Thesis Organization - 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

Chapter 1 Introduction

1.4 Thesis Organization

The remaining thesis is organized as follows. Chapter 2 defines the machine capacity allocation problem in a re-entrant line and models the re-entrant line into an open queueing network model. Parametric decomposition method is introduced in Chapter 3 and two simplified models with different characterization of variability, QNA and JNA, are also discussed. In Chapter 4, motivated by the deficiencies of traditional simulation-based optimization approaches, ordinal transformation (OT), an OO-based approach, facilitates us significantly reduce the computation effort is described. Then, we analyze the goodness of using QNA or JNA as the simplified model for OT. By bound and ranking analysis (BRA), we investigate the probability of correct ranking in single GI/G/m queue. In Chapter 5, we extend the BRA of single GI/G/m queue to general queueing networks. An experiment of five-workstation re-entrant line with hundreds of designs is conducted in Chapter 6 and the ranking performances of QNA and JNA are compared. Chapter 7 concludes this thesis.

Chapter 2 Conveyor Problem: Re-entrant Line Capacity Allocation

In this chapter, machine capacity allocation problem in re-entrant lines will be defined as our conveyor problem for ordinal optimization. At first, re-entrant lines are characterized by the inclusion of feedback loops that allow products to visit some workstations more than once at different stages of processing. The challenges of machine capacity allocation in re-entrant lines are introduced and the complexity is also investigated. Then, we model the re-entrant lines as an open queuing network (OQN) with multiple product types, shared service workstations, and deterministic routing for each product. At last, the reasons of using re-entrant line capacity allocation as the conveyor problem for ordinal optimization are discussed.

2.1 Problem Description and Complexity Analysis

Re-entrant lines consist of multiple products, shared workstations with several identical machines, and predetermined processing routing through the network for each product. At each workstation, there not only the external arrival flows but also the internal re-entrant flows cause the fierce competition of machine capacity. It is

important to determine how best to allocate service capacity to each workstation so as to optimize various performance measures, such as the total cycle time(or makespan).

In actual manufacturing systems, there are several machine groups with different characteristics. Our research focuses on the machine allocation of one machine group and assume all machines in a machine group are identical and without the quality concern. Even though actual manufacturing systems have various machine groups and the assumption of identical machine here is not realistic, the major effect to this investigation is only the generation of all possible designs.

This scenario is often seen in daily operation planning of flexible manufacturing systems. Managers are responsible for meeting daily production goals and it is important to decide how allocate available machines to each workstation. However, the number of allocation designs depends on how many workstations in the network and how many available machines to be allocated. If there are M workstations and N machines, total number of possible allocation design grows in a combinatorial way. It is known that determining the optimal machine capacity allocations in re-entrant lines is a NP-hard problem. Therefore, how to determine the optimal allocation design of the machine capacity to each workstation in re-entrant lines is a significant and challenging problem and need an efficient methodology to solve [27].

2.2 Mathematical Abstraction of Machine Allocation Problem

Consider a production line illustrated in Figure 2.1. There are 𝑀 workstations and 𝐼 products. Machines in one workstation are identical and independent. The process routing of each product among workstations is predetermined. It is assumed that there is an infinite buffer for each workstation. As each workstation can be visited more than once at different processing stages of one product, the line is re-entrant.

Figure 2.1 Re-entrant Lines

2.2.1 Open Queuing Network Modeling

For analysis of a re-entrant line, we use an open queuing network(OQN) to model the re-entrant line. In OQN modeling, a workstation 𝑚 of 𝑀_𝑚 identical

machines is represented as a service node of parallel machines with infinite buffer, 𝑚=1,2,…,𝑀. The total number of machines is 𝑁, 𝑀₁+ 𝑀₂+ ⋯ + 𝑀_𝑀 = 𝑁.

