Mathematical models for the productivity rate of

0 downloads 0 Views 1MB Size Report
calculations are different from the actual productivity. Manufacturers require robust ... Idle time due to common mechanisms and units θw. Machine work time θi.
Int J Adv Manuf Technol DOI 10.1007/s00170-015-7005-6

ORIGINAL ARTICLE

Mathematical models for the productivity rate of automated lines with different failure rates for stations and mechanisms R. Usubamatov 1 & T. C. Sin 1 & R. Ahmad 1

Received: 13 February 2013 / Accepted: 9 March 2015 # Springer-Verlag London 2015

Abstract Automated lines with complex structures consist of stations and mechanisms with different levels of reliability. Most of the publications that present mathematical models for productivity of multi-station automated lines are based on simplifications that enable researchers to derive the approximate equations of productivity. This simplification is accepted in a form in which all stations in the automated lines are characterised by one level of reliability and the balancing of technological processes on the stations is conducted evenly. This approach yields simplified mathematical models for the productivity rate of the automated lines, but the results of these calculations are different from the actual productivity. Manufacturers require robust and clear mathematical models that enable them to calculate and predict productivity of the automated lines with high accuracy. High accuracy of mathematical model of prediction yield is important to meet customer demand. Profit of a company would decrease due to inaccurate prediction of production which does not meet demand. This paper presents an analytical approach to the productivity rate of automated lines with stations and mechanisms that each display different failure rates and processing times. The mathematical models allow for the output of the automated lines to be modelled with different failure rates for the stations and mechanisms and yields results that are close to the actual productivity values.

Keywords Productivity . Reliability . Automated lines

* R. Usubamatov [email protected] 1

School of Manufacturing Engineering, University Malaysia Perlis, Pauh Putra, 02600 Arau, Perlis, Malaysia

Nomenclature A Availability of a machine T Cycle time for the machining of one product QT Cyclic productivity of a machine b Total number of the automated line failures fm Correction factor of the machining time fi Correction factor of the ith station or mechanism failure rate mw Mean time between failures mr Mean repair time ni Number of failures of ith mechanism or station p Number of parallel products ps Number of parallel stations pτ Number of stations located on the transport zone τ of the transport rotor pγ Number of stations located on the zone γ of the working rotor q Number of serial stations mechanisms and units ta Auxiliary time tav Average machining time tmb Machining time at bottleneck station tmo Machining time tc Time losses with respect to one product due to reliability of common control system ts Time losses with respect to one product due to reliability of one station or machine ttr Time losses with respect to one product due to reliability of transport mechanism tτ Time losses with respect to one product due to reliability of one transport rotor z Number of products machined per observation time θ Observation time θc Idle time due to common mechanisms and units θw Machine work time θi Idle time due to technical failures of a machine θs Idle time of one station θt Idle time of one transport rotor λ Failure rate

Int J Adv Manuf Technol

λc λs λtr g

∑ t ei i¼1

Failure rate of common control system Failure rate of station or machine Failure rate of transport mechanism Total time losses with respect to one product due to reliability issues of g machine units

1 Introduction The analysis and synthesis of engineering solutions is based primarily on criteria (productivity rate, quality of products, cost and system flexibility, etc.) that describe the most important parameters of the manufacturing processes. Mathematical models for the productivity rate of industrial machines with complex designs are primary attributes that enable evaluation of the efficiency of the manufacturing system; without accurate estimations of the productivity rate, the industrial machines and systems are essentially useless. The methodology for calculation of the system productivity was presented in several key publications that considered all aspects of the manufacturing processes [1–4]. The intensification of industrial processes based on manufacturing systems with complex designs has increased the request to enhance the reliability of all units and mechanisms. Today, the efficiency of expensive production systems such as automated lines with complex designs depends on the reliability of the main mechanisms and units. A decrease in the reliability of automated manufacturing systems leads to a dramatic decrease in productivity. The reliability problem in industry is not new; the associated engineering attributes have been thoroughly presented and make it possible to describe manufacturing problems analytically [5–7]. Many engineering problems exist that require mathematical models based on the reliability attributes of industrial machines and systems. The theory of reliability provides standard attributes and modes of calculation that can be used for mathematical modelling of the reliability of the manufacturing machines. However, the known attributes and equations of reliability describe the properties of an industrial machine separately from its productivity rate, economic efficiency and other indices. Additionally, the theory of reliability is based on principles derived from electrical and electronic systems that cannot be applied for industrial machines with complex designs whose operating principles are quite different [5–7]. The literature contains publications describing the failure of models of manufacturing systems based on a probabilistic approach [8, 9]. Several publications have been dedicated to study of the productivity and reliability attributes of automated lines with different designs [10–13]. Analysis of the productivity equations for multi-station automated lines with different structures has shown that the attributes of reliability for the stations are accepted in terms of average magnitude for simplification purposes. This simplification cannot produce

exact mathematical models of automated line productivity and will always results in differences between the theoretical and actual productivity values [14, 15]. The trends in engineering progress have produced industrial machines with increasingly complex designs whose productivity should be predicted with high accuracy. The reason to obtain high accuracy of prediction yield from mathematical model is to meet customer demand. If the actual productivity is less than planned, customers face a shortage problem and may refuse to order products anymore. This example obviously presents the impact of robustness and accuracy of productivity mathematical model on the profit of an industry. Mathematical models of industrial machine productivity rates are derived according to the level of consideration of the manufacturing processes. This article presents equations of productivity that include the technological and technical aspects of the manufacturing systems and does not consider the maintenance aspects of the machines or the prescribed planned overhaul repair time. The concept of maintenance addresses the machine when it is nonworking, i.e. planned repair and service of machines are conducted during nonworking overhaul periods that do not include random stoppages. Reliability theory represents the indices using the following attributes: the machine failure rate λ=1/mw, the mean time to work mw, the mean time to repair mr and availability A. These attributes of reliability are used to develop analytical equations for the productivity rate of machines with complex designs such as automated lines with different structures. The known mathematical models for the productivity rate of automated lines include technological parameters (machining time), technical parameters (auxiliary and idle time, capacity of buffers, etc.) and structural parameters (number of serial and parallel stations and number of sections of automated lines) [15]. These analytical models for the productivity of multi-station automated lines contain one simplification, that the reliabilities of all stations are equal on average. Hence, the failure rates of all stations are assumed equal, but this assumption is incorrect. This simplification results in productivity rate equations that cannot yield accurate results of the calculation. However, for the short production lines with more or less equal reliability attributes of the stations, the mathematical model of the productivity rate [15] gives acceptable results. This statement is validated by practice [16, 17]. This paper presents contents mathematical models for the productivity rate of multi-station automated lines in which the station failure rates are different and the results of the productivity calculations produce values close to the actual output.

