Two Simplified Models for Re-entrant Line: QNA and JNA

Chapter 3 Parametric Decomposition Method for OQN

3.5 Two Simplified Models for Re-entrant Line: QNA and JNA

= 𝜆_𝐸 ∑ 𝑉_𝑚(𝜏_𝑚+ 𝐸[𝑊_𝑚])

𝑀

𝑚=1

= 𝜆_𝐸× 𝐸[𝑇]. (3.19)

Then, expected cycle time of individual products can be obtained because the parametric-decomposition approximation views each workstation as an independent node, the expected total cycle (or sojourn) time for a product is the summation of expected cycle time of each node in the routing of that product. The expected cycle

time for product 𝑖 is

𝐸[𝑇_𝑖] = ∑^𝑆_𝑘=1^𝑖 (𝜏_𝑖𝑘+ 𝐸[𝑊_𝑚_𝑖,𝑘]). (3.20)

where 𝐸[𝑊_𝑚_𝑖,𝑘] is the expected waiting time of the 𝑘^𝑡ℎ processing step of product 𝑖

at node 𝑚_𝑖,𝑘. And the cycle time variance for product 𝑖 can be calculated by

𝑉𝑎𝑟[𝑇_𝑖] = ∑^𝑆_𝑘=1^𝑖 𝜏_𝑖𝑘² 𝐶_𝑖𝑘² + ∑^𝑆_𝑘=1^𝑖 𝑉𝑎𝑟[𝑊_𝑖,𝑘]. (3.21)

3.5 Two Simplified Models for Re-entrant Line: QNA and JNA

We discuss two simplified models developed according to the parametric decomposition method: one is queueing network analyzer (QNA) and the other is

Jackson network approximation (JNA). JNA assumes the OQN follows Markovian routing using Equation (3.10) and inter-arrival time and service time of each node are exponentially distributed (CV=1). Thus, each node is treated as an independent M/M/m queue. Under these assumptions, the decomposition approximation is equivalent to Jackson network approximation. JNA is a simplified model of OQN which uses one parameter (mean) to characterize each node with unity SCVs.

Unlike JNA, QNA is a free software package which first developed at Bell Laboratories to calculate approximate performance measures for general (non-Markov) open queuing network [13]. QNA describes the variability parameter of network flows by Equation (3.11), (3.12), and (3.13), so each node has its own specific characterization of variability. Thus, QNA is another simplified model of OQN, which uses two parameters (mean and SCV) to characterize each node with heterogeneous SCVs. Most importantly, JNA can be regarded as a special case of QNA while each node assumes unity SCV.

Both QNA and JNA are mathematical models developed based on the same theoretical basis, parametric decomposition method. The major difference between these two simplified models is the characterization of variability parameter. It is a meaningful comparison to investigate how modeling heterogeneous variability in a simplified model affects ranking for ordinal optimization.

Chapter 4 Ordinal Transformation and BRA

In this chapter, we introduce the ordinal transformation (OT) which exploits a simplified model to quickly determine rough performance of designs and their relative orders instead of finding accurate performance. The goodness of ranking by adopting a simplified model for performance approximation is quantified and analyzed in terms of rank correlation.

Queueing network analyzer (QNA) is mainly investigated in the following analysis because QNA utilizes both mean and variance to characterize network flows and JNA is a special case of QNA while assumes unity SCVs. Due to no analytical solution to mean cycle time performance of a general queueing network, we develop a bound and ranking (B&R) analysis to investigate the relation between the bounds of true and approximated performances, and analyze the probability of correctly ranking by a simplified model under some assumptions of true performance between the bounds. In this chapter, we use BRA to take the first step to analyze single GI/G/m queue with 2 designs.

4.1 Ordinal Transformation

In many complex optimization problems, the objective function is a black box

and seems impossible to apply traditional optimization approaches. Discrete-event system (DES) simulation becomes increasingly important in the analysis and design of such complex systems. Simulation optimization methods are often exploited to tackle such type of problems. For those simulation optimization approaches, they have a detailed simulation model to obtain accurate performance measure of the system but such a detailed simulation model usually has high computation cost and may be very time-consuming. With a large-scale design space, it is impossible to evaluate all designs by a detailed simulation model and find the optimal design.