Each product type 𝑖 has a total of 𝑆_𝑖 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, … , 𝑆_𝑖}. Step (𝑖, 𝑘) is processed by the service node 𝑚_𝑖𝑘 ∈ {1,2, … , 𝑀}. The service time of each node varies with processing steps

and service times of individual product types at a given node are assumed independent of each other. Arrival processes of individual product types to the network are assumed independent and identically distributed with a general distribution. First-Come-First-Serve (FCFS) is the service discipline of each node.

The re-entrant line is modeled as an open queuing network as follows:

(1) Multiple server nodes: Each workstation with multiple machines is modeled as one multiple-server node.

(2) External arrivals: Products loaded into the re-entrant line for processing constitute the arrival to the network.

(3) General arrival processes: Product arrival processes to each node are described by the specific probability distribution of product inter-arrival times. Inter-arrival times of different product types are assumed individual and identically distributed.

(4) General service processes: Service processes are specified by general distributions.

Service time distribution varies with different processing steps of different products. And the service time distribution at a node is independent of the other.

(5) Deterministic routing: Processing routings of individual product types are fixed.

(6) Service discipline: The service discipline of each node is FCFS.

2.2.2 Formulation: Nonlinear Integer Programming

Let us formalize the machine capacity allocation problem in re-entrant lines.

Suppose we have 𝑀 workstations, 𝑁 machines to be allocated, and 𝐼 product types, our objective is to find an optimal machine allocation design which minimizes the average of mean cycle time of each product.

Definition: Machine allocation design

A machine allocation design is specified by the number of machines in each workstation and represented by a set with 𝑀 elements. The 𝑖^𝑡ℎ element in this set stands for that there are 𝑀_𝑖 machines to be allocated at workstation 𝑖. And every machine must be allocated to a workstation. Therefore, the machine allocation design

indexed by 𝑘 can be written as 𝐷_𝑘 = {𝑀₁^𝑘, 𝑀₂^𝑘, … , 𝑀_𝑀^𝑘} where 𝑀₁^𝑘+ 𝑀₂^𝑘+ ⋯ + 𝑀_𝑀^𝑘 = 𝑁.

Definition: Machine capacity allocation problem

Our objective is to find an optimal machine capacity allocation design 𝐷_𝑘^∗ = {𝑀₁^𝑘^∗, 𝑀₂^𝑘^∗, … , 𝑀_𝑀^𝑘^∗}, to minimize the average of mean cycle time of each

product, where

𝑘^∗ = argmin

𝑘

𝐼∑ MCT_𝑖(𝐷_𝑘)

𝐼

𝑖=1

, 𝐷_𝑘 ∈ 𝐷

and MCT_𝑖(𝐷_𝑘) denotes the mean cycle time of product type 𝑖 under a specific allocation design 𝐷_𝑘.

Because MCT_𝑖(𝐷_𝑘) is nonlinear with respect to 𝐷_𝑘, each element in 𝐷_𝑘 is the number of machines in each workstation and must be an integer. Therefore, this machine capacity allocation problem is formulated as a nonlinear integer programming problem.

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.

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 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.

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

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 𝑆_𝑖 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, … , 𝑆_𝑖}. Step (𝑖, 𝑘) is processed by the service node 𝑚_𝑖,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:

𝜆_𝑖: external arrival rate of product type 𝑖;

𝐶_𝑖²: inter-arrival time SCV of product type 𝑖;

𝛿_𝑖𝑗 = { 1, product type i externally entering the network at node j.

0, otherwise.

𝜆_𝐸 : aggregate external mean arrival rate;

𝐶_𝐸² : inter-arrival time SCV of aggregate external arrivals;

𝜆_𝑚𝑛 : aggregate mean arrival rate from node 𝑚 to node 𝑛;

𝜏_𝑚 : aggregate mean service time at node 𝑚;

𝐶_𝑠𝑚² : aggregate service time SCV at node 𝑚;

𝜏_𝑖𝑘 : mean service time at 𝑘-th step of product type 𝑖;

𝐶_𝑖𝑘² : service time SCV at 𝑘-th step of product type 𝑖;