2 Analytical approach The actual reliability of an industrial machine or station is high, and the probability of failure is low. However, the

Int J Adv Manuf Technol

reliability of multi-station automated lines decreases with an increasing number of stations characterised by hard linking. Any random failure of a single station leads to stoppage of the entire automated line and the other workable stations. The random stoppages in multi-station automated lines that consist of independently operated stations are calculated using probability theory. Naturally, the reliability level and hence the attributes of the manufacturing machines, units and systems are all different. However, in many cases and for certain reliability attributes, the repair time required for machine or station stoppages is short and is accepted as component of the average normative repair time. These regulations are used in mathematical models for productivity of manufacturing system with different station failure rates. If the idle time due to managerial and organisational problems is not considered, the productivity rate of a machine or station with discrete processes is described by the following equation [3, 15]: Q¼

z z 1 1 ¼ ¼ ¼ g θw θi θ θw þ θi X þ Tþ t ei z z

ð1Þ

i¼1

where z is the number of products machined per observation time and θ is the observation time. The observation time θ=θw +θi is presented as the sum of the machine work time θw and the idle time due to technical failure of the machine θi. Algebraic transformation of Eq. 1 yields the ratio θw/z=T, which represents the cycle time for the machining of one product. The cycle time T=tmo +ta is the sum of the machining time tmo (minutes per part) and the total technological process and auxiliary time ta (minutes per part), which includes the time required for loading/unloading of work-pieces in the machining area, clamping/unclamping of those work-pieces and the forward/backward motion of the support to bring the cutters to/from the machining area. Quick and precise execution of these steps is important to minimise the auxiliary time. The continuous-action industrial machines work without the need for auxiliary time. Hence, the cycle time is presented by the following equation, T=tmo. The idle time is expressed by the equation θi =mrb, where b is the number of machine stops. The work time is expressed by the equation θw =mwb, where b is the number of machine time work events that equal the number of machine stops; all g

other parameters are specified above. The ratio of ∑ t ei ¼ θi =z i¼1

is the total time losses with respect to one product due to reliability issues in g machine units and mechanisms (the expression “time losses” is used for simplicity). The total time losses due to reasons related to the reliability of machine units n can be expressed by the sum of time losses of separate units, θ=θ1 +θ2 +θ3 +…+θn. In the manufacturing area, time losses

are classified into the following three groups: losses due to reliability reasons in units and mechanisms, losses due to machining of defective products and losses due to managerial, logistic and organisational problems. The third group is not considered in this article. The mathematical models for the productivity of industrial machines and systems with complex structures are represented by different equations. The typical designs of a multistation automated line are presented as three types of structures: serial action, parallel action and serial–parallel action. All of these designs can be presented in linear, circular or rotary arrangements. 2.1 Productivity of a serial automated line containing stations and mechanisms with different failure rates In general, machines with complex designs (such as those of automated lines) consist of different stations (drilling head, milling head, boring head, power head, control station, etc.) and common units and devices such as a control system or a transport mechanism that serve the entire automated line (Fig. 1). Figure 1 displays a typical serial automated line with hard-linked stations and units. Failure of any mechanism leads to stoppage of the entire line. The productivity equation of a serial automated line with q stations is represented by the following equation [15]: Q¼

1  T



1 g X i¼1

. t ei

1 1 ¼t  g  . t mo X þ ta mo þ ta T 1þ t ei q q i¼1

ð2Þ

where tmo is the total machining time of a product, ta is the auxiliary time for handling work-pieces in the machining area (load, clamp, remove, etc.), T=(tmo/q)+ta is g

the cycle time, q is the number of serial stations, ∑ t e:i i¼1

is the time losses due to the n mechanisms and units of an automated line referred to one product, Q=1/T is the cyclic productivity and A is the availability of the automated line. The time losses for each station and mechanism are different. Hence, the equation for the time losses of the stations, the common mechanisms and the units can be represented by the following equation: n X i¼1

θs:q θc θtr X θs:1 θs:2 θs:3 θs:i þ þ þ…þ …þ þ þ ¼ t s:i þ t c þ t tr z z z z z z z i¼1 q

t ei ¼

ð3Þ where θs.i is the idle time of a single i station, θc is the idle time of the common control system, θtr is the idle time of the comq  q mon transport mechanism, the ratio of ∑ t s:i ¼ ∑ θs:i =z i¼1

i¼1

measures the time losses with respect to one product due to the

Int J Adv Manuf Technol

Fig. 1 Picture (a), schematic (b) and symbolic (c) presentations of an automated line with q serial stations

reliability of q stations, the ratio tc =θc/z measures the time losses with respect to one product due to the reliability of a common control system, the ratio ttr =θtr/z displays the time losses with respect to one product due to the reliability of a common transport mechanism, and all other parameters are as specified above. Substitution of Eq. 3 into Eq. 2 and subsequent transformation yield the following equation for the productivity rate of a machine: 1 Q¼ t  mo þ ta 1þ q

