Chapter 2 Conveyor Problem: Re-entrant Line Capacity Allocation
2.3 Conveyor Problem for Ordinal Optimization
For machine capacity allocation problem in re-entrant lines, the original solution space is discrete and high-dimensional, which may have multiple local optimums and be hard to search in such a solution space. The re-entrant behavior poses the challenge to analyze the effects of re-entrant flows since that the interactions or dependencies among workstations are hard to describe and also difficult to exactly analyze the system in re-entrant lines [28]. If we would like to obtain accurate evaluation of a machine allocation design in re-entrant lines, we have to exploit the discrete event simulation (DES) but discrete event simulation suffers from the high computational cost. Due to its nature of complexity and combinatorial solution space, machine capacity allocation problem in re-entrant lines is extremely complicated and the use of simplified models is necessary. Thus, this re-entrant line capacity allocation problem is suitable to be the conveyor problem for investigating the model selection of ordinal optimization.
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Chapter 3
Parametric Decomposition Method for OQN
In this chapter, we introduce the parametric-decomposition approximation method of OQNs and the related extensions to improve its accuracy of approximations. According to the parametric decomposition method, there are two simplified models developed for OQNs, one is with unity SCVs and the other is with heterogeneous SCVs. The simplified model with unity SCVs is essentially based on the assumption of Jackson networks, and also called “Jackson network approximation (JNA)”. The simplified model with heterogeneous SCVs is developed by Whitt[13], and names as “Queueing Network Analyzer (QNA)”. It is a meaningful comparison because both of them are developed in a same theoretical basis, and the only difference between JNA and QNA is characterization of SCVs. Then, node-level measures and system-level measures of simplified models can be obtained. At last, we summarize how the comparison of these two simplified models relates to the model selection problem in OO.
3.1 Introduction
T
he parametric-decomposition approximation method first proposed by Reiser and Kobayashi[10] is a useful method to analyze the steady-state performance of23
OQNs. The main idea is to approximately characterize the arrival or service processes of each node by two parameters: mean and variability, approximate the relationship among nodes in the network, and then analyze the individual nodes separately.
Parametric decomposition method treats each node as an independent GI/G/m queue with m identical machines, infinite buffer for waiting, FCFS discipline and using two parameters to describe its general inter-arrival time distribution and general service time distribution respectively.
A standard decomposition approximation assumes Markovian routing of products after the service at each node in an OQN, which is the basic property of Jackson network. Bitran and Tirupati [14] observed that the SCV of departure of a product calculated under the assumption of Markovian routing is distorted by the presence of other products at a node. Bitran and Tirupati proposed the approximation of the SCV of inter-departure times at each node for each product and showed that the SCV of inter-departure times can be refined as the sum of two terms: the first reflects the queuing effect at the node, and the second captures the effect caused by inter-arrival time distributions of other products. Then, Segal and Whitt[16] proposed the refined approximation of the SCV of inter-departure times for aggregated product flows in re-entrant lines with deterministic routing of products. Numerical results in [16] showed that the refined approximations have relative errors of about 5-20% in
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estimating the inter-departure SCVs.
3.2 Class Aggregation
First recall that the notations of a multiple-product OQN model. Each product type 𝑖 has a total of 𝑆𝑖 processing steps, 𝑖 =1,2,…, I. Let (𝑖, 𝑘) be the k-th processing step of type-i product. The process routing of type-i follows a deterministic route {(𝑖, 𝑘), 𝑘 = 1,2, … , 𝑆𝑖}. Step (𝑖, 𝑘) is processed by the service node 𝑚𝑖,𝑘 ∈ {1,2, … , 𝑀}. Then, multiple types of products are aggregated into a single product in the OQN model. The aggregation procedure follows the work of Whitt[13] and summarizes in the following.
Define some notations:
𝜆𝑖: external arrival rate of product type 𝑖;
𝐶𝑖2: inter-arrival time SCV of product type 𝑖;
𝛿𝑖𝑗 = { 1, product type i externally entering the network at node j.
