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Single and multiple period layout models for automated manufacturing systems
300
European Journal of Operational Research 52 (1991) 300-314 North-Holland
Theory and Methodology
Single and multiple period layout models for automated manufacturing systems Panagiotis Kouvelis
Management Department, University of Texas, Austin, TX 78712, USA A l i S. K i r a n *
Department of Industrial and Systems Engineering, Unioersity of Southern California, Los Angeles, CA 90089, USA Abstract: Layout models for automated manufacturing systems are considered. The changes in product mix, part routings and process plans are incorporated into a stochastic single period layout model. A solution method, based on generating non-dominated layouts, is developed and illustrated. A dynamic programming formulation is also developed to capture the dynamic layout decisions in an automated manufacturing environment. A state space reduction approach is proposed for solving this multi-period model. Keywords: Machine layout, quadratic assignment, flexible manufacturing
I. I n t r o d u c t i o n
The complex interplay of engineering design and analysis is nowhere more apparent than in the design of automated manufacturing systems. The designer of such a system faces the difficult task of developing a system that is capable of handling a variety of products with variable demands at a reasonable cost. Alternate and probabilistic routings, and schedule and inventory constraints further complicate the design task. The designer uses analysis tools to ensure that the selected resource levels are adequate enough to provide the output requirements in a wide range of operating conditions. Making the designer's task even more difficult is the fact that decisions regarding the different subsystems cannot be made in isolation. This has been illustrated by Kouvelis and Kiran (1990) as the interactions between the layout, material handling system and storage requirements of an automated manufacturing system. Optimal design of the physical layout is one of the most important issues that must be resolved in the early stages of the manufacturing system design. Cost consequences of the decisions related to layout of the machines can be observed not only during the implementation but also in the operation of the system. Good solutions to these problems provide a necessary foundation for effective utilization of the system. Tompkins and White (1984) emphasized the importance of layout decisions for effective material handling by pointing out that 20% to 50% of the total operating expenses in manufacturing are attributed to material handling and layout related costs. * Present address: Department of Industrial Engineering, Dai-Yeh Institute of Technology,Taiwan, ROC Received March 1989; revisedAugust 1989 0377-2217/91/$03.50 © 1991 - ElsevierSciencePublishers B.V. (North-Holland)
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In this paper we address the layout decisions of an automated manufacturing system in realistic scenarios. Our work is motivated by the fact that the product mix that will be operational, is highly uncertain during the design phase. Usually at the time layout decisions are made, the products and the process plans have not been completely determined. Furthermore, the product mix (may be given as part production rates) is subject to changes because of forecasting errors and fluctuations. The designer's task is to develop a satisfactory layout design in this highly nondeterministic environment. We consider this problem in two different scenarios: First, we consider the cases where once chosen, a layout cannot be changed in the foreseeable future. During the layout design phase the product mix is uncertain. However once realized, the product mix is expected to remain stable over the planning horizon. Such a scenario is applicable in automated environments where layout changes are not feasible because of very high changeover costs and uncontrollable changeover times. Automated paint lines, coordinate measurement machines, automated washing and deburring stations are examples of the facilities with high installation costs and times. In the second case, we consider layout decisions in a dynamic way. The planning horizon for the automated manufacturing system design is divided into smaller planning periods. The product mix for each planning period, though uncertain during the design phase, remains stable over that period. The designer is allowed to specify different layouts, one for each planning period, over the planning horizon. This second case is motivated by Flexible Manufacturing System (FMS) installations. Such systems consist of cells of physically identical machines that perform a variety of operations. At the beginning of each planning period (week, month), operations allocation decisions are made. These decisions cause a significant change in the operational product mix of the cell in each period, and in most cases imply sub-optimality of a static layout configuration. The paper is organized as follows. The next section introduces the notation and provides a brief background on the previous work in the area. In Section 3, we present the effects of product mix changes on layout decisions. Section 4 deals with the above scenario 1: A single period layout decision model handling the product mix uncertainty is developed and a solution approach for the model is also described. In Section 5, we present a multiperiod dynamic layout model, able to handle the uncertainty of product mix at each planning period (i.e., scenario 2). In Section 6, we summarize our conclusions.
