Assignment of tools to machines in a flexible manufacturinf system

8/10/2019 Assignment of tools to machines in a flexible manufacturinf system

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European Journal of Operational Research 81 (1995) 115-133 115

North-Holland

T h e o ry a nd e t h o d o l o g y

s s i g n m e n t o f t oo l s to m a c h i n e s

in a f le x i b l e m a n u f a c t u r i n g s y s t e m

T h o m a s H D A l f o n s o a n d J o s e A . V e n t u r a

Dep ar tme n t o f I ndus t r ia l and M anag eme n t Sys t em s Engineer ing

Universi ty Park PA 16802 US A

The Pennsy l van ia S ta t e

University

Received July 1992

Abstract: Subgradient optimization is employed to solve the problem of assigning tools to machines in a

flexible manufacturing system (FMS). Machines in the FMS have a limited number of slots in the tool

magazine, and tools may require multiple slots. Tools are grouped based upon a pairwise similarity

coefficient that indicates the frequency of successive operations that require both tools. Typical solution

strategies have included graph theoretic heuristics. Lagrangian relaxation is utilized in the algorithm

developed in this article. The p roblem is formu lated as a linear i nteger program. Aft er dualizing two sets

of constraints, two integer subproblems are formed in which the first is further decomposed into several

knapsack subproblems. The second subproblem can be solved by a linear network code. A subgradient

algorithm is developed to solve the dual problem. The algorithm is compared to a graph theoretic

heuristic that utilizes cluster analysis. For most test problems, the subgradient algorithm is superior, but

is sensitive to convergence parameters.

Keywords:

Scheduling; Flexible manufacturing systems; Subgradient optimization; Lagrange multipliers;

Heuristics

1 I n troduct ion

This article investigates the application of group techno logy (GT) to a special manufactu ring

environment. GT is the science of grouping objects together based upon similar characteristics. Hyer

(1984) described GT as a way of organizing information about similarity of objects. Gall agher and Knight

(1986) went one step further, and described GT as a way of taking advantage of similarities of parts or

processes in all stages of design and manufa cturing systems. GT is a means of organizing both the facto ry

layout and production flow. These definitions imply that GT has had broad applications in engineering.

Production flow analysis requires a classification scheme for organizing parts in the manufacturing

process. McAuley (1972) published one of the first GT studies in this area. The focus was on analyzing

characteristics of parts to be manufactured. McAuley developed a relationship factor between machines

Correspondence to: Prof. J.A. Ventura, Department of Industrial and Management Systems Engineering,The Pennsylvania State

University, UniversityPark, PA 16802, USA.

0377-2217/95/ 09.50 1995 - Elsevier ScienceB.V. All rights reserved

SSDI 0377-2217(93)E0135-K


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6 T .H . D A l f o n s o , J ,4 . V e n t u r a / A s s i g n m e n t o f t o o ls t o m a c h i n e s

that manufactu red components. The number of components that require both or either machine in a pair

is used to determine the similarity coefficient.

GT is considerably versatile. A successful application requires basically two things: a classification

scheme for objects to be grouped and an objective that can be optimized by examining possible

groupings. Ng (1989) classified both parts and machines into families and cells, respectively. Grouping

efficiency was quantified, and the method can improve machine utilization.

In this paper, GT is applied to a flexible manufacturing system (FMS) environment with special

machine constraints. The objective is to minimize production travel between machines that are capable

of performing specified operations. Various parts are manufactured that require tooling sequences.

When a tool is not included in the tool magazine of the station that is producing a part, the part must

travel to another station. Each FMS machine is restricted to a limited number of tools it may carry.

FMSs are considerably complex, and with each new objective or constraint considered, the model may

change significantly, and so too the solution algorithm. Cluster analysis, graph theory, and dual

algorithms are not new solution strategies to manufacturing models.

Two solution algorithms are used in this study. The first is a subgradient algorithm based upon a

Lagrangian Dual (LD) formulation. The second solution algorithm is a graph theoretic heuristic that

combines techniques supported in separate published papers. Computational results for several FMS

environments are presented. Kusiak (1990) asserted that GT has become one of the most important

methods applied to intelligent manufacturing systems. Designing rows of machines, determining machine

cells, organizing part information, and designing layouts for new factories and renovations of existing

factories, can all benefi t from GT. Wi th each new application, new algorithms may need to be developed.

This need has been highlighted by Ng (1989), Kusiak (1990), Mulvey and Beck (1984), and by Ventura

and Hsu (1989) and Ventura et al. (1990).

2 Rev iew o f recent op t imizat ion t echn iques for group techno logy

GT models have been optimized by graph theoretic methods such as cluster analysis, various

heuristics, and nonlinear and integer programming. McCormick et al. (1972) applied a graph theoret ic

heuristic to an airport design problem. Similarity coefficients between various services of an airline were

determined, and service stations were established. For example, passenger and baggage check-in are

highly related operations in an airport, and these services almost always occur at the same station. In

1975, Lenstra and Rinnooy Kan showed that the airport design problem could be modelled as a traveling

salesperson problem and solved it with existing graph theoretic algorithms.

Ng (1989) developed a heuristic especially for the machine cell problem. Machines are grouped into

ceils, parts are grouped into families, and a grouping efficiency measure de termines how they are joined.

The key to Ng s heuristic is a maximal spanning tree algorithm. Spanning trees, whether minimal or

maximal, are very effective in joining nodes (machines) from a graph based on a simple objective

function. The maximal spanning tree algorithm works as follows. Given a network, repeatedly select the

arc with the highest weight, provided a cycle is not formed by doing so. In a complete network with n

nodes, n - 1 arcs will be selected to form a spanning tree (Gould, 1988).

Lagrangian relaxation is an effective approach to solving complicated mathematical models. In a very

simple description, this approach identifies constraints that are difficult to meet and incorporate them

into the objective function. Also called Lagrangian dual (LD) methods, these techniques use subgradi-

ents to optimize over the modified, less complicated, expanded solution space. Part of the objective,

then, is to systematically approach feasibility in the complicated constraints. The term subgradient refers

to the fact that the objective function is nondifferentiable, and subgradients are used in the optimization

algorithm rather than gradients.

In GT modelling, only a few research publications have included Lagrangian relaxation as the solution

algorithm. Mulvey and Crowder (1979) developed one of the first such models. Ventura et al. (1990)

developed a LD formulation of a GT application to FMSs. In this model, production flow data are


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T.H. D Alfonso, J.A. Ventura / As sig nm en t of tools to machines

Table 1

Tool sequence and daily demands for production of seven parts

117

Part Tool sequence Daily demand

1 2 1 3 4 25

2 2 1 4 3 100

3 4 2 3 5 25

4 5 2 1 50

5 1 5 2 25

6 3 2 5 1 25

7 4 2 3 1 100

considered and tools and parts are grouped so as to minimize the interdependencies among the groups.

Uppe r and lower bounds were placed on the number of tools and parts per FMS station. Th e

formulat ion used by Ventur a et al. 1990) is similar to the formulat ion consid ered in this article. The

complete formulation is given in the next section.

3 P r o b l e m f o r m u l a t i o n

The FMS scenario considered in this article is one in which machines are capable of operating several

tools stored in a tool magazine. Parts to be manu fact ured require a certain sequence of tooling

operations to be performed, and many different parts can be pro duced by the system. The problem is to

determine which tools are assigned to which machine, while considering

1) the sequence of tools each part requires,

2) the demand for each part, and

3) the number of tools each machine magazine can hold.

Furthermore,

4) some tools may take up more than one magazine slot, and

5) the nu mber of groups machines) is predetermined.

Consider the infor mation in Table 1 pertain ing to a set of parts that are to be manufa ctured . Let i, j)

be defined as opera tion i immediately preceding operat ion j. Fo r example, 5, 2) occurs in the

prod ucti on seque nce of Part 4, which has a daily dem and of 50. Orde red pair 5, 2) also occurs in the

prod ucti on seque nce of Part 5 with a dem and of 25. Ordere d pair 5, 2) occurs nowher e else in the

prod ucti on plan. Because daily dem and measures the nu mber of times a particular part will be

manufa cture d, t he total n umb er o f times per day that sequence 5, 2) occurs is 75. It is our objective to

minimize the production flow between machines; therefore, the amount of travel between each ordered

pair is an important measurement. Table 2 contains a matrix in which each cell shows daily production

flow between ordered pairs.

Because the objective is to minimize flow between machines, and oij represents the daily production

flow from tool i to tool j, a better performance measure is to consider the total flow between one tool

Table 2

Daily production flow

oi j )

between ordered pairs of tools

1 2 3 4 5

1 0 0 25 100 25

2 175 0 125 0 25

3 100 25 0 25 25

4 0 125 100 0 0

5 25 75 0 0 0


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118

T . H . D ' A l f o n s o , J . A . V e n t u r a / A s s i g n m e n t o f t o o ls t o m a c h i n e s

Table 3

Daily production flow ( a i j ) between pairs of tools

1 2 3 4 5

1 - 175 125 100 50

2 - - 150 125 100

3 - - - 125 25

4 . . . . 0

5

a n d a n o t h e r f o r a g i v e n d a i l y p r o d u c t i o n s c h e d u l e . T h a t i s , c o n s i d e r a i j = O i j [ - O f f , t h e f l o w i n b o t h

d i r e c t io n s . T h e m a t r i x A = [ a i ] ] i s s y m m e t r ic , a n d c a n b e c o n s t r u c t e d b y O + O w, w h e r e O = [oil]. T h e

u p p e r t r i a n g u l a r p a r t o f s u c h a m a t r i x is g i v e n i n T a b l e 3 . F i n a l ly , th e n u m b e r o f m a g a z i n e s l o t s e a c h t o o l

r e q u i r e s i s g iv e n in T a b l e 4 . T h i s p r o p e r t y w il l c o n s t r a i n t h e c h o i c e o f t o o l a s s i g n m e n t s .