Besides detailed DES simulation model, there are actually some simplified (or approximation) models for complex systems, i.e. QNA for OQN, whose computation costs can be ignored in comparison with DES simulation. Such simplified models are much faster but usually have large biases between actual performance measures.

For the sake of finding the optimal design, relative ranking orders among designs are much important than exact differences between their performances. If we find a

good simplified model whose relative ranking orders among designs are highly

correlated with actual ranking orders, then we can make use of such a simplified model to improve the efficiency of searching the optimal design. That is the main idea of OT. In the following, we investigate the basic two steps of OT:

(1) Ranking in terms of approximation by a simplified model

(2) Transform original solution space into ordinal space according to the rankings determined in step (1)

4.1.1 Ranking in terms of Approximations by Simplified Model

We first define some notations below.

𝐷: a design space;

n: total number of designs in D, i.e., n = |D|;

𝑥: a design in design space 𝐷;

𝑥^∗: the optimal design (according to the detailed DES simulation);

𝑓(𝑥): performance of design 𝑥 ∈ 𝐷 evaluated by detailed DES simulation model

f(^.);

𝑔(𝑥): performance of design 𝑥 ∈ 𝐷 evaluated by the simplified model g(^.);

𝛿(𝑥): (𝑥) − 𝑔(𝑥)

𝐹(𝑥): ordinal rank of a design 𝑥 ∈ 𝐷 based on {𝑓(𝑦),

𝑦 ∈ 𝐷} in ascending

order;

𝐺(𝑥): ordinal rank of a design 𝑥 ∈ 𝐷 based on {𝑔(𝑦),

𝑦 ∈ 𝐷} in ascending

order;

𝐹⁻¹(𝑖): the 𝑖^𝑡ℎ best design in 𝐷 according to {𝑓(𝑦),

𝑦 ∈ 𝐷};

𝐺⁻¹(𝑖): the 𝑖^𝑡ℎ best design in 𝐷 according to {𝑔(𝑦),

𝑦 ∈ 𝐷};

We represent the relationship between the detailed model and the simplified model by

𝑓(𝑥) = 𝑔(𝑥) + 𝛿(𝑥)

where 𝑓(𝑥) is the accurate performance measures evaluated by detailed model, 𝑔(𝑥) is the performance measures approximated by simplified model, and 𝛿(𝑥) is

the bias term. Actually, how accurate the performance measure of the simplified model is not significant because the idea of OT focuses on the ranking orders among all designs. Therefore, instead of obtaining a simplified model with small 𝛿(𝑥) (accurate performance measures), we would like to find a simplified model whose relative ranking orders among designs in term of an approximate performance measure by the simplified model are highly correlated with actual ranking orders. In

short, we desire a simplified model whose rankings among designs are highly correlated with actual rankings,𝐺(𝑥)~𝐹(𝑥), ∀𝑥 ∈ 𝐷, rather than high accuracy in

performance, 𝑓(𝑥)~𝑔(𝑥), ∀𝑥 ∈ 𝐷. The definition of evaluation between rankings will be defined in subsection 4.1.3.

Next is a small numerical example to illustrate the basic idea and potential benefits of OT. The detailed model is

Example 4.1

𝑓(𝑥) = 𝑠𝑖𝑛⁴(2𝑥) − 3𝑠𝑖𝑛³(2𝑥) + 𝑠𝑖𝑛²(2𝑥) + 4

= [𝑠𝑖𝑛²(2𝑥) + 𝑠𝑖𝑛 (2𝑥) + 1][𝑠𝑖𝑛²(2𝑥) − 4𝑠𝑖𝑛 (2𝑥) + 4],

and the simplified model is

𝑔(𝑥) = 𝑠𝑖𝑛²(2𝑥) − 4𝑠𝑖𝑛 (2𝑥) + 4

The possible designs are 𝑥 ∈ D ={-0.5, -0.25, 0, … , 2.5}, and n = |D|=13. We show the performance measures by f and g in Figure 4.1 and Table 4.1. We can observe that the simplified model, 𝑔(𝑥), approximately captures relative performance among designs even through some biases are significant, for example, f(0.75)=3.007 and

g(0.75)=1.005 but both lead to design

𝑥=0.75 as the best.