0, otherwise.
𝜆𝐸 : aggregate external mean arrival rate;
𝐶𝐸2 : inter-arrival time SCV of aggregate external arrivals;
𝜆𝑚𝑛 : aggregate mean arrival rate from node 𝑚 to node 𝑛;
𝜏𝑚 : aggregate mean service time at node 𝑚;
𝐶𝑠𝑚2 : aggregate service time SCV at node 𝑚;
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𝜏𝑖𝑘 : mean service time at 𝑘-th step of product type 𝑖;
𝐶𝑖𝑘2 : service time SCV at 𝑘-th step of product type 𝑖;
𝑄={𝑞𝑚𝑛}: routing matrix, and 𝑞𝑚𝑛 is ratio of routings from node 𝑚 to node 𝑛;
𝟏H(x): an indicator function of the set H, 𝟏H(x) = {1, 𝑖𝑓 𝑥 ∈ 𝐻.
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
First, we obtain the aggregate external arrival rates by adding up mean arrival rate of
each product at node 𝑛,
𝜆𝐸𝑛 = ∑𝐼𝑖=1𝜆𝑖𝛿𝑖𝑛. (3.1)
As the external arrivals of products are independent, the inter-arrival time SCV of
aggregate external arrivals is 𝐶𝐸𝑛2 = ∑ 𝐶𝑖2 𝜆𝜆𝑖𝛿𝑖𝑛
𝐸𝑛
𝐼𝑖=1 . (3.2)
The aggregate mean arrival rate from node 𝑚 to node 𝑛 is
𝜆𝑚𝑛 = ∑𝐼𝑖=1∑𝑆𝑘=1𝑖−1𝜆𝑖𝟏{𝑚𝑖,𝑘 = 𝑚, 𝑚𝑖,𝑘+1 = 𝑛}, ∀m ≠ n, m, n = 1,2, … , M. (3.3)
And the ratio of routings from node 𝑚 to node 𝑛 can be calculated as
𝑞𝑚𝑛 = 𝜆𝑚𝑛
∑𝐼𝑖=1∑𝑆𝑖𝑘=1𝜆𝑖𝟏{𝑚𝑖,𝑘=𝑚} (3.4)
The aggregate service time of a step at node 𝑚 is composed of service times of each step of each product that routed to be served by node 𝑚. The aggregate mean service
time at node 𝑚,
𝜏𝑚 = ∑𝐼𝑖=1∑𝑆𝑖𝑘=1𝜏𝑖𝑘𝜆𝑖𝟏{𝑚𝑖,𝑘=𝑚}
∑𝐼𝑖=1∑𝑆𝑖𝑘=1𝜆𝑖𝟏{𝑚𝑖,𝑘=𝑚} , (3.5)
and the corresponding SCV at node 𝑚 is
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𝐶𝑠𝑚2 =∑ ∑ 𝜏𝑖𝑘𝟐(𝐶𝑖𝑘
𝟐+𝟏)𝜆𝑖𝟏{𝑚𝑖,𝑘=𝑚}
𝑆𝑖𝑘=1 𝐼𝑖=1
∑𝐼𝑖=1∑𝑆𝑖𝑘=1𝜏𝑚𝟐𝜆𝑖𝟏{𝑚𝑖,𝑘=𝑚} − 1. (3.6)
3.3 Parametric Decomposition Method
T
he parametric-decomposition approximation method first proposed by Reiser and Kobayashi[10] is a useful method to analyze the steady-state performance of OQNs. The main idea is to approximately characterize the arrival or service processes of each node by two parameters: mean and variability, approximate the relationship among nodes in the network, and then analyze the individual nodes separately. The decomposition approximation can be comprised of the basic three steps:(1) analysis of the relationships between arrival, service, and departure processes at a node;
(2) analysis of the dependency among nodes of the network;
(3) approximation of performance measures of the whole network.
