Multi-Job Production Systems: Definition,Problems, Analysis, and Product-MixPerformance Portrait of Serial Lines

P. Alaviana, P. Dennob, and S. M. Meerkova

aDepartment of Electrical Engineering and Computer ScienceUniversity of Michigan

Ann Arbor, MI, 48109-2122bSystems Integration Division, Engineering Laboratory

National Institute of Standards and TechnologyGaithersburg, MD 20899-1070

Abstract

Multi-job production (MJP) is a class of flexible manufacturing systems intended to produce dif-

ferent products (job-types) according to a given product-mix and build-schedule. In MJP systems,

all job-types are processed by the same sequence of manufacturing operations, but with different

processing times at some or all machines. To characterize MJP, we introduce the work-based (rather

than the traditional part-based) model of production systems, which is “insensitive” to whether a

single- or multi-job manufacturing takes place. Using this model, we develop a method for per-

formance analysis of MJP serial lines with the emphasis on their throughput and bottlenecks as

functions of the product-mix. We show, in particular, that for the so-called conflicting job-types,

there exists a range of product-mix, where the throughput of MJP is larger than that of any individ-

ual job-type involved. To characterize the global behavior of MJP systems, we introduce the notion

of Product-Mix Performance Portrait, which represents the system throughput and bottlenecks for

all feasible product-mixes. Finally, we apply the results obtained to a section of the underbody

assembly system at an automotive assembly plant, calculate its performance portrait, and evaluate

the efficacy of potential continuous improvement projects.

Keywords: Multi-product manufacturing; Flexible production systems; Product-mix; Serial lines;

Exponential machines; Finite buffers; Work-based model; Throughput; Bottleneck; Performance por-

trait; Automotive assembly.

1 Introduction

1.1 Multi-job production systems: description and definition

Multi-job production (MJP) is a class of flexible manufacturing systems, intended to produce different

products (or job-types) within the same production system. These systems are often used in product

assembly, e.g., in automotive assembly plants, engine and battery plants, computer and appliance

assembly, etc.

To illustrate MJP operation, consider an automotive assembly plant manufacturing two car models,

A and B. In each area of the plant, i.e., body shop, paint shop, and final assembly, each job-type follows

the same sequence of manufacturing operations. Let r � prA, rBq be the product-mix, where rA is the

fraction of automobiles A to be manufactured and rB � 1�rA is that of B. The jobs are released one-

by-one into the body shop in a sequence defined by a build-schedule and then proceed to the paint shop

and final assembly. For instance, a segment of a release sequence may be � � �BAAABABAAB � � � ,

generated by the product-mix with rA � 2{3 and rB � 1{3. The jobs are transported from one

operation to another (typically, by conveyors, which serve also as buffers) in the sequence of release.

Each job-type is processed by the machines (operations or stations) with zero (or practically zero)

setup time, but requires different processing time at some or all machines.

Based on the above, the class of MJP systems is defined as follows:

• The jobs are released one-by-one, according to a given product-mix and build-schedule.

• All jobs undergo identical sequence of manufacturing operations, but require different amount

of work at some or all operations.

• The setup times are zero.

• In-process buffers are non-dedicated (i.e., store different job-types in the sequence of arrival).

• The jobs are processed on the first-come first-served basis.

• The machines are unreliable and experience random breakdowns.

• The processing time of each machine is deterministic and job-dependent.

The MJP systems can be classified into two groups: serial lines and assembly systems (see Figures

1.1 and 1.2, where the circles represent the machines, the rectangles are the buffers, and J1 and J2

2

Figure 1.1: MJP serial line

Figure 1.2: MJP assembly system

denote the two job-types being manufactured). While the serial lines may (and, often do) comprise

assembly operations, it is assumed that either no starvations by subassemblies take place or the

probabilities of starvations are given along with other system parameters.

In MJP assembly systems, the subassembly components are manufactured by subassembly lines,

which are a part of the overall system at hand and which, as shown in Figure 1.2, may operate in a

single-job production (SJP) or MJP regime and have either dedicated finished goods buffers (FGB) for

different job-types or non-dedicated ones. Therefore, in addition to the above, MJP assembly systems

are characterized by the following:

• MJP assembly systems consist of a main assembly line (MA) and subassembly lines (SAi); while

the former operates in MJP regime, the latter may be either MJP or SJP.

• Subassembly lines may have dedicated or non-dedicated FGBs.

• Job release in the main and subassembly lines may be coordinated (i.e., synchronized) in accor-

dance with the build schedule or not; the former case is referred to as build to schedule and the

latter build to finished goods buffer.

It should be pointed out that the term MJP is not a standard one. It can be viewed as a part of the

3

general field of flexible multi-product manufacturing. However, the term flexibility is used in several

different meanings, including production system flexibility, machine tool flexibility, material handling

flexibility, routing flexibility, etc. To emphasize that we address only production system flexibility

with job-dependent work-requirements and variable product-mix, we use the term MJP.

1.2 MJP problems and goal of the paper

In practice, MJP systems often perform far below their capacity: throughput losses of 15% and more

are not unusual. Recovering these losses is an important industrial problem. Unfortunately, very

little engineering knowledge is available on how it can be achieved. The literature on MJP is quite

limited (see Subsection 1.4 for a review). Therefore, owing to their industrial importance, rigorous

engineering methods for analysis, continuous improvement, and design of MJP systems are necessary

from the point of view of both theoretical and practical considerations.

To develop these methods, the following problems must be addressed:

• Performance analysis: Given the machine, buffer, and job parameters as well as the product-mix,

calculate the system performance characteristics (e.g., throughput, work-in-process, probabilities

of machine blockages and starvations, etc.)

• Continuous improvement: Given the machine, buffer, and job parameters as well as the product-

mix, determine the system bottlenecks and indicate a way to alleviate them and quantify the

resulting performance improvement.

While these problems arise in SJP systems as well, the next five problems are MJP-specific:

• Product-mix feasibility: Given an MJP assembly system, quantify the set of achievable (i.e.,

feasible) product-mixes.

• Product-mix optimization: Given an MJP system, quantify the product-mix leading to the

throughput maximization.

• Product-mix assignment: Assuming that a company has several plants capable of manufacturing

the products in question and given the desired company-wide product-mix, assign a product-

mix to each plant so that the overall throughput is maximized, while the required company-wide

product-mix is maintained.

4

• Job sequencing: Develop a method for selecting a build-schedule, which achieves the throughput

corresponding to the assigned product-mix, while satisfying sequence-constrained conditions, if

any.

• Structural analysis: Investigate robustness of MJP assembly systems with respect to starvations

by subassembly lines. In particular, determine the optimal structure of subassembly lines leading

to starvation minimization of the main line.

The goal of the current paper is to address the first two problems (i.e., performance analysis and

continuous improvement) in the framework of MJP serial lines; for MJP assembly systems, these

problems will be considered elsewhere.

1.3 Approach

The approach of this paper is based on a novel work-based model, instead of the traditional part-based

model, of production systems. This implies that, unlike the traditional approach, where the analyses

are carried out in terms of ‘parts produced’, in this work the research is carried out in terms of the

‘work produced’, which is insensitive to whether a single- or multi-job manufacturing takes place.

Given the work produced, the throughput of each job-type and other performance metrics can be

calculated using the product-mix.

More precisely, in the work-based model (see Section 2 for details), the machines are defined by

the amount of work they can carry out per unit of time. The jobs are defined by the amount of work

they require at each machine. For instance, a welding operation is defined by the maximum number of

welds it can carry out per unit of time, and the jobs are specified by the number of welds they require

per job-type. The reliability characteristics of the machines in the work-based model remain the same

as in the part-based case, i.e., defined by distributions of up- and downtimes. The buffer capacity in

the work-based model also remains the same as in the part-based case.

This model provides a foundation for solving the problems addressed in this paper.

1.4 Related literature

The literature related to MJP systems can be classified into two groups: multi-class queuing systems

and flexible multi-product manufacturing systems. Each of these groups is briefly characterized below

and variances with the research addressed in this paper are indicated.

5

Representative papers of the vast literature on multi-class queuing systems are [1]-[10]. In these

publications, jobs of various types arrive in a random manner (typically, defined by Poisson distri-

bution) and are processed by servers having a random (typically, exponential) processing time. The

servers are connected according to various topologies. Problems considered are performance analysis,

job scheduling, and system-theoretic properties (e.g., stability, monotonicity, and reversibility). Since

the systems topology, job classes, and statistics of jobs arrival and processing are motivated by com-

munication networks, the results obtained are not applicable to MJP systems defined in Subsection

1.2.

The literature related to the second group, devoted to flexible multi-product manufacturing, is

also quite extensive, with the burst of activity occurring in 1980-2000 (see reviews [11]-[20]). By the

end of this period, it became clear that the ideas of flexibility have not been adopted in industry as

widely as originally expected, with the exception of flexibility in assembly. This happened, perhaps,

because flexibility in assembly is relatively easy to implement (due to innately short or even zero

set-up times), while flexibility in machining requires sophisticated and expensive equipment in order

to ensure sufficiently short set-ups. Nevertheless, for the sake of completeness, we briefly overview

below the results obtained to-date in various areas of research on production flexibility.

The publications on flexible multi-product systems can be classified into two subgroups: design

of flexible systems and their performance analysis. Representative papers on design include [21]-[29]

and several chapters of monographs [30] and [31]. The main issues addressed are machine layout,

flexible material handling, structures of inventory storage, parts grouping, flexible planning, etc. The

approaches are typically based on optimization. The results obtained provide guidance for designing

flexible machining and assembly operations.

