Introduction

The flexibility of assembly systems is essential in the assembly industry in order to be able to respond to the changeable characteristics of market demands. These demands, represented by increasing customisation, the shortening of a product lifecycle, and a high product variety produced in small batches (Shishir 2010). For this reason, it has become necessary to develop dynamic, flexible and reconfigurable assembly systems. The flexible labour line (or flexible assembly line), is one of the promising techniques to create the most effective and productive assembly systems to respond to the challenges in manufacturing environments (Moslemipour and Lee 2012). On the other hand, mixed-model assembly lines enable manufacturers to assemble multiple models in any order within a short delivery time and with the lowest possible cost. As a consequence, considering flexible workforce in designing a mixed-model assembly line is an innovative solution.

One attractive form of flexible workforce in mixed model assembly lines is the use of cross-trained workers who can perform all the assembly processes of products and also shifting their capacity to where it is needed (Sawhney 2012). This paper explores one such system that of staffing a line of assembly processes for mixed models of a product, with fewer workers than workstations on the line. This dynamic, flexible and reconfigurable system is so-called mixed model walking worker assembly line. In this type of assembly line, each worker travels along the line to carry out all required tasks (Wang et al. 2005). Walking worker assembly line (WWAL) results in a series advantages over a traditional lines, that is, fixed worker assembly lines (FWAL), where traditionally each worker has an assigned fixed task and continuously repeats that assigned task (see Fig. 1). In this context, rearranging assembly lines from the FWAL to the WWAL by a number of companies has led to achieve the followings (Wang et al. 2005):

  • Greater ease of line balancing whilst contributing to reducing the number of buffers required;

  • An adjustable number of line workers according to demand requirement; and

  • A minimisation of labour and tooling costs.

Certainly, design of such assembly lines has significantly impacted workforce performance and consequently is more complex in comparison with single fixed worker assembly lines. This is because there is an essential need to consider human worker capabilities and their limitations in operational designs for these systems (Alzuheri et al. 2010). Also, due to the sources of complexities in manual assembly systems such as processing times at each workstation and skills of workers who perform these tasks, most of these systems are stochastic rather than deterministic (Longo et al. 2012). Considering the nature of the system thus, physical workers capabilities as well as the “multiple components” of WWAL can lead to large design space solutions and it is more difficult (if not impossible) to use exact mathematical methods such as integer-programming to solve problem design of such system.

Fig. 1
figure 1

Layout of walking worker line (Wang et al. 2005)

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Therefore, this difficulty requires the use of meta-heuristic approaches to solve the problem. The resulting design solutions are adjusted to create an acceptable design and the best resulting solution is implemented. Consequently, the design process must be regarded as an iterative, generative and test-involving process (Scholl et al. 2010).

For this purpose genetic algorithms (GAs) are proposed to solve the WWAL optimisation design problem, aiming at yielding optimal (or near-optimal) solution in large search spaces quite faster than any other optimisation technique owing to their parallel searching feature (Chambers 2010). Currently, GAs required for such a demonstration does not exist in literature on the topic. This paper describes development of a novel customised GAs that are used for solving the large scale optimisation problem that is typically can found in manual assembly line with walking workers.

In the next section of this paper, a literature review is conducted for research work that considers WWAL. Section “Design problem: mixed model assembly system optimisation problem”, introduces the design problem that this paper attempts to solve. Section “The mathematical model” includes a summarised background about the mathematical model, which drives GAs comprising the objective functions. In “Genetic algorithms” section, GAs are reviewed as the approach for optimisation selected to handle the design problem presented in this paper, and there follows a description of the developed structure for this approach. Then the results of the GAs application are presented and analysed in “The results and analysis of design optimisation problem” section. Finally, in “Conclusions” section, the conclusion of this paper is presented.

Literature review

In last 15 years, research has investigated the subject of cross-trained walking (moving) workers performance in production systems. Mileham et al. (2000) and Nakade and Nishiwaki (2008) give a summary of this research. However, all this research was limited to the application of moving cross-trained workers in a cell in linear or U-shaped production line in the industry. In addition, most of the research referred to systems including such applications by names other than WWAL. The term WWAL is a recent concept (Bley et al. 2007; Mileham et al. 2008). Publications have focussed little on the subject of this type of assembly line, and even then, it has only been on the initial stages of the design phase. Table 1 summarises the number of research papers published by topic.

Table 1 Summary of the research papers conducted on WWAL
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As the results of Table 1 show, most of the research carried out using both simulation technique and integer-programming approach for designing WWAL, so far has considered the single model type at early stage of the design phase. As a result, the design process was relatively simple due to the considerably large amount of extant research examining solution techniques offered on the problem structure of FWAL, which is also applicable in solving simple WWAL problems. Although WWAL is an important consideration in assembling different models as noted previously, there has been no research reported on the operational aspects of mixed model assembly line design with walking workers.

Since cross-trained workers have become a necessary requirement of WWAL, effective operational design is critical for enhancing productivity and maximising the usefulness of a given assembly system by optimising the utilisation of workers effectively. This improvement in system performance should be concerned with investigating the ergonomics consequences of any improvement processes, in terms of exposure to risk factors for work-related musculoskeletal injuries. This results in a multi-objective design problem.

However, as noted from the review of literature, none of the researchers focused on operational design problems of mixed model lines adopted from the WWAL approach. Also, because of the nature of WWAL, and because the design problem is a multi-objective one, it is expected to be a large-scale one. This paper explores one such problem that of optimising operational design for mixed model assembly lines with walking workers in an industrial environment for made-to-order assemblages with some specific objectives, constrains and considerations about productivity and ergonomics.

