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Analytical analysis of a generic assembly system

Liesje De Boeck and Nico Vandaele

HUB RESEARCH PAPER 2008/44. SEPTEMBER 2008

Analytical analysis of a generic assembly system Liesje De Boeck ab, Nico Vandaele cd a

Centre for Modelling and Simulation, University College Brussels (HUBrussel), Stormstraat 2, 1000 Brussels, Belgium b Affiliated with the Faculty of Business and Economics, K.U.Leuven, Naamsestraat 69, 3000 Leuven, Belgium [email protected] c Faculty of Business and Economics, K.U.Leuven-Campus Kortrijk, Etienne Sabbelaan 53-bus 00000, 8500 Kortrijk, Belgium d Faculty of Business and Economics, K.U.Leuven, 3000 Leuven, Belgium [email protected] Abstract The assembly of components is a key element in manufacturing facilities. Nevertheless, literature analysing them remains scarce. The paper narrows this gap by studying a generic first-come first-serve assembly system, consisting of two generally-distributed component input streams. The field of probability theory serves as a means to derive good approximations for the waiting time of the components for a kit and the inter-arrival time of the kits at the assembly facility. Apart from providing insight in generally accepted managerial principles, this analysis also constitutes a required step in obtaining the performance of an open queueing network including assemblies. Keywords: Assembly; Performance analysis; Synchronization time; Probability theory

1. Introduction Since the eighties, manufacturing played an increasing role in the responsiveness and the competitiveness of a firm. This created a need for analysing and improving manufacturing systems (Suri et al., 1993). At the same time, the focus changed from cost reduction to lead time reduction, an evolution which was strengthened by the introduction of time-based competition in the nineties. As a result, the supply of goods and services faster than the competitors became a central theme, inducing the need for quantifying lead times in a fast way. Nowadays, this concept of time based competition appears beyond the borders of the production system. Taking care of this evolution during the last decades, the supply chain necessitates the right information on the right time. The right information stands for taking into account the dynamic and complex character, the stochastic nature, the variability and the capacity constraints present in the supply chain environment (Vandaele and De Boeck, 2002; Vandaele and De Boeck, 2003; Vandaele and Lambrecht, 2003; Van Nieuwenhuyse et al., 2007) whereas the right time deals with the responsive ability of the tools for quick decisionmaking. Literature reveals that many models have already been developed in order to achieve this goal, for instance open queueing models (Vandaele and De Boeck, 2002) including optimal lot sizes (Vandaele, 1996), incorporating sources of variability (Hopp and Spearman, 2000), accounting for transfer batches (Van Nieuwenhuyse, 2004), etc. This paper offers a next step in refining the existing open queueing network models of Vandaele and De Boeck (2002) by incorporating assembly systems. In this way it contributes to the development of a tool providing the right information in a quick way. On top of the lack of open queueing network models which incorporate assembly systems, there are two other reasons for modelling assembly systems. The first reason covers the practical relevance of assembly systems. Indeed, assembly systems are widespread in many supply chain systems. A second reason accounts for the scarce literature on the modelling of assembly systems. Compared to other queueing topics which have been thoroughly studied in the literature, assembly systems have received much less attention.

1

The paper is organised as follows. We will start in Section 2 with definitions and assumptions, followed by a review of the literature, a discussion of the methodology and the notation. The assembly analysis itself as well as the results will be developed for a generic assembly system in Section 3. This section ends with simulation results to validate the performance of our approximations. Section 4 will conclude with the main results of this paper. 2.

Definitions, assumptions, literature, methodology and notation

2.1. Definitions and assumptions We define assembly systems as systems where more than one component (class) is brought together with the intention to jointly undergo an (assembly) operation. The set which contains all components in the right quantities needed for assembly is defined as a kit. Note that such a system induces two additional performance measures compared to a traditional queueing system: • the synchronization time which is the additional waiting time of the components for one another in order to form a kit, • an arrival process of kits at the assembly station, which is clearly distinguished from the arrivals of components at the assembly system. In view of our final goal of embedding the assembly model in the open queueing network model of Vandaele and De Boeck (2002), we must impose the same assumptions on our assembly system as those that are imposed on the open queueing network model of Vandaele and De Boeck (2002): • the system only deals with discrete products, • the system is an open system with infinite capacity buffers where the FCFS-discipline applies, • the individual component arrivals are assumed to be independent with generally distributed IID inter-arrival times. In what follows, we assume a generic assembly system. Taking into account the above assumptions, we can define it as an open discrete production system with infinite buffers and two independent input streams having generally distributed IID component inter-arrival times. A kit consists of exactly one component of each input stream. Both components and kits obey the FCFS-discipline. In what follows, we will always assume the same average inter-arrival times for both components as we want to avoid non-convergence of the synchronization time due to unequal average component inter-arrival times. However, the variance and in fact the entire distribution for both component inter-arrival times may differ. A representation of the generic assembly system is given in Fig. 1.

Input 1

Input 2

Fig. 1. Representation of the generic assembly system with a synchronization buffer and an assembly buffer

2

The two component input processes are represented by the first two squares. The first buffer, the synchronization buffer, contains the individual components which wait to form a complete kit. The second buffer, the assembly buffer, holds the ‘collected’ kits which wait for their operation on the assembly station. The two buffers have an infinite capacity. The assembly station is represented by the last square. This buffer split can also be found in Lipper (1986). The reason for focusing on a generic assembly system is for clarity reasons. The analysis can however be easily extended to general assembly systems (i.e. assembly systems with more than two input streams or assembly systems with kits existing of more than one unit of each component). For this analysis, we refer to De Boeck (2003). 2.2. Literature and methodology Relevant literature on assembly systems as described in Subsection 2.1 includes: Harrison (1973), Crane (1974), Latouche (1981), Bhat (1986), Lipper and Sengupta (1986), Bonomi (1987), Hopp and Simon (1989, 1993), Baker et al. (1990, 1993), Simon and Hopp (1991, 1995), Duenyas and Hopp (1992, 1993), Som et al. (1994), Duenyas and Keblis (1995), Baker and Powell (1995), Powell and Pyke (1998), Wilhelm and Som (1999), Takahashi et al. (2000), Sabuncuoglu et al. (2002) and Ramachandran and Delen (2005). The above literature reveals that modelling an assembly system is very hard. The main difficulties trace back to the system state space, the mathematical complexity and the instability of the assembly systems when buffers have an infinite capacity and input processes are independent renewal processes. Indeed, Harrison (1973) proved that the synchronization time does not converge (when the input processes are independent renewal processes) unless buffers are finite. Therefore, there is always some kind of input control to preserve system stability in the existing assembly literature. This input control can take on many different forms. • When the maximum buffer capacity is reached, components are removed from the input processes (e.g. Lipper and Sengupta, 1996) or the input process is shut down (e.g. Simon and Hopp, 1991; Hopp and Simon, 1993; Som et al., 1994; Simon and Hopp, 1995). • The component arrivals depend on the excess of the number of one component in the system over the others (e.g. Latouche, 1981). • The number of each component allowed in the system is limited (e.g. Bhat, 1986). • The component arrival process is stopped when the number of units of that component exceeds the number of units of all the other components with a certain value (e.g. Bonomi, 1987). • Control mechanisms as bins (e.g. Hopp and Simon, 1989), conwip (e.g. Duenyas and Hopp, 1993) or kanban (e.g. Duenyas and Keblis, 1995) are used. • … Hence, only if the number of components in the system cannot exceed a predefined limit, infinite buffers are allowed. That is why we will assume a finite number of kits after which the system is ‘emptied’. In other words, we control the system by imposing a limit on the total throughput. We do believe that the independency assumption of the component input streams is justified when components are processed in a stochastic production environment before assembly takes place. As an example, we can think of the component input processes in Fig. 2 being variable flow lines. Although one can synchronize both components in the beginning of those flow lines, variability will cause both departure streams out of these flow lines (i.e. the arrival streams at the assembly system) to be almost independent. If we examine exact relations versus approximations in the assembly literature, we clearly observe the absence of exact relations. Hopp and Simon (1989) state that in the absence of exact relations, only approximations and analytical bounds are available for calculating 3

