MULTIOBJECTIVE OPTIMIZATION OF MANUFACTURING SYSTEMS

As production goals in manufacturing activities to be taken into consideration in this study, three production goals in terms of the production time or production rate, the production

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MULTIOBJECnVE OPTIMIZATION OF MANUFACTURING SYSTEMS

The Pennsylvania Stale University Ph.D 1982

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University Microfilms International

Multiobjective Optimization of Manufacturing Systems

Doctor of Philosophy

March 1982

© 1982 by Souemon Takakuwa

I grant The Pennsylvania State University the nonexclusive

right to use this work for the University's own purposes and

a non-for-profit basis if copies are not otherwise available,

Souemon Takakuwa

Date of Signature:

/ * / 2 / sy

h )

rofessor of gineering, Chairman

of Committee, Thesis Adviser

Allen L Soyster, Proffessor of Industrial Engineering, Head of the Department of Industrial and Management Systems Engineering

William E Biles, Professor of Industrial Engineering

frYYlXj P. v2.

/ CkYYljU

-Barnes P Ignizio, Industrial Engineering

Kochenberger, Professor of Management Science

objective optimization, optimization analyses of manufacturing systems

are made First, single-product, single-stage manufacturing systems

are treated and analyzed Then, the discussion is extended to multi­

product manufacturing systems, and finally to multi-stage manufactur­

ing systems.

As production goals in manufacturing activities to be taken into

consideration in this study, three production goals in terms of the

production time or production rate, the production cost, and the

profit rate are introduced and utilized for multiobjective optimiza­

tion Through the study, nonlinear goal programming is mainly applied

to multiobjective optimization of machining conditions because of its

suitability and flexibility for application to problems considered in

this study.

In optimization analyses of single-product, single-stage manu­

facturing systems, the nondominated solution sets for both uncon­

strained and constrained optimization are examined In unconstrained

optimization, characteristics and sensitivity of functions of perform­

ance measures are discussed, and the concept of optimal machining

speeds under multiple objectives is introduced Also, multiobjective

optimization of both machining speed and feed rate is considered.

In optimization analyses of multi-product, single-stage manufactur­

ing systems, first, optimizing machining speeds under fundamental

evaluation criteria is examined Then, multiobjective optimization of

the machining condition on manufacturing systems under a GT environment

proposed for obtaining a multiobjective optimal machining condition.

In optimizing analyses of multi-stage manufacturing systems, a

basic nonlinear multiobjective (goal-programming) model for the flow-

type multi-stage manufacturing system is built and analyzed in an

attempt to determine optimal machining conditions to be set on pro­

duction stages.

Some modification of the nonlinear goal programm ing via pattern

search is made, particularly in exploratory moves for obtaining

multiobjective optimal solutions.