The literature on performance analysis is less extensive. Namely, references [32]-[39] use queuing

theory methods to investigate the performance of flexible manufacturing systems and quantify the

effects of flexibility. Papers [40]-[42] use a decomposition approach to compute the throughput and

buffer occupancy in linear and nonlinear multi-product manufacturing systems. Finally, [43]-[46] use

the aggregation approach to investigate the throughput and bottlenecks of multi-product serial lines

and report an application at a furniture manufacturing plant.

In summary, while the results reported in [1]-[46] provide powerful methods for designing and

6

investigating flexible multi-product systems, the issue of product-mix and its effect on the system

performance has not been explored. This issue, i.e., the performance characteristic of flexible system

as a function of product-mix, is addressed in the current paper.

1.5 Paper outline and abbreviations/notations

The remainder of this paper is structured as follows: Section 2 introduces the work-based model. In

Section 3, this model is used to investigate certain properties of SJP systems, which are not explicitly

revealed by the part-based model. In Section 4, the work-based model is used for throughput and

bottleneck investigation of MJP lines for a given product-mix. In Section 5, qualitative and quantita-

tive behavior of the throughput and bottlenecks as functions of product-mix is investigated. Section 6

introduces the notion of the Product-Mix Performance Portrait, which is intended to represent global

properties of MJP systems and thereby enhance the decision-making process by operations manage-

ment. An application of the results obtained to a section of the underbody assembly system at an

automotive assembly plant is described in Section 7. Finally, Section 8 formulates the conclusion and

topics for future research. All proofs are included in the Appendix.

Throughout this paper, the following abbreviations and notations are used:

Abbreviations: BN � bottleneck; FGB � finished goods buffer; JPH � jobs per hour; MA � main

assembly line; MJP � multi-job production; PP � performance portrait; PSE � production systems

engineering; SA � subassembly line; SJP � single-job production.

Notations: b � buffer; BL � probability of blockage; c � machine capacity; e � machine efficiency;

λ � machine breakdown rate; M � number of machines in a serial line; m � machine; µ � machine

repair rate; N � buffer capacity; PR � production rate; r � rr1, � � � , rSs � product-mix; S � number

of job-types; smci � system-modified capacity of machine i; ST � probability of starvation; τ �

machine cycle time; TPj � throughput of job j; TPv � throughput of work-equivalent virtual job; tpi

� stand-alone throughput of machine i; Wi � work capacity of mi; wij � work-requirement of job j

at mi; WIPi � work-in-process in bi.

2 Work-Based Model

The work-based model of a serial line operating in MJP regime is defined as follows:

7

(i) Each machine mi, i � 1, � � � ,M , is characterized by its work capacity, Wi (in units of work/min).

(ii) In addition, each machine is characterized by its breakdown and repair rates, λi and µi (in units

of 1/min), respectively; this implies that the machines are exponential with the average up- and

downtime given by Tup,i � 1λi

and Tdown,i � 1µi. Therefore, the machine efficiency is ei � µi

λi�µi.

(iii) The buffers are not dedicated and each buffer, bi, i � 1, � � � ,M�1, is characterized by its storing

capacity, Ni (finite or infinite).

(iv) Each job-type, Jj , j � 1, � � � , S, is characterized by its work-requirements wij , i � 1, � � � ,M ; j �

1, � � � , S, (in units of work/job), i.e., by the vector of work-required, wj � rw1j , � � � , wMjs. The

set-up time of each job is zero.

(v) The jobs are released one-by-one according to a build-schedule, characterized by the vector

of product-mix, r � rr1, � � � , rSs,°Sj�1 rj � 1, where rj is the fraction of job-type Jj to be

manufactured. The release sequence is formed by releasing each job-type j with probability rj ,

j � 1, � � � , S.

(vi) Machine mi, i � 2, � � � ,M is starved, if buffer bi�1 is empty; machine mi, i � 1, � � � ,M � 1 is

blocked, if buffer bi is full and machinemi�1 does not take the material from this buffer. Machine

m1 is never starved and mM is never blocked.

Discussion: (a) The machine work capacity, Wi, is defined by the technological operation it carries

out. In addition to the welding operation mentioned in Subsection 1.3, it can be the feed rate of a

cutting instrument in turning, milling or drilling; the rate of etching or material deposition in semi-

conductor manufacturing; the number of assembly steps carried out in a robotic or manual assembly

operation per unit of time, etc. The units of work in job work-requirements, wij , are the same as in

the corresponding machines (but in terms of work per job-type, rather than work per unit of time).

(b) Although assumption (ii) calls for exponential machines, other reliability models, e.g., Weibull,

gamma, and log-normal with coefficients of variation, CV ¤ 1, can be considered as well (using the

approach developed in [47]).

(c) As it follows from (i) and (iv), the time necessary to process a job-type j on machine i (i.e.,

the cycle time of machine i for processing job j) is

τij �wijWi

, i � 1, � � � ,M ; j � 1, � � � , S. (2.1)

8

While in the part-based model the cycle time is an independent variable, (2.1) indicates that in the

job-based model it is not: wij and Wi are the independent variables. This allows for investigating the

effect of the job work-requirements on the system’s throughput and bottleneck.

(d) The random release, mentioned in assumption (v), is introduced for simplicity. Also, for

simplicity we assume that there are no sequence-dependent constraints in job release. The case study

considered in Section 7 of this paper satisfies these assumptions.

(e) The performance metrics of production systems in the framework of work-based model are as

follows: TPj , j � 1, � � � , S � the average number of jobs of type j produced by the last machine per

unit of time; TP �°Sj�1 TPj � the average total number of jobs produced by the last machine per

unit of time; WIPi, i � 1, � � � ,M � 1 � the average number of jobs in buffer bi; STi, i � 2, � � � ,M �

the probability that machine mi is starved; BLi, i � 1 � � � ,M � 1 � the probability that machine mi

is blocked. Additionally a performance metric, applicable to synchronous SJP, is PR � the average

total number of jobs produced by the last machine per cycle time.

(f) The model (i)-(vi) can be used for SJP systems as well. In this case, S � 1 and wij � wi. If

wiWi

� const,@i, (2.2)

the SJP system is called synchronous; otherwise, it is asynchronous.

3 Performance and Bottleneck Analysis of SJP Serial Lines Using

Work-based Model

In this section, we apply work-based model (i)-(vi) to performance analysis of SJP lines. The reason

is two-fold: First, it reveals properties of serial lines not exposed by the part-based model (see,

for instance, monographs [31], [48]-[57]). Second, we use the results of this section for performance

investigation of MJP systems described in Section 4.

3.1 Performance analysis

Theorem 3.1. Consider a synchronous two-machine SJP line defined by assumptions (i)-(vi). Assume

for simplicity that Wi � W,wi � w, i � 1, 2. Then the performance metrics of this line as functions

9

of the job work-requirement are given by:

PR pwq � e2

�1 �Q

�λ1, µ1, λ2, µ2, N

w

W

�, (3.1)

ST2 pwq � e2Q�λ1, µ1, λ2, µ2, N

w

W

, (3.2)

BL1 pwq � e1Q�λ2, µ2, λ1, µ1, N

w

W

, (3.3)

where

Q�λ1, µ1, λ2, µ2, N

w

W

�

$''''&''''%

p1�e1qp1�φq1�φe�βN

wW, if e1 � e2,

λ1pλ1�λ2qpµ1�µ2qpλ1�µ1qrpλ1�λ2qpµ1�µ2q�λ2µ1pλ1�λ2�µ1�µ2qN

wWs , if e1 � e2,

(3.4)

and

ei �µi

µi � λi, φ �

e1p1 � e2q

p1 � e1qe2, β �

pλ1 � λ2 � µ1 � µ2qpλ1µ2 � λ2µ1q

pλ1 � λ2qpµ1 � µ2q. (3.5)

Proof: Follows the proof of formulas (11.13)-(11.17) in [57, Section 11.1] and is omitted here. The

expression for WIP pwq can also be derived, which turns out to be the same as (11.14) in [57], but

with N wW instead of N .

Discussion: (a) Using (3.4), it is possible to show that function Q is monotonically deceasing in w.

Thus, as it follows from (3.1), PR is an increasing function of w. Similar result can be obtained for

two-machine synchronous lines with non-identical machine work-capacities and jobs having different

work-requirements, as long as (2.2) is satisfied.

(b) Using the aggregation procedures of [57, Section 11.1] with Q from (3.4), it is possible to

show that the above effect takes place for synchronous lines with M ¡ 2 as well (see Figure 3.1

for an illustration). Thus, the work-based model demonstrates that serial lines are more effective

producing parts with larger work-requirements than those with smaller ones. This phenomenon takes

place because the machine downtime in units of job cycle time is smaller for jobs with larger work-

requirements.

(c) The above conclusion does not necessarily imply that TP is an increasing function of w (since

TP is proportional to Ww PR pwq, and, therefore, TP may be decreasing even if PR is increasing).

However, the efficacy of buffers, as a tool for disturbance rejection, is indeed increasing with w.