T o f o r m u l a t e t h e m a t h e m a t i c a l e x p r e s s i o n f o r t h e o b j e c t iv e f u n c ti o n , c o n s i d e r t h e v a r ia b l e Xik, t h a t

i n d i ca t e s t h e m e m b e r s h i p o f to o l i i n m a c h i n e g r o u p ) k :

1 i f t o o l i ls w i t h m a c h i n e k , f o r i = l k = l

Xik

= o t h e r w i s e ,

n ,

p .

9

N o w , i f t o o l i a n d t o o l j a r e n o t i n t h e s a m e g r o u p , t h e d a i ly p r o d u c t i o n f l o w w i ll b e a i j . A n e w

var iab le Xijk, c a n b e i n t r o d u c e d s o t h a t ~

1

i f t o o l s i a n d j a r e w i t h m a c h i n e

k ,

X i j k = 0 o t h e r w i s e ,

f o r i = 1 . . . . , n - l , ] = i l , . . . , n , k = l . . . . . p .

W i t h t h is v a r i a b l e , i t is e a s y t o d e r i v e t h e o b j e c t i v e f u n c t i o n .

F i r s t, p r o d u c t i o n f l o w o f aij b e t w e e n m a c h i n e s o c c u r s w h e n E k X ij k = 0 a n d a i j ~ O. I t i s t h e r e f o r e t h e

\

o b j e c t i v e t o m i n i m i z e t h e a g g r e g a t e fl ow ; n a m e l y ,

m i n ~ _ a i j - - x i j k . 1 )

i=1 j= i+ l

E x p r e s s ion 1 ) s imp l i f i e s t o t h e ob j e c t ive f u n c t ion t h a t i s u se d in th i s ar ti cl e:

n-- 1 ~ P

m a x E E a i j X i j k 2 )

i=1 j= i+ l k= l

T h e o b j e c t iv e , 2 ) , i s i n t e r p r e t e d a s d e t e r m i n i n g t o o l g r o u p s t h a t m a x i m i z e a g g r e g a t e d a i ly p r o d u c t i o n

Table 4

Number of magazine slots required by each tool

tool i) magazine slots

( r )

1 2

2 1

3 1

4 2

5 1


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T.H. D Alfonso, ZA. Ventura /Assignment of tools to machines

119

f lo w a m o n g m e m b e r s w i t h i n t h e g r o u p s i s e q u i v a le n t t o m i n im i z i n g a g g r e g a t e p r o d u c t i o n f l o w b e t w e e n

g r o u p s m a c h i n e s ) . T h e o b j e c t i v e f u n c t i o n c a n a l so b e e x p r e s s e d i n a n o n l i n e a r f o r m :

n--1 ~ P

m a x E ~_,

aijXikXjk. 3 )

i = 1 j = i l k = l

I n t h i s st u d y , f u n c t i o n 2 ) i s u s e d a s t h e o b j e c t i v e , b u t t h e X i k - v a r ia b l e s a r e a l s o u s e d in t h e c o n s t r a i n t

s e t t h a t w i l l b e d e v e l o p e d n e x t .

B e c a u s e e a c h t o o l c a n o n l y b e a s s i g n e d t o o n e m a c h i n e , t h e f o l lo w i n g c o n s t r a i n t is n e c e s s a r y :

P

E Xik = 1 f o r i = 1 , . . . , n . 4 )

k = l

B e c a u s e t h e o b j e c t i v e 2 ) a n d c o n s t r a i n t s e t 4 ) d i f f e r i n v a r i a b l e ty p e s , c o n s t r a i n t s e ts 5 ) a n d 6 ) a r e

i n t r o d u c e d :

x i j k

~ Xik f o r i = 1 . . . . , n - 1 , j = i + 1 . . . . . n , k = 1 . . . . , p , 5 )

Xi ik


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120 T.H.

D Alfonso, .A.Ventura Assignment f ools o machines

X i j k ~ { 0 , 1 } f o r i = l . . . . . n - - l , j = i + l , . . . , n , k = l , . . . , p , ( 1 6 )

Y i k ~ { O , r i }

f o r i = l . . . . . n , k = l , . . . , p . ( 17 )

T w o a l g o r i t h m s w il l b e e x a m i n e d t o s o l v e t h is p r o b l e m . T h e f i r s t u s e s s u b g r a d i e n t o p t i m i z a t i o n , a n d

t a k e s a d v a n t a g e o f th e s t r u c t u r e o f L I P . T h e s e c o n d i s a g r a p h t h e o r e t i c h e u r i s t i c t h a t f o l lo w s t h e

r e c o m m e n d a t i o n s g i ve n i n r e s e a r c h p a p e r s o n s i m i la r G T p r o b l e m s .

4 L a g r a n g i a n d u a l f o r m u l a t i o n

B e c a u s e p r o b l e m L I P i s d i f fi c u lt , a s t r a t e g y t o s i m p l if y t h e s o l u t i o n p r o c e s s i s e m p l o y e d . L a g r a n g i a n

r e l a x a t i o n is u s e d t o d u a l i z e c o n s t r a i n t s (1 3 ) a n d ( 14 ) . T h e L a g r a n g i a n f u n c t i o n b e c o m e s

n - - 1 p

ZD (W ,U)= m ax E ~ ~-~ [aijXijk+Wijk(Yik--riXijk)+Uijk(Yjk--FjXijk)]. 1 8 )

i =1 j= i + l k = l

T h e v a r i a b l e s

wiy k

a n d

viyk

a r e t h e L a g r a n g e m u l t i p l i e r s t h a t c o r r e s p o n d t o c o n s t r a i n t s ( 1 3 ) a n d ( 1 4 ) ,

r e s p e c t i v e l y .

T h e L a g r a n g i a n d u al p r o b l e m t h e n b e c o m e s

( C D P )

M i n

Z ~ ( w , ~ )

( 1 9 )

s u b j e c t t o w , v > 0 . ( 2 0 )

E q u a t i o n ( 1 8) c a n a l so b e e x p r e s s e d i n t h e f o r m

n--1 n P ~ ~ [( ~ ) (i--j~ll

) ]

Z D ( W , v ) = m a x ~ ~ , ~ , ( % - - r iW i j k - - r j V i j k ) X i j k + W is k + V iik Y i k

i =1 j = i + l k = l k = 1 i = 1 j = i + l

( 2 1 )

I n t h i s fo r m , i t is e a s y t o d e t e r m i n e a d e c o m p o s i t i o n o f ( 2 1) in t o f u n c t i o n s o f

x i j k

a n d

Y i k ,

a n d t o a s s i g n

c o n s t r a i n t s w i t h t h e c o r r e s p o n d i n g v a r i a b l e s . S u c h a d e c o m p o s i t i o n f o r m s tw o s u b p r o b l e m s , S P 1 a n d

S P 2 . A c o m p r e h e n s i v e d e v e l o p m e n t o f e a c h s u b p r o b l e m f o ll ow s .

( S P 1 )

n - -1 n P

M a x E E E ( a i y - - r i W i j k - - r y V i y k ) X i j k

( 2 2 )

i =1 j = i + l k = l

s u b j e c t t o

P

~ _ , X i j k < _ l

f o r i = l . . . . , n - - l ,

j = i + l , . . . , n ,

( 2 3 )

k = l

X i j k ~ { 0 , 1 }

f o r i = l . . . . . n - - l , j = i + l . . . . . n , k = l , . . . , p . ( 24 )

C o n s t r a i n t ( 2 3) is a d d e d t o t h e m o d e l t o r e d u c e t h e s iz e o f t h e e x p a n d e d s o l u t io n s p a c e. T h e

c o n s t r a i n t w o u l d b e r e d u n d a n t i n L I P d u e t o c o n s tr a i n ts ( 1 2 ) -( 1 4 ) . SP 1 c a n b e f u r t h e r d e c o m p o s e d i n t o

n ( n -

1 ) 0 - 1 k n a p s a c k ( K S P ) s u b p r o b l e m s . T h i s f o l l o w s f r o m t h e f a c t t h a t 1 + 2 + + n - 1

( i , j )

p a i r s e x i st i n S P 1 . E a c h K S P s u b p r o b l e m ( i , j ) c a n b e s o l v e d b y t h e f o l l o w i n g a l g o r i th m :

S t e p 1. L e t k ' ~ a r g m a x { a i j - r i w i j x - r j v i i k l k = 1 . . . . p } .

S t e p 2 .

I f

a i j - r iw i j k , - r j v i j k , >_ O,

t h e n

x i*k , = 1 a n d x i~k

= 0 fo r a l l k :/: k . O t he r w i s e , s e t

x * k

= 0 f o r

k = 1 . . . . p .