performance measures of assembly systems. Such bounds and approximations can be found in almost all articles of the above mentioned literature. When more realistic settings are studied, analysis is limited to simulation studies (Baker et al.,1990; Baker et al.,1993; Baker and Powell, 1995). At the same time, since the lead times in an assembly system are the result of system dynamics, static models are not useful. Rules of thumb and spreadsheets lack realism and are therefore not sophisticated enough to tackle the ‘right information’ goal. Also discrete-event simulation and physical modelling are not appropriate (Suri et al., 1995) as those tools are too complex, time–consuming and detail-minded to efficiently proceed with the assembly analysis. In this way they do not provide the decision maker with the information in a timely manner. Therefore, we like to rely on the development of approximate models. In Section 3, we will focus on the synchronization time of components for kits and the interarrival time of kits at the assembly station. That is because those two performance measures are the only unknown parameters in an assembly system (see Subsection 2.1) and need to be quantified before the performance of the entire assembly system can be obtained. We will rely on the field of probability theory to find approximations for the expected value and the variance of those performance measures. We focus on the average value and variance of the performance measures because these are sufficient to characterize a flow process (Whitt, 1982) of components and kits in an open queueing system. Once those measures are determined, the performance of the entire system can be obtained (De Boeck, 2003) and can be incorporated in the open queueing network model of Vandaele and De Boeck (2002). 2.3. Notation We introduce the following notation as represented in Table 1. Table 1 Notation used in Sections 2 and 3 Symbol W|i Wi Ti MAXAT|i E(X) V(X)

µX σX σ²X

SCVX fX(x) FX(x) ATic ITc Lb Logn(µ,SCV) AR SR

Meaning the synchronization time of the components for kit i the synchronization time of the components for the first i kits the inter-arrival time for the first i kits the maximum of the component arrival times for the components of kit i the expected value of the variable X the variance of the variable X the average value of the variable X the standard deviation of the variable X the variance of the variable X the squared coefficient of variation of the variable X (=σ²X/µ²X) the density function of the variable X the distribution function of the variable X the arrival time of component c for kit i the inter-arrival time of component c Lowerbound the lognormal distribution with average value equal to µ and squared coefficient of variation equal to SCV the approximate results the simulation results

The symbol ‘|i’ stands for ‘for a specific kit i’ whereas the subscript ‘i’ refers to ‘for the first i kits (i.e. irrespective of a kit)’. In Section 3 it will become clear that through the way the results for the synchronization time of the components are obtained, the results only account for a specific kit i (W|i) whereas for the inter-arrival time of kits at the assembly station the results are only valid for the first i kits (Ti). However, in order to find the performance of the 4

complete assembly system we need the expected value and variance of the performance measures irrespective of a kit. This means that we need formulas to translate the results for W|i into results for Wi. This will be explained in Section 3. Note also that the reason why we express the formulas of the performance measures (the first four symbols in Table 1) as a function of the index of the kit (i) is because we will work with a finite number of kits (see supra). As such, depending on the number of kits that are produced, the performance of the system will change. 3.

Analytical analysis, results and validation

3.1. Analytical analysis and results As already mentioned in the previous subsection, to obtain the performance of a generic assembly system, we should first obtain the synchronization time of components and the inter-arrival time of kits. Therefore, we will concentrate in this section on developing good approximations for those two performance measures by relying on probability theory. Therefore we will proceed in three steps as represented in Fig. 2.

Component 1

at11

at21

at31

at41

Step 1

Component 2

at12

at22

at32

at42

Step 2 Kit

w|1

w|2

w|3

maxat|2

maxat|3

w|4

t4 t3 t2 Step 3

Kit

t1

maxat|1

maxat|4

Fig. 2. Representation of the three steps in finding approximations for the synchronization time of components and the inter-arrival time of kits

Note from Fig. 2 that the first step focuses on finding the arrival times (atic) of both components for kits. Once these are known, we can proceed with steps 2 and 3. In step 2 we calculate the synchronization time (w|i) of the components for each kit i as the absolute value of the difference of the arrival time of both components in forming kit i. In a third step we develop an approximation for the inter-arrival time (ti) of the first i kits as the maximum of both component arrival times (maxat|i) divided by i.

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3.1.1. Step 1: Arrival time of the components We first look for information on the arrival time of all components c for each kit i. This information is exactly determined by the component arrival time density function of all components c for kit i. In general, we have that the arrival time of component c for kit i (ATic) equals the sum of i inter-arrival times (ITc): ATic = iITc = AT( i −1 )c + ITc .

(1)

As AT(i-1)c and ITc are independent, the density function of ATic can be written as a convolution of the density functions of AT(i-1)c and ITc: i=1 f ITc ( it c ) + ∞ f ATic ( at ic ) = ∫ f ( at ic − at( i −1 )c ) f ( at ( i −1 )c )d ( at ( i −1 )c ) i > 1, Lb

(2)

where Lb should be calculated. However, the calculations will become more complex as i grows which makes the exact analysis intractable (De Boeck, 2003). That is why we will rely on an approximation for the arrival time density function of the components for kit i. This approximation can be found by relying on the central limit theorem. ATic is the sum of i IID random variables (see first equation in formula (1)) and so approaches a normal distribution:

f ATic ( atic ) ≅

−( at ic − µ ATic ) 2

1

σ AT

ic

2π

e

2 2σ AT ic

- ∞ ≤ ATic ≤ +∞

(3)

2 with µ ATic = iµITc and σ AT = iσ IT2 c . ic

As the exact density function is only defined between [0,+∞] in case of generally distributed component inter-arrival times, the above approximation can be improved by truncating. We will not proceed on these lines for reasons that will be explained in Subsection 3.2. Once we know when both components for kit i arrive, we can obtain formulas for the synchronization time of the components for kit i as well as for the inter-arrival time of the first i kits at the assembly station.

3.1.2. Step 2: Synchronization time of the components for kits Synchronization time appears when one component arrives later than the other component in forming kit i. Remark from Fig. 2 that the synchronization time equals the absolute value of the difference of both component arrival times. Therefore, the synchronization time of the components for kit i equals the absolute value of the difference of both component arrival times for kit i. The expected value and variance of W|i follows by the definition of the expected value and variance of the continuous random variable W|i: +∞

E( W i ) = ∫ ( w i ) fW i ( w i )d ( w i ),

(4)

0

6

+∞

V ( W i ) = ∫ ( w i )2 f W i ( w i )d ( w i ) − [E( W i )] . 2

(5)

0

Here, fW|i(w|i) is the density function of the absolute value of the difference of the arrival time density functions of component 1 and component 2 for kit i. For the synchronization time we should take the absolute value of the difference of both component arrival times:

W i = ATi1 − ATi 2 .

