Log-linear learning model for predicting a steady-state manual assembly time

Abstract. This paper presents the method for estimating the parameters of a two parameter learning curve (LC). Different values of parameters and different sample sizes are used for this estimation. Based on the experimental data an adequate mathematically grounded LC model is proposed for a manual assembly process of automotive wiring harness. The model enables us to determine the LC parameters αε (slope coefficient) and the learning rate stabilization point xc, i.e. to completely restore LC and predict the production process. The propositions that ground the model application correctness are proved. The model adequacy is estimated, based on concrete production process monitoring data. The criterion that determines production process without stabilized learning rate is proposed.


Introduction
Learning curves (LC) are mathematical models used to estimate efficiencies gained when an activity is repeated.LC have already been considered for quite a long time, but their application is urgent up till now, as far as many enterprises are striving to apply the LC to determine their production process time, but they face various problems [1] most important of which are errors due to the wrong LC application.
When applying the LC model in practice there are three basic things that defined the accuracy of the applied model: 1. Adequacy to the specific character of production under consideration; 2. Rather exact method of LC parameters restoration; 3. Sufficient quantity of the production made according to which the producing time is predicted.At present, very many LC models are applied that are widely reviewed in [2] and other papers.In this work, the LC model for learning phase modeling of the production process was used that meets the following requirements: c Vilnius University, 2014 1) it is simple (minimal number of parameters); 2) it well approximates the whole length of LC; 3) it is able to define the stabilization point of the learning rate; 4) monitoring of producing process for the necessary data for restoring parameters of the model is rather cheep.Several learning curve models have been proposed, but only two Wright's [3] and Crawford's [4,5] models are in widespread use.By Wright's model the log-linear equation is the simplest and more common equation and is valid for a wide variety of processes where y W is the average time of all units produced up to the xth unit, the parameter α W is a slope coefficient, β is the number of direct labor time required to produce the first unit.Crawford's model is as follows [4,5]: where x is the unit number, y c is the number of direct labor time required to produce the xth unit, the parameter α c (α c > 0) is a slope coefficient, β is the number of direct labor time to produce required to produce of the first unit.Some authors also use the Plateau model [5][6][7][8]: where x, y p , α p , and β are the same as in (1), γ (γ > 0) is the constant that describes when the steady-state is reached after the learning is concluded or when machinery limitations block workers improvement.Baloff [6] studied this plateauing phenomenon and found it to be extensively present in machine intensive manufacturing.This work investigates the LC application to the manual assembly process of automotive wiring harness.Production data for Crawford's model have been collected.Observing the production process, certain tendencies of assembly time dynamics have been noticed: 1) assembly time mostly decreases for the first products; 2) assembly time of produced in large amounts and constantly is stable and does not decrease.The stability is described in the paper by the learning rate (parameter ε) that is determined by the steady-state time T c .
By means of experiments (monitoring of the manual assembly process) LC was obtained as one or several random samples.Note that such a way, by fixing each complete cycle is most frequently used in the investigation in LC parameters restoration [9,10].However, it is also most expensive since it requires more expenditure for monitoring the production process: scanning equipment, its management and administration, because one group of researchers can tackle only one experiment at a time.
If learning curves are obtained experimentally by only one sample then the statistical methods, such as the least squares method maximum likelihood method and others, are not suitable here.Though a statistical method for the case with one random sample available is offered in the paper [11], in this research a deterministic method of the empirical function that approximates LC, based on the obtained monitoring data will be used, i.e. of one random sample.Basing on the tendencies observed in most experiments done, a premise that in a assembly process there exists a Plateau phase in the sense of assembly rate stabilization is made, i.e. the assembly time is decreasing to a certain limit T c until the steady-state assembly time is reached.
The aim of this work is to propose, to an adequate, mathematically grounded LC model, that meets the above mentioned requirements on the basis of experimental data, and which will enable us to define the LC parameters α ε and steady-state point x c , i.e. to restore LC completely and predict the assembly process.