7/19

T . H .D A l fo n s o , . 4 . Ve n t u r a A s si g n m e n t f t o o l s o m a c h in e s 121

T h i s a l g o r i th m m a y b e e m p l o y e d b e c a u s e , f o r e a c h

i , j ) ,

t h e s u m o f

xi j k

o v e r e a c h k m u s t b e l e ss th a n

o r e q u a l t o 1 , a n d t h e o b j e c t i v e i s m a x i m i z e d o v e r e a c h

i , j )

b y c h o o s i n g t h e l a r g e s t

aij - r i w i y k - r j v i j k .

C o n s t r a i n t ( 2 3 ) i s i m p l i c it ly m e t b y t h i s m e t h o d .

T h e s e c o n d s u b p r o b l e m i s a s f o l l o w s .

( S P 2 )

P ~ [ ( = ~ i+ lW ij ) ( i - - j ~ = l ) ]

ax ~ + Vjik Yik

( 2 5 )

k=l i=1 j

P

s u b j e c t t o

~

Y i k : r i f o r i = 1 . . . . , n , ( 2 6 )

k= l

L < ~ y ~ k < U

f o r k = l . . . . . p , ( 2 7)

i = 1

Y ik ~ { O , r i }

f o r i = l . . . . , n , k = l . . . . , p . ( 2 8)

B y m a k i n g t h e y /k - v ar ia b le s c o n t i n u o u s t h i s m o d e l b e c o m e s a l i n e a r t r a n s p o r t a t i o n p r o b l e m w i th

l o w e r a n d u p p e r b o u n d s o n t h e r e q u i r e m e n t s . T h e o r i g i n al n o d e s r e p r e s e n t t o o l s, t h e i n p u t f lo w s a r e t h e

r e q u i r e d n u m b e r o f sl ot s f o r e a c h t oo l , a n d t h e d e s t i n a t io n n o d e s r e p r e s e n t m a c h i n e s . T h e r e q u i r e m e n t s

a t th e d e s t i n a t i o n n o d e s a r e b o u n d e d b y L a n d U . B y m a k i n g

Yik

c o n t i n u o u s , h o w e v e r , t h e o p t i m a l

s o l u t i o n t o S P 2 d o e s n o t n e c e s s a r i l y s a ti s fy c o n s t r a i n t ( 2 8 ). T h i s a n d o t h e r i s s u e s a r e a d d r e s s e d i n th e

n e x t s e c t io n .

5 Subgradient optimization

B e c a u s e t h e o b j e c ti v e f u n c t io n o f L D P is n o n d i f f e r e n t ia b l e , a n d g e n e r a l ly c o n ta i n s m a n y v a r ia b l es ,

s u b g r a d i e n t o p t i m i z a t i o n i s a g o o d c h o i c e fo r a s o l u ti o n a l g o r it h m . T h e t e c h n i q u e e i t h e r p r o d u c e s a n

o p t i m a l o r p r o v i d e s a g o o d b o u n d o n t h e o p t i m a l s o l u t i o n . T h e s u b g r a d i e n t a l g o r i t h m i s a n i t e r a t i v e

p r o c e d u r e t h a t p r o d u c e s a s e q u e n c e o f L a g r a n g i a n m u l t i p li e r s ( H e l d e t a l ., 1 9 7 4) . T h i s s e q u e n c e c o n t a in s

a s u b s e q u e n c e t h a t c o n v e r g e s to t h e o p t i m a l m u l t i p l ie r s ( G o f f i n , 1 9 77 ) . T h e f o l l o w i n g s u b g r a d i e n t

a l g o r i th m w a s e m p l o y e d t o s o l ve L D P .

Subgradient algorithm

Step 1. Initialization.

P i c k t h e s t o p p i n g c r i t e r i o n , e > 0 . S e t w 1

= v 1 = 0,

L B = 0 ,

n- -1

U B = ~ ~

aiy,

i = 1 j = i l

a n d r = 1 . T h e v e c t o r s w a n d v a r e d u a l m u l t i p l i e r s f o r c o n s t r a i n t s e t s (1 3 ) a n d ( 1 4 ), re s p e c t i v e l y . L B i s

t h e l o w e r b o u n d , a s s o c ia t e d w i th t h e p r i m a l p r o b l e m , a n d U B is t h e u p p e r b o u n d , a s s o c ia t e d w i th t h e

d u a l p r o b l e m .

Step 2 . Solve

S P 1

a n d

S P 2 . U s i n g t h e c u r r e n t L a g r a n g i a n m u l t i p l e r s w r a n d v r, l e t x r b e a n o p t i m a l

s o l u t i o n t o S P 1 w i t h o b j e c t i v e f u n c t i o n v a l u e S P r , a n d l e t y ~ b e a n o p t i m a l s o l u t i o n t o S P 2 w i t h o b j e c t i v e

f u n c t i o n v a l u e S P 2 r. S e t Z D ( W , V )

=

S P Y + S P 2 r .

Step 3. Update upper bound.

L e t U B r = m i n { ( U W - 1 ,

ZD W ~,

V ~)}. U B r i s t h e u p p e r b o u n d , n a m e l y t h e

b e s t c u r r e n t d u a l s o l u t io n , Z D -

Step 4. Generate primal feasible solution.

Step 4 .1 .

I f ( X r ,

y~)

i s f e a s i b l e t o L I P , t h e n s e t x x ~ = x r , y y~

= y r ,

a n d g o to S t e p 5 . O t h e r w i s e ,

Step 4 .2 .

F o r e a c h i , l e t

K r i )

= a r g

max{Y[klk = 1 . . . . p} .

S tep 4 .3 .

F o r e a c h i , i f

I g r i ) l =

1, l e t

k

1 , r a n d o m l y

s e l e c t

k


8/19

1 2 2 T . H .

D ' A l f o n s o, J . A . V e n t u ra / A s s i g n m e n t o f t o ol s t o m a c h i n e s

S t e p 4 . 4 . L e t

yyrik = { riO otherwise,ifk=k i) f o r i = 1 . . . . , n ,

{~i i f b o t h y y / ~ = r i a n d y y r = r y ,

r __ jk

X X i j k - -

o t h e r w i s e ,

k = 1 . . . . , p ,

f o r i = 1 , . . . , n - 1 , j = i + l , . . . , n ,

k = l . . . . . p .

T h e v e c t o r

X X , y y r )

m a y n o t b e p r i m a l f e a s i b l e ) .

S t e p 5 . U p d a t e l o w e r b o u n d . I f x x r , y y ~ ) is p r i m a l f e a s i b l e , L B ~ = m a x { L B r - l , Z x x r , yy~)}; o t he rw i s e ,

L B r = L W -1 . L B r i s t h e l o w e r b o u n d , n a m e l y t h e c u r r e n t b e s t p r i m a l s o lu t io n , Z .

S t e p 6 . C h e c k s t o p p i n g c r i t e r i o n . I f U B r - - L B r ) / L B r < e , t h e n s t o p . O t h e r w i s e , g o to S t e p 7 .

S t e p 7 . C a l c u l a t e s u b g r a d i e n t s .

G ~ k = r i x r y k - - y i r k i = 1 . . . . . n - 1 ,

r _ r r

H i j k - - r y x i i k - Y j k , i = 1 . . . . , n - - 1 ,

S t e p 8 . C o m p u t e s t e p s iz e .

[ f r U B r -

L B r ) ]

t r : v ~ n - l ~ n v i~ p [ [ G r ~ 2 ( n i r l . k ) 2 ]

z - ' i ~ l z - ' j = i + l Z k = l [ [ , i j k } 4 -

S t e p 9 . U p d a t e L a g r a n g e m u l t ip l i e rs .

r t r G r I

wi~.~ 1 = m a x { 0 , w i j k + u ~ J , i = 1 , . . . , n - 1 ,

r l r r r

viyk = m a x { 0 , v i j k + t H i j k } , i = l . . . . , n - l ,

S t e p 1 0 . I n c r e m e n t r b y 1 a n d g o b a c k t o S t e p 2.

j = i + l . . . . . n ,

k = l , . . . , p ,

j = i + l . . . . . n , k = 1 , . . . , p .

j = i + l . . . . , n , k = 1 , . . . , p ,

j = i + l , . . . , n , k = l . . . . . p .

I t s h o u l d b e p o i n t e d o u t t h a t S t e p 4 c a n p r o d u c e a p r i m a l f e a s i b l e s o l u ti o n , w h i c h i s n o t a l w a y s a t

h a n d f o r L a g r a n g i a n a l g o r it h m s . S t e p 4 . 3 c al ls f o r th e k - t h g r o u p t o b e s e l e c te d r a n d o m l y i f m o r e t h a n

o n e g r o u p i s a c a n d i d a t e . T h i s f e a t u r e h e l p s t o p r e v e n t c y c li n g i n t h e a l g o r i t h m .

S u b g r a d i e n t s a r e s i m i l a r to g r a d i e n t s , w h i c h a r e d i r e c t i o n s o f s t e e p e s t a s c e n t f o r a v e c t o r fu n c t i o n .