1 X q

t s:i þt c þ t tr

!  . t

 mo

q

i¼1

1 q X i¼1

t s:i þt c þ t tr

!  . t



 q

f m þ ta

¼ QT  A ð5Þ Equation 5 contains two components: the equation QT = 1/ T = 1/[(tmo/q)fm + ta] is the cyclic productivity of an automated line, and A is its availability. The availability equation of a

q

1 g X

i¼1

i¼1

! ¼ . X t s:i þt c þ t tr T

1 g X

¼ 1þ

þ ta

mo

1



ð4Þ

where all parameters are as specified above. In practical terms, after balancing, an automated line will contain a bottleneck station whose machining time is the longest. The accepted average machining time tm.o/q=tav of one section presented above should be replaced by the machining time tm.b. of the bottleneck station. Hence, the machining time, tm.b, of the bottleneck station can be represented by the equation tm.b =fmtm.o/q, where fm is a correction factor that expresses the difference in the machining time between the bottleneck station and the average machining time of the station. In these connections, Eq. 4 is corrected and presented as in the following equation: 1 Q¼ t  mo f m þ ta 1þ q

serial automated line can be represented by the idle symbols and work times in Eq. 3:

1þ 1 g X

¼ θi

i¼1

ðθw =T ÞT



θi

zT ð6Þ

θi

i¼1

θw

where θi =mrni is the idle time of the ith mechanism, mr is the mean repair time, ni is the number of failures of the ith mechanism, θw =mwb is the work time of the automated line, mw is the mean work time between two failures, and b is the total number of failures. The total number of failures of the automated line is the sum of the failures of the single mechanisms and units, i.e., b ¼ ns:1 þ ns:2 þ ns:3 þ … þ ns:i þ …þ ns:q þ nc þ ntr ¼ g

∑ ni , where ns.1, ns.2, ns.3, and ns.i are the respective number of i¼1

failures of stations 1, 2, 3 and i, and nc, and ntr are the number of failures of the control and transport systems, respectively. Substitution of the defined parameters into Eq. 6 and subsequent transformation gives the following equation of availability: 1 mr ns:1 þ mr ns:2 þ mr ns:3 þ … þ mr ns:i þ … þ mr ns:n þ mr nc þ mr ntr mw b 1   ¼ mr ns:1 þ ns:2 þ ns:3 þ … þ ns:i þ … þ ns:q þ nc þ ntr 1þ mw b





ð7Þ

1   ¼ ns:1 þ ns:2 þ ns:3 þ … þ ns:i þ … þ ns:q þ nc þ ntr 1 þ mr λ b 1   ¼ 1 þ mr λ f s:1 þ f s:1 þ f s:1 þ … f s:i … þ f s:q þ f c þ f tr

Int J Adv Manuf Technol

where λ=1/mw is the average failure rate of the automated line, and fi.i =ni.i/b is the coefficient of correction that shows the ratio of the number of failures of the ith mechanism to the total number of failures of the automated line. Equation 7 consists of components λfi.i =λi or the failure rate of the ith mechanism (station, control unit, etc.). Next, the mathematical model for the productivity of a serial automated line with stations, mechanisms, and units with different failure rates is represented by the following equation: 1 1 ! Q¼ t  q mo X f m þ ta λs:i þ λc þ λtr 1 þ mr q

ð8Þ

i¼1

where all parameters are as specified above. Equation 8 represents the productivity of a serial automated line via the symbols of reliability of each station and mechanism as well as the machining and auxiliary times of the bottleneck station. The analytical result and the failure rate λi of selected mechanisms are represented as the product of the average failure rate λ and the correction factor fi, (λi =λfi), which will be used in the other equations for the productivity of automated lines with different designs. The productivity rate equation for a serial automated line with the average failure rates of the stations is expressed by the following equation [15]: 1 1 Q¼ t  mo 1 þ m ð qλ r s þ λc þ λtr Þ f þ ta q m

ð9Þ

where all parameters are as specified above. The productivity rate equation based on the average failure rates of the automated lines units can be effectively used for analytical optimisation analysis of the automated line parameters. Equation 9 gives the extremum of the function, i.e. the local maximum productivity rate for the defined number of serial stations. This extremum of the function is defined by differentiating of Eq. 9 with respect to the change of variable q when other parameters are held constant: 3 2 6 d6 4t dQ ¼ dq

1 mo

q

f m þ ta

*

7 1 7 1 þ mr ðqλs þ λc þ λtr Þ 5 dq

which yields: t mo − 2 f m ½1 þ mr ðλc þ λtr Þ þ t a λs mr ¼ 0; q

¼ 0;

and solution of this equation produces the following: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h .  i u u t mo f 1 m þ λ þ λ r c tr m t ð10Þ qopt ¼ t a λs where all parameters are as specified above. Equation 10 presents the optimal number of serial stations that yield the maximum productivity rate.

2.2 Productivity of automated lines with parallel actions, serial–parallel actions and linear arrangements with different failure rates for the stations and mechanisms In practical situations, manufacturers in different industries produce automated lines with different numbers of parallel p and serial q stations, and the automated lines with parallel stations are designed according to the peculiarities of the technological processes. A multi-station automated line with parallel action contains identical parallel stations with equal reliability. Each station is equipped with the transport and workpiece feed mechanism. Increasing the number of parallel stations in the line leads to an increase in the productivity rate, but the reliability attributes of lines with complex designs impose restrictions on the machine design. Optimisation of the structure for an automated line with parallel stations is not a simple process and should be considered using economic approaches. The picture and schematic diagram of such a line is represented in Fig. 2. The productivity rate equation for an automated line with parallel stations is expressed as follows [15]. Q¼

p 1  t m þ t a 1 þ mr ½ps ðλs: þ λtr Þ þ λc 

ð11Þ

where p is the number of products machined, ps is the number of parallel station arrangements and p=ps, λtr is the failure rate of the transport with the work-piece feed mechanism, λc is the failure rate of the control system that serve the entire line, and all other parameters are as specified above. Equation 11 enables us to find the maximum productivity rate by taking the mathematical limit of the equation with respect to the variable p when other parameters are held constant:   p 1 LimQ ¼ Lim  p→∞ t m þ t a 1 þ mr ðps λs: þ λc þ λtr Þ p→∞ ð12Þ 1 Qmax ¼ ðt m þ t a Þmr ðλs: þ λtr Þ The maximum productivity of an automated line with parallel stations depends on the machining times, auxiliary times, failure rates of the stations and the transport with the work-