Although there are many alternative heuristic methods for solving multiple objective design problems, there is still no clear guidance available regarding the optimal choice for a particular problem. GAs seem particularly desirable to solve multi-objective optimisation problems because they deal simultaneously with a set of possible solutions (the so-called population) which allows to find an entire set of Pareto-optimal solutions in a single run of the algorithm, instead of having to perform a series of separate runs as in the case of the traditional mathematical programming techniques. On the other hand, GAs have been applied successfully in a wide variety of optimisation problems (Li et al. 2007). Therefore, developed GAs has been used to solve the problem presented in this paper. At the same time, a mathematical model is proposed for driving developed GAs. A detailed description of this model can be seen in Al-Zuheri et al. (2013). Thus, this paper focused mainly on the development of genetic algorithms to optimise multi-objective functions of performance measures in terms of productivity and ergonomics of mixed model assembly line using WWAL approach for assembling a plastic electrical box.

Design problem: mixed model assembly system optimisation problem

Let’s assume, hypothetically speaking, an assembly cycle of a plastic electrical box produced in a design and manufacturing company. Figure 2 shows the plastic box assembly process and the models. The product is assembled on a flow line with seven workstations.

Fig. 2
figure 2

Illustration of plastic electrical box assembly process; a the assembly line, b model A, c model B

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The line is used to assemble two different model types. To assemble model A requires processing on workstations (1, 3, 4, 6, 7) and model B requires processing on workstations (1, 2, 4, 5, 6, 7). Therefore, the models are nearly identical except that with model B process 2 is substituted for process 3 in model A. In addition, model B requires process 5 while model A does not. The processes must be run and completed in the exact sequence given for each model,

Table 2 gives average working time, and average expenditure for the metabolic energy requirement schedules of the two model types needed to complete the assembly tasks. The average working time are determined through method time measurement (MTM) and hence require re-calculation according to the variable requirements in the model proposed in Al-Zuheri et al. (2013).

Table 2 Task descriptions, average working time, required metabolic energy expenditure and the weight of the each model of product at each workstation on the line
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Also Table 2 lists the product model data in terms of task descriptions and the weight of the product of each model at each workstation on the line.

Let’s assume, that on this particular day the “make to order” WWAL is asked to assemble of 80 and 100 products from model A and model B, respectively. In this design problem, the assumptions that the available production time per that day for assembly of both models is one eight hour shift per day (480 min). Each model gets exactly half of this available time, then each model will have 90–240 min of assembly time to produce the required rate. The available production time is split between the two models. The required production demand from model A is assembled first and then production demand of model B. The workstations are located to ensure an efficient layout, and the most efficient assembly process arrangement maximises productivity of workers.

The design problem is to determine which number of workers and their skill levels (low and high) to select and which walking speed of workers (slow or fast) to assign so as to meet production demand requirements for the two product models as well as optimising the system operational design as mentioned in the objectives in “Design problem: mixed model assembly system optimisation problem” section. Also, the design problem includes determining the best available assembly time for both models to match the set design objectives.

Rather than the specific data that are given in Table  2, the parameters related to structure of the system and for workers gender, weight, and their cost, according skill levels are shown in Table 3, while the operational design variables that will affect the objective design functions are:

Variable \(x_{1}\)::

skill level of slow walking workers;

Variable \(x_{2}\)::

number of slow walking workers;

Variable \(x_{3}\)::

walking speed of workers;

Variable \(x_{4}\)::

the available assembling time for model A; and

Variable \(x_{5}\)::

the available assembling time for model B.

Table 3 Input parameters of the WWAL and their levels
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These five critical variables are set to a different level as shown in Table 4. Based on these settings, there are a total of hundred thousands of different combinations and for each combination the model has a fitness function value is also equal to performance measure (objective function value).

Table 4 Operational design variables and their levels for the WWAL design
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The mathematical model

The model completes two objective functions for each of productivity and ergonomics. The four objective functions as follow:

  • Minimise the balance of labour blockage,

  • Minimise the shift time labour cost,

  • Minimise the metabolic energy consumption, and

  • Maximise the rate of exposure variability

A detailed formulation of these objective functions with assumptions, input parameters, variables and constraints can be seen in Al-Zuheri et al. (2013).

In conclusion, following the end-user equations that are used to describe these objective functions.

  • Minimise the balance of labour blockage:

The balance of labour blockage \(\eta \), is the ratio of in-process waiting time divided by the overall cycle time:

$$\begin{aligned} f_1 \left( x \right)&= \eta \end{aligned}$$
(1)
$$\begin{aligned} Min.f_1 \left( x \right)&= Min\left( \frac{t_{Im}^{k_b } }{T_{ca} (n)}\right) \end{aligned}$$
(2)

Subject to:

$$\begin{aligned} P_{Shift} =\mathop \sum \limits _{m\in M} P_m \ge \left( {\frac{T_{Shift} }{t_{Im}^{k_b } }} \right) \end{aligned}$$
(3)

This constraint ensures that every candidate solution has production rate equal or greater than to that with traditional assembly line FWAL.