2. Background A plant layout problem in a conventional manufacturing setting is often defined as determination of the relative location of a given number of machines among the candidate locations. A typical objective is the total material handling cost, which is defined as the product of material movement frequencies and distances between the machines. The problem is then modelled as a Quadratic Assignment Problem (QAP), which is NP-complete (Sahni and Gonzalez, 1976). Gilmore (1963) and Lawler (1963) were the first to develop optimal branch and bound solution procedures for the problem. Bazaraa and Elshafei (1979) proposed a branch and bound algorithm for the QAP which is based on the stage by stage assignments of single facilities to unoccupied locations. Another branch and bound algorithm was recently suggested by Kaku and Thompson (1986). Cutting plane methods for the solution of the QAP are presented in Bazaraa and Sherali (1980) and Burkard and Bonninger (1983). Armour and Buffa (1963), Hillier (1963), Hillier and Connors (1963), Zoller and Adendorff (1972), Neghabat (1974), Lewis and Block (1980), Heragu and Kusiak (1988), and Golany and Rosenblatt (1989) proposed heuristic procedures. Computerized packages that solve the plant layout problem are available, to mention a few, CRAFT (Buffa et al., 1964), COFAD (Tompkins and Reed, 1976), ALDEP (Seehof and Evans, 1967), and CORELAP (Lee and Moore, 1967). Surveys of algorithms on the quadratic assignment problem can be found in Burkard (1984) and Kusiak and Heragu (1987). Extensive computational results comparing the various algorithms are reported in Burkard and Stratman (1978), Burkard (1984), and Levany and Kalchik (1985). Rosenblatt (1988) developed a dynamic programming formulation for a multiperiod plant layout problem. Kouvelis and Kiran (1990) define the
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layout p r o b l e m for a u t o m a t e d m a n u f a c t u r i n g systems as the selection of locations for M stations subject to a given part p r o d u c t i o n rate requirement. Their modified f o r m u l a t i o n of the Q u a d r a t i c A s s i g n m e n t P r o b l e m ( M Q A P ) is given below: M
(MQAP)
M
z(N, X) = E
Min N,X
M
M
Y'~ Y] E ajlkmXjlXkm + h ( N ) ,
j=l
l=1 k=l
(1)
m=l
M
s.t.
~
x/=
1,
l=1 ..... M,
(2)
j=l
(3)
j=l M
Y] x / t = 1,
. . . . . M,
1=1
T H ( N , X ) > V, x/={O,
1},
(4)
Ninteger,
where
= [ F ( j , k ) d ( l , m) a/kt'~ [ F ( j , j ) d ( j , j)+Cjl
ifj=gkorl--/=m, ifj=kandl=m,
(5)
P
V= E Vp,
(6)
p=l
x/=
1
if workstation j is assigned to location l,
0
otherwise.
All relevant n o t a t i o n is explained in T a b l e 1. M o d i f i c a t i o n of the G i l m o r e - L a w l e r b r a n c h a n d b o u n d procedure h a n d l e s the above p r o b l e m successfully for usual sizes of a u t o m a t e d m a n u f a c t u r i n g systems Table 1 Notation P:
p:
s~: M: j:
eje: #j:
s/ = (s+)~=1: N: d(I, m): Op(i):
R?(i, i + 1): Tp(i, i +1): Wpi: X: q: Qp.
