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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 .
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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
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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
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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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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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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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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 .
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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.
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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
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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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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.
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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=
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~.
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.
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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 =
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 Zx~.
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