LIST OF TABLES viii

LIST OF F I G U R E S ix

LIST OF S Y M B O L S xi

ACKNOWLEDGEMENTS xv

CHAPTER 1: GENERAL INTRODUCTION 1

1.1 Introduction 1

1.2 Previous Related Studies 2

1.3 Outline of T h e s i s 4

CHAPTER 2: MANUFACTURING S Y S T E M S 6

2.1 Manufacturing Systems - Functions and Procedures 6

2.2 Basic Mathematical Models of Manufacturing Systems 10

2.2.1 Basic mathematical models of manufacturing systems 10

2.2.1.1 Unit production time 10

2.2.1.2 Production rate 11

2.2.1.3 Unit production cost 11

2.2.1.4 Unit p r o f i t 12

2.2.1.5 Profit r a t e 13

2.2.2 Manufacturing models as a function of machining speed 13

2.2.3 Manufacturing models as a function of machining speed and feed rate 15

2.3 Production Goals in Manufacturing Activities 16

2.3.1 Three fundamental evaluation criteria 16

2.3.1.1 Maximum-production-rate or minimum-time criterion 17

2.3.1.2 Minimum-cost criterion 17

2.3.1.3 Maximum-profit-rate criterion 17

2.3.2 Aspiration levels on production goals 18

CHAPTER 3: OPTIMIZATION OF SINGLE-PRODUCT, SINGLE-STAGE MANUFACTURING SYSTEMS 19

3.1 Determining Optimal Machining Speeds - Unconstrained O p t i m i z a t i o n 19

3.1.1 Characteristics of functions of performance m e a s u r e s 19

3.1.1.1 Unit production time 19

3.1.1.2 Unit production cost 20

3.1.1.3 Profit 20

3.1.1.4 Profit r a t e 20

3.1.2 Optimal machining speeds under fundamental evaluation criteria 21

o b j e c t i v e s 27

3.1.4 Sensitivity of single-stage manufacturing systems 32

3.2 Determining Optimal Machining Speeds and Feed Rates -Constrained Optimization 33

3.2.1 Introduction 33

3.2.2 Basic measures of p e r f o r m a n c e 43

3.2.3 The nondominated solution s e t 50

3.2.4 Nonlinear multiobjective (goal-programming) m o d e l 55

3.2.5 Applicability of nonlinear goal programming for the multiobjective optimization of manufacturing systems 58

3.3 C o n c l u s i o n s 64

CHAPTER 4: OPTIMIZATION OF MULTI-PRODUCT, SINGLE-STAGE MANUFACTURING SYSTEMS 65

4.1 Multi-Product, Single-Stage Manufacturing Systems 65

4.1.1 Introduction 65

4.1.2 Optimal machining speeds under fundamental evaluation criteria 65

4.2 Determining Optimal Machining Speeds - Unconstrained Optimization of Manufacturing Systems Under a GT Environment 72

4.2.1 Introduction 72

4.2.2 Problem d e f i n e d 72

4.2.3 Basic mathematical models 73

4.2.3.1 Job production t i m e 73

4.2.3.2 Group production time 74

4.2.3.3 Total production time 74

4.2.3.4 Unit production cost 74

4.2.3.5 Group production cost 74

4 2 3 6 Total production cost 75

4.2.3.7 Profit * 75

4 2 3 8 Profit r a t e 75

4.2.4 Nonlinear multiobjective (goal-programmin g) m o d e l 75

4.2.4.1 Baseline m o d e l 75

4.2.4.2 Nonlinear goal-programming m o d e l 77

4.2.5 Numerical examples and consideration 80

4.3 Determining Optimal Machining Speeds and Feed Rates -Constrained Optimization of Manufacturing Systems Under a GT E n v i r o n m e n t 91