S u b g r a d i e n t s e x i s t w h e n t h e f u n c t i o n i s n o n d i f f e r e n t i a b l e . S i n c e t h e p r i m a l p r o b l e m i s a l i n e a r i n t e g e r

p r o g r a m , t h e d u a l f u n c t i o n is n o n d i f f e r e n t i a b l e . T h e s t e p s iz e d e t e r m i n e s h o w f a r a l o n g a s u b g r a d i e n t

t h e n e x t v a r i a b l e v a l u e w i ll b e . T h e n u m e r a t o r c o e f f i c i e n t , f r is a v a l u e t h a t m a y b e c h a n g e d e a c h

i t e r a t i o n . I t w il l b e s h o w n i n a l a t e r s e c t i o n t h a t t h e c h o i c e o f t h e s t e p s i z e p a r a m e t e r ,

f r

f o r e a c h

i t e r a t i o n i s v e r y i m p o r t a n t t o f i n d i n g a g o o d s o l u t i o n . I n t h i s s t u d y a n i n i t ia l f r i s c h o s e n , a n d i s

p e r i o d i c a l l y h a l v e d e v e r y 1 , 2 , 3 , o r 4 i t e r a t i o n s ) .

S P 2 w a s so l v e d w i t h a n e t w o r k f l ow c o m p u t e r p r o g r a m c a l le d N E T F L O K e n n i n g t o n a n d H e l g a s o n ,

1 9 8 0) . A c o n s t r a i n t w a s a d d e d t o t h e n e t w o r k f l o w p r o b l e m r e q u i r i n g t o o l 1 t o b e a s s i g n e d t o g r o u p 1 ,

t o o l 2 t o b e a s s i g n e d t o e i t h e r g r o u p 1 o r g r o u p 2 , a n d s o o n , u n t i l th e p - t h t o o l a n d a b o v e , t h a t c a n b e

a s s i g n e d t o a n y g r o u p . T h i s m o d i f i c a t i o n r e d u c e d t h e n u m b e r o f e x i st in g o p t i m a l s o l u ti o n s a n d g r e a t l y

i m p r o v e d t h e p e r f o r m a n c e o f t h e s u b g r a d i e n t a l g o r i th m .

6 G r a p h t h e o r et i c h e u r i s t ic

T h e s u b g r a d i e n t a l g o r i t h m d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n h a s i t s d r a w b a c k s . A s t h e n u m b e r o f

t o o l s a n d m a c h i n e s i n c r e a s e s , t h e p r o b l e m s i ze g e ts si g n i fi c a n tl y l a r g e . F u r t h e r m o r e , t h e s u b g r a d i e n t

a l g o r i t h m r e q u i r e s c a r e f u l s e l e c t i o n o f t h e s t e p s i z e f o r e a c h i t e r a t i o n . T h i s w il l b e d e m o n s t r a t e d i n t h e

n e x t s e c t i o n . F i n a l l y , L a g r a n g i a n d u a l t e c h n i q u e s d o n o t g u a r a n t e e a p r i m a l f e a s i b l e s o l u t i o n . B e c a u s e


9/19

T.H. D Alfonso, J.A. Ventura / Assignment of tools to machines 123

c o n s t r a i n t s a r e d u a l i z e d , s o l u ti o n s g e n e r a t e d b y t h e a l g o r i t h m m a y b e i n f e a s i b l e in t h e s e c o n s t r a in t s . T h e

s u b g r a d i e n t a l g o r i t h m h a s t h e o b j e c t iv e o f m i n i m i z i n g i n f e a si b il it y , b u t n o t n e c e s s a r i l y e l i m i n a t i n g

i n fe a s ib i li ty . I n t h i s c a s e , h o w e v e r , w e h a v e b e e n a b l e t o d e v e l o p a h e u r i st i c a l g o r i t h m t h a t m a y p r o d u c e

a p r i m a l f e a s i b l e s o l u t i o n a t e a c h i t e r a t i o n .

M a n y h e u r i s ti c s h a v e b e e n d e v e l o p e d a n d a p p l ie d t o p r o b l e m s s i m i la r to L I P . M a n y o f t h e s e h e u r is t ic s

a r e s i m p l e t o p r o g r a m , a n d f i n d g o o d f e a s i b l e s o l u t i o n s . T h e h e u r i s t i c d e s c r i b e d i n t h i s s e c t i o n i s c a l l e d

C h o p t h e M a x i m a l S p a n n i n g T r e e C M S T ) . I t c o m b i n e s a t e c h n i q u e s u g g e s t e d b y N g 1 9 89 ) t o f o r m a

m a x i m a l s p a n n i n g tr e e M S T ) o f th e t o o l p r o d u c t io n f l o w m a t r ix , a n d a n a l g o r i th m f o r d e t e rm i n i n g a

s o l u t io n t h a t m e e t s t h e m a c h i n e c a p a c i t y c o n st r a in t s . T h e l a t t e r p a r t o f t h e h e u r i s t i c is s i m i l a r t o

m o d i f i c a ti o n s m a d e t o t h e d e n d r o g r a m t e c h n i q u e o f A u g u s t o n a n d M i n k e r 1 9 70 ) b y S e i fo d d in i 1 9 88 )

a n d S h t u b 1 9 88 ). S h r u b s u g g e s t e d to s t a r t w i t h o n e g r o u p , a n d f o r m a d d i t i o n a l g r o u p s b y r e m o v i n g

m e m b e r s s o a s t o m e e t f e a s i b il it y c o n s t r a i n t s w h i le r e d u c i n g d i ss i m i la r it y . T h e C M S T a l g o r i t h m

d e s c r i b e d h e r e f o l lo w s t h e a d v i c e o f S h t u b , b u t t h e f e a s ib i l it y h e u r is t i c i s u n i q u e .

Chop the Maximal Scanning Tree Algorithm

S t e p 1 . F o r m a m a x i m a l s p a n n i n g t re e . T h e t o o l p r o d u c t i o n f l o w m a t r i x is s y m m e t r i c . I t c a n b e

c o n s i d e r e d a c o m p l e t e g r a p h , i n w h i c h t h e n o d e s a r e t o o l s, a n d a r c s a r e w e i g h t e d b y a ij . W h e n a t o o l

r e q u i r e s m o r e t h a n o n e s lo t , s a y r o f t h e m , t h e n r n o d e s a r e u s e d t o r e p r e s e n t t h is t o o l. T h e w e i g h t

b e t w e e n t h e s e n o d e s i s i n f in i te , e n s u r i n g th e i r s e le c t i o n i n t h e m a x i m a l s p a n n i n g t r e e M S T ) . A n M S T

c a n b e f o r m e d f r o m t hi s a r r a n g e m e n t . G o u l d 1 9 88 ) p ro v i d e d a n a l g o r i t h m f o r a c c o m p l is h i n g t hi s.

S tep 1 .1 . R e p e a t e d l y s e l e c t th e l a r g e s t a r c t h a t d o e s n o t f o r m a c y cl e u n t il a s i n gl e tr e e i s f o r m e d .

S t e p 2 . Cr e a t e a f o r e s t m a i n t a i n i n g l o we r b o u n d fe a s i b il i ty . T h e M S T m u s t h a v e p - 1 a r c s r e m o v e d t o

f o r m p s u b g r a p h s . R e m o v i n g a r c s f r o m a t r e e r e s u l t s in w h a t i s c a l l e d a f o r e s t. E a c h t r e e i n th e f o r e s t

c r e a t e d b y C M S T m u s t b e b e t w e e n a m i n i m u m a n d a m a x i m u m s iz e. W h e n c h o o s in g a n a r c t o b e

r e m o v e d , t w o t h in g s m u s t b e c o n s i d e r e d . F i r s t, t h e f o r e s t m u s t m e e t s iz e r e q u i r e m e n t s , a n d s e c o n d , t h e

o b j e c t i v e f u n c t i o n m u s t b e k e p t i n m i n d . T h e s e s t e p s a r e f o l l o w e d :

S tep 2 .1 . S o r t t h e a r c s b y n o n d e c r e a s i n g w e i g h t .

S tep 2 .2 . R e m o v e a r c s f r o m t h e t o p o f t h e l is t, m a i n t a i n i n g l o w e r b o u n d L B ) fe a s ib i li ty . I f r e m o v i n g a n

a r c v i o l a t e s L B f e a s i b i l i ty c o n s i d e r t h e n e x t a r c i n t h e l i st .

S tep 2 .3 . C o n t i n u e u n t il p - 1 a r c s h a v e b e e n r e m o v e d . I t m a y b e n e c e s s a r y t o b a c k t r a c k , r e p l a c e a n a r c

t h a t h a s b e e n r e m o v e d , a n d c o n t in u e . T h e w o r s t - c a s e s ce n a r io , c o m p l e t e e n u m e r a t i o n , h a s i t e r a ti o n s

t o t a l l i n g p 2 1 ) .