Product and its assembly time
As typical manual assembly process wiring harness production will be studied.Since wiring harness performs electrical circuit function, its manufacturing is pure assembly process: manual, semi automatic, automatic, depending on production volume [12].
According to wiring harness definition [13], the main wiring harness components are terminated cables and wires (circuits), housings and connectors as other wrapping and protecting material (tapes, cable ties, hoses and tubes).These components constitute wiring harness layout with main trunk, branches, break-outs, legs (see Fig. 1).When protective materials such as hoses are used, additional components like adapters and manifolds are added.The main wiring harness manufacturing steps [14]: 1. Wire preparation, when wires are being cut ant terminals mounted.2. Installation, cables and branches are being placed on assembly board according the certain layout.3. Securing, cables and wires are wrapped together; protective hoses are pulled on branches and legs.4. Attachment, cables and wires with mounted terminals are being assembled into housings, connectors, splices and etc.
The first step representing wiring preparation is mostly automatic process and next three steps are assembly of the wiring harness performed by operator on the assembly jig.In this research only manual assembly operations are studied, so only last tree steps will be considered.
During the assembly of wiring harness, all components must be installed and assembled into the final product.To determine assembly time of the assembly the total assembly time could be measured and also time of each operation could be measured.
Each component of the wiring harness has specific mounting operation or several operations and each operation has certain processing time i.e. to pick and place component, pull the hose, assemble terminal into housing, wrap the cable tie and etc.Even different Fig. 1.The main wiring harness components and terminology [13].
wiring harness have different number of different components, the basic operation time of specific mounting procedure will be the same, for all.This basic operation time represents necessary -steady state operation time, which is reached, after operator has finished the learning phase.
There two groups of component assembly operations, simple and complex.Simple operation represents single process and depends only on number of components used i.e. the assembly of terminal, fixing of cable tie.The complex assembly operation depends on number of components and number of sub-operations or component length.For instance, to pull the hose, constant assembly time is needed, and variable assembly time depending on the length of the hose is needed as well.Analogical situation is with the sub operations: the number of wires in the splice, the number of wires in the branch and etc.
After regression analysis [14], linear relationship between number of sub-operations/ length and assembly time was determined: , where t const is constant operation, t add is additional time for sub-operation or for additional meter, s op is number of sub-operations, l op is length of component.
To determine the total assembly time of the product the sum of the operating time need to be calculated.So in general the vector of all operations with certain operation times is defined: where t i is operation time of the particular operation, m is total number of all possible operations.t i is determined as a time measured during time study of each operation.During time study, every operation was thoroughly evaluated, not only on single, but on several different products and different type of components and performed up to the several hundred times, to warrant that pure steady state operational time is reached.
Each different wiring harness product will have different numbers and quantities of operations.This information is easily collected from product structure.So for every product operation quantity vector is determined: where a i is the quantity of i operation.If particular unit does not have the certain operation the quantity is obviously 0. When fully defined vectors D and W are given, the total assembly time is calculated as a scalar product: After ( 2) is applied, the total assembly time of selected wiring harness product is determined.Since T c is the sum of the steady state operation times, T c as well represent the steady state assembly time of the wiring harness.

Learning curve model
Let one experiment is done and data be obtained.We shall use only "cheep" data, i.e.
Proof.It follows from Proposition 1 that x c (T c , α(ε)) > 0, therefore, derivatives of implicit function x c (T c , ε) are of constant sign: Thus after finding α ε (there is no analytical solution), we completely restore LC ( 5) and the learning rate stabilization point x c (T c , α ε ), using only two experimental data (x 0 , y 0 ) and T c .

Results and discussion
With a view to verify the adequacy of the method proposed 11 respective measurements have been performed in which only x q (see ( 1)) and T (k) c have been selected from here k is the number of experiment, n k is the number of the kth experiment data.In line with this parameters, the parameters α ε ) obtained values with the complete experimental data (11).Function (12) was chosen as adequacy criterion (average percent relative error) q , y In Table 1, the data of the first experiment are presented.The calculation results are given in Fig. 2, and, for all experiments, in Table 3. Fig. 3 shows the experimental data in log-log scale.

c
have been calculated.According to manufacturer recommendations ε = 0.0016 T (k) c is selected in all the calculations.A comparison of the LC Y (x, x

Fig. 4 .
Fig. 4. Dependency of the average relative error on ε for the data set from Table 3, k = 1.

Fig. 5 .
Fig. 5. Learning curve based on the proposed model (line) and a graph for the data set from Table 3, k = 3 (circles).Point of the stabilized learning rate xc.

Table 1 i x
Fig. 2. Learning curve based on the proposed model (line) and a graph of the data set from Table 3, k = 1.