10

w

1 2 3 4 5

PR(w

)

0.82

0.83

0.84

0.85

0.86

0.87

0.88

Figure 3.1: Production rate of 5-machine synchronous exponential SJP line as a function of job work-requirement pλi � 0.1, µi � 0.9,Wi � 1, i � 1, � � � , 5 and N � r2, 2, 2, 2sq

(d) While Theorem 3.1 addresses synchronous lines, a similar theorem can be proved for asyn-

chronous lines as well. In this case, the performance of two-machine SJP line is characterized by

expressions (11.40), (11.41), (11.43), (11.45) of [57] with ci replaced by Wiwi

.

(e) The performance of SJP asynchronous lines with more than two machines, can be analyzed

using the aggregation procedure of [57, Section 11.2], leading to a conclusion similar to that of item

(b) above.

Note that all performance metrics of SJP synchronous and asynchronous lines in the framework of

work-based model can be evaluated using the PSE Toolbox, [58]. Indeed, although this toolbox has

been developed for the part-based model, it can be used for the work-based model as well by modifying

Ni into NiwiWi

for synchronous lines and using Wiwi

instead of ci for asynchronous ones.

3.2 Bottlenecks and their identification

In [57, Section 13.2], the bottleneck machine (BN) of an SJP serial line has been defined as the machine,

mi, having the largest effect on the system throughput quantified as

BTP

Bci¡

BTP

Bcj, @j � i, (3.6)

where ck � 1{τk is the capacity of machine mk, and τk is its cycle time.

Since in the work-based model τk � wkWk

, and the only variable, which characterizes the machine is

Wk, expression (3.6) becomes

wiBTP

BWi¡ wj

BTP

BWj, @j � i, (3.7)

This implies that the bottleneck is defined not only by the machines (i.e., their work capacity), but by

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(a) w1 � r1.0, 1.2, 1.1, 1.0, 1.2s (b) w2 � r1.0, 1.0, 1.1, 1.2, 1.2s

Figure 3.2: Bottlenecks of SJP line as a function of job work-requirements.

the job work-requirements as well. In other words, in the same system, different job-types may have

different bottlenecks.

It turns out, however, that for a given job-type, the BN identification in SJP systems can be carried

out using the same procedure as that of [57, Section 13.2] for the part-based model. This procedure

consists of the following:

• Calculate or measure on the factory floor blockages and starvations of all machines in the system,

BLi, i � 1, � � � ,M � 1, and STi, i � 2, � � � ,M .

• Place arrows between machines mi and mi�1 according to the following rule: If BLi ¡ STi�1,

the arrow is directed from mi to mi�1, i � 1, � � � ,M � 1; if BLi   STi�1, the arrow is directed

from mi�1 to mi.

• If there is a single machine with no emanating arrows, it is the BN. If there are multiple machines

with no emanating arrows, the one with the largest severity is the BN, where the severity of the

bottleneck, Si, is defined as

S1 � |BL1 � ST2|,

Si � |BLi�1 � STi| � |BLi � STi�1|, i � 2, � � � ,M � 1, (3.8)

SM � |BLM�1 � STM |.

Example 3.1. Consider a five-machine SJP line with λi � 0.1, µi � 0.9, i � 1, � � � , 5, W �

r1.2, 1.0, 1.1, 1.05, 1.2s, and N � r3, 5, 1, 1s. If this line produces job J1 with work-requirements

w1 � r1.0, 1.2, 1.1, 1.0, 1.2s, the bottleneck is machine m2 (see Figure 3.2(a)); if job J2 with work-

requirement w2 � r1.0, 1.0, 1.1, 1.2, 1.2s is manufactured, the bottleneck is machinem4 (Figure 3.2(b)).

This illustrates the effect of w on the bottleneck position (as quantified in (3.7)).

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4 Performance and Bottleneck Analysis of MJP Serial Lines Using

Work-based Model

4.1 Approach

The approach to performance analysis of MJP serial lines, developed in this paper, is based on reducing

MJP to SJP. This is accomplished by introducing the notion of virtual job and analyzing the resulting

virtual SJP line (denoted as SJPv) using the method of Section 3.

Specifically, given an MJP serial line defined by model (i)-(vi), this approach is described by the

following procedure:

Stage 1: Calculate the work-requirement of the virtual job,

wi,v :�S

j�1rjwij , (4.1)

and introduce the virtual SJP line (denoted as SJPv) consisting of the original machines and

buffers, but manufacturing the virtual job.

Stage 2: Estimate the performance metrics of SJPv line, using the method of Section 3 (based on the

aggregation procedure of [57, Section 11.2]), i.e., evaluateyTP v,{WIP i,v, xST i,v, andyBLi,v.Stage 3: Estimate the performance metrics of the original MJP line according to:

yTP j � rjyTP v, j � 1, � � � , S,

{WIP i �{WIP i,v, i � 1, � � � ,M � 1,

xST i � xST i,v, i � 2, � � � ,M, (4.2)

yBLi �yBLi,v, i � 1, � � � ,M � 1.

Each of these stages may introduce errors in the performance metrics evaluation. These errors are

investigated next.

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Figure 4.1: Flow diagram of MJP performance evaluation and the corresponding errors

4.2 Accuracy

4.2.1 Preliminaries

Figure 4.1 illustrates the above procedure by the three-stage diagram. As mentioned, using (4.1),

Stage I transfers the original MJP line (with the performance metrics TP �°Sj�1 TPj , WIPi, STi,

BLi) into the virtual line SJPv (with the performance metrics TPv, WIPi,v, STi,v, BLi,v). Using

the recursive aggregation procedure of [57, Section 11.2], Stage II transfers SJPv into ySJPv (with

the performance metrics yTP v, {WIP i,v, xST i,v, yBLi,v). Finally, using (4.2), Stage III transfers ySJPvinto zMJP (with the performance metrics yTP �

°Sj�1yTP j , {WIP i, xST i, yBLi). We denote the errors

introduced by each stage as εIX , εII

X , and εIIIX (see Figure 4.1), where the subscript ‘X’ stands for one

of the four performance metrics. In order to quantify the errors introduced by Stage III, we evaluate

the errors denoted as εTX (see Figure 4.1), where ‘T’ stands for the total error between the metrics of

MJP and zMJP. For Stage I, these errors are defined as follows:

εITP �

|TP � TPv|

TP� 100%,

εIWIP �

1M � 1

M�1¸i�1

|WIPi �WIPi,v|

Ni� 100%,

εIST �

1M � 1

M

i�2|STi � STi,v|, (4.3)

εIBL �

1M � 1

M�1¸i�1

|BLi �BLi,v|.

Errors εIIX , εIII

X , and εTX are defined similarly. In this section, we evaluate these errors using simulations

for Stage I and simulations/calculations for Stages II and T; then εIX , εII

X , and εTX , provide information

about the errors of Stage III.

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In addition to evaluating errors in performance metrics, we evaluate the discrepancies in bottleneck

identification of the first three systems in Figure 4.1. (Note that, as it follows from (4.2), the bottleneck

of the forth system is the same as that of the third one.) This is accomplished as follows: Denote

these bottlenecks as BN, BNv, andyBNv, respectively, identify them using the arrow-based method of

Section 3, and quantify the discrepancies among them by:

εIBN �

°Kk�1 I

Ipkq

K� 100%,

εIIBN �

°Kk�1 I

IIpkq

K� 100%, (4.4)

εTBN �

°Kk�1 I

Tpkq

K� 100%,

where k P t1, � � � ,Ku is the index of the line analyzed and IIpkq is the indicator function taking value

0 when BN � BNv and 1 otherwise. The indicator functions IIIpkq and ITpkq are defined similarly

in terms of the discrepancies between BNv and yBNv and between BN and yBNv, respectively. The

analysis of these discrepancies is also carried out in this section.

Specific MJP lines, for which these evaluations are carried out, have been constructed as follows:

The values of M and S have been selected from the sets

M P t2, 3, 4, 5, 10u, S P t2, 3, 4u. (4.5)

For each pair pM,Sq from these sets, 400 MJP serial lines have been constructed by selecting their

parameters randomly and equiprobably from:

Tup,i P r20, 100s, ei P r0.80, 0.99s,Wi P r0.75, 1.25s, wij P r1.0, 1.5s,

Ni � tkiWiTdown,iu� 1, where ki P t1, 2, 3, 4, 5u, (4.6)

rj P r0.1, 0.9s so thatS

j�1rj � 1,

where ki � NiµiWi

represents the number of average downtimes the buffer of capacity Ni protects

machine i. Thus, the total of 6000 lines have been constructed and evaluated using the following

simulation procedure: In each simulation run, the first 20,000 units of time were considered as warm-

up period, and the subsequent 180,000 units of time were used to statistically evaluate TP sj , WIP si ,

15

ST si , and BLsi , where s is the index of the simulation run. For each line, 20 simulation runs have been

carried out, leading to the expected values denoted as TPj , WIPi, STi, and BLi, with 95% confidence

intervals less than 0.002 for TP , less than 0.1 for WIPi, and less than 0.005 for STi and BLi.

The results of these analyses for performance metrics and bottlenecks are presented in Subsections

4.2.2 and 4.2.3, respectively.

4.2.2 Accuracy of performance metrics evaluation

The average values of εIX , εII

X , and εTX are shown in Tables 4.1-4.3. Examining these data, we arrive

at the following:

Observation 4.1.

• Stage I induces practically no errors in all four performance metrics for all M and S considered.