  • Minimise the shift time labour cost:

The cost of shift time labour requires calculation for each design of WWAL. The cost considered is direct labour cost. Given the number of workers, their efficiencies, their cost per hour and the shift time hours, the labour cost in terms of these variables are:

$$\begin{aligned} f_2 \left( x \right)&= C_{TI} =\mathop \sum \limits _{m\in M} \left( Cost \,of\, worker\, per\, hour\, shift\,time \right. \nonumber \\&\left. \times \, Shift\, time\, hours \right) \end{aligned}$$
(4)

Mathematically, the objective function is to minimise the total shift time labour cost:

$$\begin{aligned} Min.f_2 \left( x \right) \!=\!Min.(C_{TI} )\!=\!Min.\left( {\mathop \sum \limits _{m\in M} \!\!\mathop \sum \limits _{j\in J} C_{mj} \times T_{Shift } } \right) \nonumber \\ \end{aligned}$$
(5)

Subject to:

$$\begin{aligned} C_{TI} \le C_{Tl} FW \end{aligned}$$
(6)

where \(C_{mj} \) is the cost of worker per shift time hour. \(C_{Tl} FW\) is the shift time labour cost for traditional assembly line-FWAL, and normally in this line the number of workers equal the number of workstations and the cost of shift time hour is low.

  • Minimise the metabolic energy consumption:

Since in WWAL, the worker needs to work standing up rather than sitting down also, he or she moves along with the work-piece. Mathematically then, the Garg’s model (Garg et al. 1978) for the prediction of average metabolic energy expenditure requirement to perform the entire assembly tasks in one cycle time \(E_m \left( n \right) \) in WWAL is as follows:

$$\begin{aligned}&f_3 \left( x \right) \!=\!E_m \left( n \right) \!=\!Min.f_3 \left( x \right) \!=\!Min.E_m \left( n \right) \end{aligned}$$
(7)
$$\begin{aligned}&Min.E_m \left( n \right) \nonumber \\&\quad =\frac{\left[ \left( E_{mp} \!\times \! T_c \left( n \right) \right) \!+\!\left( {\sum _{k\in K} \Delta E_{Om}^k } \right) \right. \left. \!+\,Min.\left( {\sum _{k\in K} \Delta E_{Wm}^k } \right) \!+\!Min.(\Delta E_{Im}^{k_b } )\!\right] }{ {T_{ca}}(n)}\nonumber \\ \end{aligned}$$
(8)

Subject to:

$$\begin{aligned} E_m \left( n \right) \le 3.2\frac{Kcal}{min.} \end{aligned}$$
(9)

This constraint ensures that in addition to the energy cost of walking or in-process waiting to work, the metabolic energy requirement to assemble one product by the worker in each candidate solution should not exceed the limitation value of energy consumption 3.2 kcal/min. in a working day (8 h) (Waters et al. 2011; Zhu et al. 2010).

  • Maximise rate of exposure variability

This measure is the ratio of overall walking time between workstations to the amount of time worker spends to perform all tasks of assembly job during the work cycle with standing position.

$$\begin{aligned}&f_4 \left( x \right) =\varphi (n)\end{aligned}$$
(10)
$$\begin{aligned}&Max.f_4 \left( x \right) =Max.\left( {\varphi \left( n \right) } \right) =\frac{W_c \left( n \right) }{S_C \left( n \right) +t_{Im}^{k_b } } \end{aligned}$$
(11)

Subject to:

$$\begin{aligned} L_w =\frac{W_C \left( n \right) \times n\times v_o }{1000}<4.32km \end{aligned}$$
(12)

The OSHA regulations (2009) related to the amount of maximum walking distance for assembly line worker cannot have more than 4.32 km (2.7 miles) at the shift time.

Genetic algorithms

GAs are meta-heuristic stochastic approach for finding the global optimal solution for a combinatorial optimisation problem (Liu et al. 2011). They mimic the mechanism of genetics and natural selection (survival of the fittest) natural genetics as described by Charles Darwin (Li et al. 2007). Holland was introduced GAs in the 1960s and 1970s as an approach to evolve an optimal solution from a population of initial feasible solutions available for solving an optimisation problem (Goldberg 2012). The GAs search for optimal or near-optimal solutions starts with the generation of a number of individuals for the initial random population of solutions. Each individual in the population is a chromosome representing a solution of the problem. The chromosome is a string of symbols or (set of genes) can be coded in different forms; binary, integer, real, etc. With taking into account specified selection rules, the initial population evolves towards a population expected to include the optimal solution, eventually accomplished through successive iterations called generations.

Within a generation, GAs undertake to select subsets (usually two) of chromosomes from the current population called parents for mating them to produce new chromosome called children (or offspring). Chromosomes are selected according to their fitness relative to the current population. Based on that, the chromosomes or solutions that are of higher fitness then, have a greater chance of selection and then “mating” with another high-fitness chromosome for producing new chromosomes. Selected chromosomes are subjected to rules of combination to yield children; genetic operators—crossover and mutation. The crossover consists of the exchange of parental genes through passing it on to the new chromosome. Through applying mutation, further genetic diversity is introduced into the chromosomes of a population, and by further mating between parents; a set of new generation chromosomes is formed. After several of generations, the algorithm tends to the optimal chromosome, which globally, has a better fitness in comparison with other chromosomes.

Design of the GAs structure for the optimisation of the WWAL design problem

GAs are responsible for inserting the changes of values for design variables into the mathematical model. For reaching optimal design of WWAL, the approach works by iteration, and every result given by the model corresponds a design solution of WWAL. The chromosome in solution spaces of WWAL includes a number of genes (design variables), that describe operational aspect of the system. These variables are as listed in “Design problem: mixed model assembly system optimisation problem” section. The WWAL design corresponding to each chromosome is characterised by its fitness, which is measured by its resultant objective function value (will be derived later in this section).