F(j, k): h(N):
number of different part types part type index, p =1,..., P number of unit operations required to produce part type p required production rate of parts of type p number of stations station index j = 1,..., M visit frequency of part type p to station j service rate of station j number of servers at station j vector representation of the station configuration number of jobs in the system at any point in time transportation cost (based on some metric of distance per unit work flow between location l and location m) i-th operation of part type p (random variable) time required to perform refixturing on part p between the successive operation Op(i) and Op(+ 1) (random variable) time required to transport parts between the work stations Wei and Wml+ 1 workstation where operation i of part type p takes place layout decision matrix, binary matrix of assignment of workstations to locations index for process plans, q = 1.... ' Qe number of different process plans for part type p cost per unit time associated directly with assigning workstation j to location l workflow per unit time between workstation j and workstation k inventory and resource allocation costs of job population N
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Table 2 Computational results of the branch and bound solution procedure for MQAP Number of Stations
(CPU sec on VAX 11/780) "
<5 5 6 7 8 > 8
<5 6.5 16.9 65.2 410.5 > 1112.7
Reported average performance over 5 randomly generated examples in each case.
(6-10 machines grouped into 5 or fewer stations, as reported in a large scale survey on American and Japanese manufacturing systems by Jaikumar (1986)). Computational results demonstrating the ability of such a method to handle real size manufacturing layout problems are presented in Table 2. In order to apply the MQAP formulation, an appropriate throughput evaluation model for the FMS is needed. Kouvelis and Kiran (1990) proposed a simple Closed Queueing Network (CQN) model for that purpose. The model captures the effects of layout decisions on the throughput of the manufacturing systems. Here we summarize the model for convenience. Let us assume that our FMS has M - 1 machining stations and a load/unload ( L / U L ) station. For reasons of convenience, assume that the L / U L station performs only refixturing operations with random refixturing Rp(i, i + 1) that follow a general distribution with mean time R m. The material handling (transportation) times of parts between stations are modelled with the random variables Tp(i, i + 1) that follow a general distribution with mean time Tp,, which is a function of the layout decision and can be estimated as
d(wp,. .I..,+,) T,, =
(7)
U~t
where UM is the average speed of the material handling system (MHS). In the Kouvelis and Kiran (1990) paper the operation of the MHS and the L / U L station is modelled as a single node, node M, of the proposed CQN model. The service time /z~t1 at node M includes both transportation delays and the refixturing operations at the L / U L station: sp-i
t~Ml= /--, --~ E ( Rp,+ p=l
r,,).
(8)
i=1
The average service time is an adequate descriptive parameter for node M. It is modelled as an M / G / ~ node of the CQN which still has a product form solution (Kelly, 1979). The other important input quantities to the proposed CQN model are the visit frequencies of parts to a station. The visit frequency of part type p to station j is given by
up s. e,, = -~ Z lj ( Wm ).
(9)
~=1
where
II(WP')={Io
if i-th°perati°n°fparttype
i takesplaceatstati°n J'i'e"
(10)
Visit frequency of all parts to station j is given by P
e,= E ejp. p=l
(11)
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Using standard results from CQN theory (e.g., Gross and Harris, 1985) the steady state probability of having n~, n 2. . . . . nM number of jobs at stations 1, 2 . . . . . M, respectively, is given by
1 P"" n2 . . . . . n i - -
(e'/"J) °'
G(N)
~ ( n j ) = t n j! Is('J S,)s t
fj(n,)
j=l
for nj ~sj,
'
j=l
(12)
. . . . . M,
(13)
M (ejlzj)'.
G(N) =
.,+.2+
E
I--I +.M=NJ=' ( ( ' J )
(14)
The throughput function of station j is given by (see e.g., Solberg, 1977) THj=e/
G ( N - 1) G(N)
(15)
Assuming e i = 1, the throughput of the system is given by TH = TH/-
G ( N - 1) G(N)
(16)
Since the normalizing constant and throughput rate are functions of the layout decisions, we use the notation G(N, X ) and T H ( N , X) respectively to express the relation between layout decisions, X, and these functions.