• Stage II does introduce errors in all performance metrics. The errors in TP are two-to-four

times smaller than those in WIP . The errors in BL and ST are practically identical. All the

errors are increasing functions of M and practically independent of S. We note that these errors

are similar to those observed in evaluating asynchronous SJP lines (see [57, Section 11.2]).

• Stage III introduces practically no errors. This follows from the fact that the values of εIIX and

εTX are almost the same.

4.2.3 Accuracy of bottleneck identification

The values of εIBN , εII

BN , and εTBN are shown in Table 4.4. Part (a) of this table considers all 6000 lines

mentioned in Subsection 4.2.1. In some of these systems there is only one machine with no emanating

arrows (i.e., a single BN). In others there are multiple machines with no emanating arrows and, thus,

Table 4.1: Average errors and confidence intervals of Stage I

M 2 3 4 5 10S 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4

εITP (%) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.1 0.2 0.2

�0.01 �0.01 �0.02 �0.01 �0.01 �0.02 �0.01 �0.02 �0.02 �0.01 �0.02 �0.02 �0.01 �0.01 �0.02

εIWIP (%)

0.5 0.8 0.8 0.4 0.6 0.7 0.4 0.5 0.6 0.3 0.5 0.5 0.4 0.5 0.5�0.15 �0.18 �0.13 �0.08 �0.13 �0.12 �0.07 �0.11 �0.11 �0.06 �0.08 �0.09 �0.05 �0.07 �0.07

εIBL

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001�0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001

εIST

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.002�0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001 �0.0001

16

Table 4.2: Average errors and confidence intervals of Stage II

M 2 3 4 5 10S 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4

εIIyTP

(%) 0.1 0.1 0.1 1.0 0.9 0.8 1.6 1.5 1.6 2.1 2.2 2.2 3.7 3.9 3.9�0.013 �0.015 �0.012 �0.163 �0.154 �0.121 �0.204 �0.179 �0.197 �0.231 �0.215 �0.224 �0.295 �0.309 �0.286

εII{WIP

(%) 0.2 0.2 0.2 4.1 4.2 4.3 6.4 6.5 7.0 8.0 8.7 8.0 12.8 13.4 13.8�0.016 �0.017 �0.018 �0.76 �0.775 �0.77 �0.903 �0.998 �1.025 �1.077 �1.104 �1.011 �1.377 �1.376 �1.464

εIIyBL

0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.03�0.001 �0.001 �0.001 �0.001 �0.001 �0.001 �0.002 �0.002 �0.002 �0.002 �0.002 �0.002 �0.001 �0.001 �0.001

εIIyST

0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.05 0.05�0.001 �0.001 �0.001 �0.002 �0.002 �0.001 �0.002 �0.002 �0.002 �0.002 �0.003 �0.002 �0.004 �0.004 �0.004

Table 4.3: Average Total errors and their confidence intervals

M 2 3 4 5 10S 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4

εTyTP

(%) 0.1 0.1 0.1 1.0 1.0 0.9 1.7 1.6 1.7 2.2 2.3 2.3 3.8 4.0 4.1�0.015 �0.016 �0.017 �0.166 �0.158 �0.125 �0.207 �0.184 �0.2 �0.234 �0.221 �0.23 �0.299 �0.314 �0.291

εT{WIP

(%) 0.5 0.8 0.9 4.4 4.6 4.8 6.6 6.8 7.4 8.2 9.0 8.3 13.0 13.6 14.1�0.152 �0.182 �0.139 �0.757 �0.783 �0.779 �0.904 �1 �1.024 �1.074 �1.105 �1.01 �1.376 �1.374 �1.461

εTyBL

0.011 0.010 0.010 0.018 0.018 0.017 0.023 0.023 0.024 0.028 0.028 0.027 0.042 0.042 0.042�0.001 �0.001 �0.001 �0.001 �0.001 �0.001 �0.002 �0.002 �0.002 �0.002 �0.002 �0.002 �0.004 �0.004 �0.004

εTyST

0.010 0.011 0.010 0.020 0.019 0.019 0.026 0.026 0.026 0.029 0.032 0.029 0.043 0.046 0.048�0.001 �0.001 �0.001 �0.002 �0.002 �0.002 �0.002 �0.002 �0.002 �0.002 �0.003 �0.002 �0.004 �0.004 �0.004

multiple BNs, one of which, with the largest severity (see (3.8)), is the Primary BN. To characterize

each of these cases, Table 4.4(a) addresses systems with single and multiple BNs, while Table 4.4(b)

considers only lines with a single BN. Based on Table 4.4, we formulate:

Observation 4.2.

• Stage I induces practically no errors in BN identification for all M and S considered.

• Stage II introduces significant errors in BN identification. These errors increase as a function of

M and are almost independent of S. The errors of BN identification in MJP lines with a single

BN are almost twice smaller as those in the case of combined single/multiple BNs.

• Since, εIIBN and εT

BN are almost the same, we conclude that Stage III introduces practically no

errors in BN identification.

Summarizing the results of the last two subsections, the following can be stated:

• Stage I, i.e., the reduction of MJP to SJPv, introduces very small errors in all performance

metrics and BN identification.

• Stage III, i.e., evaluation of zMJP based on zSJPv, also introduces very small errors.

17

Table 4.4: Errors of BN identification

(a) Combined single/multiple BNs

M 2 3 4 5 10S 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4

εIBN (%) 0.0 0.3 0.0 0.0 0.3 0.5 0.3 0.5 0.5 0.5 0.5 0.8 0.9 0.9 0.9εIIBN (%) 1.5 1.3 0.5 2.8 2.8 3.5 7.8 4.3 7.3 8.3 11.0 8.8 19.8 22.1 22.4εTBN (%) 1.5 1.0 0.5 2.8 2.5 3.5 8.0 4.8 7.8 8.8 11.3 9.0 19.5 22.7 22.4

(b) Single BNs only

M 2 3 4 5 10S 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4

εIBN (%) 0.0 0.3 0.0 0.0 0.3 0.5 0.0 0.3 0.6 0.0 0.3 0.3 1.1 1.1 1.2εIIBN (%) 1.5 1.3 0.5 1.8 1.6 3.4 4.5 2.5 4.9 5.8 5.8 6.4 10.4 8.0 7.9εTBN (%) 1.5 1.0 0.5 1.8 1.3 3.4 4.5 2.8 5.5 5.8 5.8 6.1 10.4 8.6 8.5# lines 400 400 400 379 375 383 352 359 348 329 328 344 183 175 165

• In contrast, Stage II, i.e., the reduction of SJPv to zSJPv, leads to significant errors. These errors

are similar to those reported in [57] for SJP systems. In spite of these errors, textbook [57]

developed a relatively complete theory for SJP serial lines and assembly systems and reported

numerous industrial applications. Based on this experience, we use the three-stage performance

analysis technique proposed in this section to develop a theory for MJP systems as well. Several

results in this direction are reported in Sections 5-7 below.

5 MJP Systems Throughput and Bottlenecks as Functions of Product-

Mix

5.1 Preliminaries

The throughput, bottlenecks, and other characteristics of SJP serial lines as functions of machine

and buffer parameters have been investigated in numerous studies (see, for instance, [31], [48]-[57]).

Since, as it is shown in Section 4, performance analysis of MJP systems can be reduced to analysis

of SJP lines (manufacturing virtual jobs), the results of these studies are applicable to MJP lines as

well. Thus, the only open issue is the behavior of the throughput and bottlenecks as functions of

product-mix. The current section is intended to investigate this issue. For simplicity, we consider

mostly the case of two job-types. Note that for S � 2, the product-mix is defined by a scalar r � r1

(since r2 � 1 � r1) and, therefore, TPvprq � TPvprq.

For N � 0 and N � 8, (where N � rN1, � � � , NM�1s), the function yTP vprq � TPvprq can be

18

evaluated quite easily � without using the aggregation procedure of [57, Section 11.2]. Indeed, for

N � 8, TP vprq can be represented as

TP vpr,N � 8q � minttp1,vprq, � � � , tpM,vprqu, (5.1)

where tpi,vprq is the virtual stand-alone throughput of machine i given by

tpi,vprq �eiWi

wi,vprq�

eiWi

rwi1 � p1 � rqwi2. (5.2)

For N � 0, TP vprq is

TP vpr,N � 0q � mintsmc1,vprq, � � � , smcM,vprqu, (5.3)

where smci,vprq is the so-called virtual system-modified capacity of machine i given by

smci,vprq �Wi

wi,vprq

M¹i�1

ei �Wi

rwi1 � p1 � rqwi2

M¹i�1

ei. (5.4)

Clearly,yTP vpr,Nq for all other N ’s is upper- and lower-bounded by (5.1) and (5.3), respectively:

TP vpr,N � 0q ¤yTP vpr,Nq ¤ TP vpr,N � 8q. (5.5)

In this section, we first analyze the behavior of (5.1) and (5.3) as functions of the product-mix and

then address the case of finite buffers in more details.

It turns out that both qualitative and quantitative properties of (5.1) and (5.3) depend on the

relationship between the jobs work-requirements. To characterize this relationship, consider an MJP

serial line producing two job-types, J1 and J2, with work-requirements, wi1 and wi2, i � 1, � � � ,M ,

respectively, and product-mix r. Let the bottleneck of this line be the machine denoted as BNJ1 when

r � 1, and BNJ2 when r � 0. Note that in all systems of Figure 4.1, the bottleneck for N � 8 is

the machine with the smallest stand-alone throughput, and for N � 0 the machine with the smallest

system-modified capacity.