(6)

Since we assume the component arrival time density functions to be normally distributed (see Subsection 3.1.1), the difference is again a normal distribution (Blumenfeld , 2001). Hence, we get the following expression for the density function of the synchronization time of the components for kit i: − ( w i − µ w i )2

fW i ( W i ) ≅

2

σ w i 2π

e

2 σ w2

i

0 ≤ w i ≤ +∞

(7)

with µ w i = 0 and σ w2 i = iσ IT2 1 + iσ IT2 2 . The reason why we do not start from the truncated normal distribution to approximate the synchronization time is because the calculations are more complicated but do not generate better results. For more evidence on this statement, we refer to De Boeck (2003). By substituting expression (7) in (4) and (5), we can simplify E(W|i) and V(W|i) to (De Boeck, 2003):

E( W i ) =

V (W i ) =

2i( σ IT2 1 + σ IT2 2 )

i( σ IT2 1

,

π + σ IT2 )( π − 2 ) . π 2

(8) (9)

In Subsection 2.3, we noted that to obtain the performance of the entire assembly system, we need the performance measures irrespective of a specific kit, i.e. E(Wi) and V(Wi). Since the expected value is insensitive to dependency (of the random variables W|i), we have the following exact relation: E( Wi ) =

1 i ∑ E( W k ). i k =1

(10)

For the variance, the covariance coefficients should be taken into account. However, we believe that by the way we calculated V(W|i), a large portion of the covariance is already included which makes the following expression a good approximation: V ( Wi ) ≅

1 i ∑V ( W k ). i k =1

(11)

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3.1.3. Step 3: Inter-arrival time of the kits at the assembly station From Fig. 2, it should be clear that the arrival time of kit i at the assembly station is determined by the component that last arrives in forming kit i. Therefore, the arrival time of kit i at the assembly station equals the maximum of both component arrival times for kit i. As the arrival time of both components for kit i is the sum of i inter-arrival times for those components, it is obvious that to obtain the inter-arrival time of the first i kits, the arrival time of kit i should be scaled with a factor i. The expected value and variance of Ti can be found by applying the definition of the expected value and variance of the continuous random variable Ti: +∞

E( Ti ) = ∫ ( t i ) f Ti ( t i )d ( t i ),

(12)

−∞ +∞

V ( Ti ) = ∫ ( t i ) 2 f Ti ( t i )d ( t i ) − [E( Ti )] . 2

(13)

−∞

Here, f Ti ( t i ) is the density function of the maximum of the arrival time density functions of component 1 and component 2, scaled by a factor i. The arrival time of kit i at the assembly station equals the maximum of both component arrival times for kit i, scaled with a factor i: Ti =

max( ATi 1 , ATi 2 ) . i

(14)

As both component arrival times are IID, we can easily calculate the density function of the maximum of the component arrival time density functions (fMAXAT|i(maxat|i)) (Papoulis, 1991): f MAXAT i ( max at i ) = f ATi 1 (max at i )FATi 2 ( max at i ) + f ATi 2 ( max at i )FATi 1 (maxta i ).

(15)

Scaling this density function with a factor i boils down to multiplying the density function with i and writing it in terms of ‘(i)ti’ instead of ‘maxat|i’:

fTi ( ti ) = if MAXAT i (( i )ti ).

(16)

From expression (3), we have that:

f ATic (max at i ) ≅

− ( max at i − iµ ITc ) 2

1

σ IT

c

2πi

e

2 2 iσ IT c

1 max at i − iµ ITc FATic ( max at i ) ≅ erf 2 2iσ IT2 c

,

(17)

+ 1 .

(18)

Therefore,

8

fTi ( ti ) ≅

− (( i )ti −iµ IT1 )2

i

σ IT 2πi

2 2 iσ IT 1

e

×

1

−(( i )ti −iµ IT2 ) 2 i 1 ( i )ti − iµ IT1 2 iσ IT + 1 + 2 e erf σ 2 2iσ IT2 IT2 2πi 1

1 ( i )ti − iµ IT2 erf 2 2iσ IT2 2

2

(19)

+ 1.

By substituting expression (19) in (12), we can simplify E(Ti) to (De Boeck, 2003): E( Ti ) = µ ITc +

σ IT2 + σ IT2 1

2

2πi

(20)

.

We do not have a formula for the variance. This is not a problem because the way in which we calculated this inter-arrival time of kits, we do not obtain the right formula for V(Ti). Note that for increasing values of i (see infra), V(Ti) converges to 0. However, since in the end (i → ∞) one component will always be waiting for the other one, the arrival stream of kits will be completely determined by the input stream of the other component which is not deterministic. Since V(T|i) converges fast to the variance of the component input streams in case both component input streams have the same variance (De Boeck, 2003), we propose the following approximation for V(Ti): V ( Ti ) ≅

1 2 2 ∑ σ IT , 2 c =1 c

(21)

where ‘2’ in ‘1/2’ and the summation sign is the number of input streams. As example, we now represent E(Wi,), V(Wi), E(Ti,) and V(Ti), for the first 20 kits for Logn(8.5,0.25) (a lognormal distribution with an average value of 8.5 and a squared coefficient of variation of 0.25) distributed inter-arrival times for both components in Fig. 3 and 4. The reason why we illustrate those performance measures for lognormally distributed component inter-arrival times is explained in subsection 3.2.

14 12 E(Wi)

10

E(Ti) 8 6 4 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 Kit

Fig. 3. E(Wi) and E(Ti) for the first 20 kits for Logn(8.5,0.25) distributed inter-arrival times for both components

9

140 120 100 80

V(Wi)

60

V(Ti)

40 20 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 Kit

Fig. 4. V(Wi) and V(Ti) for the first 20 kits for Logn(8.5,0.25) distributed inter-arrival times for both components

The figures clearly reveal that E(Wi) and V(Wi) increase for increasing values of i. Also formulas (10) and (11) show that both performance measures evolve to ∞ for i → ∞. In other words, the synchronization time of components for kits for a generic assembly system does not converge. That is why we will assume a finite number of kits. E(Ti) and V(Ti) however, decrease for increasing values of i. Formula (20) shows that E(Ti) converges to µ ITc for i → ∞. In other words, the inter-arrival time of kits at the assembly station does converge. 3.2. Validation To validate the approximations obtained in subsection 3.1, we rely on simulation. Since we assume that components are processed in a stochastic production environment preceding the assembly system, we assume both component inter-arrival times to be lognormally distributed. Since this distribution is a good approximation for machine processing times (Law and Kelton, 2000) and we assume machines to have a relatively high utilization (around 90%), the distribution of the inter-departure times out of the machines (or the distribution of the interarrival times at the machines that follow) approximates the processing time distribution. To investigate the influence of equal and different variances and small, moderate and highly variable inter-arrival times for both components, we combine the following distributions for both component inter-arrival times resulting in 6 scenarios represented in Table 2. Remark that small (SCVX < 0.5625), moderate (0.5625 ≤ SCVX < 1.7689) and high variability (SCVX ≥ 1.7689) of a variable X is determined by the squared coefficient of variation (SCVX) (see Hopp and Spearman, 2000). Table 2 The different scenarios for validating the approximations for E(Wi), V(Wi), E(Ti) and V(Ti) Logn(8.5,0.25) Logn(8.5,1.25) Logn(8.5,2) Logn(8.5,0.25) Scenario 1 Scenario 2 Scenario 3 Logn(8.5,1.25) Scenario 4 Scenario 5 Logn(8.5,2) Scenario 6

We assume a finite number of kits in our simulation model equal to 1000 and perform 100 replications for each scenario. The approximate results (indicated by the symbol AR) for E(Wi) (see formula (10)), V(Wi) (see formula (11)), E(Ti) (see formula (20)) and V(Ti) (see formula (21)) and the simulation results (indicated by the symbol SR) are represented in Table 3.