Figure 4.1 An illustrative example of OT

Table 4.1 Ranking order among designs of Example 4.1

In Table 4.1, in spite of the large biases, ordinal rank of design 𝑥 based on 𝑔(𝑥) is almost the same as the ordinal rank of design 𝑥 based on 𝑓(𝑥), 𝐹(𝑥)~ 𝐺(𝑥).

0 2 4 6 8 10

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

x

^f(x) ^g(x)

4.1.2 Transformation to Ordinal Space

We utilize a simplified model to quickly approximate the performances of all designs and rank designs according to their approximated performances. With ranking orders in terms of approximations, we transform the original solution space D to an ordinal space, so this kind of transformation is called ordinal transformation (OT). OT substitutes the original space, 𝑥, by the ordinal space of the simplified model, 𝐺(𝑥).

And we use the numerical results in 4.1.1 to demonstrate OT in Figure 4.2.

Figure 4.2 Transformation to ordinal space

In the ordinal space, designs with similar performances are ranked and positioned nearby together and either better or worse designs are easily differentiated.

Thus, after OT, we can search the optimum in a better space, which increases the efficiency of subsequent optimization processes.

4.1.3 Performance Index: Rank Correlation

The benefit of OT crucially depends on the quality of a simplified model but for

0 2 4 6 8 10

1 2 3 4 5 6 7 8 9 10 11 12 13

G(x):ranking in g(x)

^f(x) ^g(x)

ordinal transformation there is not a standard index to quantify the quality of a simplified model. Here we introduce a meaningful index, rank correlation, to quantify the goodness of ranking performance of a simplified model. Rank correlation is first developed by Kendall [46] to measures the similarity of the orderings of data when ranked by each of the quantities. Rank correlation is a statistic of pair-wise comparisons which corresponds to the idea of OT which comparing the relative order among designs, rank correlation is therefore adopted to measure the concordance of pair-wise comparisons in true and approximated performances.

Definition: Kendall rank correlation coefficient

There are 𝑁 designs, labeled as 𝑥₁, 𝑥₂, … , 𝑥_𝑁. Let (𝐹(𝑥_𝑖), 𝐺(𝑥_𝑖)) be a rank observations of design 𝑥_𝑖 in the detailed model and the simplified model. Any pair of

observation (𝐹(𝑥_𝑖), 𝐺(𝑥_𝑖)) and (𝐹(𝑥_𝑗), 𝐺(𝑥_𝑗)) are concordant, if both 𝐹(𝑥_𝑖) >

𝐹(𝑥_𝑗) and 𝐺(𝑥_𝑖) > 𝐺(𝑥_𝑗) or if both 𝐹(𝑥_𝑖) < 𝐹(𝑥_𝑗) and 𝐺(𝑥_𝑖) < 𝐺(𝑥_𝑗). They are

discordant, otherwise.

The Kendall rank correlation coefficient is defined as:

RC =Number of concordant pairs − Number of discordant pairs Total number of pairs =1

2 𝑁(𝑁 − 1)

The denominator is the total number of pairs, so the coefficient must be in the range,

−1 ≤ RC ≤ 1. If RC = 1, relative ranking orders among designs in both the detailed

and simplified model are completely the same. If RC = −1, relative ranking orders among designs in the detailed and simplified model are totally reverse. If RC = 0, then the rankings in the detailed and simplified model are uncorrelated.

4.2 BRA of QNA and JNA in single GI/G/m queue

Now consider using QNA and INA as simplified models for ranking capacity allocation designs over their mean cycle time performance. It has been pointed out in section 3.5 that JNA is as a special case of QNA, where the inter-arrival and service times of each node assume unity SCVs. In the following discussions of this sub-section, we focus on the analysis of goodness of ranking by using QNA as a simplified model.

Bound analysis investigates the relation between the bounds of true and approximated performance. Under some assumptions of the distribution of true performance, ranking analysis focuses on the conditional probability of correctly ranking given approximated performances. We start our BRA to analyze the probability of correctly ranking by using QNA evaluations from analyzing the case of single GI/G/m queue with 2 designs. BRA is mainly composed of two analyses: (1) bound analysis and (2) ranking analysis.