Definition 5.1. Given a serial MJP line defined by assumptions (i)-(vi), jobs J1 and J2 are called

19

non-conflicting if BNJ1 � BNJ2. Otherwise the jobs are conflicting.

In Subsections 5.2 and 5.3 below, we characterize the behavior of the throughput and bottlenecks

of MJP lines for non-conflicting and conflicting jobs, respectively.

5.2 Throughput and bottlenecks of MJP serial lines with non-conflicting jobs

Theorem 5.1. Consider an MJP serial line defined by assumptions (i)-(vi) and producing two non-

conflicting jobs, J1 and J2, with BNJ1 � BNJ2 � mk. Then, if all buffers are of infinite or zero

capacity,

(a) mk is the BNvprq for all r P r0, 1s;

(b) TP vprq is given by

TP vprq �1

rTPJ1

� 1�rTPJ2

. (5.6)

(c) TPvprq is:

• strictly monotonically increasing if TPJ1 ¡ TPJ2;

• strictly monotonically decreasing if TPJ1   TPJ2;

• constant if TPJ1 � TPJ2.

Proof: See Appendix.

For finite buffer capacity,N , statements (a) and (c) of this theorem have been verified numerically.

Specifically, we have constructed 25,000 non-conflicting MJP lines with five machines producing two

job-types. The machine and job-type parameters have been selected randomly and equiprobably from

the sets defined in (4.6). For each of these lines,yTP vpr,Nq andyBNvpr,Nq have been evaluated using

the method of Section 4 for r P t0, 0.01, 0.02, � � � , 1u. As a result, we arrive at the following:

Observation 5.1. Among the 25,000 lines with non-conflicting jobs analyzed:

• yBNvpr,Nq, is the same machine for all r P r0, 1s in 99.2% of cases;

• yTP vpr,Nq is monotonic in 92.8% of cases.

Statement (b) of Theorem 5.1 shows that TPvprq for N � 0 and N � 8 can be represented not

only by (5.1) and (5.3), but by (5.6) as well. This representation allows to evaluate the throughput

20

for any buffer capacity. Indeed, motivated by (5.6), introduce the following approximation for the

throughput of a serial MJP line with arbitrary buffer capacity vector N :

}TP vpr,Nq �1

ryTPJ1pNq

� 1�ryTPJ2pNq

, (5.7)

whereyTP J1pNq andyTP J2pNq denote the system throughput operating in the SJP regime, which can

be evaluated using the method of Section 3. Note that while }TP v represents the throughput for any

r P r0, 1s, it requires calculations only for r � 0 and r � 1.

To investigate the accuracy of approximation (5.7), we used the same 25,000 MJP lines as before,

along with theiryTP vpr,Nq. In addition, we computed}TP vprq using (5.7) and quantified its accuracy

by

ε}TP

� max0 r 1

$&%���yTP vpr,Nq �}TP vpr,Nq

���yTP vprq

,.- � 100%. (5.8)

As a result, we arrive at the following:

Observation 5.2. Among the 25,000 lines with non-conflicting jobs analyzed, ε}TP

¤ 1% in 95.8% of

cases and ε}TP

¤ 2% in 98.5% of cases.

Thus, expression (5.7) offers an efficient way for calculating the throughput of MJP lines with

non-conflicting jobs for any buffer capacity.

Example 5.1. Consider an MJP line with five identical machines, defined by λi � 0.01, µi � 0.09,

Wi � 1, producing two jobs, J1 and J2, with the following work-requirements: w1 � r1.3, 3.0, 2.3,

1.9, 1.9], w2 � r1.9, 4.0, 1.6, 1.4, 2.6s. For N � 8 and N � 0, the jobs are non-conflicting and the

common BN is m2. Thus, forN � 8, according to (5.1), TP vprq � tp2,vprq for all r P r0, 1s; the graph

of TP vprq along with the stand-alone throughput of non-bottleneck machines are shown in Figure

5.1(a). Similarly, for N � 0, TP vprq � smc2,vprq for all r P r0, 1s; the graph of TP vprq, along with

the system-modified capacities of other machines is shown in Figure 5.1(b). Finally, for N � 2, the

graph of}TP vpr,Nq, calculated according to (5.7), is shown in Figure 5.1(c) along with its upper- and

lower-bounds, TP vpr,N � 8q, TP vpr,N � 0q. Clearly, in this MJP system, even small buffers of

capacity 2 lead to the throughput within 12% of that for N � 8 for all r. �

21

r0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

TPv(r) = tp2,v(r)tpi,v(r) of non-BN machines

(a) N � 8

r0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

TPv(r) = smc2,v(r)smci,v(r) of non-BN machines

(b) N � 0

r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP

v(r)

0

5

10

15

20

25

30

35

N = ∞

N = 2N = 0

(c) N � 2

Figure 5.1: Throughput of a five-machine MJP line with non-conflicting jobs

5.3 Throughput and bottlenecks of MJP serial lines with conflicting jobs

The behavior of MJP serial lines with conflicting jobs is more complex than that with non-conflicting

ones. Therefore, we begin with the simplest case of two-machine lines and then generalize the results

to systems with M ¡ 2.

5.3.1 Two-machine lines

Theorem 5.2. Consider an MJP serial line defined by assumptions (i)-(vi) and producing two con-

flicting jobs, J1 and J2, having BNJ1 � m1 and BNJ2 � m2. Then, if the buffer is of infinite or zero

capacity,

(a) BNvprq has one switch on the interval r P r0, 1s:

BNvprq �

$'&'%m2, if r P r0, r�q,

m1, if r P pr�, 1s,(5.9)

where r� is the unique solution of tp1prq � tp2prq for N � 8 or smc1prq � smc2prq for N � 0.

(b) TPvprq is given by:

TPvprq �

$''&''%

r�

rTPvpr�q

� r��rTPJ2

, if 0 ¤ r   r�,

1�r�r�r�

TPJ1� 1�rTPvpr�q

, if r� ¤ r ¤ 1.(5.10)

(c) This function of r is:

• non-monotonic if tw11 ¡ w12, w21   w22u, so that TPvpr�q ¡ maxtTPJ1, TPJ2u;

22

• strictly monotonically increasing if tw11 ¡ w12, w21 ¡ w22u and decreasing if tw11   w12, w21   w22u;

• non-strictly monotonic if wi1 � wi2 for either i � 1 or 2, so that TPvpr�q � TPJ1 or TPvpr�q �

TPJ2.

Proof: See Appendix.

The first bullet in part (c) of this theorem implies that there exists a range of product-mix, where

the throughput of MJP line is larger than the throughputs of each individual job-type manufactured

in SJP regime. This phenomenon occurs because SJP regimes of J1 and J2 overload their respective

BNs, whereas MJP with the “right” product-mix leads to a more balanced allocation of work on both

BNs and, thus, to a higher throughput.

For finite buffer capacity, statements (a) and (c) of Theorem 5.2 have been verified using the same

approach as in Subsection 5.2. Specifically, we have constructed 25,000 MJP two-machine lines with

two conflicting job-types and evaluated their yTP vpr,Nq andyBNvpr,Nq using the method of Section

4 for r P t0, 0.01, 0.02, � � � , 1u. As a result, we have obtained the following:

Observation 5.3. Among the 25,000 lines with conflicting jobs analyzed:

• yBNvpr,Nq, r P r0, 1s, switches once in 99.1% of cases.

• If w11 ¡ w12 and w21   w22, then there exist r1 and r2 such thatyTP vpr,Nq ¡ maxrtyTP J1pNq,yTP J2pNqu

for all r P rr1, r2s in 76% of cases.

• If the difference between job work-requirements is sufficiently large, so that w11�w120.5pw11�w12q

¡ 0.1

and w22�w210.5pw22�w21q

¡ 0.1, the inequalityyTP vpr,Nq ¡ maxrtyTP J1pNq,yTP J2pNqu, r P rr1, r2s, takes

place in 96.7% of cases.

Statement (b) of Theorem 5.2 can also be generalized for arbitrary buffer capacity. This is ac-

complished as follows: Consider a two-machine MJP line with a buffer of capacity N and r� defined

by tp1prq � tp2prq. Using the methods of Sections 3 and 4, evaluate yTP vpr,Nq for the following four

values of r: r � 0, r�2 ,1�r�

2 , and 1. Introduce the estimate of throughput, }TP vpr,Nq, @r P r0, 1s, as

follows:

}TP vpr,Nq � min0¤r¤1

$&% 0.5r�

ryTP p0.5r�,Nq

� 0.5r��ryTPJ2pNq

,0.5p1 � r�q

r�0.5p1�r�qyTPJ1pNq

� 1�ryTP p0.5p1�r�,Nqq

,.- (5.11)

Quantifying the accuracy of (5.11) by the error (5.8), we obtain:

23

r0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP(r)

14

15

16

17

18

19

20

21

22

23

24

r∗

TPv(r,N = ∞)

TP v(r,N = 10)

TP v(r,N = 10)TPv(r,N = 0)

(a) w1 � r3.0, 2.0s, w2 � r2.0, 3.0s

r0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP(r)

14

15

16

17

18

19

20

21

22

23

24

r∗

TPv(r,N = ∞)

TP v(r,N = 10)

TP v(r,N = 10)TPv(r,N = 0)

(b) w1 � r2.0, 2.3s, w2 � r3.0, 2.6s

r0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP(r)

14

15

16

17

18

19

20

21

22

23

24

r∗

TPv(r,N = ∞)

TP v(r,N = 10)

TP v(r,N = 10)TPv(r,N = 0)

(c) w1 � r2.5, 2.0s, w2 � r2.5, 3.5s

Figure 5.2: Patterns of throughput behavior for a two-machine line producing two conflicting jobs

Observation 5.4. Among the 25,000 lines with conflicting jobs analyzed, ε}TP

¤ 2% in 95.6% of cases

and ε}TP

¤ 3% in 99.9% of the cases.