10

Logn(8.5,0.25)

Logn(8.5,1.25)

Logn(8.5,2)

Table 3 The approximate and simulation results for E(Wi), V(Wi), E(Ti) and V(Ti) Logn(8.5,0.25) Logn(8.5,1.25) Logn(8.5,2) AR SR AR SR AR E(Wi) 101.18 97.82 175.24 182.06 214.63 V(Wi) 6570.10 4484.53 19710.35 13886.53 29565.55 E(Ti) 8.57583 8.58 8.63133 8.61 8.66085 V(Ti) 18.06 18.40 54.19 59.56 81.28 E(Wi) 226.235 246.82 257.95 V(Wi) 32850.60 23928.35 42705.75 E(Ti) 8.66955 8.66 8.69332 V(Ti) 90.31 96.67 117.41 286.17 E(Wi) V(Wi) 52561.00 E(Ti) 8.71447 V(Ti) 144.5

SR 215.53 21143.67 8.64 67.91 266.59 32071.54 8.73 127.26 277.45 35835.60 8.70 169.94

We observe from Table 3 that the approximations for E(Wi) perform very well. These results together with additional simulation results for squared coefficients of variation equal to 0.5, 0.75, 1, 1.5 and 1.75 (we did not include those results because it would make Table 3 unreadable) reveal an average relative deviation of the simulation results in terms of the approximate results of 0.02% and a relative deviation within [-7.88%,9.10%]. The results for V(Wi) however do not perform well but are robust in the sense that the approximations always overestimate the simulation results. Here we have an average relative deviation of the simulation results in terms of the approximate results of -31.39% and a relative deviation within [-46.06%,-16.26%]. The results for E(Ti) and V(Ti) perform well and give average relative deviations of the simulation results in terms of the approximate results of -0.06% and 5.28% respectively and relative deviations within [-0.65%,0.45%] and [-5.99%,17,60%] respectively. In general we can state that all our approximations perform well except for V(Wi). These last results are however robust in the sense that they always overestimate the simulation results by around 30%. Finding a better approximation is an issue for further research.

4. Conclusions In this paper we assumed a generic assembly system consisting of two independent component input streams having generally distributed inter-arrival times. We assumed the same average arrival rate for both input streams and each input process to be a renewal process. A kit consisted of one unit of each component. For such a system, we obtained the following unknown parameters: the synchronization time of the components for kits and the inter-arrival time of the kits at the assembly station. These performance measures are not encountered in a traditional queuing system. The results revealed increasing values for the synchronization time of the components for kits which is confirmed by the existing literature (Harrison, 1973). That is why we introduce a limitation on the number of kits in our system. We also observed converging values for the inter-arrival time of the kits at the assembly station. These performance measures are a required step in obtaining the performance measures of the entire assembly system and the performance of an assembly system embedded in an open queuing network model. From a managerial point of view, we can derive some take-aways for the generic assembly system. These take-aways result from the formulas of the synchronization time (see formulas (10) and (11)) of components for kits and the inter-arrival time (see formula (20)) of kits. • More reliable supply (with the same frequency) leads to less synchronization stock and improved productivity. Reliable supply indicates less variable component inter-arrival 11

• •

times. Less synchronization stock translates in lower synchronization times of components for kits and improved productivity translates in smaller inter-arrival times of kits at the assembly station. More frequent supply (with the same reliability) does not lead to less synchronization stock. More frequent supply is obtained by increased average component inter-arrival times. Supply has to be synchronised and supply needs a cap in order to shutdown the input streams, at least temporarely. Note that this translates in a limitation on the number of kits in our assembly system.

References Baker, K.R., Powell, S.G., Pyke, D.F., 1990. Buffered and unbuffered assembly systems with variable processing times. Journal of Manufacturing and Operations Management, 3 (3), 200223. Baker, K.R., Powell, S.G., Pyke, D.F., 1993. Optimal allocation of work in assembly systems. Management Science, 39 (1), 101-106. Baker, K.R., Powell, S.G., 1995. A predictive model for the throughput of simple assembly systems. European Journal of Operational Research, 81 (2), 336-345. Bhat, U.N., 1986. Finite capacity assembly-like queues. Queueing Systems, 1 (1), 85-101. Blumenfeld, D., 2001. Operations research calculations handbook. CRC Press. Bonomi, F., 1987. An approximate analysis for a class of assembly-like queues. Queueing Systems, 1 (3), 289-309. Crane, M.A., 1974. Multi-server assembly queues. Journal of Applied Probability, 11 (3), 629-632. De Boeck, L., 2003. Performance analysis of assemblies in open queueing systems. Dissertation, University of Antwerp, Belgium. Duenyas, I., Hopp, W.J., 1992. Conwip assembly with deterministic processing and random outages. IIE Transactions, 24 (4), 97-109. Duenyas, I., Hopp, W.J., 1993. Estimating the throughput of an exponential CONWIP assembly system. Queueing Systems, 14 (2), 135-157. Duenyas, I., Keblis, M.F., 1995. Release policies for assembly systems. IIE Transactions, 27 (4), 507-518. Harrison, M.J., 1973. Assembly-like queues. Journal of Applied Probability, 10, 354-367. Hopp, W.J., Simon, J.T., 1989. Bounds and heuristics for assembly-like queues. Queueing Systems, 4 (2), 137-156. Hopp, W.J., Simon, J.T., 1993. Estimating throughput in an unbalanced assembly-like flow system. International Journal of Production Research, 31 (4), 851-868. Hopp, W.J., Spearman, M.L., 2000. Factory physics. McGraw-Hill. Latouche, G., 1981. Queues with paired customers. Journal of Applied Probability, 18 (3), 684-696. Law, A.M., Kelton, W.D., 2000, Simulation modeling and analysis. McGraw-Hill. Lipper, E.H., Sengupta, B., 1986. Assembly-like queues with finite capacity: bounds, asymptotics and approximations. Queueing Systems, 1 (1), 67-83. Papoulis, A., 1991. Probability, random variables and stochastic processes. McGraw-Hill. Powell, S.G., Pyke, D.F., 1998. Buffering unbalanced assembly systems. IIE Transactions, 30 (1), 55-65. Ramachandran S., Delen, D., 2005. Performance analysis of a kitting process in stochastic assembly systems. Computers and Operations Research, 32 (3), 449-463. Sabuncuoglu, I., Erek, E., Kok, A.G., 2002. Analysis of assembly systems for interdeparture time variability and throughput. IIE Transactions, 34 (1), 23-40.