4.3.1 Introduction 91

4.3.2 Problem d e f i n e d 92

4.3.3 Algorithmic procedure for determining optimal machining speeds and feed rates 93

4.3.3.1 Determining optimal feed rates and machining-speed range (Phase I ) 93

5.1 Introduction 105

5.2 Assumptions for Analysis of Multi-Stage Manufacturing Systems 106

5.3 Determining Optimal Machining Speeds - Unconstrained Optimization 107

5.3.1 Basic mathematical m o d e l s 107

5.3.1.1 Stage production time 107

5.3.1.2 Cycle time and waiting t i m e 107

5.3.1.3 Stage production cost 108

5.3.1.4 Total production cost 109

5.3.1.5 Profit 110

5.3.1.6 Profit r a t e 110

5.3.2 Manufacturing models as a function of machining c o n d i t i o n s 110

5.3.3 Determining bottleneck stage and machining-speed r ange 112

5.3.4 Some considerations of the multi-stage manu­ facturing system 115

5.3.4.1 Optimization under the minimum-cycle-time criterion 115

5.3.4.2 Production cost 118

5.3.5 Nonlinear multiobjective (goal-programming) m o d e l

5.3.5.1 Baseline m o d e l 119

5.3.5.2 Nonlinear goal-programming m odel 122

5.3.6 Numerical example and consideration 124

5.4 C o n c l u s i o n s 135

CHAPTER 6 : C O N C L U S I O N S 137

6.1 Summary and C o n c l u s i o n s 137

6.2 Suggestions for Future R e s e a r c h 139

APPENDIX A: NONLINEAR GOAL PROGRAMMING VIA PATTERN SEARCH 141

A 1 N L G P / P S 141

A 2 NLGP/PS-M1 and N L G P / P S - M 2 143

A 3 Considerations 144

APPENDIX B : COMPUTER CODES F OR MULTIOBJECTIVE OPTIMIZATION OF MANUFACTURING S Y S T E M S 147

REFERENCES 165

Table Page

3.1 Basic Data for Numerical Computation of Machining

Speeds for the Single-Stage Manufacturing System 23

3.2 Basic Data for Numerical Computation of

S e n s i t i v i t y 34

3.3 Impact of Changes of Time and Cost Parameters on

Optimal Machining Speeds Under Three Kinds of Evaluation Criteria 42

3.4 Basic Data for Numerical Computation of Machining

Speeds and Feed Rates for the Single-Stage Manufacturing System 45

4.1 Basic Data for Numerical Computation of Example 4.1 69

4.2 Basic Data for Unconstrained Optimization of a

4.3 Basic Data for Constrained Optimization of a

5.1 Basic Data for Optimization of a Four-Stage

Manufacturing System 125

A.l CPU Times Required for Solving Numerical Examples 145

2.2 Block Diagram of Process Planning Stage 8

2.3 General Flowchart of Optimization of Machining Conuitions 9

3.1 Unit Production Cost, Unit Production Time, and Profit Rate Versus Machining Speed 24

3.2 Functions of Performance Measures and Their Derivatives in the High-Efficiency Speed Range 26

3.3 Speed Range Satisfying Priority 1 in Example 3.1 28

3.4 Speed Range Satisfying Priorities 1 and 2 in Example 3 1 29

3.5 The Multiobjective Optimal Solution in Example 3.1 30

3.6(a) Change in t£ (tool-replacement time, min/edge) 35

3.6(b) Change in tp (preparation time, min/pc) 37

3.6(c) Change in k, (direct labor cost and overhead, $/min) 38

3.6(d) Change in km (machining overhead, $/min) 39

3.6(e) Change in k £ (tool cost, $/edge) 40

3.6(f) Change in r^ (gross revenue, $/pc) 41

3.7(a) Contours of the Unit Production Time and V-Minimum L i n e 46

3.7(b) Contours of the Production Rate and V-Maximum Line 47

3.7(c) Contours of the Unit Production Cost and V-Minimum L i n e 48

3.7(d) Contours of the Profit Rate and V-Maximum Line 49

3.8 The Optimal Region for Absolute Objectives 52

3.9(a) The Nondominated Solution Set (a) 53

3.9(b) The Nondominated Solution Set (b) 54

Figure Page

3.10 Graphical Presentation of a Nonlinear Multi­

objective M o d e l 56

3.11 Optimal Solution of Example 3 3 62

4.1 Profit Rate in Two-Product Manufacturing 71

4.2 Optimal Machining Speeds Under Achievement Functions

in Terms of Different Sets of Weighting Factors (Sj and W 2 ) 86

4.3 Three Performance Measures Under Achievement

Functions in Terms of Different Sets of Weighting Factors (W^ and W ^ 87

4.4 Computational Process of Phase I in Example 4.3 100

5.1 Schematic Model of a Flow-Type Multi-Stage

Manufacturing System 106

5.2 Relationships Among Machining Speed, Stage Produc­

tion Cost, and Cycle Tim e 116

5.3 Stage Production Cost 120

5.4 Computational Process of Machining-Speed Ranges 127

5.5 Optimal Machining Speeds at Corresponding Cycle

T i m e 132

5.6 Total Production Cost and Profit Rate at

Corresponding Cycle Time 133

k^ = overhead cost ($/min)

k^ = direct labor and overhead cost ($/min)

km = overhead cost during machining ($/min)

k t = tool cost per cutting edge ($/edge)

K = stage number governing the cycle time; used as subscript

K' = stage number having the longest stage production time at v ^

N

= Z = total direct l^.bor and overhead cost for N-stage manufactur-

J =1 ing system ($/min)

Simplified notation and nomenclature in Appendix A not included.