Example 5.2. Consider a two-machine MJP line with λi � 0.01, µi � 0.09,Wi � 1, i � 1, 2, and with

two conflicting jobs having work-requirements corresponding to the three cases of part (c) of Theorem

5.2. For each set of the work-requirements, Figure 5.2 shows TPvprq, evaluated for N � 0 and N � 8

using (5.10). As one can see, under the conditions of the first case, there exists a set R of r’s (which

includes the point of the bottleneck switch r�), such that TPvprq ¡ maxtTPJ1, TPJ2u,@r P R, and

TPvpr�q is 22% larger than TPvp0q and TPvp1q. Parts (b) and (c) of Figure 5.2 illustrate the behavior

of TPvprq for two other cases. The behavior ofyTP vpr,N � 10q (evaluated using the method of Section

4) and}TP vpr,N � 10q (calculated using (5.11)) are also shown in Figure 5.2. Clearly, buffer capacity

10 ensures almost the same performance as the infinite buffer. �

5.3.2 M ¡ 2-machine lines

For MJP lines with more than two machines and conflicting jobs, numerous patterns of TPvprq behavior

are possible and, therefore, the theorem below quantifies these patterns in a qualitative manner.

Theorem 5.3. Consider an MJP serial line defined by assumptions (i)-(vi) with M ¡ 2 machines

producing two conflicting jobs, J1 and J2. Then, if all buffers are of infinite or zero capacity,

(a) BNvprq has at most M � 1 switches in the interval r P r0, 1s; each machine can be a bottleneck

only in a single interval of r0, 1s;

(b) TPvprq has the following properties:

• if the number of switches of BNvprq is 1 ¤ K ¤ M � 1, then function TPvprq, r P r0, 1s has

24

r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TPv(r)

80

100

120

140

160

180

200

220

r∗

N = ∞

N = 1N = 0

(a) w1 � r30, 1, 10, 10sw2 � r5, 25, 10, 10s

r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TPv(r)

80

100

120

140

160

180

200

220

r∗

r∗∗

N = ∞

N = 1N = 0

(b) w1 � r30, 1, 20, 10sw2 � r5, 25, 15, 10s

r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TPv(r)

80

100

120

140

160

180

200

220

r∗

r∗∗

r∗∗∗

N = ∞

N = 1N = 0

(c) w1 � r30, 1, 20, 15sw2 � r5, 25, 15, 19s

Figure 5.3: Patterns of throughput behavior for a four-machine line producing two conflicting jobs

K � 1 intervals of continuous differentiability; the BNvprq switches occur at the values of r,

where TPvprq is non-differentiable (the value of K is referred to as the order of conflict);

• if wBNJ1,1 ¡ wBNJ1,2 and wBNJ2,2 ¡ wBNJ2,1, then there exist r1 and r2such that TPvprq ¡

maxtTP1, TP2u,@r P pr1, r2q.

Proof: See Appendix.

Note that a characterization of the throughput similar to (5.11) is possible for M ¡ 2 as well.

However, this characterization becomes too involved, and, therefore, is not pursued here.

Example 5.3. To illustrate this theorem, consider a four-machine MJP line with λi � 0.01, µi � 0.09,

Wi � 1, manufacturing two jobs with work-requirements indicated in Figure 5.3. Using (5.1) and (5.3)

we evaluate TPvprq forN � 8 andN � 0. The results are shown in Figure 5.3, illustrating three types

of the order of conflicts and the resulting patterns of TPvprq behavior. For 0   N   8, yTP vpr,Nq

can be evaluated for various r P r0, 1s using the method of Section 4. The results for N � 1 are shown

in Figure 5.3 as well. �

6 Product-mix Performance Portraits

In System Science, the global behavior of dynamical systems is often represented by a state-space

portrait (SSP), [59], which is a set of system trajectories for various initial conditions. The SSP

succinctly represents the main properties of the system � its steady states, stability, limit cycles, and

even strange attractors � in a single picture. Control system engineers often use SSP in order to

design a controller, which forces the system operation in a desired regime.

Similar to SSP, it is possible to represent the global behavior of MJP systems by their portraits

25

Figure 6.1: PP of MJP serial line with two conflicting job-types

with respect to the product-mix. We refer to this representation as the Product-mix Performance

Portrait or just Performance Portrait (PP). It consists of two graphs: the throughput graph, which

shows yTP v as a function of the product-mix and the bottleneck graph, which shows yBNv, also as a

function of the product-mix. The purpose of this section is to discuss the PPs and outline their utility

for operations management and control.

We have created a software tool for calculating and displaying PPs. The calculations are based on

the method of Section 4. Several screenshots of this tool are shown in Figures 6.1 and 6.2 for S � 2

and S � 3, respectively. Each of these figures is elucidated below.

Figure 6.1 represents the PP of the MJP serial line with the following parameters:

λ � r0.05, 0.05, 0.06, 0.04, 0.09s, µ � r0.95, 0.95, 0.94, 0.96, 0.91s,

W � r1, 1, 1, 1, 1s, N � r1, 2, 1, 3s,

w1 � r2, 1, 1, 1, 1s, w2 � r1, 1.75, 1.75, 1, 1s.

As indicated at the top of this figure, it shows PP with J1 selected as the primary job; the PP with

J2 as the primary job is the symmetric image of the one shown. For the purposes of explanation, we

have added to the PP in Figure 6.1 two lines, AB and CD.

The shaded area in the throughput graph is the feasibility domain: for every r, it represents all

attainable yTP v’s. The line AB indicates all product-mixes, for which yTP v of at least 30 JPH can

26

be obtained. The line CD indicates all the throughputs, which can be attained for the product-mix

corresponding to point C. Finally, r� is the product-mix, for whichyTP vprq is maximized.

The bottleneck graph represents the bottleneck machine for each product-mix. It indicates the

most efficient way for system improvement (as far as the machines are concerned). For example, if

the desired product-mix is rd P r0, r�q , the most effective way of system improvement is to improve

operation OP30; if rd P pr�, 1s, OP10 should be improved.

For MJP systems with S ¡ 2, an additional feature is introduced in PP, which enables to modify

the ratio of non-primary job-types. Figure 6.2 provides an illustration for the system with S � 3 and

the parameters given by

λ � r0.05, 0.05, 0.06, 0.04, 0.09s, µ � r0.95, 0.95, 0.94, 0.96, 0.91s,

W � r1, 1, 1, 1, 1s, N � r1, 1, 1, 1s,

w1 � r1, 3, 1, 1, 2.1s, w2 � r1, 1, 3, 1, 2.1s, w3 � r2, 1, 2.5, 1, 2.1s.

The rows of Figure 6.2 show the system performance for the primary job-type being, J1, J2, and J3,

respectively, while the columns show the performance for various allocations of the non-primary job-

types (0.2/0.8, 0.5/0.5, and 0.8/0.2, respectively). The meaning and the significance of the throughput

and the bottleneck graphs, remain the same as in the case of S � 2.

As it follows from the above, the utility of PP for operations management is that for any assigned

product-mix (which often changes daily), the manager can see what level of throughput can be achieved

and, if necessary, which operation(s) should be most profitably improved, if a higher throughput is

required to meet the daily production schedule.

7 Application

The methods developed in this paper have been used in a six-month project devoted to analysis and

potential improvement of a section of the underbody assembly system at an automotive assembly

plant. This section, which we refer to as Line MA (where MA stands for the Main Assembly), has

been the bottleneck of the body shop for a long time, consistently producing about 15% less than its

daily target. The goal of the project was to identify the reasons for these losses and suggest steps for

their elimination.

27

Figure 6.2: PP of MJP serial line with three job-types

28

Table 7.1: Line MA modified machines and jobs parameters for Week 1

Operation i 1 2 3 4 5 6 7 8 9 10 11λi

�1

min

0.007 0.023 0.015 0.001 0.062 0.083 0.037 0.078 0.009 0.002 0.006µi

�1

min

0.339 2.857 0.541 20.000 0.282 0.571 0.377 0.408 0.952 20.000 0.267ei 0.98 0.99 0.97 1.00 0.82 0.87 0.91 0.84 0.99 1.00 0.98τi1 �

wi1Wi

(sec) 150.0 1.0 39.5 30.3 30.6 44.9 45.5 42.8 40.7 23.8 33.9τi2 �

wi2Wi

(sec) 1.0 75.0 43.3 24.5 27.3 42.7 44.5 43.8 38.2 22.5 35.1

Line MA consists of 11 automated welding operations and a conveyor material handling system.

Although it is capable of producing four different products, over 98% of the product-mix (which

changes daily) is comprised of two job-types. Therefore, we assume S � 2. Operations 3, 4, 7, and 8

require subassemblies. While the system is closed (with respect to the palettes, transporting jobs from

one operation to another), during the six months of the study no blockage or starvation by palettes

have been reported; therefore, Line MA is modeled as an open line.