4.2.1 Bound Analysis of QNA and JNA

We begin with the well-known upper bound of expected waiting time of single

GI/G/m queue derived by Kingman [30]:

𝐸[WT_GI/G/m] ≤^𝐶^𝑎²^+𝑚²_{2𝜆(1−𝜌)}^𝜌𝐶^𝑠²^{+(𝑚−1)𝜌}² (4.1)

The lower bound of waiting time from Brumelle and Marchal [31][32] is

𝐸[WT_GI/G/m] ≥^𝜌²^𝐶_{2𝜆(1−𝜌)}^𝑠²^{−𝜌(2−𝜌)}−^{(𝑚−1)(𝐶}_2𝑚^𝑠²^+1)𝜏 (4.2)

QNA approximation of expected waiting time WT_QNA is (Eq. 3.15) 𝐸[𝑊𝑇]_QNA≅ ^𝐶^𝑎𝑖²^+𝐶₂ ^𝑠𝑖²𝐸[WT_M/M/m] =^𝐶^𝑎²^+𝐶₂ ^𝑠²(^{C(𝑚,𝜆,𝜏)}_{𝑚/𝜏−𝜆})

We know that the cycle time consists of waiting time and service time, so the

upper bound of expected cycle time of one GI/G/m queue is

𝑈𝐵[ECT_GI/G/m] =^𝐶^𝑎²^+𝑚²_{2𝜆(1−𝜌)}^𝜌𝐶^𝑠²^{+(𝑚−1)𝜌}²+ 𝜏 (4.3)

The lower bound of expected cycle time of one GI/G/m queue is

𝐿𝐵[ECT_GI/G/m] =^𝜌²^𝐶_{2𝜆(1−𝜌)}^𝑠²^{−𝜌(2−𝜌)}−^{(𝑚−1)(𝐶}_2𝑚^𝑠²^+1)𝜏+ 𝜏 (4.4)

And, QNA approximation of expected cycle time is

𝐸[𝐶𝑇]_QNA =^𝐶^𝑎²^+𝐶₂ ^𝑠²(^{C(𝑚,𝜆,𝜏)}_{𝑚/𝜏−𝜆}) + 𝜏 (4.5)

In the following, we prove that the upper bound and lower bound of a GI/G/m queue are also the upper bound and lower bound of QNA approximations in terms of expected cycle time.

Theorem 4.1: The upper bound of expected cycle time of a GI/G/m queue is also the

upper bound of expected cycle time of QNA approximation.

Proof :

The expected cycle time of QNA approximation, 𝐸[𝐶𝑇]_QNA is 𝐸[𝐶𝑇]_QNA =^𝐶^𝑎²^+𝐶₂ ^𝑠²(^{C(𝑚,𝜆,𝜏)}_{𝑚/𝜏−𝜆}) + 𝜏 because there is at least one machine in each workstation, 𝑚 ≥ 1, and the utilization

of each workstation is limited to smaller than one to maintain the stability of the

network. So,

𝑈𝐵[ECT_GI/G/m] >^𝐶^𝑎²^+𝐶₂ ^𝑠²(_{𝜆(1−𝜌)}^𝜌 ) + 𝜏 > 𝐸[𝐶𝑇]_QNA

Q.E.D.

To discuss the lower bound of QNA approximation, let us derive three lemmas regarding function C(𝑚, 𝜆, 𝜏) in QNA approximation.

Theorem 4.2: The lower bound of expected cycle time of a GI/G/m queue is also the

lower bound of expected cycle time of QNA approximation.

Proof :

47 𝜌𝐶𝑠2𝜌^𝑚

2𝜆(1−𝜌)+ 𝜏 >^𝜌𝐶^𝑠²_{2𝜆(1−𝜌)}^{[1−𝑚(1−𝜌)]}+ 𝜏 > 𝐿𝐵[𝐸CT_GI/G/m], From Lemma 4.1, C(𝑚, 𝜆, 𝜏) > 𝜌^𝑚,

𝜌𝐶𝑠2C(𝑚,𝜆,𝜏)

2𝜆(1−𝜌) + 𝜏 >_{2𝜆(1−𝜌)}^𝜌𝐶^𝑠²^𝜌^𝑚 + 𝜏 > 𝐿𝐵[ECT_GI/G/m].