Define more notations for each node in OQN:
𝑀𝑚 : number of machines in node 𝑚;
𝜆𝑎𝑚 : mean total arrival rate to node 𝑚;
𝐶𝑎𝑚2 : inter-arrival time SCV at node 𝑚;
𝐶𝑑𝑚2 : inter-departure time SCV at node 𝑚;
𝐶𝑚𝑛2 : inter-departure time SCV for the flow transiting from node 𝑚 to node 𝑛;
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Because of the flow relationship among nodes in the network, the total arrival rate of node 𝑛 is the summation of external arrivals to node 𝑛 and internal arrivals to
node 𝑛, represented as
𝜆𝑎𝑛 = ∑𝐼𝑖=1𝜆𝑖𝛿𝑖𝑛+ ∑𝑀𝑚=1𝜆𝑚𝑛 = ∑𝐼𝑖=1𝜆𝑖𝛿𝑖𝑛+ ∑𝑀𝑚=1𝜆𝑎𝑚𝑞𝑚𝑛 (3.7)
Where 𝛿𝑖𝑛=1 if product type 𝑖 entering the network at node 𝑛. Equation (3.7) is known as the traffic rate equations with 𝜆𝑚𝑛 as defined by equation (3.3). In
equation (3.7), there are 𝑀 equalities with 𝑀 unknown variables { 𝜆𝑎𝑛, 𝑛 = 1,2, … , 𝑀}, so the 𝜆𝑎𝑛 can be solved by these 𝑀 simultaneous equations. After
obtaining the total arrival rate to node m, the average utilization of node 𝑛 can be
calculated by, 𝜌𝑛 =𝜆𝑎𝑛𝑀𝜏𝑛
𝑛 (3.8)
Segal and Whitt [14] pointed out that the resulting utilization of each node is exact. To ensure the stability of networks, the average utilization should be limited below the
capacity of the line,
𝜌𝑛 < 1, 𝑛 = 1,2, … , 𝑀.
By utilizing the procedure of Whitt[13], the inter-arrival time SCV of an
aggregate arrival process can be obtained as 𝐶𝑎𝑛2 = 1 − 𝜔̃ + 𝜔𝑛 ̃𝑛𝜆𝐸𝑛𝜆𝐶𝐸𝑛2
𝑎𝑛 + 𝜔̃ ∑𝑛 𝜆𝜆𝑚𝑛
𝑎𝑛
𝑀𝑚=1 𝐶𝑚𝑛2, (3.9)
where 𝜔̃ = [1 + 4(1 − 𝜌𝑛 𝑛)2(𝑣𝑛− 1)]−1 and 𝑣𝑛 = [∑ (𝜆𝜆𝑚𝑛
𝑎𝑛)2
𝑀𝑚=1 ]−1.
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However, the approximate variability parameter of the sub-flow from node 𝑚 to node 𝑛, 𝐶𝑚𝑛2, is related to the routing criteria and the dependency among nodes.
Note that the arrivals of a node are aggregated by the departures of its upstream nodes.
And the departures out of a node is split into several sub-flows of different downstream nodes according to the routing matrix 𝑄={𝑞𝑚𝑛}. Thus different routing criteria will influence the characteristics of nodes and the properties of networks. In the following we discuss two kinds of routing criteria, Markovian routing and deterministic routing, and obtain the approximate variability parameter of the sub-flow from node 𝑚 to node 𝑛, 𝐶𝑚𝑛2, under different routing criteria.
3.3.1 Markovian Routing
The Markovian routing means that each product completes service at node 𝑚
and proceeds to node 𝑛 with probability 𝑞𝑚𝑛, which is independent of the current state and history of the network. The routing matrix 𝑄={𝑞𝑚𝑛} interprets as the
independent probabilities of going to node 𝑛 after completed at node 𝑚. The approximate variability parameter of the sub-flow from node 𝑚 to node 𝑛, 𝐶𝑚𝑛2, under Markovian routing is proposed by Whitt[13],
𝐶𝑚𝑛2 = 𝑞𝑚𝑛𝐶𝑑𝑚2+ (1 − 𝑞𝑚𝑛) (3.10)
Because of the independency of Markovian routing, if all the external arrival
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processes are Poisson and products follow Markovian routings, then all internal arrival processes are also Poisson. If we assume that there is a single product and service time distributions are exponential, then parametric decomposition method is consistent with Jackson network and most significantly provides exact performance measures of Jackson network [13].