Int J Adv Manuf Technol

Fig. 2 Picture (a), schematic (b) and symbolic (c) presentations of a linear multi-station automated line with parallel p parallel stations

piece feed mechanism, and the mean time required to repair the automated line. A multi-station automated line with serial–parallel actions and a linear design produces the highest productivity rate. The design specifications for this type automated lines are presented by the following properties: All parallel and serial stations begin work simultaneously and after finishing operations; the failure rates of the serial stations are the same for other parallel arrangements; the control and transport systems serve all serial and parallel stations, and all products are transported by the linear design of the transport mechanism to the subsequent serial–parallel stations. This automated production line can be considered as a system for a collection of serial and parallel stations arranged according to a certain structure that depends on the technological process of the machined parts. Any failure in either the serial or parallel stations leads to a stoppage of the entire automated line due to the mechanical hard-linking of all mechanisms. In the case of the serial–parallel automated line, one principle is accepted for the serial and parallel stations: The system fails if at least one of the serial or parallel stations fails. The schematic diagram of such a line is shown in Fig. 3. Equation 9 for the productivity rate of the serial automated line can be transformed and presented for an automated line with serial–parallel actions and a linear arrangement, which

contains its own specificity of design, structure and work processes as described above. This automated line contains p parallel stations, which are components of the productivity equation. Based on mathematical considerations, the failure rates for the automated line considered above can be represented by a mathematical model for the productivity rate and the availability of automated lines with serial–parallel action using the following equation: p 1 ! Q¼ t  q mo X f m þ ta 1 þ m r ps λs:i þ λc þ λtr q i¼1

where p is the number of products machined, ps is the number of parallel station arrangements and p=ps,, λc and λtr are the respective failure rates of the control and transport systems that serve the entire line, and all other parameters are as specified above. In the metal cutting industry, automated lines with parallel– serial action and linear arrangements contain two to three parallel lines (ps =2; 3) due to the complexity of the design. However, in the electronics industry, automated lines with parallel–serial action can contain a larger number of parallel stations. The productivity rate equation for a serial–parallel automated line with an average station failure rate is expressed by the following equation [15]: p 1 Q¼ t  mo 1 þ m ð p qλ r s s þ λc þ λtr Þ f þ ta q m

Fig. 3 Symbolic presentation of a linear multi-station automated line with q serial and p parallel stations

ð13Þ

ð14Þ

where all parameters are as specified above. The productivity rate equation based on the average failure rates of the automated lines units can be effectively used for analytical optimisation analysis of the automated line parameters. Equation 14 yields the extreme of the function, i.e. the

Int J Adv Manuf Technol

maximum productivity rate for the defined number of serial stations. This maximum is defined by differentiating of Eq. 14 with respect to the change of variable q when other parameters are held constant. The process of solution is same as for Eq. 9, which yields the following result: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h .  i u u t mo f 1 mr þ λc þ λtr m t q¼ ð15Þ ps t a λ s

one transport and work-piece mechanism. The picture (a), schematic (b) and symbolic presentation of a rotor-type machine with circular (c) and linear (d) arrangements are shown in Fig. 4 The productivity rate equation for the rotor-type machine is expressed as follows [15]:

where all parameters are as specified above. Equation 15 represents the optimal number of serial stations that yields the maximum productivity rate for a serial–parallel automated line with a given number of parallel stations.

1 ð16Þ 1 þ mr ðps λs þ λc þ λtr Þ   p −1 where 2 þ p γ−p is the cycle time displacement factor, pγ is



p pγ −1 ðt m þ t a Þ 2 þ ps −pγ 

s

2.3 Productivity of a rotor-type automatic machine and automated lines with serial–parallel actions and different failure rates for stations and mechanisms The rotor-type automated production line belongs to the category of serial–parallel action machines (Fig. 4). This rotor-type arrangement is applied in many branches of industry, including pressing, coining, filling of bottles and cans, and others. The work process of a rotor-type automated line is well known, and the main peculiarity of this type of automatic machine is that the machining process for the part is executed with displacement in time at each station. The single rotor-type automatic machine is a multi-station machine with parallel action containing identical parallel stations that perform identical machining processes. Each single rotor-type automatic machine is equipped with a

!

γ

the number of stations located in the idle zone γ of the rotor-type machine (Fig. 4), and all other parameters are as specified above. The productivity rate equation for a rotor-type automatic machine with parallel stations does not have an extreme of the function. The maximum productivity rate is defined by taking the mathematical limit of Eq. 16 with respect to the change of variable ps, when other parameters are held constant. 1

0 B B LimQ ¼ Lim B p→∞ B p→∞ @

p ðt m þ t a Þ 2 þ

pγ −1 ps −pγ

!

C C 1 C 1 þ mr ðps λs þ λc þ λtr Þ C A

1 !

