# A Weighted Multiobjective Optimization Method for Mixed-Model Assembly Line Problem

Authors: Şükran Şeker; Mesut Özgürler; Mehmet Tanyaş
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

Mixed-model assembly line (MMAL) is a type of assembly line where several distinct models of a product are assembled. MMAL is applied in many industrial environments today because of its greater variety in demand.  This paper considers the objective of minimizing the work overload (i.e., the line balancing problem) and station-to-station product flows. Generally, transportation time between stations are ignored in the literature. In this paper, Multiobjective Mixed-Integer Programming (MOMIP) model is presented to optimize these two criteria simultaneously. Also, this MOMIP model incorporates a practical constraint that allows to add parallel stations to assembly line to decrease higher station time. In the last section, MOMIP is applied to optimize the cycle time and transportation time simultaneously in mixed-model assembly line of a real consumer electronics firm in Turkey, and computational results are presented.


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## Body

## 1. Introduction

Mixed-model assembly lines are one of the important parts of mass production systems and generally useful in small product variety and high-volume industries such as automotive, electronics, and machinery. An assembly line comprises several successive workstations where a set of parts for one or more product types are assembled [1]. The assembly line balancing problem (ALBP) has had significant industrial importance since Henry Ford’s introduction of the assembly line. ALBP is the assignment of product tasks to different stations with considering precedence relations between tasks. In spite of the high cost of settings and operating an assembly line, manufacturers often simultaneously produce one model with different features or several models on a single line [2]. Assembly lines can be classified as “single model,” “multimodel,” and “mixed model” with respect to the number of different products assembled on an assembly line. We can see difference between single-model, mixed-model, and multimodel assembly lines in Figure 1.Difference between single-model, mixed-model, and multi-model assembly lines.
(a)
Single-model assembly line
(b)
Mixed-model assembly line
(c)
Multimodel assembly lineInsingle-model assembly lines, only one model product is produced in the same line. The assembly lines where products of one similar model are assembled with batches are called multimodel assembly lines. Mixed-model assembly lines are the lines on which in the simultaneous production products of more than one model are assembled [3]. In order to meet the customers’ demand of individual and different models of product, mixed-model assembly lines are used.  Thus, there can be produced various models but small quantity of products without setup operations on the same line. This is required to respond to the demand of market quickly and adjust to the change of market environment in time. Mixed-model assembly lines have a strong impact to improve the goal of Just in Time (JIT) and balanced production [4].Mixed-model assembly has had a big importance because of its diversity. Although other goals of mixed model assembly lines such as low costs, high productivity, and standardization are in contrast with diversity, the success of a company is related to its ability to deal with complex products and process designs [5].There are two types of mixed-model assembly line problems (MMALPs), which are referred as dual problems in the literature.(i)
MMALBP-I minimises the number of workstations for a given cycle time.(ii)
MMALBP-II minimises the cycle time for a given number of workstations.In type I problems, the cycle time (the time elapsed between two consecutive products at the end of the assembly line) and the production rate have to be prespecified. Thus, it is more frequently used in the design of a new assembly line for which the demand can be easily forecasted. Type II problems deal with the maximisation of the production rate of an existing assembly line.  For instance, the MMALBP-II is applied when changes in the assembly process or in the product range require the line to be redesigned [6].Material movement is an important factor in the production line. Although it is an important factor in nonvalue-added cost function, it increases the cost of production. So, in a well-designed system, work and storage areas should be near to their point of use. In an assembly line material handling system, transportation of materials and storage devices require huge capital investment. However, transportation does not add value to the final product.Transportation time is usually added to the processing time or negligible in assembly line systems. However, minimization of assembly parts from a machine to another decreases the completion time and cost of products. But, minimization of part unit movements can cause workloads imbalance. This leads to unequal total assembly times assigned to stations and longer intermediate queues and bottlenecks in the system as a result. In order to have the best assembly system capabilities, balancing of the line and transfer times between stations should be taken into account simultaneously. Routings of part units are completely defined by the sequence of machines they have to visit.Although there is a vast literature on the mixed-model assembly line balancing and sequencing problem, transportation times between stations consist of process routings of products and balancing problem generally is not considered in an integrated fashion. In this paper, we propose a mixed-integer programming model to tackle this problem. Also, we assume that there can be parallel machine in some stations, and we add this as a criterion to mathematical model.With this point, this paper includes two main objectives.(1)