3. Effects of product mix changes on layout decisions In this section we explore the interactions of product mix and other uncertain input factors (e.g., operation allocation policies, alternate process plans for part types) with the optimal layout decisions. The understanding of such interactions is a step towards the development of design models that will capture the relations between the components of automated manufacturing systems. Using these models, integration of the various subsystems into an optimum system design could be possible. Usually the designer starts with a set of part types and a rough operation process chart for each part type. In modelling the part processing operations, we will use 'unit operation' as a basic unit. A 'unit operation' is defined as an elementary transformation of an input object into an output object with no interruption. Since unit operations are defined to be specific to a part type, we order the index of unit operations to be such that the first n I unit operations correspond to part type 1, the subsequent n 2 unit operations correspond to part type 2, and so on. For a particular part type P ' , let Np, be the cumulative number of unit operations for part types 1 to P ' - 1, i.e., P'--I
Np, = ~
n p,
and
Nl=0.
P=I
The unit operations of part type p are organized as an operations graph Qp (Ip, Ap) where the node set lp corresponds to unit operations of the part type p and the directed arc s e t Ap captures all technological precedence relations between the unit operations. We denote a process plan by Sqp, q = 1 . . . . . Qp, and the associated probability of utilizing process plan q by 0qp with ~Q£10qp = 1. The product mix to be processed in the automated manufacturing system can be described by production volumes cp, p = 1 . . . . . P, over a planning horizon. Let yj, be the percentage of time (during the
P. Kouvelis, A.S. Kiran / Layout models for automated manufacturing systems
305
planning horizon) that unit operation i is allocated to machine j. A long term operation allocation policy can be summarized by matrix Y = [y,]. Then the workflow over the planning horizon between stations j and k is given by ?
F(j, k)=
9.
(17)
E '9 E Oq; E Yj,Yk,,+, p-1
q~l
i~S,;'
and the visit frequencies of parts to workstations are given by Up Qp
(18) p=l
p=l
q=l
i~S~'
From equation (1), (4), (5), (16), (17), and (18) the dependency of layout decisions on the product mix, selection of process plans and long term allocation policy of operations to machines is apparent. To indicate such dependency, we introduce the following parametric notation:
a,kt,,,=a,k,,,(v, O, Y ) ,
T H ( N , X; v, O, Y)
and
z ( N , X; v, O, Y ) ,
where
v = ( v e , p = l ..... P),
O=(OP, q = l ..... Qp, p = l ..... P ) .
It is easy to observe that a new part type introduction or volume changes for existing parts will affect (17) and (18). Any technological changes or changes in process plans may change Qp and 0qp, q = 1 Qp, hence affecting (17) and (18). In the layout model developed in the next section, these interactions are captured in the values of the visit frequencies that are input to CQN representation of the system. The CQN model of the system is an integral part of the MQAP formulation through the production requirement constraint. The methodology developed in the remainder of the paper explicitly considers the effect of product mix changes in layout decisions. It is, however, easy to apply this methodology to any of the previously mentioned interactions between layout decisions and other operational decisions (e.g., process selection, product introduction, operation allocation policies, etc.). . . . . .
4. Single period model
A major advantage of an automated manufacturing system is its capability to process highly variable product mixes and consequently respond effectively to demand changes. Our purpose is twofold in addressing the single-period stochastic case. First, for the systems with very high relocation costs, the single-period stochastic model may be sufficient. Secondly, the properties of the single-period model will be used in subsequent sections for the development of a multi-period model. Assume that the manufacturing system will produce a variety of products in different product mixes. Let M ~ ~ A, be the product mix index where A is the set of all possible product mixes. A specific product mix can be defined as vx = (rex, p = 1. . . . . P) where vpx is the volume of part type p to be produced over the planning period. Let o x be the probability that product mix vx will be the operational product mix. Let o = (ox, ~ = 1, ..., ~,) be the probability vector and vx = Ep=lV)~. e x Alternatively o x can be considered as a percentage of time that vx be operational during the planning period. Assuming each part is associated with a revenue, Rp, a modified QAP formulation for the static (i.e., single-period, stochastic) layout
P. Kouvelis,A.S. Kiran /Layout modelsfor automatedmanufacturingsystems
306
problem is given as follows: (MQAP-S)
Max
if(N, X ) = E {
RpVXp-FC(X; vX)-h(N))
~
(19)
p=l M
E Xjz=l,
s.t.
l = 1 . . . . . M,
(20)
j=l
(21)
j=l M
Y~Xj,=I,
. . . . . M,
/=l
T H ( N , X; v X l > v x,
Xjl =
{0, 1 },
X~A,={X~A:
ox>0},
(22)
N integer,
where M
FC(X; vx) = E
M
M
M
E
E
E
aj*,m(vX)Xj,X*m•
j = l 1=1 k = l m = l
Also to be consistent with the previous notation call,
z(N, X;
vX)= FC(X;
vX)+h(N).