Table 5

Com putational results for five-tool test prob lem

Test 1 Test 2

Number of Tools n) 5 5

Number of Groups p) 2 2

Lower Bound L) 2 2

Upper Bound U) 5 4

Subgradient Algorithm:

Solution 625 450

G ro up l {1,2,3,5} {1,2,3}

Gr oup 2 {4} {4,5}

Dual Lowe r Bound 975 975

Iterations 3 3

CM ST Algorithm:

Solution 625 450

G ro up l {1,2,3,5} {1,2,5}

Gr oup 2 {4} {3,4}

L iterations 2 2

U iterations 1 2


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124 T.H. D Alfonso, J.A. Ventura / Assignment of tools to machines

Table 6

Description of control variables for airport design problem

Item Space Description

01 1 Passen ger check in

02 1 Baggage check in

03 2 Baggage claim

04 1 Baggage moving system

05 2 Intra airport transportation

06 2 Cargo terminal

07 2 Close in parking lot

08 3 Remo te parking lot

09 2 Main access road

10 2 Circulation roads

11 1 Renta l car service area

12 2 Rent al car parking lot

13 2 Curb space for unloading

14 2 Curb space for loading

15 2 Waiting areas at gates

16 2 Transportati on stations

17 2 Aircraft loading system

18 1 Concessions

19 1 Renta l car desk

20 3 Runway capacity

21 2 Number of gates

22 1 Passenger information

23 2 Cargo trans fer

24 3 Air traffic control

25 2 Refus e removal

26 2 Flight crew facilities

27 2 Aircraft service on the apron

Table 7

Depend ency coefficients for airport design problem

0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7

0 1 3 0 2 2 0 1 1 0 0 0 0 2 0 0 0 0 1 0 0 0 3 0 0 0 0 0

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

0 3 0 0 1 1 0 0 0 0 2 0 0 1 0 0 1 0 0 3 0 0 0 0 0

3 1 0 1 1 0 0 0 0 0 3 0 1 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0

0 2 2 1 1 0 0 0 0 0 3 0 0 1 0 3 1 0 0 0 0 0

0 0 2 1 0 0 0 0 0 0 1 0 0 1 0 0 3 0 0 1 2

2 2 3 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0

2 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 2 2 0 0 0 0 1 0 0 0 0 0 0 0 0

3 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0

1 1 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 1 0 0 1 0 0 2 0 0 0 0 0

0 1 0 0 1 0 0 0 0 0 0 0 0

1 1 1 0 0 3 1 0 0 0 0 1

1 1 1 0 3 2 0 0 0 0 0

0 0 1 1 0 1 0 0 0 3

0 0 0 1 0 0 2 0 0

0 0 1 0 0 0 0 0

2 0 0 3 0 1 0


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T.H. D Alfonso, J.A. Ventura / Assignment of tools to machines

125

175 ( ~

F i g u r e 1 . G r a p h t h e o r e t i c r e p r e s e n t a t i o n o f t e s t p r o b l e m

Step 3. Redistribution of nodes to obtain upper bound feasibility, p

s u b t r e e s a r e f o r m e d i f S t e p 2 i s

s u c c e s s f u l . N o d e s a r e r e d i s t r i b u t e d a m o n g g r o u p s i n t h e r e m a i n d e r o f t h e h e u r i s t i c t o s a t i s f y u p p e r

b o u n d c o n s t r a i n t s .

T a b l e 8

C o m p u t a t i o n a l r e s u l t s f o r a i r p o r t d e s i g n p r o b l e m a

fr I I p = 3 p = 4 p = 5

U = 2 0 U = 3 0 U = 1 5 U = 2 0 U = 15 U = 2 0

0 . 5 0 1 9 4 1 2 0 8 1 8 6 6 1 8 4

0 . 5 0 2 9 6 1 1 9 7 3 8 5 7 7 + b 7 8

0 . 5 0 3 1 0 6 1 2 0 7 9 8 1 7 4 7 7

0 . 5 0 4 9 1 1 1 8 8 4 8 8 5 5 7 1

0 . 7 5 1 1 0 6 + 1 1 4 7 0 8 7 5 7 8 1

0 . 7 5 2 9 0 1 1 9 7 3 9 3 6 7 8 5

0 . 7 5 3 9 5 1 2 0 7 4 9 9 + 6 9 7 7

0 . 7 5 4 9 6 1 2 3 + 8 3 8 3 6 2 9 1 +

1 . 0 0 1 9 6 1 2 3 7 4 8 8 6 9 7 2

1 . 0 0 2 9 6 1 1 9 7 1 8 3 6 5 7 6

1 . 0 0 3 9 3 1 2 0 7 7 8 3 6 4 8 1

1 . 0 0 4 9 1 1 1 9 7 8 8 1 6 8 8 1

2 . 0 0 1 9 8 - - c 1 1 9 - - 7 1 - - 7 8 - - 6 9 - - 8 3 - -

2 . 0 0 2 9 8 1 1 8 6 7 9 1 6 5 8 2

2 . 0 0 3 9 1 1 1 9 8 1 8 6 6 4 8 6

2 . 0 0 4 9 3 1 1 8 8 5 + 8 8 5 2 6 7

Z ~ TE R 6 9 6 8 5 6 7 0 4 7 6 9

Z ~ 1 4 3 1 4 6 1 4 0 1 4 3 1 4 0 1 4 3

Z D I T E R

9 3 9 5 9 1 1 0 0 9 0 1 0 1

C M S T * I N F 1 0 4 I N F 9 1 8 6 8 6

L I T E R

6 6 8 8 10 10

U I T E r

2 5 2 3 1 7 4 9 1 1 1

a p n u m b e r o f g r o u p s , U = m a g a z i n e t o o l c ap a c i ty , f r = i n it i al s t e p - s ~ e c o e f f ic i e n t, II = m u l t i p l e o f i t e r a ti o n s t o h a lv e s t e p -s i z e

c o e f f ic i e n t, Z * I T E R = s m a l l e s t n u m b e r o f it e r a t i o n s r e q u i r e d t o o b t a i n t h e b e s t p r i m a l s o l u t i o n f o r s u b g r a d i e n t a l g o r it h m ,

Z D r r E R s m a l l e s t n u m b e r o f i t e r a ti o n s r e q u i r e d t o o b t a i n b e s t d u a l s o l u ti o n fo r s u b g r a d i e n t a l g o r i th m , C M S T * = b e s t s o l u ti o n

d e t e r m i n e d b y C h o p t h e M a x i m a l S p a n n i n g T r e e h e u r i st i c , L IT E R = n u m b e r o f lo w e r b o u n d i t e r a ti o n s f o r h e u r i st i c , U IT ER =

n u m b e r o f u p p e r b o u n d i t e r a t i o n s f o r h e u ri s ti c , I N F = n o f e a s i b le s o l u t io n w a s d e t e r m i n e d .

b + : T h i s s o l u t i o n p r o d u c e s l a r g e s t p r i m a l o b je c t iv e v a l u e Z * ) .

c _ : T h i s s o lu t i o n p r o d u c e s s m a l l e s t Z * .


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D Alfonso, J.A. Ventura / Assignment of tools to machines

Step 3.1. C h e c k f e a s i b il it y . I f a ll s u b t r e e s h a v e n o m o r e t h a n U B n o d e s , s t o p , t h e h e u r i s t i c is su c c e s sf u l .

O t h e r w i s e , e x a m i n e t h e s u b t r e e w i t h t h e l a r g e s t n u m b e r o f n o d e s . S e l e c t t h e l i g h t e st - w e i g h te d a r c t h a t

d i s c o n n e c t s a s i n g l e n o d e . U B i s t h e u p p e r b o u n d o n t h e g r o u p s i z e .

Step 3.2.

R e c o n n e c t t h e n o d e t o a d if f e r e n t su b t r e e w i th l es s t h a n U B n o d e s . T h e n o d e i s c o n n e c t e d b y

t h e l a r g e s t w e i g h t e d a r c b e t w e e n i t a n d a c a n d i d a t e s u b t r e e . G o t o S t e p 3 .1 .

7 C o m p u t a t i o n a l r e s u l ts

Five too l t e s t p rob lem

T h e 5 - t o o l p r o b l e m t h a t w a s d e s c r i b e d e a r l i e r p r o v i d e s t h e f i r s t t e s t re s u l ts . R e c a l l f r o m T a b l e 4 t h a t

t o o ls 1 a n d 4 r e q u i r e 2 s lo ts o f t h e F M S m a c h i n e m a g a z i n e . F i g u r e 1 s h ow s th e g r a p h r e p r e s e n t i n g

t h e t o o l p r o d u c t i o n f l o w m a t r i x o b t a i n e d f r o m T a b l e s 3 a n d 4 . T h e p r o b l e m is to g r o u p t h e s e f i v e t o o ls

i n t o 2 g r o u p s w i t h a t l e a s t 2 s lo t s fi ll e d , b u t n o m o r e t h a n 5 s l ot s fi ll e d . W i t h t h e s e b o u n d s , b o t h s o l u t i o n

a l g o r it h m s f o u n d t h e o p t i m a l s o l u ti o n . W h e n t h e u p p e r b o u n d o n m a g a z i n e c a p a c i t y is lo w e r e d f r o m 5 t o

4 , t h e c u r r e n t s o l u t i o n i s i n f e a s ib l e . B o t h a l g o r i th m s a g a i n f o u n d t h e o p t i m a l s o l u t io n . R e s u l t s o f b o t h

T a b l e 9

R e o r d e r e d m a t r i x f o r a ir p o r t d e s i g n te s t p r o b l e m

T o o l T o o l

2 3 4 5 7 8 9 3 6 9 2 2 6 1 5 1 7 2 0 2 1 2 3 2 4 2 6 2 7 1 0 1 1 1 2 1 4 1 8 2 5