The workload balancing assigns tasks to stations in order to equalize station workloads.(2)
The total amount of transfers of components from a station to another is minimized. In other words, we look for balancing routings of part units to minimize transportation times that cause the increase of completion time of the product.To reach these objectives, some constraints are added to the mathematical model. If it is required, we add parallel machine to minimize cycle times of stations. MOMIP model also includes a finite workspace constraint that is considered with line balance. Finite working space might be subject to technical restrictions and space requirements of assigned station. The limited workspace capacity restricts the number of tasks which are assigned to each station.In Section2, there is a review of literature on MMALP. Then in Sections 3 and 4, MOMIP model is proposed with results of experiments and real-world application in consumer electronics in Turkey. In Section 5, conclusions are presented.


## 2. Review of the Existing Literature in MMAL

The mixed-model assembly line deals with solving two primary problems in a production line. The first problem is the design and balancing of the production line, whereas the second problem is the determination of the production sequence for different models [7]. The line balancing comprises the assignment of tasks to stations and determination of the work content and model type per station.  However, the production sequence is the model mix arranged with regard to minimum overloads on the assembly line. Balancing and sequencing problems are known as an NP-hard class of combinatorial optimization problems in mixed model assembly line literature.Several researchers have studied MMAL balancing problems. Thomopoulos [8] and Macaskill [9] are the initial researchers in solving this problem. Macaskill [9] and Chakravarty and Shtub [10] have also studied the line balancing for traditional mixed model straight lines (MMSLs).  Also, [11] presents a balancing methodology for mixed-model lines with deterministic task times. The objective is minimizing the total cost of stations (essentially the regular time labor cost) and work overload. Erlebacher and Singh [12] proposes a method to allocate a fixed total processing time variance among multiple stations and to minimize the total expected work overload. Zhang and Gen [13] present random key-based representation method with adapting Genetic Algorithm to assign the suitable task to the suitable station and the allocation of the proper worker to the proper station. The objective is to minimizate the variation of workload and the total cost under the constraint of precedence relationships.The mixed model assembly line sequencing is investigated in [14] for the first time. Various objectives are reported in the literature in determining the optimal sequence for a mixed model assembly line. The common objectives are minimizing the overall line length [15–19], minimizing the risk of stopping a conveyor [20], minimizing the total utility work [21, 22], and keeping a constant rate of part usage [23–27]. Also, Bard et al. [15], Scholl [28], and Yano and Bolat [29] present several procedures for different versions of the mixed-model-sequencing problems [30].However, Celano et al. [31] investigate the sequencing of MMAL assuming that the parts usage smoothing is the goal of the sequence selection. Mirzapour Al-E-Hashem et al. [7] present a sequencing problem with a bypass subline with the goals of leveling the part usage rates and reducing line stoppages. To solve this problem, a novel hybrid algorithm incorporating genetic algorithm and event-based procedure is developed to solve the problem.However, there are a couple of papers dealing with simultaneous assembly line balancing and sequencing [32–37]. Hu et al. [38] also analyze the balancing of mixed model assembly lines and design a new algorithm based on the process exchange according to its attributes and characters. The algorithm could be utilized to make further optimization of mixed model assembly lines on the basis of the best production sequence.Merengo et al. [39] develop a new balancing and production sequencing method for manual mixed-model assembly lines. Minimization of the number of stations is provided by the balancing method, and a uniform part usage is obtained by the sequencing method. Kim et al. [40] present a new method using a coevolutionary algorithm that can simultaneously solve balancing and sequencing problems in mixed-model assembly lines. Karabatı and Sayın [41] propose the MMALBP with the objective of minimizing the total cycle time by combining the cyclic sequencing information. They propose a mathematical model and an alternative heuristic approach to minimize the maximum subcycle time [42]. Fattahi and Salehi [43] consider a mixed-integer programming model with a variable rate launching interval between products on the assembly line, to minimize the idle and utility time cost, with optimization sequencing and launching interval for each sequence.Although transportation of units is very important in assembly systems, there are few studies that take into account that part units’ movements in assembly lines.  Transportation time is usually added to the processing times or negligible in assembly systems. However, minimization of assembly parts movements from a machine to another decreases the completion time of products.In our study, we propose a mixed-integer programming that can simultaneously deal with both balancing and sequencing problems in MMAL, and identical parallel machines are allowed at each stage of the serial system. With this approach, we obtain the best task sequences for the models by taking into consideration the minimization of transportation of models between stations; we also solve the balancing problem in the line at the same time. The mathematical model intends for balancing of workloads and routing of unit parts.