Then
p=l
z'NX v '1
Equation (19) is equivalent to a cost minimization objective: min z(N, X; vX). In our presentation we preferred to follow a profit maximization objective. If there is only a single product mix, i.e., A a = (%0}, then (MQAP-S) reduces to (MQAP) (to be r~ p ~0 is a referred to as (MQAP-%o) to show the dependency to ?%). This is obvious by recalling that ~7'P Z~p=11XpO constant, E{ z(N, X; vx° } = z(N, X; vxo) and the only active throughput constraint is T H ( N , X; 'vx°) > vx°. Let z rXo denote the objective function value of the r-th best layout for the given volume vector vxo. The optimal objective value of (MQAP-h0) can also be defined as zlx°= z(N*x0, X*'xo, vX°) where (Nx0, XX*o)is the optimal solution (MQAP-~0). Let (N* *, X* *) be an optimal solution to (MQAP-S). Obviously, if a solution is optimal with respect to each X, it is the optimal solution to (MQAP-S), i.e., if 3X: X = Xx* V~, the X = X* * and N * * = N x VX. Furthermore, P
,(N**, X**)= E XEA
Z
Z
p=l
XEA
If ~IX: X = Xx* V% then we define an upper bound on the objective value of the optimal solution as P (/)max =
Z Oh Z A~A p=l
RpOXp- Z ~,~A
ohZ~l >~q ~ ( g * * ,
X**).
P. Kouvelis, A.S. Kiran / Layout models for automated manufacturing systems
307
For a specific layout X, we define the following quantities of interest: N x ( X ) = { m i n N : T H ( N , X; v X ) > v x}
and
N(X)=
max{Nx(X)}, X~A,,
R(X)
=
{r,(X),..
. ,
rx(X ) , . . .,rlx,,l(X)} = {rx(X ) , ) ~ A , , : z ( N x ( X ) , X; vX) -- -' a~ r x ( . ~ ) } , and
Definition 1: Layout X~ dominates layout X2 iff
eP(N( X1), X , ) > * ( N ( X 2 ) ,
X2)-
Proposition 2: Layout X 1 dominates layout X 2 iff E X~ ~,,/~(
O)~[Zrhx(X2)--Z~I] > ¢ m a x - ¢ ( N ( X , )
" X1)-
X 2)
Proof. First observe that (N(Xa),)(1) is a feasible solution to the (MQAP-S) since if satisfies (20), (21) and (22). Then observe that N(X) for a specific layout X is the minimum N that will satisfy constraints (22) and consequently ~(N(X), X) is the best objective value for layout X. But P
E
~b(N(X2), )(2)= y" o x y~ RpvXp-
O).Zrx(X2)
,k~A
p=l
X~ A
;k~ A
p = 1
~ A ( X2)
XcA
p=l
=
~ ~ A ,,/A ( X 2 )
A ~ A , , / A ( X2)
X~A(X2)
X ~ A ~ / A ( X2)
+
Y'X ~ ,4 ,,/A
( X 2)
P
E ~,~A
E RA- zoo:, X~A,,
p=l
x
Zx
XEX./A(X2)
So
+( N(X2), X 2 ) : + .... -
2
°x[z,X~
X ~ ) % / A ( X 2)
From the definition of a dominance relationship between two layouts we have O(N(X2), X2)
X,) ¢~
Y'.