1 = additional feed length (mm)

1 = lot size of J (pcs/lot)

N = total number of jobs to be processed

= number of jobs in group i

P = profit rate or profit per unit time interval ($/min)

P = positive deviation from desired performance level in goal-

programming model

P = profit rate ($/min) (Chapts 4 and 5)

P = motor power (W)

P^ = sum of total job production time of G (min)

P „ = total job production time of J „ (min/lot)

q = production rate (pc/min)

= group production time of (min)

R = nose radius of cutting tool (mm)

R = surface roughness goal (y)

r = r - m = unit revenue minus material cost ($/pc)

r^ = unit revenue ($/pc)

r^, = total revenue per unit time interval ($/min)

s = feed rate (mm/rev)

= group set-up time for G^ (min)

S _ = set-up time for (min/lot)

Ce = tool-replacement time (min/pc)

t = machining time (min/pc)

t = preparation time (min/pc)

Cw = waiting time (min/pc)

t CT = cycle time (min/pc)

(c)

t = unit production time at the minimum-cost machining speed (min/pc)

t ^ = unit production time at the maximum-production-rate m a c h i n i n g

speed (min/pc)

T = total production time (min) (Chapt 4)

T = tool life (min/edge)

u^, = total production cost per unit time interval ($/min)

U = total production cost ($) (Chapt 4)

UT = total production cost ($) (Chapt 5)

(c)

v = minimum-cost machining speed (m/min)

(e)

v - a high-efficiency machining speed (m/min)

v ^ = maximum-profit-rate machining speed (m/min)

v ^ = maximum-production-rate machining speed or minimum-time machining speed (m/min)

= group production cost for ($)

a = constant in power function

3 = constant in power function

A = constant dependent on work size and machining conditions

Aq = machining constant

E = sum in regard to J from 1 to N, exclusive of K

J^K

Appreciation is also expressed to the other members of his

doctoral committee, composed of Drs J P Ignizio, C D Pegden,

W E Biles, and G A Kochenberger.

Finally, the author would like to express gratitude to Dr K

Hitomi for his continuous encouragement.

GENERAL INTRODUCTION

1.1 Introduction

The optimization analysis of manufacturing systems has been

studied for the past few decades In the field of economics of

machining, the machining, as run by a single tool (i.e., single-stage

manufacturing), has been discussed frequently Optimal machining

conditions, particularly optimal machining speed, have been analyzed

from the standpoint of the minimum-cost criterion (1), the maximum-

production-rate criterion or minimum-time criterion (1), and the

maximum-profit-rate criterion (2,3) ln practice, however, there

exist various constraints: i.e., feed-rate constraints,

machining-speed constraints, power constraints, rouehness constraints, and so on.

Recent developments in material-removal technology relying upon

computers have resulted in the advancement of machining information

systems which provide effective information, particularly, optimal

machining conditions for a certain criterion such as those mentioned

earlier.

Ideally, optimization should be applied to a total manufacturing

system, such as material fabrication, part-machining, product assembly,

and so on In this study, however, optimization for part-machining is

to be made for both single-stage and multi-stage manufacturing systems

Practically, 'machining' is a complex physical phenomenon; hence, a

simplified mathematical model should be constructed for the optimiza­

tion analysis of machining conditions.