## 3. Model Description

This paper introduces a mathematical model for MMAL, and it can be categorized as MOMIP model. This mathematical model is integrated to real-world application. Our mathematical model is based on Sawik’s integer programming models about mixed model assembly line [44, 45]. The problem objectives are the determination of allocation of assembly tasks among the stations, where they have parallel machines and selection of assembly sequences and assembly routes for a set of products simultaneously to balance station workloads and to minimize total transportation time in assembly line.We give weights to objectives to solve this multiobjective problem. This problem is solved using ILOG OPL optimization software [46]. The proposed model is used for a mixed model assembly line problem from real world in a Turkish consumer electronics firm. Due to the NP-hard nature of the problem, the size of our model would be large to obtain optimal solutions for problems of realistic sizes. Also, the model suggested in this paper presents a significant improvement relative to the models in the literature.The assumptions of the proposed model are as follows.(1)
Each assembly task must be assigned to at least one station (alternative assignments are allowed).(2)
There are parallel machines in some stations.(3)
Total space required for the tasks assigned to each station must not exceed the station’s finitework space available.(4)
Each product must be routed to the stations subject to precedence relations defined by its assembly plan.(5)
Revisiting of stations is not allowed.(6)
Each station can perform at most one task at any given time.(7)
Transfer times between stations are not negligible.NotationIndices
i: assembly station i∈I, I={1,…,m}
j: assembly task j∈J={1,…,n}
p: parallel machine at station i{h=1,…,mi}
k: product, k∈K={1,…,v}
s: assembly sequence, s∈S={1,…,w}.Input Parameters
a
i
j: working space of station i for task j
b
i: working space for station I
m
i: number of parallel machines in station I
p
j
k: process time for task j of model k
q
i
l: transportation time from station i to station l
I
j: the set of stations capable of performing task j
J
k: the set of tasks required for product k
R
s: the set of immediate predecessor-successor pairs of tasks (j,r) for assembly sequence s∈S such that task j must be performed immediately before task r
S
k: the set of assembly sequences available for product k
T
s: the set of tasks in assembly sequence s.The following decision variables are introduced to model the loading and routing problem:α: the weight factor (0<α<1),M: a big number.Decision Variables
u
s
=
1, if assembly sequence s∈S is selected; otherwise 0;
x
i
j
=
1, if task j is assigned to station i∈Ij; otherwise xij=0;
z
s
j
i
h
=
1, if task j is assigned to parallel machine h in station i∈Ij; otherwise zsihj=0;
y
i
l
j
s
=
1, if product in s sequence is transferred from station i after the completion of task j to station l to perform next task; otherwise yiljs=0;
P
max is the maximum station workload (cycle time),
Q
sum represents the weighted sum of total assembly and transportation time.We can state the problem formally as follows:(1)Min(αPmax+(1-α)Qsum)
subject to
(2)∑i∈Ij∑l≥iyiljs=us;s∈S,j∈Ts,(3)∑l≤iylijs-∑l≥iyilrs=0;i∈Ir,(j,r)∈Rs,s∈S,(4)∑k∈K∑s∈Sk∑j∈Ts‍pjkzsjih≤Pmax;i∈I,h∈H,(5)∑i∈Iyiljs≤∑h=1mizsjih;i∈I,j∈J,s∈S,(6)∑s∈S∑h=1mizsjhi≤mi*∑i∈I∑s∈Syiljs;i∈I,j∈J,(7)∑s∈S∑j∈Ts∑i∈I∑l∈Iqilyiljs=Qsum,(8)∑i∈Ijxij≥1;j∈J,(9)∑j∈Jaijxij≤mibi;i∈I,(10)∑j∈Jaijzsjih≤bi;i∈I,h∈H,s∈S,(11)∑s∈S∑i∈I∑h>mi∑j∈J‍zsjih=0,(12)z[s][j][i][h]≤u[s];i∈I,j∈J,h∈H,s∈S,(13)yiljs≤xij;i∈Ij,l≥i,l∈lr,(j,r)∈Rs,s∈S,(14)yiljs≤xlr;i∈Ij,l≥i,l∈lr,(j,r)∈Rs,s∈S,(15)yiljs≤us;i∈Ij,l≥i,l∈lr,(j,r)∈Rs,s∈S,(16)∑s∈Skus=1;k∈K,(17)∑i∈I∑l<i∑s∈S∑j∈Ts‍yiljs=0,(18)∑s∈S∑j∈J∑i∈I∑i∉IJ‍yiljs=0,(19)∑j2∈J∑h2≠h‍zsj2ih2≤M(1-zsjih);s∈S,j∈J,i∈I,h∈H.