OX[ZrXx(X2)--ZXl]> ~ma×--dp(N(X,), )(1). []
X ~ A . / A ( X 2)
An easy case in detecting dominance to two layouts is given in Figure 1. We refer to that case as
absolute dominance or 'path non-crossing' (see Figure 1 for motivation of the terminology). Definition 3. Layout X1 absolutely dominates layout X2 iff x~
~-r;~ ( .vl ) ~
z rx(x2) x2
V X ~ A.
The previously stated proposition supports the development of the following algorithm for the (MQAP-S).
P. Kouvelis,A.S. Kiran / Layoutmodelsfor automatedmanufacturingsystems
308
Product Mix
Rank Order Solutions (High to Low Profit)
7,.2
%,I
v
V
V
(N:x, JAal (Xl),Xl) /
(N~.I(X,),X1)
/
/ (Nx2(X1),Xl)
/ ( N x I(X2),
(N;k2 (X2),X2).
/
(NxIAaI(X2),X2)
/
2)
Figure 1. An exampleof absolutedominance('Paths non-crossing')
Algorithm A1. Step 1. Select an initial good feasible solution (N(X0), X0). Step 2. For each A ~ h a determine ~0x(X0):=r,
where
ox[zX~-zX] <~q~max-q~(N(Xo), Xo)
and
ox[zX~- zX~] > ~max-- q~(N(X0), X0).
Step 3. Compare all layouts X that have rx(X)<~o~x(Xo)
V~cA~.
Eliminate absolutely dominated layouts using Definition 3. For the remaining pairs of layouts apply Proposition 2. The remaining non-dominated layout(s) are the optimal layouts. Let X* * be a non-dominated layout given by Algorithm A1. We define
N ( X * * ) = max {min N: T H ( N , X**; vx) > vX}. Theorem 4. Algorithm A1 provides an optimal solution to ( MQAP-S).
Proof. First observe that for all layouts
qXo ~ A,:
rxo(X) <~tOxo(Xo)
and the following holds even if A a / A ( X ) = { h0 }: E
°h[ zhrx(X) -zlx] ~Oho[Zhrx°tX)--2#°] >~max-d~(N(Xo), Xo)
)~eA./A( X)
Based on Proposition 2, we can imply that layout Xo dominates such layouts and consequently layouts eliminated in Step 3 are clearly dominated.
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309
Table 3 Part routings and processing times Part type
1
Operations
Req'd station
Proc. time
II Req'd station
Proc time
Req'd station
Proc time
Req'd station
Proc. time
1 2 3 4
1 2 .
3 1 -
2 1 4
1 3 1
3 2 4
2 1 2
4 2 1 3
1 2 1 1
.
.
III
.
.
IV
.
Then observe that N(X**) is the minimum number of parts that satisfies the constraints (22) for layout X**. Due to the monotonicity of the objective function with respect to N, we can conclude that N ( X * *) is the optimal N and ( N * * , X* *) is the optimal solution to (MQAP-S). []
We have implemented Algorithm A1 on a multi-user VAX 750 system. The following example illustrates the model and the solution procedure. The computational requirements for solving such an example, is less than 120 CPU seconds. For manufacturing systems up to 8 workstations, the computational requirements of our procedure do not exceed 25 CPU minutes. Such a requirement is reasonable during the initial design phase of a manufacturing system. Example. We have considered a system with 4 workstations and a load/unload/refixturing station. We assumed that four part types are produced in the system. Part routing and processing information is given in Table 3, and location costs, distances and workflow in Table 4. Product mixes considered are given in Table 5. The best six layouts for each product mix are shown in Figure 2. As can be seen from Figure 2, the algorithm eliminates layout 2 / 4 / 3 / 5 / 1 and 2 / 4 / 5 / 3 / 1 since they are dominated by layout 5 / 4 / 1 / 2 / 3 .