3.3.2 Deterministic Routing
As the observation of Bitran and Tirupati in [14] that if in the multiple product types, their arrivals do not follow Poisson distributions and the routings are deterministic, the use of Equation (3.9) to describe the approximate variability parameter of the sub-flow from node 𝑚 to node 𝑛 may not perform well due to the independency assumption of Markovian routing. Bitran and Tirupati identified the distortion in the SCV of a given product because of the presence of other products and refer to this distortion as the interference effect. Following the work of Bitran and Tirupati, Segal and Whitt proposed the refined calculation of the approximate
variability parameter of the sub-flow from node 𝑚 to node 𝑛,
𝐶𝑚𝑛2 = 𝑞𝑚𝑛𝐶𝑑𝑚2+ (1 − 𝑞𝑚𝑛)𝑞𝑚𝑛𝐶𝑎𝑚2+ (1 − 𝑞𝑚𝑛)2𝐶𝑒𝑚2, (3.11)
where 𝐶𝑒𝑚2 is an average of the external arrival-process variability parameters, 𝐶𝑒𝑚2 = ∑ 𝐶𝑖2(∑∑𝑆𝑘=1∑ 𝜆𝑖1{(𝑖,𝑘):𝑚𝜆 𝑖,𝑘=𝑚}
𝑆 𝑖
𝑘=1 1{(𝑖,𝑘):𝑚𝑖,𝑘=𝑚}
𝐼𝑖=1 )
𝐼𝑖=1 . (3.12)
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And Whitt[13] suggested that the SCV of the departure process at node 𝑚 by 𝐶𝑑𝑚2 = 1 + (1 − 𝜌𝑚2) (𝐶𝑎𝑚2− 1) +𝜌𝑚
2(max{𝐶𝑠𝑚2 ,0.2}−1)
√𝑀𝑚 . (3.13)
The experiments conducted in [14] and [16] if the network is multiple-product and deterministic routing, then apply Equation (3.11) to capture the interaction among stations instead of using Equation (3.10). Numerical results in [16] showed the refined approximations have relative errors of about 5-20% in estimating the inter-departure SCVs.
In this thesis, we focus on re-entrant lines with multiple products, deterministic routing, general (non-Poisson) arrivals, and general service time distributions.
Therefore, instead of Equation (3.10), we approximate the variability parameter of the sub-flow from node 𝑚 to node 𝑛 , 𝐶𝑚𝑛2, by Equation (3.11). By substituting
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approximate the relationship among the inter-arrival time SCV of all nodes.
Finally, from the parametric-decomposition approximation analysis, we can obtain the four parameters (𝜆𝑎𝑚, 𝐶𝑎𝑚2 , 𝜏𝑚, 𝐶𝑠𝑚2 ) of each node by Equation (3.7), (3.14),
(3.5), and (3.6) respectively to describe the characteristics of each node and approximate the performance measures of node-level and system-level as follows.
3.4 Performance Measures
Once we obtain the arrival and service parameters, (𝜆𝑎𝑚, 𝐶𝑎𝑚2 , 𝜏𝑚, 𝐶𝑠𝑚2 ), of each
node, we can exploit them to calculate many performance measures. In this section, we would describe how to approximate the performance measures by utilizing the results of the parametric-decomposition approximation analysis. Assume that all service nodes are highly utilized, which is usually realistic in industry.