Herein, a generation consisting of evolving chromosomes of the previous population and new chromosomes are made through reproduction process by means of crossover, mutation, and selection of their parent’s chromosomes. Figure 3 shows optimisation procedure using GAs and mathematical model. Details are presented on the designed GAs components in the forthcoming subsections of this paper.

Fig. 3
figure 3

A schematic diagram of the integrated GAs and the mathematical for conducting the design problem

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Chromosome representation

For the problem under consideration in this paper, real coded value variables are adopted to present chromosomes. Using real coded variables makes it possible to use large domains (even unknown domains) for variables. There is also a chance to increase the efficiency of GAs by exploiting the incremental nature of the functions with continuous variables. Herein, each chromosome (solution) of WWAL design comprises a vector of real design variables, and it is very close to the real structure of the practical problem. In this regard, considering the alternative design solution is \(a\), let \(n\) the number of design variables that are allocated to design solution, the whole chromosome code which represents all design variables, is shown by the following equation:

$$\begin{aligned} x_a =[x_a (1),x_a (2),\ldots .,x_a (n)],\qquad a=1,\ldots \ldots .A \end{aligned}$$
(13)

where \(A\) is the number of available feasible solutions to the problem. The design variables \(x_n (n=1,2,3,\ldots ,n)\) are operational variables. Also these variables is not unconstrained and must be within a level from \(x_{min}\)to \(x_{max} \).

Generation of the initial population

Initialisation of the population of chromosomes (set of possible solutions to the problem) may be randomly created, or created according to heuristic procedures (Zhan and Zhang 2013). Yu and Yin (2010) have been indicated that the performance of the GAs algorithms is not as good from well-adapted (seeded) population as it is from a random start. Based on that, in this paper the chromosomes in an initial population are randomly generated ones. The number of chromosomes within the population is called population size \(Pz\).

Fitness evaluation of the chromosomes in the population

Once the initial population is developed, fitness evaluation for each chromosome is performed. Fitness function is the key performance index in GAs application and can determine which of the chromosomes will reproduce and survive into the next generation. The better performing chromosomes are selected as candidates for evolving them using the genetic operations. In general, derivation of fitness function is through the objective functions, and is used in successive genetic operations (Garg 2010). Since the objectives \( f_1 \left( x \right) ,\,f_2 \left( x \right) \) and \(f_3 \left( x \right) \) must be minimised and \(f_4 \left( x \right) \) must be maximised (as stated in “Design problem: mixed model assembly system optimisation problem” section), it is then required to convert these multiple objectives into a single overall objective function, called a resultant objective function. The resultant objective function is considered as a fitness function requiring final optimisation. The desirability function approach (Derringer and Suich 1980) is used to achieve the converting from multiple objectives to single objective in two stages, as described below.

Stage 1—defining individual desirability functions for objective functions: as a result from computational experiments, assuming that there are \(\hbox {a}\) different alternative solution for the candidate problem, each objective (productivity or ergonomics) have desirability at \(\hbox {a}\) solution. Accordingly \(\eta _a,C_a , E_a \) and \(\varphi _a \) denote, respectively the \(\eta (n), C_{TI},E_m \left( n \right) \) and \(\varphi (n)\) objectives values for solutions \( (a=1,\ldots .,A)\). In this paper, the target is for three objectives to be minimised and one objective to be maximised (for the one-sided case), as in the case when the objective is productivity measure (balance of labour blockage), the individual desirability \(d_a \) for \( \eta _a \) is defined as follows:

$$\begin{aligned} d_a(\eta _a)=\left\{ \begin{array}{lll} 1,&{} \eta _a\le L_n&{}\\ \left( \frac{\eta _a-L_\eta }{U_\eta -\eta _a}\right) ^{W1}&{} L_\eta \le \eta _a\le U_\eta &{} \,a=1,\ldots \ldots ,A\\ 0,&{}\eta _a\ge U_n\\ \end{array}\right. \nonumber \\ \end{aligned}$$
(14)

For second productivity objective (shift time labour cost), \(C_a\):

$$\begin{aligned} d_a(C_a)=\left\{ \begin{array}{lll} 1,&{} C_a\le L_C&{}\\ \left( \frac{C_a-L_C}{U_C-C_a}\right) ^{W2}&{} L_C\le C_a\le U_{C}&{} \,a=1,\ldots \ldots ,A\\ 0,&{} C_a\ge U_C \end{array}\right. \nonumber \\ \end{aligned}$$
(15)

For ergonomics objective (average metabolic energy expenditure),  \(E_a \):

$$\begin{aligned} d_a(E_a)=\nonumber \\&\left\{ \begin{array}{lll} 1,&{} E_a\le L_E&{}\\ \left( \frac{E_a-L_E}{U_E-E_a}\right) ^{W3}&{} L_E\le E_a\le U_E&{} \quad a=1,\ldots \ldots ,A\\ 0,&{} E_a\ge U_E \end{array}\right. \nonumber \\ \end{aligned}$$
(16)

For second ergonomic objective (rate of exposure variability),\( \varphi _a \):

$$\begin{aligned} d_a(\varphi _a)=\left\{ \begin{array}{lll} 1,&{} \varphi _a\le L_\varphi &{}\\ \left( \frac{\varphi _a-L_\varphi }{U_\varphi -\varphi _a}\right) ^{W4}&{} L_\varphi \le \varphi _a\le U_\varphi &{} a=1,\ldots \ldots ,A\\ 0,&{} \varphi _a\ge U_\varphi \end{array}\right. \nonumber \\ \end{aligned}$$
(17)

where \(U_\eta (L_\eta ),\, U_C (L_C ),\, U_E (L_E )\) and \(U_\varphi (L_\varphi )\) are the upper (lower) limits of the four objective functions, respectively. The \(w1,\,w2,w3\) and \(w4\) indicates the weight. In the design problem of this paper, the assigned weights imply that balance of labour blockage \(\eta _a \) and average metabolic energy expenditure \(E_a\) are twice as important compared with other objectives important \(C_a\) and \(\varphi _a \), the shift time labour cost and rate of exposure variability.