Qmax ¼ 2þ

pγ −1 ðt m þ t a Þmr λs: ps −pγ

ð17Þ

Fig. 4 Picture (a), schematic (b) and symbolic circular (c) and linear (d) presentations of a rotor-type automatic machine with p parallel stations

Int J Adv Manuf Technol

The maximum productivity of the rotor-type automatic machine with parallel stations depends on the machining processes, auxiliary times and failure rates of the stations and the mean time to repair of the automated line. Analysis of Eqs. 12 and 17 shows that the maximum productivity rate of the rotor-type machine is almost two times less than the maximum productivity rate of an automated line with parallel action in a linear arrangement. The rotor-type automated line is designed in combination with q serial rotor-type machines according to the technological process for machining the products. The picture, p1

QT ¼ T1

pγ þ pτ −1 2þ ps:1 −pγ pi

¼ Ti 2 þ

pγ þ pτ −1 ps:i −pγ

p2

!¼ T2

pγ þ pτ −1 2þ ps:2 −pγ

Tq 2 þ

p3



pq

!¼…¼

pγ þ pτ −1 ps:q −pγ

schematic and symbolic presentation of the rotor type automated line are demonstrated in Fig. 5. Each serial rotor-type machine may be characterised by a different machining and cycle time. The problem of balancing the technological processes for serial stations in a rotor-type automated line does not create difficulty. The differences in the cycle times of the serial stations are compensated by the addition or subtraction of a number of parallel stations in the rotor-type machines. However, the cyclic productivity of each rotor-type machine in the automated line should be the same. This statement is expressed by the following equation [15]:

T3

pγ þ pτ −1 2þ ps:3 −pγ

!¼…

!

stations, where pi is the number of the parallel   Ti is the cycle p þp −1 time of the rotor-type machine, 2 þ pγ −pτ is the cycle time s:i γ displacement factor, pτ is the number of stations located in the transport zone τ of the transport rotor, pγ is the number of stations located in the idle zone γ of the rotor-type machine (Fig. 3), and all other parameters are as specified above. The mathematical model for the productivity rate of the rotor-type automated line is represented by an equation with

ð18Þ

an equal number of parallel stations on each rotor-type machine and equal failure rates [15]. In practical terms, the serial rotor machines in an automated line can have different numbers of parallel stations, and the reliability of the serial rotortype machines can differ. The failure of common mechanisms of machines with complex designs is considered separately. Therefore, the productivity rate of the rotor-type automated line should be considered in terms of the rotor-type machine

Fig. 5 Picture (a), schematic (b) and symbolic (c) presentations of a rotor-type automated line p parallel and q serial stations

Int J Adv Manuf Technol

with the maximum number of parallel stations, i.e. the arrangement that will have the highest failure rate and is a bottleneck rotor-type serial machine. However, this is not a regular practice; the rotor-type machine with the highest failure rate may be another machine with a smaller number of parallel stations. This rotor-type machine may have a high frequency of rotation and workstations that can lead to a high failure rate. Hence, any rotor-type machine with a different number of parallel stations can be a bottleneck machine in the automated line, but which rotor-type machine is the bottleneck is a function of its reliability level. Based on the known mathematical models and logical approaches presented above, the productivity rate equation for the rotor-type automatic line with serial–parallel action and different reliabilities in the parallel and serial stations can be described by the following equation [15]: Q¼



t mo f þ ta q m " 1 þ m r ps



p pγ þ pτ −1 2þ ps −pγ

!

1 q X

#

ð19Þ

ðλs:i þ λtr Þ þ λc

i¼1

where p is the number of products machined on the bottleneck station, ps. is the number of bottleneck stations (each station carries one product, and hence the number of products is equal to the number of stations, (p=ps)), q is the number of serial rotor-type machines, tmo is the machining time of the total technological process, ta is the auxiliary time, (tmo/q)=tav is the average machining time, fm =(tmo/q)/tav is the correction factor, and all other parameters are as presented above. Equation 19 contains standard attributes for the reliability of mechanisms (mr, λi), and parameters for the structure and design of the rotor-type automated line (ps, q). These equations allow calculation of the productivity rate for a rotor-type automated line with different levels of reliability for the stations and mechanisms. The productivity rate equation for a serial–parallel automated line with an average failure rate of the stations is expressed by the following equation [15]: Q¼



t mo f þ ta q m



p pγ þ pτ −1 2þ ps −pγ

!

1 1 þ mr ½ps qðλs:i þ λtr Þ þ λc 

where all parameters are as specified above.

ð20Þ

The productivity rate equation based on the average failure rates of the automated lines stations can be effectively used for analytical optimisation and analysis of the rotor type automated line parameters. Equation 20 describes the extreme of the function, i.e. the maximum productivity rate for the defined number of serial stations for the given number of parallel stations. This maximum is defined by differentiating Eq. 20 with respect to the change of variable q when other parameters are held constant. The process of solution is same as for Eq. 9, which yields the following result: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h .  i u u t mo f þ λ 1 m r c m t qopt ¼ ð21Þ ps t a ðλs þ λtr Þ where all parameters are as specified above. Equation 21 represents the optimal number of serial stations that results in the maximum productivity rate for a rotor-type automated line with a given number of parallel stations.

3 Working examples The productivity rates for automated lines of different designs are calculated using the equations represented above. The following sections describe the analysis of the productivity rate for typical designs of manufacturing machines and systems. For simplicity of calculation, it is accepted that the failure rates for certain groups of stations are equal. An engineering problem is defined to calculate the productivity rates and state the change in productivity rate in terms of a change in the number of stations. 3.1 Automated line with serial action and a linear arrangement The automated serial-action line can be designed with several variants in terms of q=10, …, 25 serial machining stations (Fig. 1). The basic technical and technological data for the automated line are presented in Tables 1 and 2. Substituting the initial data (Tables 1 and 2) into Eqs. 8 and 9 and performing the calculations yields Eqs. 22 and 23. The Table 1

Technical data for a serial automated line

Title

Data

Total machining time, tmo (min) Auxiliary time, ta, (min) Number of stations, q Correction factor for machining time of the bottleneck station fm

35 0.3 10, …, 25 1.2

Int J Adv Manuf Technol Table 2

Reliability indices for an automated line

Title

q

λs.i (k/min)

Failure rate of the stations q, λs

1–5 6–10 11–15 16–20 21–25 26–30 Average

7.0×10−2 5.0×10−2 8.0×10−2 6.0×10−2 9.0×10−2 7.0×10−2 7.0×10−2 8.0×10−4 4.0×10−5

Failure rate of the control system, λc Failure rate of the transport system, λtr Mean repair time mr =3.0 min

calculated results represented in Table 3 are used to depict the diagram of changes in the productivity rate versus the number of stations with different failure rates (Fig. 4). 1 1 ! Q¼ t  q mo X f m þ ta λs:i þ λc þ λtr 1 þ mr q i¼1