Table 4 Cost of locating station i to location j 400 500 500 500 600
700 700 800 700 500
600 500 600 600 400
800 400 400 600 500
600 500 700 800 600
60 110 0 40 90
80 20 40 0 40
60 90 90 40 0
40 80 0 0 80
60 120 0 0 40
20 60 80 40 0
Distance b / w location i and location j 0 130 60 80 60
130 0 110 20 40
Workflow b / w station i and j per unit time 0 120 40 60 20
120 0 80 120 60
Desired throughput Velocity of transportation system h(N ) Refixturing time
17.00 900.00 1000N 0
P. Kouvelis, A.S. Kiran / Layout models for automated manufacturingsystems
310 Table 5 Product mixes
Part type t [ox]
I
II
III
IV
1 (0.3) 2 (0.3) 3 (0.1) 4 (0.3) Rp
20 60 30 30 250
60 20 50 60 250
80 80 70 70 250
40 40 50 40 250
Layout 3 / 4 / 5 / 2 / 1 is dominated by layout 5 / 4 / 3 / 2 / 1 . Applying Proposition 2 to the remaining 3 layouts, layout 5 / 4 / 1 / 2 / 3 is chosen as the optimal layout. We have generated the best 6 solutions by modifying the branch-and-bound procedure given in Kouvelis and Kiran (1990). This can be done via the introduction of cutting planes that eliminate certain feasible points. Let us consider (MQAP-S). After finding z rx and defining
At= {(;, J')" X,j=l),
B,= {(;,
X,.j=0},
the following cutting plane is introduced as a constraint in the formulation: ~j-
~
(i, j ) E A r
~j=M-2.
(i, j ) ~
The programming implementation of such a step is significantly easier than its explanation in optimization terms.
Product
Rank Order Solutions (High to Low Profit)
i
Mix
2
5/4/I1213 8600 \
3
/4/3/2/1 12100 %
4
5/4/1/2/3 __5/4/1/2/3 8300
8900
2/4/3/5/1\ / 3/4/5/2/1 8100 ~ ; l 11600
3/4/1/2/5 8100
3141112/5 8300
3
3/4/i/2/5~ 8000 /
~ 3/4/1/2/5 [~ 10400 /
2/4/5/3/1 7700
2/4/3/5/1 8000
4
2/4/5/3/1 / 7800
5/4/1/2/3 10200
2/4/3/5/1 7600
5
5/4/3/2/1 7700
2/4/5/3/1 10200
\5/4/3/2/1 1 7200
214151311 7700
6
3/4/5/2/1 6400
2/4/3/5/1 9700
3/4/5/2/1 6100
3/4/s/2/1
1
/
\
\
Figure 2. Solution of example problem
//
s14131211 7900
6600
P. Kouvelis, A, S. Kiran / Layout models for automated manufacturing systems
311
5. A dynamic programming model for multi-period problems In this section, we will consider product mix changes from one period to another. For example in a real manufacturing setting, a favorite product mix in one year may not be run for even a small percentage of time another year because of the introduction of new products and market dynamics. We have formulated this and similar situations into a dynamic programming model. Recall that for each possible product mix we defined an associated probability o x. Here we extend the definition as a time dependent one, i.e.,
where o k is the prior probability that product mix v x will be operational during period t. Furthermore, we update these probabilities at the end of each period. The transition mechanism to update the probabilities is denoted by ~'x(° ''+1, 0') where 0 ''+1 is the updated probability vector for period t + 1, updated at the end of period t. Note that the previous notation will be used, with appropriate modifications throughout the remainder of the paper. Additionally, we define: T = number of planning periods (planning horizon), fl = discount factor, t = planning period index t = 1 . . . . . T, A,(N '-~, X' 1, N', X ' ) = c o s t of changing the configuration of the manufacturing system from ( N ' 1 X , - I ) to (N', X') at the end of period t - 1/beginning of period t; cost elements included are: additional buffer spaces, additional pallets, layout changeover costs, etc., Ft(N, 1, X,-1; at) = optimal cost function for periods t through T given that ( N ' - I , X,-1) is the layout at the end of period t - 1 and o' is the probability vector for period t, (N'*, X'*) = optimal solution to (MQAP-S) for period t. Then, our functional equation is
[ (POMDP)
max |q~t(N', X ' ) + A t ( N '-1, X '-l, N', X t)
F,(Nt 1, X ' - ' ; o ' ) =
( N', X')
[
A
+fl ~
'
,+1 .