Stage 2—calculation of the overall desirability: an overall desirability function \(D_a\) can be obtained by using the geometric mean of the individual desirability \(d_a\) for each measure. It reflects the composite desirable grade of the \(a_{th} \) alternative solution desirability with respect to the objective functions.

$$\begin{aligned} D_a =\left( {\prod \limits _{v=1}^v d_{a,v}^{r_v } } \right) ^{1/{\sum r_v }}a=1,\ldots \ldots .A \end{aligned}$$
(18)

where: \(d_{a,v} =\) the \(v_{th}\) objective function of alternative solution \((a)\); and

\(r_v\) = the relative importance that is assigned subjectively and, respectively to each objective function. In problem design of this paper, \(v=4\) and for more simple form of the equation, the expressions for the individual desirability of each objective function \(:d_a \left( {\eta _a } \right) = \hat{\eta } _a ,\, d_a \left( {C_a } \right) = \hat{C}_a , \quad d_a \left( {E_a } \right) = \hat{E}_a \), and \( d_a \left( {\varphi _a } \right) = \hat{\varphi } _a \). Accordingly, the overall desirability functions as in the following:

$$\begin{aligned} D_a =\left( {\hat{\eta } _a \times \hat{C} _a \times \hat{E}_a \times \hat{\varphi } _a } \right) ^{1/4}\qquad a=1,\ldots \ldots .A \end{aligned}$$
(19)

In GAs of this paper, the resultant objective function (the overall (or total) desirability, \(D_a \) given in formula 19, is adapted as a fitness function. Then, the goal of WWAL design optimisation problem is to find the levels of design variables that maximise the overall desirability \(D_a \), in the design level of interest, that is;

$$\begin{aligned} Max.(D_a )&= Max. \left[ {\left( {\hat{\eta }_a \times \hat{C}_a \times \hat{E}_a \times \hat{\varphi }_a } \right) ^{1/4}} \right] \nonumber \\&\qquad a=1,\ldots \ldots .A \end{aligned}$$
(20)

Subject to:

$$\begin{aligned} x_{min} \le x \le x_{max} \end{aligned}$$
(21)

where, formula 21 can be stated as the primary or initial design levels. \(x_{min} ,\,x_{max} \) are the minimum and maximum design levels of operational variables, respectively. This constraint of optimising design of WWAL is the last constraints to be met in a set of feasible solutions before consider one of them as superior choice that meets the previous four objectives.

Selection strategy

Since the best chromosomes (solutions) in the current generation are used to reproduce the next generation, this requires establishing a strategy to select those chromosomes. When implementing GAs, there are two selection strategies in choosing the chromosomes for reproduction of new chromosomes; elitist and non-elitist. Elitist strategy is used herein to avoid losing best solutions in population by saving a number of them and later copying it into the next generation for the process of evolution.

Genetic operators

Several types of operators are known currently. Please note however, that these operators are beyond the scope of this paper. Following is a brief background about the rule of each operator in evolving process of GAs, along with the types of operators selected for this paper.

Selection process: Darwin’s evolutionary theory of the survival of the fittest is applied in the parent selection process for chromosomes in GAs applications. Based on this law, chromosomes with high fitness function values are selected for genetic operations using stochastic selection process (Goldberg 2012). Many types of selection techniques are suggested with which to select the best chromosomes to be copied into the next generation (Martinec and Bundzel 2013). The biased roulette wheel technique is the most popular from these techniques. This paper applies the roulette wheel technique because it is easy to implement and mimics nature more faithfully and consequently is much more appealing (Tang and Tseng 2012).

Crossover: crossover or recombination is the process most widely used by which a chromosome pair recombines to generate a new chromosome pair (offspring). As the name implies, this process involves swapping some parts (genes) between the pair selected chromosomes. The crossover operators have a very big chance of reproducing the parent chromosomes’ desirable features, and consequently expect to improve the solution quality of the problem undertaken. Probabilistic recombination is commonly used to improve or repair the offspring, rather than using complex or intelligent heuristics for this purpose (Kucukkoc and Yaman 2013). In this regards, the recombination process is not applied to every pair of chromosomes selected for recombination. The process is controlled by a crossover probability \((P_c )\) (Javadian et al. 2011). This probability indicates the number of chromosome pairs that will be involved in the crossover operation. Typically, the probability for crossover ranges from 0.6 to 0.95. From several types of crossover, single-point crossover technique is used in this paper.