¼

1 35 *1:2 þ 0:3 q

1

 1 þ 3:0

q X

! −4

−5

λs:i þ 8:0*10 þ 4:0*10

i¼1

ð22Þ 1 1 !¼ Qav ¼ t  q mo X f m þ ta 1 þ mr λs:i þ λc þ λtr q

Fig. 6 Productivity rate of a serial automated line versus the number of stations with different failure rates

with the average failure rate of the stations, gives the following result: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t mo f m ½ð1=mr Þ þ λc þ λtr  q¼ t a λs sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi 35:0*1:2* ð1=3:0Þ þ 8:0*10−4 þ 4*10−5 ≈26 ¼ 0:3*7:0*10−2 The difference in the results calculated from Eqs. 8 and 9 is considerable. Hence, it is preferable to calculate the productivity rate of the industrial machines and systems based on the real technical data. A simplified approach in the calculations yields a large optimal number of serial stations that cannot be accepted for the final solutions.

i¼1

1

35 *1:2 þ 0:3 q



1   1 þ 3:0 q*7:0*10−2 þ 8:0*10−4 þ 4:0*10−5

ð23Þ Figure 6 demonstrates the change in the productivity rate for the automated serial-action lines with a different number of stations q and the failure rates. The maximum productivity rate Q=0.085 prod/min describes the automated line with q=20 serial stations (curve Q). The curve Qav represents the change in the productivity rate calculated with the average failure rates of the stations. For an optimal number of the serial stations, Eq. 10 determines the maximum productivity rate of the serial automated line (Qmax =0.82 prod/min) and, together

3.2 Productivity of automated lines with parallel and serial–parallel actions and a linear arrangement The multi-station automated line with parallel action can be designed using several numbers of parallel stations (Fig. 3). The basic technical and technological data of the automated line are presented in Tables 4 and 5. Substituting the initial data (Tables 4 and 5) into Eqs. 11 and 12 and performing the calculations produces the diagram of changes in the productivity rate versus the number of parallel stations (Fig. 7).

Table 4 Table 3

Productivity rate for a serial automated line with q stations

q

5

10

15

20

25

30

Q (prod/min) Qav (prod/min)

0.056 0.056

0.079 0.071

0.080 0.077

0.085 0.08

0.080 0.082

0.078 0.08

Technical data for an automated line with parallel action

Title

Data

Machining time, tmo, (min) Auxiliary time, ta, (min) Number of parallel stations, p

1.0 0.3 2, …, 40

Int J Adv Manuf Technol Table 5

Reliability attributes for an automated line with parallel action

Table 6

Technical data for a serial–parallel automated line

Title

λs. (k/min)

Title

Data

Failure rate of the station λs Failure rate of the control system, λc Failure rate of the transport with work-piece feed mechanism, λtr Mean repair time mr =3.0 min

7.0×10−2 8.0×10−3 5.0×10−4

Total machining time, tmo (min) Auxiliary time, ta (min) Number of serial stations, q Number of parallel stations, p Correction factor for machining time of the bottleneck station fm

15 0.3 2, …, 14 3 1.2

The maximum productivity rate of the automated line with parallel stations is calculated by Eq. 12: Qmax

1 ¼ ðt m þ t a Þmr ðλs: þ λtr Þ ¼

1 ¼ 3:63prod=min ð1:0 þ 0:3Þ*3:0*ð0:07 þ 0:0005Þ

The diagram (Fig. 7) shows that the productivity rate grows significantly for a small number of parallel stations and asymptotically reaches the limit of Qmax =3.66 prod/min. The optimal number of parallel stations should be defined by economic approaches for the automated line with parallel stations. The automated line with serial–parallel action can be designed with variants of q=5,…,15 serial machining stations and three parallel stations (Fig. 2). The basic technical and technological data for the automated line are presented in Tables 6 and 7. Substituting the initial data (Tables 6 and 7) into Eqs. 11 and 12 and performing the calculations yields Eqs. 24 and 25. The calculations and results represented in Table 8 are used to depict the diagram of changes in the productivity rate versus the number of serial and parallel stations with different failure rates (Fig. 8). p 1 !¼ Q¼ t  q mo X f m þ ta λs:i þ λc þ λtr 1 þ mr ps q 3

15 *1:2 þ 0:3 q

i¼1



1 ! q X −3 −4 λs:i þ 8:0*10 þ 5:0*10 1 þ 3:0 3

p 1 ¼ Qav ¼ t  mo f m þ t a 1 þ mr ðps qλs þ λc þ λtr Þ q 3 1    −2 15 þ 8:0*10−3 þ 5:0*10−4 1 þ 3:0 3q*6:666*10 *1:2 þ 0:3 q

ð25Þ Figure 8 demonstrates the changes in the productivity rate for automated lines for serial action with different numbers of serial q and parallel p stations and the failure rates. The maximum productivity rate Q=0.022 prod/min produces an automated line with q=8 serial stations (curve Q). The curve Qav represents the change in productivity rate calculated using the average failure rates of the stations. For an optimal number of serial stations, Eq. 13 determines the maximum productivity rate of the serial automated line (Qmax =0.202 prod/min), which, together with the average failure rate of the stations, gives the following result: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h .  i u u t mo f 1 m þ λ þ λ r c tr m t qopt ¼ ps t a λ s vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h .  iffi u u 15:0*1:2* 1 3:0 þ 8:0*10−3 þ 5:0*10−4 t ¼ 3*0:3*0:0666 ¼ 12

i¼1

ð24Þ

Table 7

Reliability indices for a serial-parallel automated line

Title Failure rate of the serial stations q, λs

λs.i (k/min)

q 1–4 5–8 9–12 13–16 17–20 21–24 Average

Fig. 7 Productivity rate of an automated line with parallel action versus the number of parallel stations

Failure rate of the control system, λc Failure rate of the transport system, λtr Mean repair time mr =3.0 min

7.0×10−2 5.0×10−2 8.0×10−2 6.0×10−2 9.0×10−2 5.0×10−2 6.666×10−2 8.0×10−3 5.0×10−4

Int J Adv Manuf Technol Table 8 Productivity rate for an automated line with q serial and p parallel stations p=3; q

4

8

12

16

20

24

Q (prod/min) Qav (prod/min)

0.173 0.182

0.22 0.201

0.202 0.202

0.202 0.198

0.183 0.192

0.185 0.185

The difference in the results calculated by Eqs. 11 and 12 is considerable. The simplified approach in the calculations produces serious deviation that cannot be accepted for the final solutions.