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Simon, J.T., Hopp, W.J., 1991. Availability and average inventory of balanced assembly-like flow systems. IIE Transactions, 23 (2), 161-168. Simon, J.T., Hopp, W.J., 1995. Throughput and average inventory in discrete balanced assembly systems. IIE Transactions, 27 (3), 368-373. Som, M.P., Wilhelm, W.E., Disney, R.L., 1994. Kitting process in a stochastic assembly system. Queueing Systems, 17 (3), 471-490. Suri, R., Sanders, J.L., Kamath, M., 1993. Performance evaluation of production networks. In: Graves, S.C., Rinnooy Kan, A.H.G., Zipkin, P.H., (Eds.). Logistics of production and inventory. North-Holland, 199-286. Suri, R., Diehl, G.W.W., de Treville, S., Tomsicek, M.J., 1995. From CAN-Q to MPX: evolution of queueing software for manufacturing. Interfaces, 25 (5), 128-150. Takahashi, M., Osawa, H., Fujisawa, T., 2000. On a synchronization queue with two finite buffers. Queueing Systems: Theory and Applications, 36 (1-3), 107-123. Vandaele, N., 1996. The Impact of lot sizing on queueing delays: multi product, multi machine models. Dissertation, Katholieke Universiteit Leuven, Belgium. Vandaele, N., De Boeck, L., 2003. Advanced resource planning. Robotics and Computer Integrated Manufacturing, 19 (1), 210-218. Vandaele, N., De Boeck, L., Callewier, D., 2002, An open queueing network for lead time analysis. IIE Transactions, 34 (1), 1-9. Vandaele, N., Lambrecht, M., 2003. Reflections on the use of stochastic manufacturing models for planning decisions. In: Shanthikumar, J.G., Yao, D.D., Zijm, W.H.M., (Eds.). Stochastic modeling and optimisation of manufacturing systems and supply chains. Kluwer Academic Publishers, 53-85. Van Nieuwenhuyse, I., 2004. Lot splitting in single-product flow shops: issues of delivery reliability, production disruptions and flow times. Dissertation, University of Antwerp, Belgium. Van Nieuwenhuyse, I., De Boeck, L., Lambrecht, M., Vandaele, N., 2007. From ERP to Advanced Resource Planning: improving operational performance by getting the inputs right. In: Sprague, R.H, (Ed.). Proceedings of the 40th Annual Hawaii International Conference on System Sciences. On CD-rom. Whitt, W., 1982. Approximating a point process by a renewal process, I: two basic methods. Operations Research, 30 (1), 125-147. Wilhelm, W.E., Som, P., 1999. Analysis of stochastic assembly with GI-distributed assembly time. INFORMS Journal on Computing, 11 (1), 104-116.

13

Lihat lebih banyak...
Liesje De Boeck and Nico Vandaele

HUB RESEARCH PAPER 2008/44. SEPTEMBER 2008

Analytical analysis of a generic assembly system Liesje De Boeck ab, Nico Vandaele cd a

Centre for Modelling and Simulation, University College Brussels (HUBrussel), Stormstraat 2, 1000 Brussels, Belgium b Affiliated with the Faculty of Business and Economics, K.U.Leuven, Naamsestraat 69, 3000 Leuven, Belgium [email protected] c Faculty of Business and Economics, K.U.Leuven-Campus Kortrijk, Etienne Sabbelaan 53-bus 00000, 8500 Kortrijk, Belgium d Faculty of Business and Economics, K.U.Leuven, 3000 Leuven, Belgium [email protected] Abstract The assembly of components is a key element in manufacturing facilities. Nevertheless, literature analysing them remains scarce. The paper narrows this gap by studying a generic first-come first-serve assembly system, consisting of two generally-distributed component input streams. The field of probability theory serves as a means to derive good approximations for the waiting time of the components for a kit and the inter-arrival time of the kits at the assembly facility. Apart from providing insight in generally accepted managerial principles, this analysis also constitutes a required step in obtaining the performance of an open queueing network including assemblies. Keywords: Assembly; Performance analysis; Synchronization time; Probability theory

1. Introduction Since the eighties, manufacturing played an increasing role in the responsiveness and the competitiveness of a firm. This created a need for analysing and improving manufacturing systems (Suri et al., 1993). At the same time, the focus changed from cost reduction to lead time reduction, an evolution which was strengthened by the introduction of time-based competition in the nineties. As a result, the supply of goods and services faster than the competitors became a central theme, inducing the need for quantifying lead times in a fast way. Nowadays, this concept of time based competition appears beyond the borders of the production system. Taking care of this evolution during the last decades, the supply chain necessitates the right information on the right time. The right information stands for taking into account the dynamic and complex character, the stochastic nature, the variability and the capacity constraints present in the supply chain environment (Vandaele and De Boeck, 2002; Vandaele and De Boeck, 2003; Vandaele and Lambrecht, 2003; Van Nieuwenhuyse et al., 2007) whereas the right time deals with the responsive ability of the tools for quick decisionmaking. Literature reveals that many models have already been developed in order to achieve this goal, for instance open queueing models (Vandaele and De Boeck, 2002) including optimal lot sizes (Vandaele, 1996), incorporating sources of variability (Hopp and Spearman, 2000), accounting for transfer batches (Van Nieuwenhuyse, 2004), etc. This paper offers a next step in refining the existing open queueing network models of Vandaele and De Boeck (2002) by incorporating assembly systems. In this way it contributes to the development of a tool providing the right information in a quick way. On top of the lack of open queueing network models which incorporate assembly systems, there are two other reasons for modelling assembly systems. The first reason covers the practical relevance of assembly systems. Indeed, assembly systems are widespread in many supply chain systems. A second reason accounts for the scarce literature on the modelling of assembly systems. Compared to other queueing topics which have been thoroughly studied in the literature, assembly systems have received much less attention.

1

The paper is organised as follows. We will start in Section 2 with definitions and assumptions, followed by a review of the literature, a discussion of the methodology and the notation. The assembly analysis itself as well as the results will be developed for a generic assembly system in Section 3. This section ends with simulation results to validate the performance of our approximations. Section 4 will conclude with the main results of this paper. 2.

Definitions, assumptions, literature, methodology and notation

2.1. Definitions and assumptions We define assembly systems as systems where more than one component (class) is brought together with the intention to jointly undergo an (assembly) operation. The set which contains all components in the right quantities needed for assembly is defined as a kit. Note that such a system induces two additional performance measures compared to a traditional queueing system: • the synchronization time which is the additional waiting time of the components for one another in order to form a kit, • an arrival process of kits at the assembly station, which is clearly distinguished from the arrivals of components at the assembly system. In view of our final goal of embedding the assembly model in the open queueing network model of Vandaele and De Boeck (2002), we must impose the same assumptions on our assembly system as those that are imposed on the open queueing network model of Vandaele and De Boeck (2002): • the system only deals with discrete products, • the system is an open system with infinite capacity buffers where the FCFS-discipline applies, • the individual component arrivals are assumed to be independent with generally distributed IID inter-arrival times. In what follows, we assume a generic assembly system. Taking into account the above assumptions, we can define it as an open discrete production system with infinite buffers and two independent input streams having generally distributed IID component inter-arrival times. A kit consists of exactly one component of each input stream. Both components and kits obey the FCFS-discipline. In what follows, we will always assume the same average inter-arrival times for both components as we want to avoid non-convergence of the synchronization time due to unequal average component inter-arrival times. However, the variance and in fact the entire distribution for both component inter-arrival times may differ. A representation of the generic assembly system is given in Fig. 1.