𝑄={𝑞_𝑚𝑛}: routing matrix, and 𝑞_𝑚𝑛 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,

𝜆_𝐸𝑛 = ∑^𝐼_𝑖=1𝜆_𝑖𝛿_𝑖𝑛. (3.1)

As the external arrivals of products are independent, the inter-arrival time SCV of

aggregate external arrivals is 𝐶_𝐸𝑛² = ∑ 𝐶_𝑖^{2 𝜆}_𝜆^𝑖^𝛿^𝑖𝑛

𝐸𝑛

𝐼𝑖=1 . (3.2)

The aggregate mean arrival rate from node 𝑚 to node 𝑛 is

𝜆_𝑚𝑛 = ∑^𝐼_𝑖=1∑^𝑆_𝑘=1^𝑖⁻¹𝜆_𝑖𝟏{𝑚_𝑖,𝑘 = 𝑚, 𝑚_𝑖,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

𝑞_𝑚𝑛 = ^𝜆^𝑚𝑛

∑^𝐼_𝑖=1∑^𝑆𝑖_𝑘=1𝜆_𝑖𝟏{𝑚_𝑖,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,

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)

and the corresponding SCV at node 𝑚 is

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)

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:

𝑀_𝑚 : number of machines in node 𝑚;

𝜆_𝑎𝑚 : mean total arrival rate to node 𝑚;

𝐶_𝑎𝑚² : inter-arrival time SCV at node 𝑚;

𝐶_𝑑𝑚² : inter-departure time SCV at node 𝑚;

𝐶_𝑚𝑛² : inter-departure time SCV for the flow transiting from node 𝑚 to node 𝑛;

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

𝜆_𝑎𝑛 = ∑^𝐼_𝑖=1𝜆_𝑖𝛿_𝑖𝑛+ ∑^𝑀_𝑚=1𝜆_𝑚𝑛 = ∑^𝐼_𝑖=1𝜆_𝑖𝛿_𝑖𝑛+ ∑^𝑀_𝑚=1𝜆_𝑎𝑚𝑞_𝑚𝑛 (3.7)

Where 𝛿_𝑖𝑛=1 if product type 𝑖 entering the network at node 𝑛. Equation (3.7) is known as the traffic rate equations with 𝜆_𝑚𝑛 as defined by equation (3.3). In

equation (3.7), there are 𝑀 equalities with 𝑀 unknown variables { 𝜆_𝑎𝑛, 𝑛 = 1,2, … , 𝑀}, so the 𝜆_𝑎𝑛 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, 𝜌_𝑛 =^𝜆^𝑎𝑛_𝑀^𝜏^𝑛

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,

𝜌_𝑛 < 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 𝐶_𝑎𝑛² = 1 − 𝜔̃ + 𝜔_𝑛 ̃_𝑛^𝜆^𝐸𝑛_𝜆^𝐶^𝐸𝑛²

𝑎𝑛 + 𝜔̃ ∑_𝑛 ^𝜆_𝜆^𝑚𝑛

𝑎𝑛

𝑀𝑚=1 𝐶_𝑚𝑛², (3.9)

where 𝜔̃ = [1 + 4(1 − 𝜌_𝑛 _𝑛)²(𝑣_𝑛− 1)]⁻¹ and 𝑣_𝑛 = [∑ (^𝜆_𝜆^𝑚𝑛

𝑎𝑛)²

𝑀𝑚=1 ]⁻¹.