3.3 Productivity rate for rotor-type automatic machines and lines with serial–parallel actions The rotor-type automatic machine with parallel action can be designed with several numbers of parallel stations (Fig. 4). The basic technical and technological data for the automatic machine are presented in Tables 9 and 10. Substituting the initial data (Tables 9 and 10) into Eqs. 16 and 17 and performing the calculations produces the diagram of changes in the productivity rate versus the number of parallel stations (Fig. 9). The maximum productivity rate for the rotor type automatic machine with parallel stations is calculated by Eq. 17: 1 1 ¼ 2ðt m þ t a Þmr λs: 2ð1:0 þ 0:2Þ*3:0*0:07 . ¼ 1:83prod min

Qmax ¼

The diagram (Fig. 9) shows that the productivity rate grows and asymptotically reaches the limit of Qmax =1.83 prod/min. The optimal number of parallel stations should be defined by economic approaches for the rotor-type automatic machine.

Technical data for a rotor-type automatic machine

Table 9 Title

Data

Machining time, tmo (min) Auxiliary time, ta (min) Number of parallel stations, p Number of idle stations, pγ

1.0 0.2 2, …, 40 3

The rotor-type automated line is designed with q=5 serial rotor machines and p=8,…,16 parallel stations in one rotor machine (Fig. 5). We assume the cyclic productivity is given as Q=2.1 prod/min. The machining time tm at each station, the number of idle stations pγ in the rotor machine, and the number of stations ptr in the transport rotor are given in Table 10. The number of parallel stations in the rotor machines is calculated using Eq. 14. Substitution of the initial data of the first serial station (Table 10) into Eq. 14 and transformation gives the following (p=ps): 2:1 ¼

p   3 þ 4−1 ðt m:1 þ 0:2Þ 2 þ ps −3

or p ¼ 4:2ðt m:1 þ 0:2Þ þ 3 For each serial rotor machine, the following parameters are calculated: – – –

The number of parallel stations, pi =4.2(tm. i + 0.2) + 3 (rounded to the nearest integer number) The machining correction factors, fm.i =tm.i/tav   p þp −1 i The cyclic productivity rate, Qi ¼ ðtm:ipþt 2 þ γp −pτ aÞ γ

The calculated and given technical and technological data for the rotor-type automated line are presented in Tables 11 and 12. We assume that the bottleneck serial station (due to reliability) is described by q = 3 with p = 16 parallel stations.

Table 10

Fig. 8 Productivity rate for an automated line versus the number of serial and parallel stations with different failure rates

s:

Reliability attributes for a rotor-type automatic machine

Title

λs. (k/min)

Failure rate of the station λs Failure rate of the control system, λc Failure rate of the transport with work-piece feed mechanism, λtr Mean repair time mr =3.0 min

7.0×10−2 8.0×10−3 5.0×10−4

Int J Adv Manuf Technol

Fig. 9 Productivity rate for a rotor-type automatic machine with parallel action versus the number of parallel stations

Substituting the initial data (Tables 11 and 12) into Eqs. 15 and 16 and performing the calculations gives Eqs. 26 and 27.

Q¼

¼

t mo þ ta q



p pγ þ pτ −1 2þ ps −pγ

!

Substituting the initial data (Tables 11 and 12) into Eq. 19 gives Eq. 26 with variable parameters q, and λs, which depend on the serial rotor machines. The correction factors fm should be omitted because the differences in the cycle times are corrected by the number of parallel stations in the serial rotors. The number of parallel stations p=16 is constant for all serial rotor machines, which contain smaller numbers of parallel stations. This assumption is correct because the intensiveness of the work of the parallel stations in other serial rotor machines is higher, and the total number of working stations is equal to the number of stations at the bottleneck rotor machine.

1

" 1 þ mr ps

q X

#

ðλs:i þ λtr Þ þ λc

i¼1

ð26Þ

16 1   " # q 8:93 3 þ 4−1 X   −5 −5 þ 0:2 2 þ 1 þ 3:0 16 λs þ 4:0*10 þ 8:0*10 q 16−3 i¼1

Q¼



p

!

1 ¼ 1 þ mr ½ps qðλs þ λtr Þ þ λc 

pγ þ pτ −1 t mo þ ta 2þ q ps −pγ ð27Þ 16 1     

8:93 3 þ 4−1 1 þ 3 16*q 40*10−3 þ 4:0*10−5 þ 8:0*10−5 þ 0:2 2 þ q 16−3 The maximum productivity rate Q=0.325 prod/min deEquations 26 and 27 enable the calculations (Table 13) and scribes an automated line with q=4 serial rotor machines result in the diagram (Fig. 10) of the change in the productivity (curve Q). The curve Qav represents the change in productivity rate for the rotor-type automated line with a different number rate calculated by the average failure rates of the rotor maof serial q and parallel p stations and the failure rates. chines. Equation 20 gives the optimal number of the serial rotor machines, which determines the maximum productivity rate of the rotor-type automated line (Qmax =0.307 prod/min), Table 11