Input 1

Input 2

Fig. 1. Representation of the generic assembly system with a synchronization buffer and an assembly buffer

2

The two component input processes are represented by the first two squares. The first buffer, the synchronization buffer, contains the individual components which wait to form a complete kit. The second buffer, the assembly buffer, holds the ‘collected’ kits which wait for their operation on the assembly station. The two buffers have an infinite capacity. The assembly station is represented by the last square. This buffer split can also be found in Lipper (1986). The reason for focusing on a generic assembly system is for clarity reasons. The analysis can however be easily extended to general assembly systems (i.e. assembly systems with more than two input streams or assembly systems with kits existing of more than one unit of each component). For this analysis, we refer to De Boeck (2003). 2.2. Literature and methodology Relevant literature on assembly systems as described in Subsection 2.1 includes: Harrison (1973), Crane (1974), Latouche (1981), Bhat (1986), Lipper and Sengupta (1986), Bonomi (1987), Hopp and Simon (1989, 1993), Baker et al. (1990, 1993), Simon and Hopp (1991, 1995), Duenyas and Hopp (1992, 1993), Som et al. (1994), Duenyas and Keblis (1995), Baker and Powell (1995), Powell and Pyke (1998), Wilhelm and Som (1999), Takahashi et al. (2000), Sabuncuoglu et al. (2002) and Ramachandran and Delen (2005). The above literature reveals that modelling an assembly system is very hard. The main difficulties trace back to the system state space, the mathematical complexity and the instability of the assembly systems when buffers have an infinite capacity and input processes are independent renewal processes. Indeed, Harrison (1973) proved that the synchronization time does not converge (when the input processes are independent renewal processes) unless buffers are finite. Therefore, there is always some kind of input control to preserve system stability in the existing assembly literature. This input control can take on many different forms. • When the maximum buffer capacity is reached, components are removed from the input processes (e.g. Lipper and Sengupta, 1996) or the input process is shut down (e.g. Simon and Hopp, 1991; Hopp and Simon, 1993; Som et al., 1994; Simon and Hopp, 1995). • The component arrivals depend on the excess of the number of one component in the system over the others (e.g. Latouche, 1981). • The number of each component allowed in the system is limited (e.g. Bhat, 1986). • The component arrival process is stopped when the number of units of that component exceeds the number of units of all the other components with a certain value (e.g. Bonomi, 1987). • Control mechanisms as bins (e.g. Hopp and Simon, 1989), conwip (e.g. Duenyas and Hopp, 1993) or kanban (e.g. Duenyas and Keblis, 1995) are used. • … Hence, only if the number of components in the system cannot exceed a predefined limit, infinite buffers are allowed. That is why we will assume a finite number of kits after which the system is ‘emptied’. In other words, we control the system by imposing a limit on the total throughput. We do believe that the independency assumption of the component input streams is justified when components are processed in a stochastic production environment before assembly takes place. As an example, we can think of the component input processes in Fig. 2 being variable flow lines. Although one can synchronize both components in the beginning of those flow lines, variability will cause both departure streams out of these flow lines (i.e. the arrival streams at the assembly system) to be almost independent. If we examine exact relations versus approximations in the assembly literature, we clearly observe the absence of exact relations. Hopp and Simon (1989) state that in the absence of exact relations, only approximations and analytical bounds are available for calculating 3

performance measures of assembly systems. Such bounds and approximations can be found in almost all articles of the above mentioned literature. When more realistic settings are studied, analysis is limited to simulation studies (Baker et al.,1990; Baker et al.,1993; Baker and Powell, 1995). At the same time, since the lead times in an assembly system are the result of system dynamics, static models are not useful. Rules of thumb and spreadsheets lack realism and are therefore not sophisticated enough to tackle the ‘right information’ goal. Also discrete-event simulation and physical modelling are not appropriate (Suri et al., 1995) as those tools are too complex, time–consuming and detail-minded to efficiently proceed with the assembly analysis. In this way they do not provide the decision maker with the information in a timely manner. Therefore, we like to rely on the development of approximate models. In Section 3, we will focus on the synchronization time of components for kits and the interarrival time of kits at the assembly station. That is because those two performance measures are the only unknown parameters in an assembly system (see Subsection 2.1) and need to be quantified before the performance of the entire assembly system can be obtained. We will rely on the field of probability theory to find approximations for the expected value and the variance of those performance measures. We focus on the average value and variance of the performance measures because these are sufficient to characterize a flow process (Whitt, 1982) of components and kits in an open queueing system. Once those measures are determined, the performance of the entire system can be obtained (De Boeck, 2003) and can be incorporated in the open queueing network model of Vandaele and De Boeck (2002). 2.3. Notation We introduce the following notation as represented in Table 1. Table 1 Notation used in Sections 2 and 3 Symbol W|i Wi Ti MAXAT|i E(X) V(X)

µX σX σ²X

SCVX fX(x) FX(x) ATic ITc Lb Logn(µ,SCV) AR SR

Meaning the synchronization time of the components for kit i the synchronization time of the components for the first i kits the inter-arrival time for the first i kits the maximum of the component arrival times for the components of kit i the expected value of the variable X the variance of the variable X the average value of the variable X the standard deviation of the variable X the variance of the variable X the squared coefficient of variation of the variable X (=σ²X/µ²X) the density function of the variable X the distribution function of the variable X the arrival time of component c for kit i the inter-arrival time of component c Lowerbound the lognormal distribution with average value equal to µ and squared coefficient of variation equal to SCV the approximate results the simulation results

The symbol ‘|i’ stands for ‘for a specific kit i’ whereas the subscript ‘i’ refers to ‘for the first i kits (i.e. irrespective of a kit)’. In Section 3 it will become clear that through the way the results for the synchronization time of the components are obtained, the results only account for a specific kit i (W|i) whereas for the inter-arrival time of kits at the assembly station the results are only valid for the first i kits (Ti). However, in order to find the performance of the 4

complete assembly system we need the expected value and variance of the performance measures irrespective of a kit. This means that we need formulas to translate the results for W|i into results for Wi. This will be explained in Section 3. Note also that the reason why we express the formulas of the performance measures (the first four symbols in Table 1) as a function of the index of the kit (i) is because we will work with a finite number of kits (see supra). As such, depending on the number of kits that are produced, the performance of the system will change. 3.

Analytical analysis, results and validation

3.1. Analytical analysis and results As already mentioned in the previous subsection, to obtain the performance of a generic assembly system, we should first obtain the synchronization time of components and the inter-arrival time of kits. Therefore, we will concentrate in this section on developing good approximations for those two performance measures by relying on probability theory. Therefore we will proceed in three steps as represented in Fig. 2.