1

2

3

4

5

7

8

9

1 3

1 6

1 9

2 2

6

15

1 7

2 0

2 1

2 3

2 4

2 6

2 7

1 0

11

1 2

1 4

1 8

2 5

- 1 3 0 2 2 1 1 0 2 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

3 - 1 0 3 0 1 1 0 2 1 1 3 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0

0 0 - 1 3 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0

2 3 3 - 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0

2 0 1 0 - 1 2 2 1 0 3 1 1 0 0 0 0 3 0 0 0 0 1 0 0 0 0 0

1 1 1 0 2 - 1 2 2 1 1 0 1 0 0 0 0 0 0 0 0 0 3 1 0 1 0 0

1 1 1 0 2 2 - 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0

0 0 0 0 1 2 2 - 1 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0

2 2 0 1 0 1 0 0 - 1 1 1 2 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0

0 1 1 0 3 1 1 0 1 - 1 1 2 0 1 1 0 3 0 0 0 0 0 0 0 1 1 0

0 1 1 0 1 0 0 0 1 1 - 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0

3 3 0 0 1 1 0 0 2 2 1 - 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 2 0 0 0 0 - 1 0 1 1 0 3 0 1 2 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 0 - 1 1 0 3 0 0 0 1 0 0 0 0 1 0

0 0 0 1 0 0 0 0 0 . 1 0 0 1 1 - 1 1 1 1 0 0 3 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 - 1 2 0 3 1 0 0 1 0 0 0 0

0 0 0 0 3 0 0 0 0 3 0 0 0 3 1 2 - 1 0 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 3 0 1 0 0 - 1 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 - 1 2 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 2 - 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 2 1 3 0 1 1 1 0 - 1 0 0 0 0 0 3

0 0 0 0 1 3 2 3 2 0 1 0 1 0 0 0 0 0 0 0 0 - 1 1 1 2 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 - 1 3 0 0 1

0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 3 - 1 1 0 0

0 0 3 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 0 1 - 1 0 0

1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 - 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 1 0 0 2 - 1

a S u b g r a d i e n t a l g o r it h m 3 m a c h i n e s m a g a z i n e c a p a c it y = 20 s l ot s .


13/19

T.It. D Alfonso, J.4. Ventura / Assignment o f tools to machines 127

t e s t p r o b l e m s a r e s u m m a r i z e d i n T ab l e 5 . T w o o p t i m a l s o l u t i o n s e x is t t o t h i s v e r s i o n o f t h e p r o b l e m a n d

b o t h a r e r e p r e s e n t e d i n t h e t a b le .

A i r p o r t l a y o u t t e s t p r o b l e m

M cCormi ck e t al . 1972) used da t a re l a t i ng se rv i ce s a t an a i rpor t . Th i s exac t t e s t p rob l em was aga i n

u s e d b y L e n s t r a a n d R i n n o o y K a n 1 97 5 ) t o d e m o n s t r a t e t r a v e l in g s a l e s m a n p r o b l e m m o d e l s . T h e

or i g i na l p rob l em was t o g rou p a i rp or t s e rv i ce s i n t o c l a s se s ba sed on a dependen cy fac t or r ang i ng f rom 0 ,

for i ndepende nt se rv i ce s , t o 3, fo r h i gh l y depend ent se rv i ce s . An exampl e of r e l a t i ve l y i ndepe ndent

s e r v i c es a r e p a s s e n g e r c h e c k - i n a n d t h e r e n t a l c a r r e p a i r s h o p . A n e x a m p l e o f h i g hl y d e p e n d e n t s e r v i ce s

i s pa ssen ge r check- i n and baggag e check- i n . In mode rn a i rpor t s r e l a ti ve l y sma l l s t a t i ons may be u t i l i z ed i f

necessa ry t o op t i mi ze cus t om er f l ow. For t h i s r ea son , g rou p s i ze s we re p e rmi t t ed t o be a s l ow a s one

uni t .

Thi s t e s t p rob l em provi des da t a t ha t wi ll cor re s pond t o an i magi na ry t oo l depend ency ma t r i x . It i s

adapt ed by a s s i gn i ng t o each se rv i ce an i n t ege r s i z e fac t or . Tha t i s, t he prob l em cons i d e red i n t h i s

r e s e a r c h r e q u i r e s t o o l s t o b e g r o u p e d i n t o F MS m a c h i n e s w i t h l i m i t e d s p a c e . Ea c h t o o l t a k e s u p a n

i n t ege r amoun t o f s lo t s . I f e ach a i rpor t s e rv i ce is a s s i gn ed an i n t ege r s i ze fac t or , and l i mi t s a re p l aced on

t h e n u m b e r a n d s i z e o f g r o u p s , t h e n t h e p r o b l e m s a r e e q u i v a l e n t . D a t a f o r t h e a i r p o r t d e s i g n p r o b l e m

are de sc r i be d i n Tabl e 6 and con t a i ned i n Tabl e 7 .

Table 10

Reord ered matrix for airport design test problem a

Tool Tool

1 22 2 13 19 4 12 3 11 14 5 21 15 16 6 9 23 10 7 8 17 27 25 18 20 24 26

1 - 3 3 2 0 2 0 0 0 0 2 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0

22 3 - 3 2 1 0 0 0 0 0 1 0 1 2 0 0 0 0 1 0 0 0 0 1 0 0 0

2 3 3 - 2 1 3 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0

13 2 2 2 - 1 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 0

19 0 1 1 1 - 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0

4 2 0 3 1 0 - 0 3 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

12 0 0 0 1 1 0 - 0 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

3 0 0 0 0 1 3 0 - 0 3 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0

11 0 0 0 0 0 0 3 0 - 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0

14 0 0 0 0 1 1 1 3 0 - 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 0

5 2 1 0 0 1 0 0 1 0 0 - 3 0 3 0 1 0 1 2 2 0 0 0 0 0 0 0

21 0 0 0 0 0 0 0 0 0 0 3 - 3 3 0 0 0 0 0 0 1 1 0 0 2 1 1

15 0 1 0 0 0 0 0 0 0 0 0 3 - 1 0 0 0 0 0 0 1 1 0 1 0 0 0

16 0 2 1 1 1 0 0 1 0 1 3 3 1 - 0 0 0 0 1 1 1 0 0 1 0 0 0

6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 2 3 1 0 0 1 2 0 0 1 0 1

9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 2 - 0 3 2 2 0 0 0 0 0 0 0

23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 - 0 0 0 1 1 0 0 0 0 0

10 0 0 0 2 1 0 1 0 1 2 1 0 0 0 1 3 0 - 3 2 0 0 0 0 0 0 0

7 1 1 1 1 0 0 0 1 1 1 2 0 0 1 0 2 0 3 - 2 0 0 0 0 0 0 0

8 1 0 1 0 0 0 0 1 0 0 2 0 0 1 0 2 0 2 2 - 0 0 0 0 0 0 0

17 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 - 3 0 0 1 0 0

27 0 0 0 0 0 0 0 0 0 0 0 1 1 0 2 0 1 0 0 0 3 - 3 0 0 1 0

25 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 - 2 0 0 0

18 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 2 - 0 0 0

20 0 0 0 0 0 0 0 0 1 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 - 3 1

24 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 3 - 2

26 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 2 -

a CMST algorithm, 5 machines, same solution for magazine capacity = 15 or magazine capacity = 20.


14/19

128 T.H. D Alfonso, J .A. Ventura /A ssig nm ent of tools to machines

Table 11

Reord ered matrix for airport design problem a

Tool Tool

1 2 4 5 7 8 9

10 22 3 11 12 13 14 16 17 19 20 21 24

26 6

15 23 18 25 27

1 3 2 2 1 1 0 0 3 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0

2 3 - 3 0 1 1 0 0 3 0 0 0 2 0 1 0 1 0 0 0 0 0 0 0 0 0 0

4 2 3 - 0 0 0 0 0 0 3 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0

5 2 0 0 - 2 2 1 1 1 1 0 0 0 0 3 0 1 0 3 0 0 0 0 0 0 0 0

7 1 1 0 2 - 2 2 3 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

8 1 1 0 2 2 2 2 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

9 0 0 0 1 2 2 - 3 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0

10 0 0 0 1 3 2 3 - 0 0 1 1 2 2 0 0 1 0 0 0 0 1 0 0 0 0 0

22 3 3 0 1 1 0 0 0 - 0 0 0 2 0 2 0 1 0 0 0 0 0 1 0 1 0 0

3 0 0 3 1 1 1 0 0 0 - 0 0 0 3 1 0 1 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 1 0 0 1 0 0 - 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0

12 0 0 0 0 0 0 0 1 0 0 3 - 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0

13 2 2 1 0 1 0 0 2 2 0 0 1 - 0 1 0 1 0 0 0 0 0 0 0 0 0 0

14 0 0 1 0 1 0 0 2 0 3 0 1 0 - 1 0 1 0 0 0 0 0 0 0 0 0 0

16 0 1 0 3 1 1 0 0 2 1 0 0 1 1 - 1 1 0 3 0 0 0 1 0 1 0 0

17 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 - 0 1 1 0 0 1 1 1 0 0 3

19 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 - 0 0 0 0 0 0 0 0 0 0

20 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 - 2 3 1 1 0 0 0 0 0

21 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 1 0 2 - 1 1 0 3 0 0 0 1

24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 - 2 0 0 0 0 0 1

26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 - 1 0 0 0 0 0

6 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 I 0 1 0 0 1 - 0 3 0 0 2

15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 3 0 0 0 - 0 1 0 1

23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3 0 - 0 0 1

18 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 - 2 0

25 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 - 3

27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 1 1 0 2 1 1 0 3 -

a Subgradient algorithm, 5 machines, magazine capacity = 15.

