from different aspects. From a physical point of view, a line can have different shapes. A line can have a straight shape if there is enough straight available space. On the other hand, it can be a Ushaped line in case of small available spaces (Baybars,
Although the literature of assembly line balancing problem is full of interesting studies, only some of its most recent studies are reported here. Lea and Gub (
In this study, as a new assembly line balancing problem, an entropybased objective function plus equipment purchasing cost and worker timedependent wage are considered to be optimized simultaneously in an assembly line. The entropybased nonlinear objective function is linearized using a bounded variable technique which gives a good approximation of the nonlinear function. As a multiobjective problem (Jablonsky,
The rest of this paper is organized as follows. Section
The assembly line considered in this study is a straight assembly line. In this type of assembly lines the stations are arranged on a straight line, so a space with enough length should be available. Any solution for this straight assembly line balancing problem must satisfy the following general conditions:
The precedence relationships among the tasks have to be satisfied among the stations too,
The summation of task times of any station should not be greater than a given cycle time of the line.
The proposed entropybased assembly line balancing problem of this paper uses the abovementioned concepts to optimize an entropy objective function and two more objective functions simultaneously. In order to further formulate this problem, we need to define some notations in Table
Notationsused for mathematical formulations.
Notation  Type  Definition 
Index/parameter  index used for task (index used for task) (the number of tasks)  
Index/parameter  index used for station (index used for station) (the number of stations)  
Parameter  processing time of task 

Parameter  predecessor set of task 

Parameter  successor set of task 

Variable  cycle time  
Variable  1, if task 
Before constructing the whole model, the objective functions of the model are individually explained and formulated here.
As the model (
Schematic representation of the bounded variable approximation method.
Considering the models (
In this section of the paper, an effective solution approach is proposed to tackle the multiobjective formulation (
The proposed solution approach of this study is a new hybrid version of fuzzy programming method to produce efficient solution to the multiobjective formulation (
Schematic representation of the fuzzy membership functions (Zimmermann,
Increase the NIS value for maximization type objective functions,
Decrease the PIS value for minimization type objective functions,
Change the given set of values for weights of the objective functions (
As the most important step of any hybrid version of fuzzy programming approach is singleobjective model phase, some advantages of singleobjective model of the proposed approach of this study (formulation (
The optimization procedure of the singleobjective model is done in one phase.
Obtaining unique or efficient solution is guaranteed.
The varying weights of the objective functions are eliminated.
Membership function values are not used in the objective function.
The goals are partially prioritized in the objective function and constraints. The weight of membership functions in the objective function depends on the number of objective functions of the main problem.
The feasibility and efficiency of the formulation (
Let’s first define the following formulation which is a part of formulation (
The efficiency of the solution of model (
In order to compare the performance of the proposed approach of this study to the other methods of the literature presented in the previous subsection, the following distance measure is used (Alavidoost
The precedence graph of test problem 1.
The proposed multiobjective formulation (
Two test problems are considered to evaluate the performance of the solution approach of this study. These are taken from the literature of assembly line balancing problem and are modified by adding some required data for the new parameters which are new in this study.
Test problem 1 (Jackson,
Test problem 2 (Mitchell,
The precedence graph of test problem 2.
In this section the performance of linearization technique used for entropy objective function (
Test problem 2 is selected for this aim.
A feasible solution is generated manually to respect the constraints (
The solution is evaluated by model (
The solution is evaluated by model (
The number of segments and their associated lengths are shown in Table
The results obtained for parameter tuning of the linearization technique.
Experiment  Number of segments 
Equal length of the segments ( 