3.4.1 Node Level Measures
According to Whitt [13][16], the expected waiting time approximation of node 𝑚 with parameter ( 𝜆𝑎𝑚, 𝐶𝑎𝑚2 , 𝜏𝑚, 𝐶𝑠𝑚2 ) as a 𝐺𝐼/𝐺/𝑀𝑚 queue based on the
heavy-traffic limit theorem is
𝐸[𝑊𝑚(𝐺𝐼/𝐺/𝑀𝑚)] =𝐶𝑎𝑚2 +𝐶2 𝑠𝑚2 𝐸[𝑊𝑚(𝑀/𝑀/𝑀𝑚)] (3.15)
where 𝐸[𝑊𝑚(𝑀/𝑀/𝑀𝑚)] is the expected waiting time for a 𝑀/𝑀/𝑀𝑚 queue,
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The expected cycle time of node 𝑚 is the sum of expected service time and expected
waiting time,
𝐸[𝐶𝑇𝑚(𝐺𝐼/𝐺/𝑀𝑚)] = 𝜏𝑚+ 𝐸[𝑊𝑚(𝐺𝐼/𝐺/𝑀𝑚)] (3.16)
From the Little formula, the expected number in node 𝑚 is 𝐸[𝑁𝑚] = 𝜆𝑎𝑚× 𝐸[𝐶𝑇𝑚(𝐺𝐼/𝐺/𝑀𝑚)],
individual nodes in the network. Average number of visits per product to node 𝑚 is 𝑉𝑚 =∑ 𝜆𝑎𝑚𝜆
𝑀 𝐸𝑛
𝑛=1 , 𝑓𝑜𝑟 𝑚 = 1,2, … 𝑀. (3.17)
Therefore, the expected cycle time of going through the network is
𝐸[𝑇] = ∑𝑀𝑚=1𝑉𝑚(𝜏𝑚+ 𝐸[𝑊𝑚]). (3.18)
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The total number of jobs in the whole network can be also obtained, 𝐸[𝑁] = 𝐸[𝑁1] + 𝐸[𝑁2] + ⋯ + 𝐸[𝑁𝑀],
= ∑ 𝜆𝑎𝑚(𝜏𝑚+ 𝐸[𝑊𝑚])
𝑀
𝑚=1
,
= 𝜆𝐸 ∑ 𝑉𝑚(𝜏𝑚+ 𝐸[𝑊𝑚])
𝑀
𝑚=1
,
= 𝜆𝐸× 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[𝑇𝑖] = ∑𝑆𝑘=1𝑖 (𝜏𝑖𝑘+ 𝐸[𝑊𝑚𝑖,𝑘]). (3.20)
where 𝐸[𝑊𝑚𝑖,𝑘] is the expected waiting time of the 𝑘𝑡ℎ processing step of product 𝑖
at node 𝑚𝑖,𝑘. And the cycle time variance for product 𝑖 can be calculated by
𝑉𝑎𝑟[𝑇𝑖] = ∑𝑆𝑘=1𝑖 𝜏𝑖𝑘2 𝐶𝑖𝑘2 + ∑𝑆𝑘=1𝑖 𝑉𝑎𝑟[𝑊𝑖,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
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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.
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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
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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
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(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 𝑖𝑡ℎ best design in 𝐷 according to {𝑓(𝑦),
𝑦 ∈ 𝐷};
𝐺−1(𝑖): the 𝑖𝑡ℎ best design in 𝐷 according to {𝑔(𝑦),
𝑦 ∈ 𝐷};
We represent the relationship between the detailed model and the simplified model by
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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
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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)40
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)41
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, … , 𝑥𝑁. Let (𝐹(𝑥𝑖), 𝐺(𝑥𝑖)) be a rank observations of design 𝑥𝑖 in the detailed model and the simplified model. Any pair of
observation (𝐹(𝑥𝑖), 𝐺(𝑥𝑖)) and (𝐹(𝑥𝑗), 𝐺(𝑥𝑗)) are concordant, if both 𝐹(𝑥𝑖) >
𝐹(𝑥𝑗) and 𝐺(𝑥𝑖) > 𝐺(𝑥𝑗) or if both 𝐹(𝑥𝑖) < 𝐹(𝑥𝑗) and 𝐺(𝑥𝑖) < 𝐺(𝑥𝑗). 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
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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.
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.