The objective function is the minimization of the weighted maximum workload Pmax and sum of transportation time Qsum. Equation (2) shows for each product and assembly sequence selected that all of its required tasks are allocated among the stations. Equation (3) is the flow of tasks for each station, for selected assembly sequence, and for successively performed tasks. Equations (4) and (7) ensure the workload of the bottleneck station with parallel machines and the total transportation time, respectively. Equations (5) and (6) define the tasks that are assigned to at least one machine and not more than all “mi” parallel machines of such a station “i” when the product moves from station “i” to station “l” to perform task  “j”. Equation (8) ensures that each task is assigned to at least one station, and by this, it admits alternative assembly routes for products. Equation (9) is the station capacity constraint. Equation (10) shows the total flexibility capacity of all parallel machines at related station. Equation (11) represents the capacity constraint of the number of parallel machines in station “i”.  Equation (12) shows that if the sequence “s” is not selected, all variables of related sequence are made zero. Equation (12) shows that if we do not select any sequence, we make all of variables in this sequence zero. Equations (13), (14), and (15) ensure that each product successively visits stations where the required tasks may be assembled subject to precedence relations defined by the assembly sequence selected. Equation (16) ensures that only one assembly sequence is selected for each product. Equation (17) eliminates upstream flow of products in a unidirectional flow system. Equation (18) eliminates assignment of tasks and products to inappropriate stations. Equation (19) ensures that all tasks, which are in the same “s” sequence, are  assigned to the same station and the same parallel machine and that also the tasks of the same product models are assigned to the same  station and the same parallel machine.Multiobjective integer programming problems may be thought as an extension of the classical single objective integer programming problem. Scalarizing functions used in subproblems are very important in solving multi-objective optimization problems [47].Different scalarization methods have been presented in the literature, some of which include the weighted sum method,ε-constraints method, hierarchical approach, weighted metrics methods, and goal attainment method [48].In this paper, there are two objective functions to be optimized with multi-objective integer programming technique in mixed model assembly line problem.We use the most convenient scalarization method with related objective functions of the problem. In this sense, the weighted sum (WS) method is used to solve multi-objective integer programming problem.  With solution procedure, we would produce the most preferred nondominated solutions. The weighted sum (WS) method involves a linear or convex combination of the objectivesfi(x), i=1,…,p. Each objective fi(x) is multiplied by a normalized weight factor wi and the product added to give the scalar objective ϕ(x,wi) as follows:
(20)φ(x,w)=∑i=1pαifi(x),
where p is the number of the objectives,  ∑i=1pαi=1, and αi>0,  i=1,…,p.Its drawbacks are also well known and discussed as follows.(i)
It misses solution points on the nonconvex part of the pareto surface.(ii)
Its diversity cannot be controlled; therefore even the distribution of weights does not translate to uniform the distribution of the solution points.(iii)
The distribution of solution points is highly dependent on the relative scaling of the objective [48].In this paper, there are two objective functions. We minimize the weighted sum of the objectives to obtain only one solution. The weight factorα∈[0,1] is used for interactive solution between two objectives. Therefore, decision makers modify the weighting coefficients after each iteration, so he/she would produce the most preferred nondominated solution.In the following section, firstly we perform experimental test problems to validate this mixed integer model, then we apply the model to real-world mixed model assembly line problem, and computational results are summarized.