When t e s t i ng t he subgr ad i en t a l gor i t hm on prob l e ms a s l a rge a s t h i s, i t was d i sco ve red t ha t t he s t ep

s i ze va l ue s i gn i f i c an t l y a f fec t s t he con ve rgen ce o f t he a l gor i t hm. Tab l e 8 i nc l udes pr i ma l fea s i b l e

so l u t i ons for s t ep s i ze numer a t or va l ue s rangi n g f rom 0 .5 t o 2, and ha l v i ng f rom eve ry t o eve ry four t h

i t e ra t i on , f r i s t he numera t o r coe f f i c i en t and I I is t he mul t i p l e o f i t e ra t i on s t o wa i t be for e ha l v i ng f t . A n

i n t e r e s t i n g r e s u l t w a s t h a t t h e b e s t p r i m a l s o l u t i o n s o c c u r r e d o v e r v a r i o u s v a lu e s o f

f ~ ,

w h i l e t h e b e s t

dua l uppe r boun ds occur r ed when f~ = 2 and t he s t ep s i ze was ha l ved eve ry i t e ra t i on . Thi s r e su l t was

repea t ed for t he o t he r t e s t p rob l ems a s we ll .

O c c a s i o n a l l y t h e h e u r i s t i c C MS T f o u n d a b e t t e r p r i m a l f e a s i b l e s o l u t i o n t h a n t h e s u b g r a d i e n t

a l gor i t hm for ce r t a i n va l ues o f f r and I I . The heur i s t i c CM ST fa i l ed to f i nd a fea s i b l e so l u t i on for t wo of

t he s i x t e s t p rob l ems . A few reord e red ma t r i ce s a re p re sen t ed i n Tabl e s 9 , 10 , and 11.

M i s c e l l a n e o u s t e s t p r o b l e m

A n a d d i t i o n a l t e s t p r o b l e m w a s c r e a t e d w i t h v a r i o u s t o o l s iz e s a n d r e s t r i c t i o n s o n t h e n u m b e r o f

m a g a z i n e s l o t s o c c u p i e d L f o r l o w e r a n d U f o r u p p e r b o u n d s ) . T a b l e 1 2 c o n t a i n s t h e i n p u t m a t r i x a n d

t wo se t s o f t oo l si z es , R1 and R2. Tabl e 13 cont a i n s comput a t i ona l r e su l t s fo r t oo l s i z e se t R1, wi t h

va r i ous g roup s i ze s and magaz i n e re s t r i c t i ons . I t i s i n t e re s t i ng t o no t e t ha t , whi l e t he be s t p r i ma l so l u t i on


15/19

T.H. D Alfonso, J.A. Ventura /Assignment of tools to machines

1 2 9

T a b l e 1 2

I n p u t m a t r i x f o r m i s c e l l a n e o u s t e s t p r o b l e m

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 R 1 R 2

0 1 - 1 0 1 5 0 1 0 0 5 0 1 0 1 5 1 0 0 5 1 0 5 5 0 1 0 2 0 5 0 0 0 5 2 1

1 0 0 1 5 0 0 5 0 0 0 10 0 0 3 1

5 0 0 0 0 0 5 0 0 0 0 2 1

0 1 0 0 5 5 1 0 1 0 0 5 0 1 3

0 1 0 5 0 1 0 0 1 5 0 1 5 1 0 0 2 2

0 5 0 5 1 5 1 5 5 0 7 5 1 2

5 0 2 5 5 0 0 0 5 1 0 0 5 0 2 1

1 0 0 7 5 5 0 1 0 0 5 0 1 0 1 0 1 0 1 2

5 5 0 5 2 0 1 0 0 1 0 0 5 0 0 1 1

2 0 5 0 1 0 0 0 0 1 0 0 7 5 1 0 0 0 5 3 1

2 5 0 1 5 1 5 1 0 0 2 5 0 1 0 0 5 1 1

0 0 1 0 1 0 5 1 0 0 7 5 2 5 3 1

1 0 1 0 1 0 0 1 0 0 5 2 5 2 2

5 1 0 1 0 0 1 0 0 5 0 7 5 3 2

. . . . 5 5 7 5 1 0 0 1 0 1 1

. . . . . 1 0 2 5 0 5 1 1

. . . . . . 0 7 5 5 0 2 2

. . . . . . . 1 0 0 3 1

. . . . . . . . 5 1 1

. . . . . . . . . 2 1

0 2 - 1 0 2 0 1 0 0 1 0 0 7 5

0 3 - 5 7 5 5 0 5 5 0 1 0

0 4 - 1 0 0 0 5 5

0 5 - - 1 0 5 0 5

0 6 - 5 5

0 7 - - 0

08 - - -

09 - - -

10 - -

1 1

1 2 - - -

13 - - -

14 - - -

15 - - -

16 - - -

17 - - -

18 - - -

19 - - -

2 0

0 0

0 5 0

5 1 0 1 0 0

5 0 1 0 0 5

0 0 1 0 0 7 5

0 0

0 5 0

0

T a b l e 1 3

C o m p u t a t i o n a l r e s u l ts f o r m i s c e l l a n e o u s t e s t p r o b l e m a

f r I I p = 3 p = 4 p = 5

L = 3 L = 3 L = 1 7 L = 1 7 L = I L = I

U = 2 0 U = 3 2 U = 2 0 U = 3 2 U = 9 U = 1 2

0 . 5 0 1 3 1 9 5 4 1 9 0 3 1 9 5 + b 3 1 9 5 + 1 3 9 5 1 6 9 0

0 . 5 0 2 2 9 4 5 4 2 0 5 3 0 9 5 3 0 9 5 1 4 3 0 1 8 1 0

0 . 5 0 3 3 0 9 5 4 1 9 0 3 0 8 0 3 0 8 0 1 5 3 5 1 6 7 0

0 . 5 0 4 3 1 9 5 4 1 4 0 2 9 9 0 3 1 8 5 1 7 3 0 1 6 7 5

0 . 7 5 1 2 9 2 0 4 2 0 0 2 9 9 0 2 9 9 0 1 4 4 5 1 5 0 5

0 . 7 5 2 3 1 3 0 4 0 1 5 2 8 7 0 - - 3 1 1 0 1 4 4 5 1 9 2 0

0 . 7 5 3 3 1 4 0 4 2 0 0 3 1 1 0 3 1 1 0 1 2 8 0 1 5 7 0

0 . 7 5 4 3 2 3 5 + 4 1 0 0 2 8 6 0 2 8 6 0 1 4 6 5 1 5 9 5

1 . 0 0 1 3 1 8 5 4 2 0 0 3 1 4 0 3 1 4 0 1 2 9 5 1 9 9 0

1 . 0 0 2 3 2 3 5 4 0 2 0 2 9 0 5 2 9 0 5 1 4 9 5 1 8 3 0

1 . 0 0 3 2 9 0 0 4 2 0 5 + 3 1 0 5 3 1 0 5 1 5 6 0 1 6 6 5

1 . 0 0 4 3 1 1 5 4 0 9 5 3 0 1 0 3 0 1 0 1 0 0 0 1 7 8 0

2 . 0 0 1 3 1 8 5 - - c 3 6 5 5 - - 2 9 9 0 2 9 8 5 - - 1 2 2 5 - - 1 8 0 0 - -

2 . 0 0 2 3 1 8 0 3 6 5 5 3 0 1 0 3 0 1 0 1 7 7 5 1 4 5 0

2 . 0 0 3 2 9 9 0 4 0 9 5 3 0 1 0 3 0 1 0 1 4 3 0 2 0 6 0 +

2 . 0 0 4 3 0 8 0 4 2 0 5 2 9 3 5 2 9 3 5 1 8 1 5 + 1 8 3 5

Z ~ T E R 5 6 4 3 8 8 88 7 3 2 2

Z ~ 4 7 4 9 4 2 1 3 4 5 0 6 4 4 4 0 4 4 1 4 4 5 5 8

Z ITE R 8 5 4 1 8 5 8 9 9 7

C M S T * 2 8 5 5 3 5 2 0 2 8 5 5 2 8 5 5 1 0 2 0 1 4 1 5

L I T E R 8 3 1 0 1 0 4 4

U I T E R 3 1 1 1 8 3 6 0

a p = n u m b e r o f g r o u p s , U = m a g a z i n e t o o l c a p a c i t y , f r = i n i ti a l s t e p - s i z e c o e f f i c i e n t , I I = m u l t i p l e O f i t e r a t i o n s t o h a l v e s t e p - s i z e

c o e f f i c i e n t , Z I * T ER = s m a l l e s t n u m b e r o f i te r a t i o n s r e q u i r e d t o o b t a i n t h e b e s t p r i m a l s o l u t i o n f o r s u b g r a d i e n t a l g o r i t h m ,

Z ~ I T E R = s m a l l e s t n u m b e r o f i t e r a t i o n s r e q u i r e d t o o b t a i n b e s t d u a l s o l u t i o n f o r s u b g r a d i e n t a l g o r i t h m , C M S T * = b e s t s o l u t io n

d e t e r m i n e d b y C h o p t h e M a x i m a l s p a n n i n g T r e e h e u r i s t ic , L I T E R = n u m b e r o f lo w e r b o u n d i t e r a t i o n s fo r h e u r is t i c , U IT E R =

n u m b e r o f u p p e r b o u n d i t e r a t i o n s fo r h e u r i s t i c , I N -F = n o f e a s i b le s o l u t i o n w a s d e t e r m i n e d .

b + : T h i s s o l u t i o n p r o d u c e s l a r g e s t p r i m a l o b j e c t i v e v a l u e Z * ) .

c _ : T h i s s o l u t i o n p r o d u c e s s m a l l e s t Z ~ .