1  5  0.200  −1.609  −1.778 
2  10  0.100  −1.748  
3  16  0.062  −1.768  
4  20  0.050  −1.770  
5  40  0.025  −1.776  
6  70  0.014  −1.777  
7  100  0.010  −1.778 
Schematic representation of the performances of the entropy objective function and its linearized form.
According to the results of Table
To measure the ability of the proposed approach the model (
The number of segments
To make a more detailed performance analysis, five combination of values for the weights (
In some of the methods, another weight (say
Different combinations of weights used to run the proposed method and the methods of literature.
Combination  
C 1  0.20  0.30  0.50 
c 2  0.30  0.40  0.30 
C 3  0.33  0.33  0.34 
C 4  0.40  0.30  0.30 
C 5  0.50  0.25  0.25 
The positive and negative ideal solutions obtained for each objective function.
Objective function  Ideal solution  Test problem  
Test problem 1  Test problem 2  
−1.544  −1.776  
0.883  0.909  
88000  144000  
150000  159600  
330  880  
675  1080 
In some of the methods, a coefficient (say
The result obtained for test problem 1.
Weights  Method  
C1  ABS  0.867  0.581  1  0.152  0.129  0.126 
DY  0.792  0.661  1  0.143  0.11  0.102  
TH  0.904  1  0.783  0.128  0.11  0.109  
SO  0.926  0.871  0.87  
The proposed approach  0.926  0.871  0.87  
C2  ABS  0.904  1  0.783  0.071  0.065  
DY  0.816  1  0.783  0.12  0.085  0.065  
TH  0.913  0.919  0.826  0.11  
SO  0.904  1  0.783  0.071  0.065  
The proposed approach  0.904  1  0.783  0.071  0.065  
C3  ABS  0.926  0.871  0.87  0.111  
DY  0.844  0.871  0.87  0.138  0.08  0.051  
TH  0.913  0.919  0.826  0.114  0.071  0.059  
SO  0.904  1  0.783  0.08  0.074  
The proposed approach  0.904  1  0.783  0.08  0.074  
C4  ABS  1  0.742  0.696  0.169  0.12  0.091 
DY  0.926  0.71  0.87  0.156  0.1  0.087  
TH  0.904  1  0.783  
SO  0.904  1  0.783  
The proposed approach  0.904  1  0.783  
C5  ABS  0.929  0.468  0.609  0.266  0.169  0.133 
DY  0.993  0.694  0.696  0.156  0.108  0.077  
TH  0.904  1  0.783  0.103  0.073  0.054  
SO  0.904  1  0.783  0.103  0.073  0.054  
The proposed approach  0.926  0.871  0.87 
The test problems were solved by submodels (
Finally, applying the abovementioned test problems, the model (
Chart of the performance of the proposed and employed approaches for test problem 1 with weight combination C1.
The result obtained for test problem 2.
Weights  Method  
C1  ABS  0.997  0.738  0.9  
DY  0.911  0.738  0.9  0.146  0.096  0.079  
TH  0.997  0.738  0.9  
SO  0.997  0.738  0.9  
The proposed approach  0.997  0.738  0.9  
C2  ABS  1  0.929  0.7  
DY  0.913  0.929  0.7  0.145  0.099  0.09  
TH  0.997  0.738  0.9  0.136  0.109  0.105  
SO  1  0.929  0.7  
The proposed approach  1  0.929  0.7  
C3  ABS  0.997  0.738  0.9  
DY  0.911  0.738  0.9  0.15  0.097  0.086  
TH  0.997  0.738  0.9  
SO  0.997  0.738  0.9  
The proposed approach  0.997  0.738  0.9  
C4  ABS  1  0.929  0.7  0.111  0.093  0.09 
DY  0.933  0.929  0.7  0.138  0.095  0.09  
TH  0.997  0.738  0.9  
SO  0.997  0.738  0.9  
The proposed approach  0.997  0.738  0.9  
C5  ABS  1  0.929  0.5  0.143  0.126  0.125 
DY  0.999  0.738  0.5  0.191  0.141  0.125  
TH  0.997  0.738  0.9  
SO  0.997  0.738  0.9  
The proposed approach  0.997  0.738  0.9 
Chart of the performance of the proposed and employed approaches for test problem 1 with weight combination C2.
According to the obtained results for test problem 1, the proposed approach is better than the other approaches of the literature in most of the weight combinations. As can be concluded from Table
Chart of the performance of the proposed and employed approaches for test problem 1 with weight combination C3.
Chart of the performance of the proposed and employed approaches for test problem 1 with weight combination C4.
As an analysis to the obtained results for test problem 2, the proposed approach is among the best performed approaches in all the weight combinations. As can be concluded from Table
Chart of the performance of the proposed and employed approaches for test problem 1 with weight combination C5.
Chart of the performance of the proposed and employed approaches for test problem 2 with weight combination C1.
A typical assembly line balancing problem with different scale objective functions was studied in this paper. An entropy function was used as an objective function in order to balance the work load of the stations of assembly line plus two more objective functions like equipment purchasing cost and worker timedependent wage. The most important limitations of this problem were its nonlinearity and its multiobjective nature. The nonlinearity of the entropybased objective function was approximated as a linear function using the bounded variable method of linear programming. A new hybrid fuzzy programming approach was developed to solve the proposed multiobjective formulation. In order to compare the efficient solutions of the problem, three distance metrics were used. The required computational experiments were performed by the proposed hybrid fuzzy programming approach and some other approaches of the literature on some test problems. According to the obtained results and using the distance metrics, the proposed solution approach performs either the same or better than the multiobjective solution approaches of the literature like ABD, DY, TH, and SO.
Chart of the performance of the proposed and employed approaches for test problem 2 with weight combination C2.
Chart of the performance of the proposed and employed approaches for test problem 2 with weight combination C3.
Chart of the performance of the proposed and employed approaches for test problem 2 with weight combination C4.
Chart of the performance of the proposed and employed approaches for test problem 2 with weight combination C5.
We are grateful to the editors and anonymous reviewers of the journal for their helpful and constructive comments that helped us to improve the quality of the paper.