For mathematical optimization analysis, there are unconstrained

optimization and constrained optimization The typical controllable

depth of cut Depth of cut can often be determined by the sizes of

the work material and the products; hence, in this research, it is

assumed constant Therefore, setting a fixed value for feed rate,

optimal machining speed will be determined in unconstrained problems.

In constrained optimization, in addition to optimal machining speed,

optimal feed rate is simultaneously determined.

The purpose of this research is to develop basic non-linear

multiobjective (goal-programming) models and make analyses for various

types of manufacturing systems Optimal machining conditions,

especially optimal machining speeds and/or feed rates, are analyzed

under multicriteria achievement functions.

1.2 Previous Related Studies

The optimization analysis of machining speeds on a single-stage

manufacturing system has been studied since Gilbert's first work (1),

in which the "maximum production rate" and the "minimum production

cost" criteria were introduced Computerized optimization and analyses

have been made by Ham (4) and Field et al (5,6) Okushima and Hitomi

(2 ) proposed a new criterion for manufacturing, i.e., "maximum profit"

in a limited time interval, which was named the "maximum profit rate"

by Armarego and Russel (3) Later, this analysis was made by Wu and

Ermer (7) and Hitomi (8 ).

On the single-objective constrained optimization, several

approaches, such as Lagrange multipliers (9) and geometric programming

(10), have been proposed Little research on the multiobjective

optimization has been done, even in a single-stage manufacturing system

The surrogate worth trade-off method (SOT Method) has been applied to

multiobjective optimization of machining conditions (11) Linear goal

programming has been formulated and analyzed by taking logarithms of

goals and constraints (12, 13) In these papers, however, the multiple

production goals are not considered simultaneously Also, nonlinear

goal programming has been applied to multiobjective optimization for a

single-stage manufacturing system (14).

Recently, in the manufacturing area the concept of Group Technology

(GT) has been introduced and has become widely used in real-life

workshops Group technology is generally considered to be a manufac­

turing philosophy or concept which identifies and exploits the sameness

or similarity of parts and operation processes in design and manufacture

In batch-type manufacturing, by grouping similar parts into part

families, based on their geometrical shapes or operation process and

also by forming machine groups or cells which process the designated

part families, it should be possible to reduce various costs (15).

From the GT standpoint, the production scheduling problem, i.e.,

"Group Scheduling", has been investigated, and several optimizing

algorithms for determining both the optimal group sequence and the

optimal job sequence (in each group) have been developed under

scheduling criteria, such as minimizing the total flow time and

minimizing the total tardiness (16, 17, 18, 19, 20, 21, 22) The

machine loading and product-mix problems under a GT environment have

been also discussed In this problem, the optimal kinds of products

(parts or jobs) to be produced within a limited time available and

the optimal machining speeds for all products are determined so as to

maximize the production amount for a given period of time (23, 24).

or product is rarely done in a single-stage manufacturing; rather, it

is usually completed by being processed successively on several machine

tools or w ork stations that constitute a multi-stage manufacturing

system Single-objective optimization for multi-stage manufacturing

systems has been analyzed (25, 26, 27, 28).

1.3 Outline of Thesis

In Chapter 2, first, functions and procedures of manufacturing

systems and the general procedure to obtain optimal machining condi­

tions are discussed Second, in order to analyze manufacturing

systems, basic mathematical models expressing production goals are

constructed, with manufacturing models expressed as a function of

machining conditions Third, production goals in manufacturing activ­

ities are defined in terms of the production time, production cost,

and profit rate These production goals might constitute integrated

production goals to be optimized in the chapters which follow.

In Chaper 3, optimization analysis of single-product, single-

stage manufacturing systems is made In unconstrained optimization,

characteristics and sensitivity of functions of performance measures

are discussed, and the concept of optimal machining speeds under

multiple objectives is introduced Next, in constrained optimiza­

tion, which is made considering various system constraints, the

nondominated solutions set is discussed Then, the nonlinear multi­

objective model is formulated and analyzed via goal programming.