## 4. Experiments and Computational Results

In this section, the experiments are used to test the performance of the mathematical model. Then, their analysis and the application to the real-world consumer electronics firm problem are described. For the experiments, the number of stations and the number of tasks are changed with the weight factor in each iteration. The problem includes optimization of balancing and transportation times of mixed model assembly line. To solve the underlying multi-objective mixed integer programming model, we use ILOG OPL optimization software. The experimental results are seen in Table1.Table 1
Experimental results.
Number of product models/Number of tasks
α
=
0 Pmax-Qsum
α
=
0.2 Pmax-Qsum
α
=
0.4 Pmax-Qsum
α
=
0.6 Pmax-Qsum
α
=
0.8 Pmax-Qsum
α
=
1 Pmax-Qsum
3/5
10-8
8-8
6-8
6-8
6-10
6-10
3/10
58-4
25-6
20-8
20-8
18-12
18-12
4/10
46-6
41-6
27-10
20-16
20-16
20-16
3/20
38-6
19-10
19-10
19-10
19-10
19-12
4/20
42-8
41-8
28-12
28-12
27-14
27-16
3/30
58-6
27-8
27-8
25-10
24-12
24-12
4/30
61-8
41-8
41-8
32-16
32-16
32-18Experimental results show thatα=0 and α=1 produce weakly nondominated solutions for the objectives. For α=0, transportation time is minimized and dominated whereas for α=1 cycle time is minimized and dominated. Other weights produce strongly nondominated solutions. As a result, while weight of α increases, cycle time decreases and transportation time increases.In addition to these test instances, we have solved a real-world problem from a consumer electronics plant in Turkey. The firm we have obtained is a household appliances’s manufacturer. It principally engages in production, marketing, and after sales service of durable goods, allied components, and consumer electronics. The assembly system of the firm is made up of 18 identical stations which produce three different LCD model TV types, and this number is constant. There are 73 tasks to produce three different model types. Some of these tasks are common. Common tasks can have different task times in each model. Thus, we assume that there is one machine in each station. According to the cycle time we have obtained, we placed one more parallel machine to the station where its cycle time is high. The objectives are the minimization of the weighted maximum workloadPmax and the sum of transportation time Qsum. The performance of the proposed algorithm is analyzed through variance between stations and balance loss of the system. In our system, each model has its own set of precedence relationships, but there is a subset of tasks common to all models. To solve this mixed model assembly line problem,  the precedence diagrams for all the models are combined, and then we use adjusted processing time to transform the mixed model line balancing problem into a single model line balancing problem and determine average task processing times for the tasks that are required by more than one model [49–52]. The average processing time of task j, tj, can be calculated by
(21)tj=∑kfkpjk,
where pjk equals the processing time of task j in model k, and fk is the relative frequency or the demand for each model. The demand for each model is assumed to be equal because of the daily production rates, so process time for each task is divided based on demand ratio. Task times of each model are shown in Table 2.Table 2
Adjusted task times.