However, the approximate variability parameter of the sub-flow from node 𝑚 to node 𝑛, 𝐶_𝑚𝑛², 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={𝑞_𝑚𝑛}. 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, 𝐶_𝑚𝑛², 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 𝑞_𝑚𝑛, which is independent of the current state and history of the network. The routing matrix 𝑄={𝑞_𝑚𝑛} 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, 𝐶_𝑚𝑛², under Markovian routing is proposed by Whitt[13],

𝐶_𝑚𝑛² = 𝑞_𝑚𝑛𝐶_𝑑𝑚²+ (1 − 𝑞_𝑚𝑛) (3.10)

Because of the independency of Markovian routing, if all the external arrival

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,

𝐶_𝑚𝑛² = 𝑞_𝑚𝑛𝐶_𝑑𝑚²+ (1 − 𝑞_𝑚𝑛)𝑞_𝑚𝑛𝐶_𝑎𝑚²+ (1 − 𝑞_𝑚𝑛)²𝐶_𝑒𝑚², (3.11)

where 𝐶_𝑒𝑚² is an average of the external arrival-process variability parameters, 𝐶_𝑒𝑚² = ∑ 𝐶_𝑖²(_∑^∑^𝑆^𝑘=1_∑ ^𝜆^𝑖^{1{(𝑖,𝑘):𝑚}_𝜆 ^𝑖,𝑘^=𝑚}

𝑆 𝑖

𝑘=1 1{(𝑖,𝑘):𝑚_𝑖,𝑘=𝑚}

𝐼𝑖=1 )

𝐼𝑖=1 . (3.12)

And Whitt[13] suggested that the SCV of the departure process at node 𝑚 by 𝐶_𝑑𝑚² = 1 + (1 − 𝜌_𝑚²) (𝐶_𝑎𝑚²− 1) +^𝜌^𝑚

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 , 𝐶_𝑚𝑛², by Equation (3.11). By substituting

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 (𝜆_𝑎𝑚, 𝐶_𝑎𝑚² , 𝜏_𝑚, 𝐶_𝑠𝑚² ) 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, (𝜆_𝑎𝑚, 𝐶_𝑎𝑚² , 𝜏_𝑚, 𝐶_𝑠𝑚² ), 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 ( 𝜆_𝑎𝑚, 𝐶_𝑎𝑚² , 𝜏_𝑚, 𝐶_𝑠𝑚² ) as a 𝐺𝐼/𝐺/𝑀_𝑚 queue based on the

heavy-traffic limit theorem is

𝐸[𝑊_𝑚(𝐺𝐼/𝐺/𝑀_𝑚)] =^𝐶^𝑎𝑚² ^+𝐶₂ ^𝑠𝑚² 𝐸[𝑊_𝑚(𝑀/𝑀/𝑀_𝑚)] (3.15)

where 𝐸[𝑊_𝑚(𝑀/𝑀/𝑀_𝑚)] is the expected waiting time for a 𝑀/𝑀/𝑀_𝑚 queue,

The expected cycle time of node 𝑚 is the sum of expected service time and expected

waiting time,

𝐸[𝐶𝑇_𝑚(𝐺𝐼/𝐺/𝑀_𝑚)] = 𝜏_𝑚+ 𝐸[𝑊_𝑚(𝐺𝐼/𝐺/𝑀_𝑚)] (3.16)

From the Little formula, the expected number in node 𝑚 is 𝐸[𝑁_𝑚] = 𝜆_𝑎𝑚× 𝐸[𝐶𝑇_𝑚(𝐺𝐼/𝐺/𝑀_𝑚)],

individual nodes in the network. Average number of visits per product to node 𝑚 is 𝑉_𝑚 =_∑ ^𝜆^𝑎𝑚_𝜆

𝑀 𝐸𝑛

𝑛=1 , 𝑓𝑜𝑟 𝑚 = 1,2, … 𝑀. (3.17)

Therefore, the expected cycle time of going through the network is

𝐸[𝑇] = ∑^𝑀_𝑚=1𝑉_𝑚(𝜏_𝑚+ 𝐸[𝑊_𝑚]). (3.18)

The total number of jobs in the whole network can be also obtained, 𝐸[𝑁] = 𝐸[𝑁₁] + 𝐸[𝑁₂] + ⋯ + 𝐸[𝑁_𝑀],

= ∑ 𝜆_𝑎𝑚(𝜏_𝑚+ 𝐸[𝑊_𝑚])

𝑀

𝑚=1

= 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

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 27-0)