Mutation: mutation is another genetic operator is applied to single chromosome for creating a modified mutant—new offspring being added to the population. In mutation operation, one or more gene values of the created chromosome, randomly chosen for replacement to create another offspring. With this new offspring, the GAs may be able to reach a more optimal solution than was previously possible using crossover operator. For real-coded genetic algorithm, like the one used in this thesis, uniform mutation is applied. In real coded GAs, uniform mutation operation replaces a gene to a real number within a fixed solutions space if a gene of a chromosome is mutated. This type of mutation is preferable in research because it can configure the searching solutions spaces with generations (Abdul-Rahman et al. 2011). Like crossover operator, mutation can be done based on heuristics or on probability \((P_m )\) (Gen and Lin 2013; Kucukkoc and Yaman 2013). Typically, this probability ranges from 0.001 to 0.01. A simplified scheme of GAs for chromosomes representing and operators is given in Fig. 4.

Fig. 4
figure 4

Evolving process by GAs operators for optimisation WWAL problem in this paper

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Mechanism of constraints handling

Before evaluating all design objectives as outcomes of modelling process it is not known if any design solution is feasible or not, especially if any conflict exists. With the mathematical model of the overall desirability function (formula 15) if any value of \(d_a \) for anyone from the objective functions (\(\eta _a ,\,C_a ,E_a \) and \(\varphi _a ) = 0\), then the overall desirability function, \(D_a = 0\). Hence, the exploration of GAs to reach an optimal solution can not include infeasible solutions that involve measure values falling outside of the acceptable limits (e.g. \(d_a=0\)). This mechanism is beneficial to the optimisation process as it ensures the optimal solutions will generally locate on the boundaries between feasible and infeasible solutions. The general form of this mechanism in this paper as follow:

$$\begin{aligned} D_{a(i)}=\left\{ \begin{array}{ll} D_{a(i)},&{}\hbox {if } a(i) \hbox {feasible};\\ 0,&{}\hbox {otherwise}; \end{array}\right. \end{aligned}$$
(22)

Stopping criteria

Usually, GAs are stopped if the quality of the solution to the problem on hand is not improved significantly per generation. Thus, the designer of the algorithms has to decide the stopping criteria. Because if the algorithms are let to run for too long, time and computational power are wasted and they will also revisit all the solutions it has previously searched out. The proposed stopping criteria are as follows:

  • Stop the algorithmic search process if it reaches the acceptable design solution for WWAL. Usually, when the GAs search continues with negligible improvement in solution quality for a number of generations, it has either found a good answer or has become trapped in the local optima.

  • The GAs search should be stopped when no significant variations are appreciated in the average of fitness function values for a certain number of generations.

Application of developed GAs to the WWAL design

As described in the preceding sections, the developed GAs are aimed to find WWAL design solution that is supposed to be optimal in terms of setting design objectives from a range of alternatives conducted by using proposed mathematical model in “The mathematical model” section. As illustrated before, finding optimal design of WWAL involves determination of the best combination of decision variables so as to optimise objective functions. By integrating the mathematical model with developed GAs, optimal design solution can be obtained. In application of GAs, the candidate design solutions would be the array \(<n_{1}, n_{2},\ldots ., n_{k}>\) in one line and number of columns equals number of design variables. Each column indicates a value to design variable.

Based on this general description, following, in Fig. 5, the basic pseudo-code of real genetic algorithms to maximise “\(D_a \)”, as an approach to solve the WWAL design problem in this paper.

Fig. 5
figure 5

The pseudo-code of GAs for maximisation of \(D_a \)

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Setting GAs parameters for the WWAL optimisation design

When applying GAs, one needs careful settings for the values of the basic parameters, such as crossover rate, mutation rate, population size, selection strategy, max generation...etc. Since the performance of a GAs is greatly dependent on setting the values of these parameters. In this regard, selecting appropriate parameters might enable the algorithm to find optimal solutions in a short time while local optimum solutions may prevail and survive throughout the algorithm run with improper parameter settings (Jäntschi 2010). Because of this, the question may be raised as to which of of the basic parameters and their level-setting values significantly influence the performance of GAs.

Factorial experimental design

Computational experiments are commonly used to determine which of the basic operators is most significant when compared with the others, and what its value is Mosadegh et al. (2012). Therefore, in this subsection, factorial experiments are conducted in order to obtain the parametric levels that will be set when implementing the GAs. Using a full factorial experiment extracts the maximum amount of unbiased information regarding the parameters affecting performance of GAs and also their interactions (Jebari et al. 2013). The experiments are conducted to investigate the effects of the population size \(N\), maximum number of generation \(G_{max}\), crossover probability \(P_{c}\), mutation probability \(P_{m}\). Rather than \(G_{max}\), the level of each parameter was adopted from previous studies. Table 5 includes the number of levels and the settings of each parameter. The selection process is used a roulette wheel with elitist strategy and setting size of two. The experiment with these five parameters and three replicates are conducted giving a total of 48 runs.

Table 5 The levels of basic parameters for GAs
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The settings of these parameters are based on characteristics of fitness function. In other words, the best parameter values must be expected to have great improvement on fitness function statistics (maximum fitness function).

The results and analysis of experimenting GAs parameters

Factorial experiments are conducted for the GAs basic parameter combinations on the design problem in this paper (presented in “Design problem: mixed model assembly system optimisation problem” section) with Table 6 showing results of these experiments. The result is the fitness function of optimal solution of operational design of the research problem.

Table 6 Computational results for combinations of GAs basic parameters
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Analysis of variance (ANOVA) is used to investigate the effects of the GAs parameters and their interactions on searching performance in terms of finding the best fitness function value. Table 7 shows these effects when GAs are applied on the design problem in this paper. \(P\) value (or significant probability) from ANOVA elements is used as an informal measure about their significant effects. If the value of “P” less than 0.05 this statistically indicates that the parameter is significant.