Number of task
Model 1
Model 2
Model  3
Number of task
Model 1
Model 2
Model  3
Number of task
Model 1
Model 2
Model  3
1
0,11300
0,11600
0,11600
26
0,10267
0,10267
0,10267
51
—
0,05533
0,02967
2
0,02100
0,02100
0,02100
27
0,01700
—
—
52
—
0,09200
0,09200
3
0,09967
0,09500
0,09600
28
0,08533
0,08933
0,08933
53
—
0,05467
0,01733
4
0,10633
—
—
29
0,05133
0,04800
0,04800
54
—
0,01967
0,01567
5
0,04800
0,04833
0,04833
30
0,15933
0,14567
0,14567
55
—
0,04533
—
6
0,01533
—
0,01533
31
0,11367
—
0,10700
56
—
0,09767
—
7
0,04333
—
—
32
0,03500
—
—
57
—
0,04267
—
8
0,09167
—
—
33
0,12967
—
0,12967
58
—
0,03500
—
9
0,06200
—
—
34
0,04367
—
—
59
—
0,05433
0,06533
10
0,07467
—
—
35
0,02500
—
—
60
—
—
0,07767
11
0,04300
—
—
36
0,07933
—
—
61
—
—
0,03200
12
0,06400
—
—
37
0,06067
0,06067
0,06067
62
—
—
0,03900
13
0,04300
—
—
38
0,07733
0,07733
0,07733
63
—
—
0,08167
14
0,03533
0,03533
0,03533
39
0,02900
—
—
64
—
—
0,04167
15
0,07767
—
—
40
0,02133
—
—
65
—
—
0,02233
16
0,11800
—
—
41
—
0,01833
—
66
—
—
0,04267
17
0,04300
—
—
42
—
0,07067
0,07067
67
—
—
0,05333
18
0,11933
0,11933
0,11933
43
—
0,06733
—
68
—
—
0,07767
19
0,02500
0,02967
0,02967
44
—
0,06533
—
69
—
—
0,01900
20
0,01733
0,01733
0,01733
45
—
0,04433
0,04433
70
—
—
0,02633
21
0,02000
0,01733
0,01733
46
—
0,06567
—
71
—
—
0,01967
22
0,01733
0,01733
0,02133
47
—
0,34000
—
72
—
—
0,07767
23
0,03600
0,03900
0,03200
48
—
0,08700
0,08567
73
—
—
0,07767
24
0,03933
0,02767
0,04067
49
—
0,08067
—
25
0,08100
—
—
50
—
0,14567
—The other variables of the proposed model are as follows.The transportation timesqil from station i to station l are 1 unit time. The set of tasks required for product k (JK) and alternative assembly sequences (plans)  (TS) are shown in Tables 3 and 4. Each product modelhas two alternative assembly plans which include task sequences. Working space of station i for task j is 2 units and working space for each stations is 10 units.Table 3
Sequencing of tasks for LCD TV product models.
Model
JK
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 39 15 16 17 18 40 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
2
1 41 42 2 3 43 44 5 45 46 47 48 49 50 14 51 52 19 53 18 54 22 21 20 23 26 55 24 28 29 56 30 57 58 59 37 38
3
1 42 2 3 60 61 62 5 6 45 63 64 48 65 66 67 51 68 14 52 69 53 70 18 54 22 19 21 20 23 71 26 24 28 29 31 33 30 72 73 59 37 38Table 4
Assembly plans for LCD TV product models.