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130 T.H.

D Alfonso, ZA. Ventura /Assignment of tools to machines

Table 14

Computational results for miscellaneous test problem a

r II L = 2 L = 13

U=16 U=2 4 U = 16 U=2 4

0.50 1 2920 4065 3170 3170

0.50 2 2950 4065 2870 2870

0.50 3 3075 4130 2875 2875

0.50 4 2980 4095 2925 2925

0.75 1 2880 4135 2870 2870

0.75 2 3010 4065+

a

2835 2835

0.75 3 3050 4135 2310 3210

0.75 4 2985 4065 3015 3015

1.00 1 3090+ 4065 3035 3035

1.00 2 2730 4130 3120 3120

1.00 3 3010 4215 3015 3015

1.00 4 3045 4110 3085- b 3085

2.00 1 2845 4215-- 3050 3050

2.00 2 2940 4065 2765 2765--

2.00 3 3085 4130 2845 2845

2.00 4 3030-- 4065 3290+ 3290+

Z~TER 95 64 26 26

Z~ 4357 4881 4611 4388

Z ITE R 76 33 31 30

CMST * 2855 3520 INF INF

LZTER 2 2 20 20

UttER 9 2 0 0

a p ~ number of groups, U = magazine tool capacity, r initial step-size coefficient, II = multiple of iterations to halve step-size

coefficient, Z~-ER= smallest number of iterations required to obtain the best primal solution for subgradient algorithm,

ZD ITER = smallest number of iterations required to obtain best dual solution for subgradient algorithm, CMST* = best solution

determined by Chop the Maximal spanning Tree heuristic, LITER= number of lower bound iterations for heuristic, UITER=

number of upper bound iterations for heuristic, INF = no feasible solution was determined.

b +: This solution produces largest primal objective value (Z*).

c _ : This solution produces smallest Z*.

varied great ly among the different s tep size parameters, al l but one of the best dual upper bounds

occurred with a numerator parameter of 2, and halving every i terat ion.

In Table 14, the second tool s ize set was examin ed with varying magazi ne restr ict ions, but only two

groups (p = 2) were fo rmed in each case. The effect of having upp er and lower bound s closer together is

evident . For these cases, the primal solut ions stayed the sa me or increased as the upp er bou nds

increased by 50 percent .

In Table 15, group sizes were increased from 2, to sizes 3, 4, and 5. Only 2 slots were required to be

fi l led, as indicat ed by the L = 2 value. On the average, primal solut ions decreased by 12 perce nt when

the num be r of groups increased from three to four, and further decreased by 20.5 with an addit ional

group. I t is no surprise that imposing more clusters results in more travel for parts.

Final ly, Table 16 cont inue d with the test probl em in Table 14 and T able 15, but f ive groups were

consid ered here. This further suppor ted the assert ion that more groups impose greater pro ducti on travel .

I t should be poin ted out tha t the subgradient a lgor i thm was super ior to the CM ST a lgor i thm in most

cases. Only when man y groups were to be formed, and bo unds o n magazi ne capacity were t ight , did the

CM ST heurist ic produ ce a bet te r solut ion. It should also be poin ted out that a rule for determ inin g step

sizes to get a good primal solut ion is not forthcoming. The number of t imes the best dual upper bound

was fou nd with a step size num era tor coefficient of 2, and halving the step size every i terat ion, should

not go unnoticed . This is a recom men dat ion for futur e applicat io n and testing.


17/19

T.H. D Alfonso, J.A. Ventura /Assignment of tools to machines

131

Table 15

Computational results for

miscellaneous test problem a

f r II p=3 p-~4 p= 5

L = 2 L = 2 L = 2 L = 2 L = 2 L = 2

U = 16 U = 24 U = 16 U = 24 U = 16 U = 24

0.50 1 2805 40 15 - c 2670 3290 1575 2705

0.50 2 2435 4260+ b 2000 3200 1950 2180

0.50 3 2655 4015 2600 2675 1640 2355

0.50 4 2320 3975 2355 3240 2430+ 1935

0.75 1 2245 3495 2455 3535 2130 1965

0.75 2 2920+ 4260 2210 3275 2230 2990

0.75 3 2780 4035 2330 3695 1665 3000+

0.75 4 2390 3900 2105 3375 2155 2725

1.00 1 2515 4115 2890+ 3535 2020 2255

1.00 2 2620 3875 2695 3355 1860 2415

1.00 3 2435 3875 2545 3215 1575 2985

1.00 4 2480 3995 2530 3345 2310 2535

2.00 1 2745 3975 2395 3560 229 5- 3415

2.00 2 2745- 4260 2385- 3840* 2185 2335-

2.00 3 2650 4115 2630 3265 2350 2850

2.00 4 2635 4115 2390 3340 2130 2745

Z~TER 43 39 71 41 27 106

Z~ 4695 4847 4619 4811 4702 4867

Z D I T E R 43 55 22 69 29 61

CMST * 2605 3045 2410 2940 2305 2685

LITER 3 3 4 4 5 5

UttER 7 1 4 1 3 1

a p = number o f groups, U = magazine tool capacity, fr = initial step-size coefficient, I I = multiple of iterations to halve step-size

coefficient, ZI*TER = smallest number of iterations required to obtain the best primal solution for subgradient algorithm,

Z I ~ I T E R

smallest number of iterations required to obtain best dual solution for subgradient algorithm, CMST* = best solution

determined by Chop the Maximal spanning Tree heuristic, LITER= number of lower bound iterations for heuristic, UITER=


b +: This solution produces largest primal objective value Z*).

c _ : This solution produces smallest Z~.

8 C o n c l u s i o n s

In this art icle, a manu fac turi ng system has bee n studied and a part icular aspect of that system has

rece ived special at tent ion. T he dec ision of locat ing tools in a f lexible manu fac turi ng system FMS) can

have t r eme ndo us influenc e on the eff iciency of the factory. Produc t ion f low is relat ed di rect ly to the

loca t ion of t oo ls necessary to manufac tur e a g iven par t . For t he F MS wi th machines capable of handl ing

several tools , pro duct ion eff iciency great ly depe nds on the p lace men t of tools .

The mathema t i ca l model t ha t has been c rea t ed to op t imize the dec i s ion of assign ing too ls i n the F MS

system can be appl ied to ot her pr oblems. The mod el i s equivale nt to grouping objects of varying sizes

into clusters wi th size speci f ica t ion s . Cluster analysis has had wide appl icat ion, and the resul t s of this

s tudy a re appl i cab le t o t hose prob lems tha t resemble the FMS problem chosen here .

In Oper a t ions Research , p rob lems a re o f t en complex enough tha t a g loba l op t imal so lu t ion is

impossible to determine. In such cases, a good mathemat ical model makes al l the di fference in solving

the pr obl em as best as possible. The complexi ty of the mo del dev elop ed in this art icle i s inde ed sizable

when many too l s a re t o be grouped in to severa l c lus t e rs . The Lagrangian dua l approach and the Chop

the Maximal Spanning Tree CMST) heur i s t i c p roved to be su i t abl e methods of de t e rmin ing good

solut ions to the problem.


18/19

132

T.H. D Alfonso, J.A. Ventura / Assignment of tools to machines

Table 16

Computational results for miscellaneous test problem a

f r

II L = 1 L = 4

U = 7 U = 9 U = 7 U = 9

0.50 1 1475 1515 1375 1280

0.50 2 1470 1515 1345 1035

0.50 3 1205 1585 1325 985

0.50 4 1270 1645 1330 1140

0.75 1 1370 1595 1530+ a 1530

0.75 2 1475 1545 1475 1205

0.75 3 1655+ 1535 1210 1445

0.75 4 1290 1615 1280 1505

1.00 1 1460 1550 1470 1210

1.00 2 1330 1675 1295 1120

1.00 3 1470 1770 1395 985

1.00 4 1400 1625 1335 1315

2.00 1 15 80 - b 1850-- 1395-- 1455

2.00 2 1410 1560 1245 1390--

2.00 3 1450 1515 1450 1380

2.00 4 1575 1860+ 1510 1545+

Z{TER 53 71 62 68

Z~ 4557 4515 4539 4510

ZDIXER 79 53 31 28

CMST * 610 1450 1700 1700

LITER 4 4 15 15

UrrER 88 60 1 1

a p = number of groups, U = magazine tool capacity,

fr

= initial step-size coefficient, II = multiple of iterations to halve step-size

coefficient, Z~rER = smallest number of iterations requ ired to ob tain th e bes t primal solution for subgradien t algorithm,

.

ZD ~WER= smallest number of iterations required to obtain b est dual solution for subgradient algorithm, CMST* = best solution

dete rmin ed by Chop the Maximal Spanning Tree heuristic, LrrER = number of lower bound iterations for heuristic, UITER =


b +: This solution produces largest primal objective value (Z*) .

c _ : This solution produces smallest Zx~.

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Assignment of tools to machines in a flexible manufacturinf system