Table 7 ANOVA table of factorial experiments of basic parameters on design problem
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From Table 7, \(P_{m}\) and \(G_{max} \) have significant effect on the performance of GAs as their P-values are below 0.05. This given conclusion from the above analysis of ANOVA applied to the experiments of the GAs parameters in the design problem of this paper can only serve as guidelines. Reasonable conclusions may need more levels of parameters for experimenting on the problem at hand and on the basis of the experimental observation an appropriate value should be selected. It is therefore advisable to run further experiments with different settings of each parameter and compare the performance of the GAs. This procedure aims to improve the stability of the algorithm and avoid setting parameters randomly so as to achieve the goal of quantifying parameter settings and significantly advance the performance of the algorithm.

Further experimentation and analysis with GAs parameters

For further experimentation analysis and consequently selecting the best values of \(P_c,\, P_m \) and population size \(N\) a comparison of the effect of changing the values of these parameters on GAs performance is conducted. Except for the changing parameter all other parameters are fixed during experimentation. In this context, in order to compare with the result of each case, basic parameters were assigned so that the size of entity population is 40, the crossover probability is 0.8, and the mutation probability is 0.005. That is, when there is a change in \(N\) value from 40 to 200 to reproduce chromosomes, the \(P_c\) is fixed to 0.8 and the \(P_m \) is fixed to 0.005. Also when there is a change in the \(P_c \) value then the \(N\) and the \(P_m\) values are fixed.

The \(G_{max}\) is not considered in further experimentation as the effect of this parameter on GAs performance was well known from previous studies—if it is large it will enhance the opportunities of obtaining the optimum solution but will augment the cost of operation and the computational time is not limited. The assumption is that the simulation model of GAs is run for 300 generations whilst taking into account the constraints of the design problem, also the dependent variable in these experiments is the maximum fitness function value. In consideration of the variation in the fitness function value that may be caused by changes to the initial population, three replications of GAs run are executed and it was decided that the GAs run showing the best performance (in terms of convergence speed and fitness function value) can be selected as an optimal GAs with respect to the values of experimented basic parameter. The goal is to select the GAs parameters in which peak fitness values are more likely to be found and also the convergence to an optimal solution occurs in a short time period.

Experimentation of crossover operator:the crossover probabilities used in this experimentation are 0.5, 0.6, 0.8, 0.9, and 1.0 and test individually. The algorithm has been run for each crossover probability \(P_\mathrm{c}\) with the other parameters fixed. The results summarised in Fig. 6 give a figure that compares only the average of best fitness function value found in a generation.

Fig. 6
figure 6

Comparison of crossover probability value on converging to best fitness function value

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As Fig. 6 shows, when \(P_\mathrm{c}\) is 0.5, the optimisation model converges to fitness a function value of 0.644749, from generation 112. In case a \(P_\mathrm{c}\) is 0.6 the model converges to value, 0.644749, after generation 56. The model converges to fitness value 0.661875, after generation 45, when a \(P_\mathrm{c} \) is 0.8. The model converges to fitness value 0.65348, after generation 145 when \(P_c \) is 0.9 and the model converges to fitness value 0.616175, after 229 generations when \(P_c\) is 1.0. The experimental result, clearly indicates that when a \(P_\mathrm{{c}}\) is 0.8 then the fastest model converges to highest fitness values of the \(P_c \) values, 0.661875.

Experimentation of mutation operator: according to the results in Fig. 7, in case a \(P_m \) is 0.005, the model converges to fitness value 0.6402472, after 43 generations while other values of \(P_m\); 0.007, 0.008, and 0.009 converged the optimisation model to the low values of fitness function, 0.6110067, 0.6329181, and 0.5445833 with more generations; 97, 84, and 48, respectively. At \(P_m\) is 0.01 the model converged to fitness values 0.621217 after 32 generations. Hence, \(P_m\) at 0.005 has the highest performance.

Fig. 7
figure 7

Comparison of mutation probability value on converging to best fitness function value

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Experimentation of population size: the tested population size in experimentation is 40, 50, 60, 80 and 200. Results in are summarised in Fig. 8 which shows the value of fitness function value per each generation after optimising the mathematical model of the design problem during generation 300. Results show that convergence to better solutions happens sooner than with bigger population sizes. The population size of 60 reached best fitness value of 0.661875 at generation 179. On the other hand, the fitness value to population size of 50, 80 reached to 0.644749 and 0.640247 during generation 136 and 221, respectively.

Fig. 8
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Comparison of the effect of population size on converging to best fitness function value

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Summary of experimentation

Examining the effects of GAs parameters values on the performance of the algorithm to find the best parameter combination for the design problem of this paper tends to conclude the best combination of parameters are the following:

  • Crossover probability is 0.8;

  • Mutation probability is 0.005;

  • Population size is 60; and

  • Maximum number of generations is 300.

The above selected parameter values for optimising the operational design of mixed model WWAL; are consistent with the results obtained from De Jong’s (1975) study for off-line setting of GAs parameters. In that study, DeJong selected low values for crossover probability and mutation probability, 0.6 and 0.001, while the selected population size was 50 and selection strategy was elitist.