TS
JK
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 39 15 16 17 18 40 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
2
1 3 2 4 5 7 6 8 9 10 11 12 14 13 39 15 17 16 18 40 19 20 21 22 23 24 26 25 27 28 29 30 32 31 33 34 35 36 37 38
3
1 41 42 2 3 43 44 5 45 46 47 48 49 50 14 51 52 19 53 18 54 22 21 20 23 26 55 24 28 29 56 30 57 58 59 37 38
4
1 42 41 2 3 44 43 5 45 46 47 48 49 50 14 51 52 53 19 18 54 22 21 20 23 55 26 24 28 29 56 30 57 58 59 37 38
5
1 42 2 3 60 61 62 5 6 45 63 64 48 65 66 67 51 68 14 52 69 53 70 18 54 22 19 21 20 23 71 26 24 28 29 31 33 30 72 73 59 37 38
6
1 2 42 3 60 62 61 5 6 45 63 64 48 65 66 67 51 68 14 52 69 53 70 18 54 19 22 21 20 71 23 26 28 24 29 31 33 30 72 73 59 37 38The balancing procedure operates as shown in Figure2.Figure 2
General flow chart of algorithm.To determine the weights of objectives in multi-objective integer programming model, we use the preferences of the decision makers. To determine the important weights of each objective, we select six participants from the production planning department. We ask them about two objectives of the problem in a meeting. Peers being aware of the optimization objectives choose one of the objectives as their preference. The weights assigned to each objective function arefg. Pfg represents the number of peers that prefer fi to the other optimization objectives. n is the number of participants in the meeting. The important weights are determined using the number of participants in the meeting. One has
(22)αfi=pfgn.
Thus, we obtain 0.8 value for the first objective (Pmax) weight and 0.2 value weight for the second objective (Qsum). Using these weights, we obtain the following results. Also to minimize the cycle time, we assign parallel machines to stations where cycle time is high. Decision of parallel machine depends on the firm’s preference because of the high costs of machines. But when we place one more machine to stations 1, 5, 6, and 10, the cycle time decreases from 0.931 to 0.481. Also, according to the results, transportation time is 51 time unit. After solving this optimization problem in ILOG OPL programming S=1,3,6 is obtained for sequences of model 1, model 2, and model 3, respectively. Selected sequences after solving mathematical model are shown in Table 5.Table 5
Sequences after solving mathematical model.
TS
JK
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 39 15 16 17 18 40 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
3
1 41 42 2 3 43 44 5 45 46 47 48 49 50 14 51 52 19 53 18 54 22 21 20 23 26 55 24 28 29 56 30 57 58 59 37 38
6
1 2 42 3 60 62 61 5 6 45 63 64 48 65 66 67 51 68 14 52 69 53 70 18 54 19 22 21 20 71 23 26 28 24 29 31 33 30 72 73 59 37 38Task assignment to stations is shown in Table6.Table 6
Assignment of task to stations.
Number of station
Assigned task
1
1, 2, 4, 42
2
2
3
2, 3, 43, 44
4
4, 5, 60, 61, 62, 73
5
6, 7, 8, 9, 45, 46, 47, 48, 63, 64
6
10, 11, 48, 49, 50, 65, 66, 67
7
12, 13, 14, 51, 68
8
52, 53, 69
9
15, 16, 17, 39, 70
10
18, 19, 21, 22, 40, 53, 54
11
20, 21, 23, 26, 71
12
22, 23, 24, 25, 26
13
24, 27, 28, 55
14
29, 30, 56, 57, 58
15
30, 31, 33, 72
16
30, 31, 32, 33, 34
17
35, 36, 37, 59
18
38After the assignment of tasks to stations with parallel machines, station time (Pih) is handled by mixed integer programming model as shown in Table 7.Table 7
Stations times (Pih) with parallel machines handled by mixed integer programming model.