Technical data for a rotor-type automated line

Title Serial station

qi

pi

tm.i, min

fm.i

1 8 0.98 0.548 2 10 1.45 0.812 3 16 2.80 1.567 4 12 1.90 1.064 5 12 1.80 1.008 Total machining time, tmo (min) Average machining time tav (min) Auxiliary time, ta (min) Number of idle stations in the rotor machine, pγ Number of transport stations, pτ

Qi, prod/min 2.11 2.12 2.16 2.14 2.25 8.93 1.786 0.2 3 4

Table 12

Reliability attributes for a rotor-type automated line

Title Failure rate of the serial rotor machine q, λs

Serial rotor machine

Failure rate of the control system, λc Failure rate of the transport system, λtr Mean repair time mr =3.0 min

q

λs.i (k/min)

1 2

40.0×10−3 35.0×10−3

3 4 5 Average

45.0×10−3 30.0×10−3 50.0×10−3 40.0×10−3 8.0×10−5 4.0×10−5

Int J Adv Manuf Technol Table 13 Productivity rate for a rotor-type automated line with q serial and p parallel stations p=16; q

1

2

3

4

5

Q (prod/min) Qav (prod/min)

0.243 0.243

0.302 0.287

0.302 0.301

0.325 0.307

0.307 0.307

which, together with average failure rate of the serial rotor machines, gives the following result: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h .  i u u t mo f 1 mr þ λc þ λtr m t qopt ¼ ps t a λ s vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h .  i u u 8:93* 1 3:0 þ 8:0*10−5 þ 4:0*10−5 t ¼ ¼4 16*0:3*0:040 There are differences in the results calculated by Eqs. 26 and 27. The simplified approach in the calculations yields a smaller maximum productivity rate than those produced from the detailed process of calculation.

4 Results and discussion The derived equations for the productivity rate of typical automated lines with complex designs and different failure rates for the stations and mechanisms enables the prediction of the real output of the production systems. The equations enable calculation of the productivity rate of the automated line as a function of the number of serial and parallel stations and the reliability attributes of the automated line mechanisms and units. The derived equations were tested on several types of automated lines and were represented by diagrams of changes in the productivity rates versus the number serial and parallel stations. The simplified approach in the calculations based on

the average technical parameters of the automated line can produce serious deviations in the results.

5 Conclusion The productivity rate of an industrial machine with a complex design is an important economic index that should be predicted and evaluated using analytical methods with accurate results to the extent possible. This paper develops mathematical models of the productivity rate for the typical automated lines with serial, serial–parallel and rotor-type designs that contain different reliability levels for the mechanisms and stations. The productivity rate equations of industrial machines with complex designs are represented as functions of the technological and technical parameters, including the number of serial and parallel stations and the reliability attributes of the typical mechanisms and units. The represented equations of the productivity rate enable the designers to analyse the output of the automated lines and produce results that are accurate and close to the actual data output. The simplified equation of the productivity rate based on the average technical parameters of the failure rates might produce considerable deviation from the real results. The productivity rate of automated lines with stations and mechanisms that each display different failure rates and processing times analytically formulated that is important for manufacturers. The mathematical models allow for the output of the automated lines to be modelled with different failure rates for the stations and mechanisms and yields results that are close to the actual productivity values. Engineers and designers of automated lines can use the mathematical models for productivity rates derived in this study in the project stage of automated line design. Production planners and engineers also can apply this model to accurately predict productivity of production line. They can also study the failure rate of each parameter and increase productivity by improving more impacted failure parameters as according to this robust mathematical model.

References 1.

Fig. 10 Productivity rate for a rotor-type automated line versus the number of serial and parallel stations with different failure rates

Chryssolouris G (2006) Manufacturing systems: theory and practice, 2nd edn. Springer, New York 2. Groover MP (2010) Fundamentals of modern manufacturing: materials, processes and systems, 4th edn. Wiley, New York 3. Volchkevich L (2005) Automation of production processes. Mashinostroenie, Moscow 4. Gershwin SB (1994) Manufacturing systems engineering. Prentice Hall, New Jersey 5. Nakagawa T (2005) Maintenance theory of reliability. Springer, New York 6. O’Connor PDT (2002) Practical reliability engineering, 4th edn. Wiley, New York

Int J Adv Manuf Technol 7.

Birolini A (2007) Reliability engineering: theory and practice, 5th edn. Springer, New York 8. Mourani I et al (2007) Failure models and throughput rate of transfer lines. Int J Prod Res 45(8):1835–1859 9. Freiheit T, Koren Y, Hu SJ (2004) Productivity of parallel production lines with unreliable machines and material handling. IEEE Trans 1(1):98–103 10. Colledany M, Tolio T (2012) Integrated quality production logistics and maintenance analysis of multi-stage asynchronous manufacturing systems with degrading machines. CIRP Ann 61(1):455–458 11. Hon KKB (2005) Performance and evaluation of manufacturing systems. CIRP Ann 54(2):675–690 12. Youseff AMA, Mohib A, ElMaraghy HA (2006) Availability assessment of multi-state manufacturing systems using universal generating function. CIRP Ann 55(1):445–448

13.

Rzepka B, Krolo M, Bertsche B (2003) Case study: influences on the availability of, machine tools. Proc Ann Reliab Maint Symp, (IEEE). Tampa, Florida, USA, 27-30 January, pp. 593–599 14. Huang SH et al (2003) Manufacturing productivity improvement using effectiveness metrics and simulation analysis. Int J Prod Res 41(3):513–527 15. Usubamatov R, Ismail KA, Shah JM (2012) Mathematical models for productivity and availability of automated lines. Int J Adv Manuf Technol. doi:10.1007/s00170-012-4305-y 16. Usubamatov R, Riza AR, Murad NM (2013) A method for assessing productivity in buffered assembly processes. J Manuf Technol Manag 24(1):123–139 17. Usubamatov R et al (2014) Analysis of buffered assembly line productivity, Assembly Automation, 34/1:34–40, Emerald Group Publishing Limited [ISSN 0144–5154], [doi: 10.1108/AA-112012-086]