Component 1

at11

at21

at31

at41

Step 1

Component 2

at12

at22

at32

at42

Step 2 Kit

w|1

w|2

w|3

maxat|2

maxat|3

w|4

t4 t3 t2 Step 3

Kit

t1

maxat|1

maxat|4

Fig. 2. Representation of the three steps in finding approximations for the synchronization time of components and the inter-arrival time of kits

Note from Fig. 2 that the first step focuses on finding the arrival times (atic) of both components for kits. Once these are known, we can proceed with steps 2 and 3. In step 2 we calculate the synchronization time (w|i) of the components for each kit i as the absolute value of the difference of the arrival time of both components in forming kit i. In a third step we develop an approximation for the inter-arrival time (ti) of the first i kits as the maximum of both component arrival times (maxat|i) divided by i.

5

3.1.1. Step 1: Arrival time of the components We first look for information on the arrival time of all components c for each kit i. This information is exactly determined by the component arrival time density function of all components c for kit i. In general, we have that the arrival time of component c for kit i (ATic) equals the sum of i inter-arrival times (ITc): ATic = iITc = AT( i −1 )c + ITc .

(1)

As AT(i-1)c and ITc are independent, the density function of ATic can be written as a convolution of the density functions of AT(i-1)c and ITc: i=1 f ITc ( it c ) + ∞ f ATic ( at ic ) = ∫ f ( at ic − at( i −1 )c ) f ( at ( i −1 )c )d ( at ( i −1 )c ) i > 1, Lb

(2)

where Lb should be calculated. However, the calculations will become more complex as i grows which makes the exact analysis intractable (De Boeck, 2003). That is why we will rely on an approximation for the arrival time density function of the components for kit i. This approximation can be found by relying on the central limit theorem. ATic is the sum of i IID random variables (see first equation in formula (1)) and so approaches a normal distribution:

f ATic ( atic ) ≅

−( at ic − µ ATic ) 2

1

σ AT

ic

2π

e

2 2σ AT ic

- ∞ ≤ ATic ≤ +∞

(3)

2 with µ ATic = iµITc and σ AT = iσ IT2 c . ic

As the exact density function is only defined between [0,+∞] in case of generally distributed component inter-arrival times, the above approximation can be improved by truncating. We will not proceed on these lines for reasons that will be explained in Subsection 3.2. Once we know when both components for kit i arrive, we can obtain formulas for the synchronization time of the components for kit i as well as for the inter-arrival time of the first i kits at the assembly station.

3.1.2. Step 2: Synchronization time of the components for kits Synchronization time appears when one component arrives later than the other component in forming kit i. Remark from Fig. 2 that the synchronization time equals the absolute value of the difference of both component arrival times. Therefore, the synchronization time of the components for kit i equals the absolute value of the difference of both component arrival times for kit i. The expected value and variance of W|i follows by the definition of the expected value and variance of the continuous random variable W|i: +∞

E( W i ) = ∫ ( w i ) fW i ( w i )d ( w i ),

(4)

0

6

+∞

V ( W i ) = ∫ ( w i )2 f W i ( w i )d ( w i ) − [E( W i )] . 2

(5)

0

Here, fW|i(w|i) is the density function of the absolute value of the difference of the arrival time density functions of component 1 and component 2 for kit i. For the synchronization time we should take the absolute value of the difference of both component arrival times:

W i = ATi1 − ATi 2 .

(6)

Since we assume the component arrival time density functions to be normally distributed (see Subsection 3.1.1), the difference is again a normal distribution (Blumenfeld , 2001). Hence, we get the following expression for the density function of the synchronization time of the components for kit i: − ( w i − µ w i )2

fW i ( W i ) ≅

2

σ w i 2π

e

2 σ w2

i

0 ≤ w i ≤ +∞

(7)

with µ w i = 0 and σ w2 i = iσ IT2 1 + iσ IT2 2 . The reason why we do not start from the truncated normal distribution to approximate the synchronization time is because the calculations are more complicated but do not generate better results. For more evidence on this statement, we refer to De Boeck (2003). By substituting expression (7) in (4) and (5), we can simplify E(W|i) and V(W|i) to (De Boeck, 2003):

E( W i ) =

V (W i ) =

2i( σ IT2 1 + σ IT2 2 )

i( σ IT2 1

,

π + σ IT2 )( π − 2 ) . π 2

(8) (9)

In Subsection 2.3, we noted that to obtain the performance of the entire assembly system, we need the performance measures irrespective of a specific kit, i.e. E(Wi) and V(Wi). Since the expected value is insensitive to dependency (of the random variables W|i), we have the following exact relation: E( Wi ) =

1 i ∑ E( W k ). i k =1

(10)

For the variance, the covariance coefficients should be taken into account. However, we believe that by the way we calculated V(W|i), a large portion of the covariance is already included which makes the following expression a good approximation: V ( Wi ) ≅

1 i ∑V ( W k ). i k =1

(11)

7

3.1.3. Step 3: Inter-arrival time of the kits at the assembly station From Fig. 2, it should be clear that the arrival time of kit i at the assembly station is determined by the component that last arrives in forming kit i. Therefore, the arrival time of kit i at the assembly station equals the maximum of both component arrival times for kit i. As the arrival time of both components for kit i is the sum of i inter-arrival times for those components, it is obvious that to obtain the inter-arrival time of the first i kits, the arrival time of kit i should be scaled with a factor i. The expected value and variance of Ti can be found by applying the definition of the expected value and variance of the continuous random variable Ti: +∞

E( Ti ) = ∫ ( t i ) f Ti ( t i )d ( t i ),

(12)

−∞ +∞

V ( Ti ) = ∫ ( t i ) 2 f Ti ( t i )d ( t i ) − [E( Ti )] . 2

(13)

−∞

Here, f Ti ( t i ) is the density function of the maximum of the arrival time density functions of component 1 and component 2, scaled by a factor i. The arrival time of kit i at the assembly station equals the maximum of both component arrival times for kit i, scaled with a factor i: Ti =

max( ATi 1 , ATi 2 ) . i

(14)

As both component arrival times are IID, we can easily calculate the density function of the maximum of the component arrival time density functions (fMAXAT|i(maxat|i)) (Papoulis, 1991): f MAXAT i ( max at i ) = f ATi 1 (max at i )FATi 2 ( max at i ) + f ATi 2 ( max at i )FATi 1 (maxta i ).

(15)

Scaling this density function with a factor i boils down to multiplying the density function with i and writing it in terms of ‘(i)ti’ instead of ‘maxat|i’:

fTi ( ti ) = if MAXAT i (( i )ti ).

(16)

From expression (3), we have that:

f ATic (max at i ) ≅

− ( max at i − iµ ITc ) 2

1

σ IT

c

2πi

e

2 2 iσ IT c

1 max at i − iµ ITc FATic ( max at i ) ≅ erf 2 2iσ IT2 c

,

(17)

+ 1 .

(18)

Therefore,

8

fTi ( ti ) ≅

− (( i )ti −iµ IT1 )2

i

σ IT 2πi

2 2 iσ IT 1

e

×

1

−(( i )ti −iµ IT2 ) 2 i 1 ( i )ti − iµ IT1 2 iσ IT + 1 + 2 e erf σ 2 2iσ IT2 IT2 2πi 1

1 ( i )ti − iµ IT2 erf 2 2iσ IT2 2

2

(19)

+ 1.