Station workload
Machine
P
ih (time)
Station
Machine
P
ih (time)
1
1
0.3206
9
1
0.294
1
2
0.205
10
1
0.4063
2
1
0.021
10
2
0.2033
3
1
0.4443
11
1
0.2436
4
1
0.4773
12
1
0.418
5
1
0.481
13
1
0.3946
5
2
0.45
14
1
0.4683
6
1
0.1176
15
1
0.460
6
2
0.4316
16
1
0.4813
7
1
0.3756
17
1
0.406
8
1
0.2203
18
1
0.2319After solving integer programming model, performance of the mathematical model is evaluated with balancing loss and variance of station workloads. Balancing loss is a measure of the line inefficiency which results from idle time due to imperfect allocation of work among stations. In this study, we measure system performance with well-balanced assembly line, when the idle time in each work station is at minimum level and the balancing loss is minimized. The balancing loss is shown withEline and calculated by
(23)Eline=n*Pmax-∑i=1Ntin*Pmax.
The total work, ti, content is the aggregate of all the work elements to be done on the line, n is the number of stations, and Pmax is the cycle time. One has
(24)Eline=n*Pmax-∑i=1Ntin*Pmax,Eline=18*0.481-7.5518*0.481=0.12.
As we can observe from the results, decision of parallel machine depends on the firm’s preference because of the high cost of machines. But, when we place one more machine to stations 1, 5, 6, and 10, cycle time decreases from 0.931 to 0.481. In addition, production rate increases from 426 units to 813 units daily. According to the results, transportation time is 51 time unit.  After solving this optimization problem in ILOG OPL programming S=1,3,6 is obtained for sequences of model 1, model 2, and model 3, respectively. The balancing loss of the system works out of 12%, and the variance between station times is computed as 0.15 which can be acceptable for MMALBP, because mixed-model assembly occurs when more than one model of the same general product are intermixed on one assembly line. This line design makes the balancing problem more complex than others and causes big balance loss in these systems.Results show that in terms of total assembly time, processing assembly of different LCD TV models in the same line is more advantageous than processing in different assembly lines. With this study, the workloads at stations are balanced, the production rate is increased, the work flow between stations is improved, and the utilization of machines and labor is increased. Also our research concentrates on testing the method with large-scale instance from real world, eventually solving mixed integer programming models.


## 5. Conclusion

Mixed model assembly lines are widely used to produce different models as per customer’s demands. The problem of mixed-model assembly line balancing is more complex than that of the single-model assembly line,  because there are different tasks in different models on mixed assembly line and the processing times for common tasks may be different. In this paper, the MOMIP of a mixed model assembly line problem is formulated.The goals of this study are to minimize station work loads and to minimize transportation time between stations. In order to achieve these goals, assembly works are assigned to stations optimally, and optimal assembly order and route is determined. The WS method is used to solve multi-objective integer programming problem. Also, to minimize the cycle time in related stations, parallelization is added as a criterion to mathematical model. An other criterion is related to the working space due to transportation times between stations.  In the literature, generally, time spent between stations for transporting goods is ignored. In this study, we aim to fill this gap and solve a real-world problem.In the first step, mixed model assembly line problem is converted to single model assembly line problem. For this, precedence diagrams for each single model are combined in combined precedence diagrams, and adjusted tasks times are computed for each task. The experiments are solved to test the performance of the mathematical model. Then, the proposed model is adapted to the real-world consumer electronics firm problem.Computational experiments show that this optimization method is capable of obtaining good results both in terms of solution quality and in terms of execution times. The main contribution of this research is the definition of a new problem with practical relevance, and the objective function models trade off between cycle time and transportation time related to selecting route for each product model.  Also, the parallelization and station length are added to the mathematical model as a constraint with balancing and sequencing, so transportation of product models is taken into account to be different from other papers in assembly sytems.After solving mixed model assembly line problem in consumer electronics firm, line efficiency is measured with balancing loss and variance between station workloads.Processing assembly of different LCD TV models in the same line is more advantageous than processing in different assembly lines in terms of total assembly time. In this way, the firm can meet the demands of different model products at the same time. This means that customer satisfaction and sales increase. Also with this study many improvements are obtained; for example, workloads at stations are balanced, the production rate is increased, the work flow between stations is improved, and the utilization of machines and labor increased. Also our research concentrates on testing the method with large-scale instance from real world, eventually solving mixed integer programming models.



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