There are several approaches to the problems with multiple objectives

(29) In this thesis, nonlinear goal programming (30) is mainly

applied to multiobjective optimization of machining conditions because

of its suitability and flexibility for application to problems

considered in this study.

In Chapter 4, optimization analysis of multi-product, single-

stage manufacturing systems is made First, optimizing machining

speeds under fundamental evaluation criteria on multi-product manu­

facturing systems is discussed Next, multiobjective optimization

of the machining condition on manufacturing systems under the GT

environment is treated in both unconstrained and constrained cases

Also, the relationships among production goals are discussed in

multiobjective optimization.

In Chapter 5, optimization analysis of multi-stage manufacturing

system is made A basic nonlinear multiobjective (goal-programming)

model for the flow-type multi-stage manufacturing system iz built

and analyzed in an attempt to determine optimal machining conditions

to be set on production stages Based upon the multiple production

goals on the cycle time (production time), the total production cost,

and the profit rate, the optimal machining speeds to be set on pro­

duction stages and the cycle time for the multi-stage manufacturing

system are analyzed, and the computational procedure for obtaining

them is developed.

The procedure of manufacturing (i.e., production of material goods)

can be divided into the product design stage, the process planning

stage, and the production implementation stage, as shown in Figure

2.1 In the product design stage* technical specifications of the

products to meet the market needs are decided through research and

development (R and D) activities; then, the product design is com­

pleted Next, in the process planning stage, an optimal process route

is decided, and machines, tools, and jigs to be used are selected,

together with optimum decisions for an operation sequence and machin­

ing conditions In the implementation stage, raw materials are

acquired, parts are machined, and then the products required are

assembled and inspected Finally, the finished products are shipped

and distributed to the market or the customers This production pro­

cedure is known as IMS (integrated manufacturing system).

Figure 2.2 shows the relationship between decision making and

associated information in the process planning stage (32) Through­

out the research, the discussion is focused on optimizing machining

conditions in this phase.

The general procedure for obtaining optimal machining conditions

is shown in Figure 2.3 The machinability data files store data for

material-removal operations, ■work materials, cutting tools, jigs,

machine tools, machining conditions, and so on On receiving input

information concerning the type of material-removal operation, work

materials, cutting and machine tools, and machining situations,

a DET E R M I N I N G MA C H I N I N G CON DI T I O N S

o M A T E R I A L A C Q U I S I T I O N

< ^ PAR T F AB RICAT ION

z PR O D U C T AS S E M B L Yz

INSPECTI ON

PR O D U C T S HI PMENT

Figure 2.1 Functions of Integrated Manufacturing Systems (IMS) (31)

Figure 2.2 Block Diagram of Process Planning Stage

M O D E LSUR FACE

ing models are mathematical models which represent the relationships

between machining conditions and variables expressing machinability,

such as tool life, surface finish, accuracy, machining power, and

others Based on those models and by the use of the machine tool,

cutting tool, and time and cost files, the optimizing processing is

performed to obtain the optimal machining conditions against produc­

tion goals, which will be defined later.

2.2 Basic Mathematical Models of Manufacturing Systems

2.2.1 Basic mathematical models of manufacturing systems

Basic mathematical models expressing production goals are to be

constructed for the purpose of analyzing manufacturing systems For

simplicity, the case of the single-product, single-stage manufactur­

ing systems is treated.

Basic mathematical models are constructed as follows (31).

2.2.1.1 Unit production time The unit production time is

the time needed to manufacture a unit of a product It is generally

assumed that unit production time is comprised of the following three

time elements:

(a) Preparation (or setup) time tp (min/pc): time necessary

to prepare for machining This includes the loading time

of work materials to a machine tool, the approach time of

a cutting tool to the workpiece, etc.

(b) Machining time tffl (min/pc): time necessary to manipulate

the machine tool for cutting.