The expected cycle time by QNA approximation in Eq. (4.5)

𝐸[𝐶𝑇]_QNA = ^𝐶^𝑎²^+𝐶₂ ^𝑠²(^{𝜌C(𝑚,𝜆,𝜏)}_{𝜆(1−𝜌)} ) + 𝜏 ≥ ^𝜌𝐶_{2𝜆(1−𝜌)}^𝑠²^{C(𝑚,𝜆,𝜏)}+ 𝜏 > 𝐿𝐵[ECT_GI/G/m]

since 𝐶_𝑎² ≥ 0.

Q.E.D.

From Theorem 4.1 and Theorem 4.2, we conclude that the approximated cycle time by QNA is bounded in the same range of true cycle time performance, which implies that QNA is an appropriate approximation on values.

D-1: Discussion About Bounds

Kingman’s upper bound for GI/G/1 is asymptotically tight in heavy traffic, but not in general [29]. Here we compare the bounds of QNA with bounds of another approximation model, JNA which assumes exponential distributions with unity SCV for inter-arrival and service times. Intuitively, QNA bounds derived by Kingman, Brumelle and Marchal are suitable for any situation no matter the level of variability or the network configuration, so JNA bounds are just a special case of QNA bounds while all SCVs are one. When all distributions are assumed unity SCV, QNA bounds are equivalent to JNA bounds.

Thus, if SCVs of actual system are smaller than one, JNA bounds are overestimated and we verify this by comparing with the existing E2/M/2 result in [30].

Figure 4.3 Comparison of upper bound between QNA and JNA while SCVs≦1 and true value obtained from table 5 in [30]

Figure 4.3 shows the tighter bounds provided by QNA while SCVs smaller than one.

If SCVs greater than one, JNA bounds are underestimated and also verify by the existing G/H₂/m result in [30].

Figure 4.4 Comparison of upper bound between QNA and JNA while SCVs>1

and true value obtained from table 7 in [30]

Importantly, true expected cycle time in [30] is not bounded by JNA bound while Ca2

=2, Cs2

=9, and utilization greater than 0.8, which shows JNA bound is not consistent for bounding the true performance because of the assumption of unity SCVs. In other words, QNA bounds are suitable for describing the range of true performance even through QNA bounds may be less tight sometimes.

Instead of JNA under the assumption of unity SCVs, QNA captures the heterogeneous SCVs to better characterize network flow, and provides more information for ranking. We show the advantage of QNA bounds with heterogeneous SCVs using the existing result in [30].

Figure 4.5 Advantage of QNA bounds with heterogeneous SCVs

It is obvious that true expected cycle time is always bounded by QNA upper bound, and most importantly, both of them grows with a similar trend which facilitates us use QNA to infer the true performance. By contrast, JNA bound is flat and this implies

JNA bound does not provide any information about the variation of true performance.

The major differences between the models with heterogeneous SCVs and unity SCVs are (1) Better bounds (2) Implicitly useful information about true performance. In the next section, we utilize the bounds to help infer the probability of correct ranking and also show why model selection in OO matters.

4.2.2 Ranking Analysis of QNA and JNA

To exploit the bounds obtained in sub-section 4.2.1 and investigate ranking among designs by QNA approximation of mean cycle times, we first consider the comparison between a pair of designs for a single GI/G/m queue, D₁ and D₂. Let their true mean cycle times be ECT₁ and ECT₂ which are random variables. The approximated mean cycle times of D₁ and D₂ by QNA are, ACT₁ and ACT₂

respectively. Note that QNA approximation describes a node using four parameters (𝜆_𝑎, 𝐶_𝑎², 𝜏, 𝐶_𝑠²) to characterize the mean and SCV of inter-arrival and service times [13][16]. We can obtain these four parameters (𝜆_𝑎, 𝐶_𝑎², 𝜏, 𝐶_𝑠²) of each node by Eq. (3.7),

(3.14), (3.5), and (3.6) respectively in Chapter 3.

Lemma 4.4: There are two designs, D

₁

and D

₂

, and there are n

₁

and n

₂

machines

allocated in node 𝑚. the SCV of service time of D

is equal to that of D

₂

at node 𝑚.

Therefore, the SCV of service time of a node is not related to the number of machine

allocated in that node.

Proof:

D₁ has n₁ machines allocated at a node 𝑚 and D₂ has n₂ machines allocated at a node 𝑚. Recall that for a node 𝑚, its service time SCV, 𝐶_𝑠𝑚² , is obtained from Eq.