By substituting expression (19) in (12), we can simplify E(Ti) to (De Boeck, 2003): E( Ti ) = µ ITc +

σ IT2 + σ IT2 1

2

2πi

(20)

.

We do not have a formula for the variance. This is not a problem because the way in which we calculated this inter-arrival time of kits, we do not obtain the right formula for V(Ti). Note that for increasing values of i (see infra), V(Ti) converges to 0. However, since in the end (i → ∞) one component will always be waiting for the other one, the arrival stream of kits will be completely determined by the input stream of the other component which is not deterministic. Since V(T|i) converges fast to the variance of the component input streams in case both component input streams have the same variance (De Boeck, 2003), we propose the following approximation for V(Ti): V ( Ti ) ≅

1 2 2 ∑ σ IT , 2 c =1 c

(21)

where ‘2’ in ‘1/2’ and the summation sign is the number of input streams. As example, we now represent E(Wi,), V(Wi), E(Ti,) and V(Ti), for the first 20 kits for Logn(8.5,0.25) (a lognormal distribution with an average value of 8.5 and a squared coefficient of variation of 0.25) distributed inter-arrival times for both components in Fig. 3 and 4. The reason why we illustrate those performance measures for lognormally distributed component inter-arrival times is explained in subsection 3.2.

14 12 E(Wi)

10

E(Ti) 8 6 4 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 Kit

Fig. 3. E(Wi) and E(Ti) for the first 20 kits for Logn(8.5,0.25) distributed inter-arrival times for both components

9

140 120 100 80

V(Wi)

60

V(Ti)

40 20 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 Kit

Fig. 4. V(Wi) and V(Ti) for the first 20 kits for Logn(8.5,0.25) distributed inter-arrival times for both components

The figures clearly reveal that E(Wi) and V(Wi) increase for increasing values of i. Also formulas (10) and (11) show that both performance measures evolve to ∞ for i → ∞. In other words, the synchronization time of components for kits for a generic assembly system does not converge. That is why we will assume a finite number of kits. E(Ti) and V(Ti) however, decrease for increasing values of i. Formula (20) shows that E(Ti) converges to µ ITc for i → ∞. In other words, the inter-arrival time of kits at the assembly station does converge. 3.2. Validation To validate the approximations obtained in subsection 3.1, we rely on simulation. Since we assume that components are processed in a stochastic production environment preceding the assembly system, we assume both component inter-arrival times to be lognormally distributed. Since this distribution is a good approximation for machine processing times (Law and Kelton, 2000) and we assume machines to have a relatively high utilization (around 90%), the distribution of the inter-departure times out of the machines (or the distribution of the interarrival times at the machines that follow) approximates the processing time distribution. To investigate the influence of equal and different variances and small, moderate and highly variable inter-arrival times for both components, we combine the following distributions for both component inter-arrival times resulting in 6 scenarios represented in Table 2. Remark that small (SCVX < 0.5625), moderate (0.5625 ≤ SCVX < 1.7689) and high variability (SCVX ≥ 1.7689) of a variable X is determined by the squared coefficient of variation (SCVX) (see Hopp and Spearman, 2000). Table 2 The different scenarios for validating the approximations for E(Wi), V(Wi), E(Ti) and V(Ti) Logn(8.5,0.25) Logn(8.5,1.25) Logn(8.5,2) Logn(8.5,0.25) Scenario 1 Scenario 2 Scenario 3 Logn(8.5,1.25) Scenario 4 Scenario 5 Logn(8.5,2) Scenario 6

We assume a finite number of kits in our simulation model equal to 1000 and perform 100 replications for each scenario. The approximate results (indicated by the symbol AR) for E(Wi) (see formula (10)), V(Wi) (see formula (11)), E(Ti) (see formula (20)) and V(Ti) (see formula (21)) and the simulation results (indicated by the symbol SR) are represented in Table 3.

10

Logn(8.5,0.25)

Logn(8.5,1.25)

Logn(8.5,2)

Table 3 The approximate and simulation results for E(Wi), V(Wi), E(Ti) and V(Ti) Logn(8.5,0.25) Logn(8.5,1.25) Logn(8.5,2) AR SR AR SR AR E(Wi) 101.18 97.82 175.24 182.06 214.63 V(Wi) 6570.10 4484.53 19710.35 13886.53 29565.55 E(Ti) 8.57583 8.58 8.63133 8.61 8.66085 V(Ti) 18.06 18.40 54.19 59.56 81.28 E(Wi) 226.235 246.82 257.95 V(Wi) 32850.60 23928.35 42705.75 E(Ti) 8.66955 8.66 8.69332 V(Ti) 90.31 96.67 117.41 286.17 E(Wi) V(Wi) 52561.00 E(Ti) 8.71447 V(Ti) 144.5

SR 215.53 21143.67 8.64 67.91 266.59 32071.54 8.73 127.26 277.45 35835.60 8.70 169.94

We observe from Table 3 that the approximations for E(Wi) perform very well. These results together with additional simulation results for squared coefficients of variation equal to 0.5, 0.75, 1, 1.5 and 1.75 (we did not include those results because it would make Table 3 unreadable) reveal an average relative deviation of the simulation results in terms of the approximate results of 0.02% and a relative deviation within [-7.88%,9.10%]. The results for V(Wi) however do not perform well but are robust in the sense that the approximations always overestimate the simulation results. Here we have an average relative deviation of the simulation results in terms of the approximate results of -31.39% and a relative deviation within [-46.06%,-16.26%]. The results for E(Ti) and V(Ti) perform well and give average relative deviations of the simulation results in terms of the approximate results of -0.06% and 5.28% respectively and relative deviations within [-0.65%,0.45%] and [-5.99%,17,60%] respectively. In general we can state that all our approximations perform well except for V(Wi). These last results are however robust in the sense that they always overestimate the simulation results by around 30%. Finding a better approximation is an issue for further research.

4. Conclusions In this paper we assumed a generic assembly system consisting of two independent component input streams having generally distributed inter-arrival times. We assumed the same average arrival rate for both input streams and each input process to be a renewal process. A kit consisted of one unit of each component. For such a system, we obtained the following unknown parameters: the synchronization time of the components for kits and the inter-arrival time of the kits at the assembly station. These performance measures are not encountered in a traditional queuing system. The results revealed increasing values for the synchronization time of the components for kits which is confirmed by the existing literature (Harrison, 1973). That is why we introduce a limitation on the number of kits in our system. We also observed converging values for the inter-arrival time of the kits at the assembly station. These performance measures are a required step in obtaining the performance measures of the entire assembly system and the performance of an assembly system embedded in an open queuing network model. From a managerial point of view, we can derive some take-aways for the generic assembly system. These take-aways result from the formulas of the synchronization time (see formulas (10) and (11)) of components for kits and the inter-arrival time (see formula (20)) of kits. • More reliable supply (with the same frequency) leads to less synchronization stock and improved productivity. Reliable supply indicates less variable component inter-arrival 11

• •

times. Less synchronization stock translates in lower synchronization times of components for kits and improved productivity translates in smaller inter-arrival times of kits at the assembly station. More frequent supply (with the same reliability) does not lead to less synchronization stock. More frequent supply is obtained by increased average component inter-arrival times. Supply has to be synchronised and supply needs a cap in order to shutdown the input streams, at least temporarely. Note that this translates in a limitation on the number of kits in our assembly system.

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