The results and analysis of design optimisation problem

The developed GAs in this paper is executed using the best parameter values gained from the experiments to solve the problem of optimising operational design for mixed model WWAL presented in “Design problem: mixed model assembly system optimisation problem” section. Figure 9 shows the convergence to best function value of optimising problem, 0.640247. It also shows the change in combinations for the design variables obtained through the operational design variables levels, which not changed the fitness function values among the solutions composing the last 258 generations. Therefore, the final optimal design variable combinations that simultaneously satisfy the requirements placed (i.e. optimisation objectives) on each one of the measures and operational design variables (i.e. multiple-objective optimisation) is determined. The final optimal operational design variable combinations of WWAL are presented in Table 8.

Fig. 9
figure 9

The optimisation process for maximizing fitness function for operational design solutions of mixed model WWAL during 300 generations

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Table 8 The optimal operational design solution of mixed model WWAL found by GAs search
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Assembly line design assessment

Although optimised operational design for mixed model WWAL are generally reliable as shown in Table 5, it has the following drawbacks:

  • Low efficiency due to high of balance of labour blockage. The value of this design objective was 0.111 indicated the optimised operational design generates blockage and starvation in the previous and successive workstations (5, 6) causing a reduction in the worker productivity and consequently in line throughput. This is because the worker productivity depends on the overall cycle time for assembling the product and as usual the line efficiency varies considerably with the cycle time.

  • High expenditure energy cost. Since the expenditure in one walk cycle was 3.28 kcal, the current design of the system where a worker carries out the assembly tasks by standing, carrying, walking and waiting during blockage in an 8 hr. workday, was found to almost exceed the physiological capacity of worker to maintain suitable physical fitness and work ability. In WWAL operational design, the worker’s metabolic response to the amount of work assigned was affected by two variables;

  • Walking distance: as expected, walking distance had the greatest effect on walking time; and

  • Behavioural factors of workers, in particular walking speed.

Therefore, implementing a WWAL approach needs “some time” involving due consideration of both productivity and ergonomics in the early design stage of manufacturing systems. Manual assembly systems can be designed using different structural designs. In general, different structural system designs have a profound impact on the performance of the system itself in terms of ergonomics and productivity. Structural system designs are determined by different variables; (1) the arrangement of the workstations (layout) (2) the number of workstations (3) the movement distance between workstations (Hu et al. 2011). Also, the structural design of the system should assure safety of movement especially in dynamic systems like WWAL where workers slipping while performing their job is a commonly reported due to flooring surface problems. These variables, therefore, should be considered satisfactorily in order to ensure that an assembly system is appropriately designed for WWAL approach implementation requirements.

The mathematical model developed in Al-Zuheri et al. (2013) and the optimisation approach developed in this paper, may allow prediction of those structural designs that maximise the flexibility in the system when future changes in product demand are required together with maximised performance and reduced costs by considering both productivity and ergonomics together.

Comparison of GAs performances using the modified desirability function versus using Derriger and Suich desirability function

To illustrate the effectiveness of using the modified desirability function, Fig. 10 illustrates a comparison of running the GAs on illustrative example problem using this desirability function and Derriger and Suich desirability function.

Fig. 10
figure 10

Comparison of running a GAs using the modified desirability function (GAs modified DF) and running GAs using Derriger and Suich desirability function (GAs D-S DF) only

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After 45 generations, the GAs with modified desirability function converged to near the global optimal design solution with value of fitness was 0.640247. While the Derriger and Suich optimal design solution in searching GAs had the fitness value of 0.551074 and converted to it after 12 generations, after that, the local search failed to increase the fitness value. The design solutions which found by GAs search with the type of Derriger and Suich desirability function is listed as follows.

$$\begin{aligned}&x_{1}=70\,\%, x_{2}=2, x_{3}=0.7, x_{4}=121,\\&x_{5}=152, y_{1}=0.227, y_{2}=637, y_{3}=2.923, y_{4}= 0.149. \end{aligned}$$

Since the Derriger and Suich desirability function cannot evaluate the relative fitness of infeasible chromosomes the GAs cannot determine which parents are better for reproduction. Evidence of this is that the GAs ran with the Derriger and Suich desirability function trapped in local optimal as shown in Fig. 9. While the GAs that ran with the modified desirability function converged to a value near the global optimal in same number of generations 300. This is because GAs searching in this case is able to consider even infeasible solutions that as possible consist of feasible regions.

Conclusions

This paper has described the development of the optimisation design approach of manual assembly line with walking worker. As GAs were selected in this paper as the appropriate optimisation approach for solving the WWAL design problem, the fundamental structure of GAs was outlined. It included chromosome representation, generation of initial population, fitness function, selection strategy, genetic operators (e.g. selection, crossover and mutation operators), handling of constraints and stopping criteria. Real number coded chromosome representation was adopted as the internal object representation for the solutions search in the GAs of this paper. The algorithm is designed to implement the developed GAs for finding a WWAL design depending on the given input parameters and design variables to make this task flexible and systematic as possible. Setting basic parameters values for developed GAs for best performance was based on a premature convergence of fitness function to optimal solution. Furthermore, the development process of GAs also included introducing a penalty function as a scheme to handling the constraints of optimisation in algorithm.

For demonstrating the effectiveness of an algorithm developed to handle the WWAL optimisation design problem, GAs were run to find an optimal operational design for the designed problem; mixed model manual assembly line intended to be run with walking workers.

Since the solution design (chromosome representation) in any GAs is quite problem domain dependant, hence the developed GAs procedure here cannot be used to effectively solve different optimisation problems elsewhere. Based on the above this paper successfully introduces a unique GAs.