(3.6)

𝐶_𝑠𝑚² =^∑ ^∑ ^𝜏^𝑖𝑘^𝟐^(𝐶^𝑖𝑘

𝟐+𝟏)𝜆_𝑖𝟏{𝑚_𝑖,𝑘=𝑚}

𝑆𝑖𝑘=1 𝐼𝑖=1

∑^𝐼_𝑖=1∑^𝑆𝑖_𝑘=1𝜏_𝑚^𝟐𝜆_𝑖𝟏{𝑚_𝑖,𝑘=𝑚} − 1. (3.6) where

𝟏H(x): an indicator function of the set H, 𝟏H(x) = {1, 𝑖𝑓 𝑥 ∈ 𝐻.

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

𝜏_𝑖𝑘 is the mean processing time at 𝑘-th step of product type 𝑖 and the 𝑘-th step

of product type 𝑖 is processed by the service node 𝑚_𝑖,𝑘. 𝜆_𝑖 is the external arrival rate of product type 𝑖.

𝜏_𝑚 is the aggregate mean service time of node 𝑚 obtained from Eq. (3.5) 𝜏_𝑚 = ^∑^𝐼^𝑖=1^∑^𝑆𝑖^𝑘=1^𝜏^𝑖𝑘^𝜆^𝑖^𝟏{𝑚^𝑖,𝑘^=𝑚}

∑^𝐼_𝑖=1∑^𝑆𝑖_𝑘=1𝜆_𝑖𝟏{𝑚_𝑖,𝑘=𝑚} , (3.5)

Eq. (3.5) is a weighted sum of mean processing time of every steps processed by node 𝑚 and it is not related to the number of machines allocated in node 𝑚.

Eq. (3.6) is also a weighted sum of SCV of processing time of every steps processed by node 𝑚 and also not related to the number of machine allocated in node 𝑚.

Therefore, for D₁, the SCV of service time of node 𝑚 is not a function of n1.

The above shows that the calculation of 𝐶_𝑠𝑚² is nothing related to the number of machine allocated in node 𝑚. Therefore, the SCV of service time of D₁ at node 𝑚 queue, Eq. (3.14) can be re-written as

𝐶_𝑎² = 𝛼 + 𝐶_𝑎²𝛽  𝐶_𝑎² = 𝛼 (1 − 𝛽)⁄ , 𝛼 and 𝛽 increases with utilization.

So, the SCV of inter-arrival time(𝐶_𝑎²) of D₁ is smaller than that of D₂.

The approximated mean cycle time of QNA is, ACT =^𝐶^𝑎²^+𝐶₂ ^𝑠²(^{𝜌C(𝑚,𝜆,𝜏)}_{𝜆(1−𝜌)} ) + 𝜏 . Because _(1−𝜌)^𝜌 is monotonically increasing with 𝜌 and C(𝑚, 𝜆, 𝜏) is as well according to Lemma 4.3, which induces that ^{𝜌C(𝑚,𝜆,𝜏)}

(1−𝜌) is also monotonically increasing with 𝜌.

Thus, we know utilization of D₁ is smaller than that of D₂, so ^{𝜌C(𝑚,𝜆,𝜏)}

(1−𝜌) of D₁ is smaller than that of D₂. In addition, the SCV of inter-arrival time (𝐶_𝑎²) of D₁ is smaller than that of D2. So, we obtain that ACT1 is smaller than ACT2.

Besides, the upper bound and lower bound of expected cycle time of a GI/G/m queue are positively related to its utilization. The growth of upper bound is faster than

ACT but the growth of lower bound is slower than ACT, as shown in Figure 4.3. So if 𝜌₁ < 𝜌₂ then both upper and lower bounds of expected cycle time of D₁ is smaller

than them of D2, noted as UB(ECT1) ≤ UB(ECT2) and LB(ECT1) ≤ LB(ECT2).

Figure 4.6 UB, ACT, LB w.r.t. utilization and number of machines

We therefore conclude that if ACT1 is smaller than ACT2, then UB(ECT1) ≤

UB(ECT2) and LB(ECT1) ≤ LB(ECT2).

Q.E.D.

Based on Lemma 4.5, a simple diagram is shown in Figure 4.7.