The system’s performance has been monitored by the Production Monitoring System, which has

been used to collect the data for this study. Based on these data, a mathematical model of Line MA

has been constructed and validated. Due to confidentiality reasons, we are displaying here modified

data, although this modification is carried out so that the qualitative features of the system at hand

are preserved.

The modified machine and job parameters for Line MA are shown in Table 7.1 for Week 1; the

data for other weeks are quite similar. The capacity of all buffers between the machines is 1. Based

on these data, the bottlenecks of Line MA in SJP regimes are BNJ1 � m1 and BNJ2 � m2. Thus,

jobs J1 and J2 are conflicting.

Using the data of Table 7.1 and the MJP toolbox mentioned in Section 6, we obtain the PP of

Line MA shown in Figure 7.1(a), where the horizontal line indicates the daily production target and

the required range of the product-mix. As one can see, the maximum throughput is 60 JPH, reached

for r P r0.25, 0.40s. When r is close to 0 or 1, throughput is 46 JPH and 24 JPH, respectively. The

bottlenecks are m2 for r P r0, 0.23s, m6 for r P r0.23, 0.39s, and m1 for r P r0.39, 1s.

Typical (modified) daily target for Line MA is 55 JPH, with the product-mix taking values in the

range r P r0.25, 0.50s. As the horizontal line in Figure 7.1(a) indicates, this target can be achieved only

for r P r0.25, 0.42s. Thus, for r P p0.42, 0.50s production is below the target, up to 15% (for r � 0.50).

29

(a) Without starvation by subassemblies (b) With starvation by subassemblies

Figure 7.1: PP of Line MA

These performance characteristics are achieved assuming that Line MA is not starved by sub-

assemblies. In reality, however, these starvations do take place. Table 7.2 shows the probabilities

of starvation by subassemblies. To take them into account, we modify the breakdown rates of the

respective operations of line MA by introducing the starvation-induced efficiency, e1i, of operation i as

follows:

e1i �µ1i

λi � µ1i�

µiλi � µi

� p1 � ST subi q, (7.1)

where µ1i is the adjusted breakdown rate and ST subi is the probability of operation i starvation by its

subassembly. From this relationship, the adjusted breakdown rate, µ1i is

µ1i �λieip1 � ST subi q

1 � eip1 � ST subi q. (7.2)

The values of µ1i and e1i are shown in Table 7.2.

Using the data of Table 7.1 with µi and ei substituted by µ1i and e1i from Table 7.2, we obtain

the PP for Line MA with the starvations by subassemblies taken into account (see Figure 7.1(b)).

As the horizontal line in Figure 7.1(b) indicates, the throughput is 13-15% below the target for all

r P r0.25, 0.50s and the bottleneck shifts to Op. 7. Thus, improvement of Op. 7 is necessary.

As it follows from Table 7.2, Op. 7 suffers from significant starvations by subassemblies. Examining

the reasons for these starvations, it has been determined that they are mostly due to late delivery

of the subassemblies by in-plant delivery trucks and by tardiness in manual loading operations. To

30

Table 7.2: Starvations of MA by subassemblies and adjusted machine parameters

Operation i 1 2 3 4 5 6 7 8 9 10 11ST subi � � 0.02 0.02 � � 0.18 0.04 � � �

e1i 0.98 0.99 0.95 0.98 0.82 0.87 0.75 0.81 0.99 1.00 0.98µ1i 0.339 2.857 0.309 0.0718 0.282 0.571 0.108 0.324 0.952 20.0 0.267

(a) Without starvation of Op. 7 (b) Without starvation of Op.7 and Op. 1 cycle time reduced

Figure 7.2: PP of improved Line MA

investigate the effect of improvement of subassembly delivery and loading, we have calculated PP

with ST sub7 � 0. The corresponding PP is shown in Figure 7.2(a). As one can see from this PP

(which is quite similar to that of Figure 7.1), eliminating ST sub7 allows for meeting the daily target

with product-mix r P r0.25, 0.42s. To meet the daily target for r P p0.42, 0.50s, one must improve

the corresponding bottleneck, which is Op. 1. Since Op. 1 has high efficiency and no starvation by

subassemblies, the only venue of improvement is to reduce its cycle time. Reducing the cycle time for

J1 of Op. 1 by 15% allows the system to satisfy its daily target for all product-mixes, as shown in

Figure 7.2(b). Note that cycle time of Op. 1 for J2 does not need to be reduced.

The above recommendations have been communicated to the plant management and found their

favorable acceptance.

8 Conclusions and Future Work

This paper addressed a class of flexible manufacturing systems referred to as Multi-job Production

(MJP). In MJP systems all job-types are processed by the same sequence of machines (operations),

31

but with job-dependent processing time at some or all machines. A characteristic feature of MJP

systems is that their performance depends not only on the machine, buffer, and job parameters, but

also on the product-mix. Therefore, the emphasis of this paper is on investigating the throughput and

bottlenecks of MJP as functions of the product-mix. The systems addressed are MJP serial lines with

exponential machines and infinite or finite buffers. Specific results obtained are:

• To characterize the MJP operation, a work-based model is introduced; unlike the traditional

part-based approach, this model is insensitive to whether a single- or multi-job production takes

place.

• Using this model, it is shown that buffers are more effective in protecting against downtime for

jobs with larger work-requirements; this phenomenon is due to the fact that the downtime in

units of the machine cycle time is smaller for jobs requiring more work.

• Performance and bottleneck analysis methods for MJP lines are developed; this is carried out

by converting an MJP line into a Single-job Production (SJP) line manufacturing a virtual

job, whose work-requirement is defined as weighted by the product-mix average of the work-

requirements of the constituent jobs.

• Using these methods, the throughput and the bottlenecks of MJP lines are investigated as

functions of the product-mix. In particular, it is shown that for the so-called conflicting jobs,

there exists a range of product-mixes, where the throughput of MJP is larger than that of SJP

of any constituent job-type; this takes place because SJP overloads the respective bottlenecks,

whereas MJP with the“right” product-mix leads to a more balanced work allocation.

• To represent the performance of MJP lines as a function of the product-mix, the so-called

product-mix performance portrait is introduced and a software tool for its calculation and user-

friendly representation is developed; this portrait is intended to help managing MJP lines having

frequent changes of product-mix.

• The methods developed have been applied to a section of the underbody assembly system at

an automotive assembly plant. Analyzing the throughput part of the resulting performance

portrait, it has been shown that the system cannot meet the daily production target for any

product-mix. Analyzing the bottleneck part of the performance portrait, improvement measures

have been suggested, resulting in the desired system operation; these suggestions have been

32

favorably accepted by the plant management.

Numerous problems related to MJP systems still remain open. These include:

• Analysis and improvement methods for hybrid SJP/MJP systems. Such systems, where some

machines operate in SJP and others in MJP regimes, are often encountered in practice.

• Analysis and improvement methods for MJP assembly systems. (Note that the underbody

assembly system analyzed in Section 7 is, in fact, an assembly system; however, in the study

reported here we reduced it to a serial lines using the measured probabilities of its starvation by

subassemblies.) Development of methods, which explicitly take into account interactions between

the main line and the subassembly lines, is an important practical and theoretical problem.

• Development of a theory for closed MJP systems. (Note that the system described in Section 7

was, in fact, a closed line; we treated it here as an open one based on the observation that no

starvation and blockages by carriers took place.) This problem also has a substantial practical

and theoretical significance.

• Robustness properties of MJP assembly systems. As illustrated in Figure 1.2, there may be

different configurations of subassembly lines supplying the main assembly. They may be SJP or

MJP, have dedicated or non-dedicated buffers, use job release for the sequence or for the buffer,

etc. Which one of these structures is the most robust with respect to various perturbations, e.g.,

machine downtime, release errors, etc.? Answering this question would lead to novel approaches

to MJP assembly systems design.

• Additional problems of importance include the issues of leanness, transients, and product quality,

which have been investigated for SJP systems (see, for instance, [57]), but not addressed yet in

MJP setting.

Solutions of the problems mentioned above will lead to a relatively complete and practical theory

of MJP systems analysis, design, and continuous improvement.

33

Appendix

Proof of Theorem 5.1: For N � 8, BN-machine for each job j, is the machine with the smallest

stand-alone throughput, given by

tpij �eiWi

wij, i � 1, � � � ,M, j � 1, 2. (A.1)

Since mk is the common BN for J1 and J2,

tpk1   tpi1, tpk2   tpi2, @i � k.

Substituting (A.1) into the above inequalities and inverting them, we have

wk1ekWk

¡wi1eiWi

, @i � k, (A.2)wk2ekWk

¡wi2eiWi

, @i � k. (A.3)

Multiplying both sides of (A.2) by r, and (A.3) by p1 � rq, and adding the inequalities, we obtain:

rwk1 � p1 � rqwk2ekWk

¡rwi1 � p1 � rqwi2

eiWi, @i � k, r P r0, 1s. (A.4)

Since the numerators in (A.4) are the work contents of the virtual jobs (see (4.1)), the above inedquality

can be rewritten aswk,vprq

ekWk¡wi,vprq

eiWi, @i � k, r P r0, 1s, (A.5)

which implies that

tpk,vprq   tpi,vprq,@i � k, r P r0, 1s. (A.6)

Clearly, expression (A.6) shows that mk is the BN machine of the virtual job for all r P r0, 1s. This

proves part (a).