(c) Tool-replacement time tg (min/pc): time required to

Denoting the time required to replace a worn cutting tool, t^,

with a new variable, tfi (min/edge), and the tool life (which is the

time length from the beginning of using a new edge until its replace­

ment) , by T (min/edge), the too1-replacement time per unit piece is:

Accordingly, the unit production time, t (min/pc), is repre­

sented by:

2.2.1.2 Production r at e The production rate is the number

of pieces produced per unit time, and is the reciprocal of the unit

production time given by equation (2.2), hence, denoting the produc­

tion rate by q (pc/min) as:

(2.3)

2.2.1.3 Unit production cost The unit production cost is a

cost required to manufacture a unit piece This cost consists of the

following six cost elements:

(a) Material cost mc ($/pc): cost of raw material per unit

piece produced.

Preparation (or

preparation time.

(d) Tool-replacement cost ug ($/pc) : cost of the

tool-replacement time.

(e) Tool cost ($/pc): cost of cutting edge required to

produce a unit product This includes purchasing and

depreciation cost of the tool and grinding wheel, direct

labor and overhead costs for grinding worn cutting edges,

etc.

(f) Overhead cost u^ ($/pc) : indirect cost necessary to

produce a unit product This includes depreciation cost

for machine tools, general administration expenses, etc.

The unit production cost, u ($/pc), is thus expressed by:

- k t — + k —

(2.4)

k l ^=kd + k i^ (S/™-11) = direct labor and overhead

km ($/min) = machining cost

2.2.1.4 Unit profit The profit is obtained by producing a

unit product The gross profit per unit piece produced, g ($/pc), is

the unit revenue or selling price, r^ ($/pc), minus the unit produc­

tion cost of u ($/pc):

(2.5) T

2.2.1.5 Profit rate The profit rate, p ($/min), is obtained

by multiplying the unit profit, g ($/.pc) by the production rate, q

(pc/min):

g r - m + k.t + (k, + k )t -

6 u L c l p 1 m m : gq = _ = -K -

2.2.2 Manufacturing models as a function of machining speed

In this case, machining speed is chosen as a representative

decision variable among three machining variables (i.e., machining

speed, feed rate, and depth of cut) An analysis is developed to

determine the optimal machining speed to be utilized on the machine

tool used For this purpose, basic mathematical models, especially

unit production time (equation (2.2)) or production rate (equation

(2.3)), unit production cost (equation (2.4)), and profit rate

(equation (2.6)), are to be expressed as a function of machining speed

The tool-life value, T (min/edge), and the machining time, tm (min/

p c ) , are the only factors in the basic models which vary with change

where X is a machining constant, and is determined, depending upon

machining patterns, as follows:

(a) For turning, boring, drilling, and reaming operations,

(b) For a milling operation,

= length of the hole to be machined (mm) for boring,

drilling, and reaming operations.

1 = additional feed length (mm) for a milling operation,

s = feed rate per tooth (mm/tooth) for a milling operation.

= feed rate per revolution (mm/rev) for other

operations.

Z = number of teeth of the milling cutter.

The relationship between tool life and machining conditions is

expressed by the tool-life equation The most typical model is the

Taylor tool-life equation (34), which shows the relationship between

tool life, T (min/edge), and machining speed, v (m/min), as follows:

where n and C are constants: n, slope of Taylor tool-life curve with

regard to machining speed, in general, between 0 and 1, usually 0.1 ^

0.4; and C, 1-min tool-life machining speed, both depending upon kinds

of cutting tools used and work material to be machined (with or

out cutting fluids), machine tools used, other machining conditions,

and so on.

Substituting tffl and T, expressed as a function of v given by

equations (2.7) and (2.10), into equations (2.2), (2.4), and (2.6),

unit production time and cost, and profit rate are expressed as a

In this case, optimal machining speed and optimal feed rate are

to be determined Therefore, a basic mathematical model for a single-

stage manufacturing system is constructed as a function of machining

speed and feed rate.