(a) Given ACT1 < ACT2 (b) Given ACT1 > ACT2

Figure 4.7 Two simple diagrams

Recall the definition of rank correlation coefficient in Section 4.1.3, RC =Number of concordant pairs−Number of discordant pairs

Total number of pairs=¹₂𝑛(𝑛−1) .

There are two kinds of pairs in the definition of rank correlation, concordant pair and discordant pair. Therefore, given two designs, D₁ and D₂, and their approximated cycle time by QNA, ACT₁ and ACT₂, there are two probable events:

(1) (ECT₁, ACT₁) and (ECT₂, ACT₂) is a concordant pair, and (2) (ECT₁, ACT₁) and (ECT₂, ACT₂) is a discordant pair.

If ACT₁ is smaller/larger than ACT₂ and ECT₁ is smaller/larger than ECT₂, then the pair of (ECT₁, ACT₁) and (ECT₂, ACT₂) is a concordant pair. Otherwise, it is discordant.

Assumption of uniform distributions

Without any prior knowledge we assume that the value lying anywhere between upper bound and lower bound has an equal probability, so distribution of ECT1 and ECT2

D-2: Discussion about probability of being a concordant pair, P

Equation (4.3) is influenced by two terms: one can be viewed as the ratio of the surely-win region in the range of ECT₂, ^(𝑈𝐵²^−𝑈𝐵¹⁾ overlapped with the range of ECT2, but Equation (4.3) is obviously greater that ¹₂ and the maximum of Equation (4.4) is ¹₂. It shows that if it is known that ACT1 < ACT2, ranking two designs according to their approximated cycle times by QNA makes sure that the probability of being a concordant pair (P_c) is not lower than 0.5.

D-3: Discussion about the effects of bounds to P

We define the ratio of surely-win region in the range of ECT₂, ^(𝑈𝐵²^−𝑈𝐵¹⁾

(𝑈𝐵2−𝐿𝐵2) as R_W, and the ratio of surely-lose region in the range of ECT₁, ^(𝐿𝐵²^−𝐿𝐵¹⁾

(𝑈𝐵1−𝐿𝐵1) as R_L, and R_W,R_L∈[0,1] and if R_W=1, then R_L=1 and if R_W=0, then R_L=0. The probability of being a concordant pair is determined by R_W and R_L, and Equation (4.3) can be written as ¹₂+¹₂R_w+¹₂R_L(1 − R_w) = 1 −¹₂(1 − R_w)(1 − R_L), visualized as Figure 4.8. (1 − R_w) is the ratio of overlapped region in the range of ECT₂ and (1 − R_L)

is the overlapped region in the range of ECT₁. If the overlapped region decreases, (1 − R_w) and (1 − R_L) decrease and the probability of correct ranking increases.

This implies the differentiation between D₁ and D₂ becomes easier.

Figure 4.8 Probability of being a concordant pair w.r.t. RW and RL

Recall that Figure 4.5 in Discussion D-1 shows that bounds of QNA with heterogeneous SCVs inhibit the similar trend with true performance and better capture the difference due to heterogeneous SCVs. In contrast, flat JNA bound whose R_W equal to zero provides no any information about the probability of being a concordant pair and this causes the probability of 0.5 like tossing a coin. Thus, it is significant that heterogeneous SCVs advantage us to correctly rank among designs with higher probability.

D-4: Discussion about distribution of actual cycle time

In our analysis, because we have no any prior information about the distribution

0 Probability of being a concordant pair w.r.t R_W and R_L

ratio of surely-lose region, R_L

of actual cycle time, the only information we known is that QNA is bounded by actual upper and lower bounds proposed by Kingman, Brumelle and Marchal. Thus, we assume actual cycle time is uniformly distributed but it may not be same as our assumption. Erlang distribution may be another alternative assumption. Because the support of Erlang distribution is greater than zero, which coincides with the nature of waiting time, and Erlang distribution is concentrated at a specific value, which is more realistic to common situations. Without uniformity, the difference between peaks of any two Erlang PDFs matters. When their peaks are getting closer, probability of correct ranking would be decreasing because of more overlapped region and smaller R_W and R_L.

In the following, we further investigate the useful information hidden behind QNA approximations and therefore derive a lower bound of probability of being a concordant pair, which must be greater than the probability of 0.5, under the

在文檔中分析不同變異考量下之簡化模型對排序佳化的影響：以迴流產線產能分配為例 (頁 45-0)