34

Using (5.1) and (5.2), part (b) is proved as follows:

TPvprq � min1¤i¤M

ttpi,vprqu � min1¤i¤M

"eiWi

wi,vprq

*�

ekWk

wk,vprq

�ekWk

rwk1 � p1 � rqwk2�

1r wk1ekWk

� p1 � rq wk2ekWk

�1

rTPJ1

� 1�rTPJ2

.

Monotonicity properties of part (c) follow directly from the derivative of (5.6) with respect to r:

BTPvprq

Br�

TPJ1 � TPJ2

TPJ1TPJ2

�r

TPJ1� 1�r

TPJ2

2 . (A.7)

For N � 0, the bottleneck and throughput are evaluated using the same steps as above, replacing

tpij by smcij �Wi{wij and using (5.3) and (5.4). �

Proof of Theorem 5.2: For N � 8, BNpr � 1q � m1 implies that tp1,vp1q   tp2,vp1q. Similarly,

BNpr � 0q � m2 implies tp1,vp0q ¡ tp2,vp0q. Since, as it follows from (5.2), tpi,vprq is a continuous

function of r P r0, 1s, there exist r�, such that tp1,vpr�q � tp2,vpr

�q. Solving for r�, we obtain

r� �w22e2W2

� w12e1W1

w11�w12e1W1

� w22�w21e2W2

, (A.8)

which is unique as long as w11 � w12 or w21 � w22. Thus, TP is characterized by

TP prq � minrttpi,vprqu �

$'&'%

tp1,vprq, if r� ¤ r ¤ 1,

tp2,vprq, if 0 ¤ r   r�.(A.9)

Using (3.7), it follows from (A.9) that BN of the line is m1 for r P pr�, 1s, and m2, for r P r0, r�q. This

proves part (a).

To prove part (b), we generalize (5.2) for any r1   r2 (rather than r1 � 0 and r2 � 1). This is

35

accomplished as follows

tpi,vprq �eiWi

rwi1 � p1 � rqwi2�

1rtpi1

� 1�rtpi2

�r2 � r1

rpr2�r1qtpi1

� p1�rqpr2�r1qtpi2

�r2 � r1

rr2�rr1tpi1

� r2�r1�rr2�rr1tpi2

�r2 � r1

rr2�rr1�r1r2�r1r2tpi1

� r2�r1�rr2�rr1�r�r�r1r2�r1r2tpi2

�r2 � r1

pr � r1q�r2tpi1

� 1�r2tpi2

� pr2 � rq

�r1tpi1

� 1�r1tpi2

�

r2 � r1r�r1

tpi,vpr2q� r2�r

tpi,vpr1q

. (A.10)

The last equality in (A.10) is obtained by taking into account that tpi,vpr1q �1

r1{tpi1�p1�r1q{tpi2and

tpi,vpr2q �1

r2{tpi1�p1�r2q{tpi2.

Each tpi,vprq in (A.9) can be rewritten using (A.10). Specifically, for tp1,vprq, set r1 � r� and

r2 � 1, and for tp2,vprq, set r1 � 0 and r2 � r�. This will prove part (b).

For part (c) differentiating (5.2) with respect to r gives us

Btpi,vprq

Br�

eiWipwi1 � wi2q

prwi1 � p1 � rqwi2q2. (A.11)

As a result, if wi1 ¡ wi2, then tpi,vprq is a decreasing function of r. Similarly, if wi1   wi2, then tpi,vprq

is an increasing function of r. Finally, if wi1 � wi2, then tpi,vprq is a constant function. With conditions

given in the first bullet of part (c), for all r P r0, r�s, TPvprq � tp2,vprq is an increasing function of

r, similarly for all r P rr�, 1s, TPvprq � tp1,vprq is decreasing. Therefore, TPvprq is non-monotonic.

Furthermore, r� yields maximum throughput, because of the monotonicity of each constituent part of

TPvprq. This proves bullet one. Other bullets can be proved similarly.

For N � 0, the proof is similar to the above with substituting tpi,vprq by smci,vprq. Switch point

in this case will be

r� �w22W2

� w12W1

w11�w12e1W1

� w22�w21W2

. (A.12)

�

To prove Theorem 5.3, we need the following auxiliary statements:

Lemma A.1. The stand-alone throughput tpi,vprq and system modified capacity scmi,vprq, i � 1, � � � ,M ,

are continuous functions of r for r P r0, 1s.

36

Proof of Lemma A.1: According to (5.2),

tpi,vprq �eiWi

rwi1 � p1 � rqwi2,

which is either a constant (when wi1 � wi2), or a hyperbolic function of r. The hyperbolic function is

continuous on all of its domain, except for r � wi2wi2�wi1

. If wi1   wi2, the discontinuity takes place for

r ¡ 1; if wi1 ¡ wi2, the discontinuity is for r   0. Thus, tpi,vprq is continuous on r P r0, 1s.

Proof for smci,vprq follows the same steps. �

Lemma A.2. Every pair of functions ptpi,vprq, tpj,vprqq, defined by (5.2), has at most one intersection

on r P r0, 1s, unless the two functions are identical in the sense that tpi,vprq � tpj,vprq,@r P r0, 1s.

Proof of Lemma A.2: If tpi1   tpj1 and tpi2   tpj2, then for all r P p0, 1q, rtpi1

¡ rtpi2

and1�rtpi1

¡ 1�rtpi2

. Thus, tpi,vprq   tpj,vprq, and, therefore, tpi,vprq and tpj,vprq do not intersect in (0,1).

Similar result takes place when tpi1 ¡ tpj1 and tpi2 ¡ tpj2.

If tpi1   tpj1 and tpi2 ¡ tpj2, then solving tpi,vprq � tpj,vprq yields

r

tpi1�

1 � r

tpi2�

r

tpj1�

1 � r

tpj2, (A.13)

which has a unique solution r� � tp�1j2 �tp�1

i2tp�1i1 �tp�1

i2 �tp�1j1 �tp�1

j2. If tpi1 ¡ tpj1 and tpi2   tpj2, solving tpi,vprq �

tpj,vprq yields similar results. �

Lemma A.3. The equality tpi,vprq � minttp1,vprq, � � � , tpM,vprqu takes place on at most one interval

of r P[0,1].

Proof of Lemma A.3: As follows from Lemma A.2, since every pair of functions tpi,vprq and tpj,vprq

intersect only once, at, say, r�, one of the following can happen:

(α) tpi,vprq   tpa,vprq, for r P r0, r�q, or

(β) tpi,vprq   tpb,vprq, for r P pr�, 1s.

If (α) takes place, tpi,vprq cannot be the minimum for any r P pr�, 1s, because at least tpa,vprq has

smaller values in this range. Similar statement holds for (β). Now, consider machine i with tpi,vprq,

for which no machine l with tpl,vprq   tpi,vprq,@r P r0, 1s exists (if such machine exists, then tpi,vprq is

37

never the minimum). Let tpa1prq, � � � , tpasprq be tp functions that intersect with tpi,v at ra1   � � �   ras ,

and satisfy item (α) above. Similarly, let tpb1prq, � � � , tpbtprq be tp functions that intersect with tpi,v

at rb1   � � �   rbt , and satisfy item (β). Then, as stated earlier, tpi,v cannot be the minimum in�sj�1praj , 1s � pra1 , 1s. It also cannot be the minimum in

�tk�1r0, rbkq � r0, rbtq. Thus, if rbt   ra1 ,

then tpi,vprq � minttp1,vprq � � � , tpM,vprqu only at the interval prbt , ra1q. �

Proof of Theorem 5.3: As it follows from Lemma A.3, there can be no more than M intervals in

which different machines are the bottlenecks. Thus, at most M � 1 switch points exist. This proves

part (a).

For part (b), let r1   � � �   rK be product-mixes at which bottlenecks switch and i0, � � � , iK the

indices of the corresponding bottleneck machines (i.e., K switches and K � 1 bottlenecks). Defining

r0 � 0, rK�1 � 1, the bottleneck is given by

BNprq �

$'''''''&'''''''%

mi0 , if r0 ¤ r   r1,

mi1 , if r1   r   r2,

� � �

miK , if rK   r ¤ rK�1,

which implies the throughput function given by

TPvprq �

$'''''''&'''''''%

tpi0prq, if r0 ¤ r   r1,

tpi1prq, if r1   r   r2,

� � �

tpiK prq, if rK   r ¤ rK�1.

Since, as stated in Lemma A.1, each tpiprq is continuous on [0,1], and at r � rk, k � 1, � � � ,K,

tpikprq � tpik�1prq, TPvprq is also continuous. Differentiability of tpiprq on [0,1] implies piecewise

differentiability of TPvprq. This proves the first bullet of part (b).

Under conditions of the second bullet of part (b), tpi0,vprq is increasing on r0, r1s and tpik,vprq is

decreasing on rrK , 1s. If TPJ1   TPJ2, then for any r P p0, r1q, TPvprq ¡ TPJ2 ¡ TPJ1. Similarly,

if TPJ1 ¡ TPJ2, then for any r P prK , 1q, TPvprq ¡ TPJ1 ¡ TPJ2. Finally, if TPJ1 � TPJ2, for any

r P p0, r1q�prK , 1q, TPvprq ¡ TPJ1 � TPJ2. This proves the second bullet of part (b). �

38

Acknowledgement

This work has been supported, in part, by the National Institute of Standards and Technology

under the Award Number 70NANB16H017.

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