The basic three measures of performance are expressed in general

form as in equations (2.2), (2.4), and (2.6) In these formulae, the

tool life, T (min/edge), and the machining time, t^ (min/pc), are

dependent upon feed rate, s (mm/rev), and machining speed, v (m/min)

As to T, the following generalized tool-life equation is employed:

where mg, n^, and Cq are constants tffi is expressed as a function

of s and v as follows, similar to equation (2.7):

X0

where Ag is a machining constant.

Substituting equations (2.14) and (2.15) into equations

(2.2), (2.4), and (2.6), unit production time and cost, and profit

rate are expressed as:

Unit production time:

2.3 Production Goals in Manufacturing Activities

2.3.1 Three fundamental evaluation criteria

The following three fundamental evaluation criteria are utilized

in the traditional single-objective manufacturing optimization (31).

2.3•1.1 Maximum-production-rate or minimum-time criterion.

The maximum-production-rate or minimum-time criterion maximizes

the amount of products produced in a unit time interval; hence, it

minimizes the production time per unit piece It is the criterion to

be adopted when an increase in physical productivity or productivity

efficiency is desired, neglecting the production cost needed and/or

profit obtained The objective is expressed as:

2.3.1.2 Minimum-cost criterion The minimum-cost criterion

refers to producing a piece of product at the least cost, and coincides

with the maximum-profit criterion if the unit revenue is constant.

This criterion is to be adopted when there is ample time for produc­

tion The objective is expressed as:

2.3.1.3 Maximum-profit-rate criterion The maximum-profit-

rate criterion maximizes the profit in a given time interval This

criterion is to be recommended when there is insufficient capacity for

a specific time interval "Profit rate" usually means "return on

investment" in the field of economics and engineering economy In the

evaluation criteria for economical production, "profit rate" means

"profit per unit time interval." Where market demands are large com­

pared to the productive capacity, a larger total profit is obtained

by producing and selling a large amount of products as a result of

reducing unit production time associated with sacrificing unit pro­

duction cost, rather than based upon the minimum-cost criterion The

objective is expressed as:

2.3.2 Aspiration levels on production goals

Depending upon the objectives of the decision maker, the pro­

duction goals may be set to achieve some levels (aspiration levels)

on the production rate (or the production time), the production cost,

and/or the profit rate In such cases, the production objectives are

expressed in aspiration-level goals (35):

The production goals are employed according to the manufacturing

objectives An optimal selection from among them should be done from

2 the managerial standpoint.

inequalities (2.24) to (2.27) are represented as three major

production goals in the baseline model of goal-programming formulation 2

Methods on priority-selection from among production goals will

not be discussed in this study For priority structure, several

references are available, for example, (36).

OPTIMIZATION OF SINGLE-PRODUCT, SINGLE-STAGE MANUFACTURING SYSTEMS

3 *1 Determining Optimal Machining Speeds - Unconstrained Optimization

3.1.1 Characteristics of functions of performance measures

Optimization analysis of the single-stage manufacturing system

is fundamental to the optimization of the integrated manufacturing

systems.

It is necessary, first, to examine the characteristics of

functions of performance measures, i.e., the unit production time (or

the production rate), the unit production cost, and the profit rate

Basic mathematical models of these performance measures as a function

of machining speed have been constructed as equations (2.2), (2.4),

and (2.6) In this section, some simplified notation is used for

simplicity of discussion.1

Definition 3 1: A controllable continuous variable which

specifies the production time and cost is the machining speed, and

it is denoted by v v is feasible if 0 < v < ®.

3.1.1.1 Unit production time The time function consists of

a fixed time and variable (inversely and positively proportional)

time elements, and can be expressed simply as

where a, b, c, and n _> 1 are positive constants.

Property 3 1 ; The time function t(v) is a positive, unimodal

twice-differentiable, and strictly convex function.

1 For a more detailed discussion, see reference (37).