o')F,+,(N',
X'; o ''+l )
] ,
X=I
(23)
FT+,(, -, ") =0. The formulation (POMDP) is extremely helpful in understanding the nature of the problem. Unfortunately, such models, though extremely powerful in accurately depicting the nature of the realistic problem, are computationally intractable as they face the curse of dimensionality of state space. In the literature, structural results for the optimal policy of the above functional equation do not exist. Also, available computational approaches for partially observed Markov Processes fail to address realistic problems. For reasons of completeness in our presentation we have outlined the above general formulation. Now we discuss a special case of the formulation (POMDP) where
o , ) = 1{o ,'+, = o , + , ) with 1{. } an indicator function. In this case we get the following functional equation (for fl = 1): (DP)
F,(N'-', X'-')=max[q,,(N', F T + , ( ' , ") = 0 .
X ' ) + A , ( N '-1, X ' - ' , N', X ' ) + F , + , ( N ' ,
X')],(24)
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The computational feasibility of the dynamic programming formulation (DP) is dependent on the number of planning periods (stages in our dynamic programming formulation) and the number of alternative solutions (potential layout configurations) considered at each planning period. The computational time required by any dynamic programming procedure increases linearly with the number of stages and exponentially with the number of states (alternative layouts). Therefore, the number of layouts to be considered at each planning period is the most critical factor from a computational point of view. The state space reduction approach about to be presented in this section tries to minimize the number of alternative layouts that are considered at each planning period. Let us consider the formulation (MQAP-S) for planning period t. Let Utr denote the objective value of the r-th best layout in period t. Then T Umax =
E Utl t=l
is an upper bound on the optimal solution of the multiperiod problem. Let L t = ( N t, X t) be a layout that satisfies (22) for period t. We define
L={L,,t=I
. . . . . t}.
Then L is a feasible solution for (DP), and T U(L)
=
E [gPt(Nt, X t ) + A t ( N t - 1 ,
x t - l , Nt, Xt)]
t=l
is a lower bound on the optimal solution of (DP).
Proposition 5. Let L be a feasible solution for (DP). Call S(t)
= Urea x - U ( L ) .
Let Rt( L ) be the number of best layouts in period t that are to be considered by (DP) for finding an optimum solution. R ( L ) satisfies U~x - Uta,(z.) <~S ( L )
and
Vtl -- UtR,(L)+I ~" S ( L ) .
Proof. Consider a period t and a layout configuration (N ~, X ~) such that Uir = ~i(N ~, X ~) and r > R i ( L ). Then the value of the best multiperiod solution containing the configuration (N i, X i) in period t is bounded by
E Utl~- Uir= E U t l - Uilq- U~r< V m a x - - a ( t ) = t~?
U(L)
t=l
so that such a configuration need not be considered for the solution of (DP).
[]
Corollary 6. For any period t and a potentially optimal layout X t we need to consider in our (DP) procedure only ( Nt( Xt), Xt) (where Nt( X t) corresponds to the definition N( X ) given in Section 4). Proposition 5 contributes to a significant reduction of the state space. After the state space reduction, by applying the standardized dynamic programming methodology on the recursive relationship, we obtain the optimal solution to the dynamic layout problem. In order for the dynamic programming procedure to be implemented we need to have an efficient algorithm for generating the s-th best solution for the formulation (MQAP-S). Below, we present a modification of Algorithm A1 to generate the s best solutions for the (MQAP-S) formulation (refer to Section 4). We call the modified algorithm A s.
P. Kouvelis, A.S. Kiran / Layout models"for automatedmanufacturingsystems Algorithm A,. Step 1. Select an initial feasible solution (N(,go), Step 2. For each X ~ A a determine o~x(X0) := r
Step
313
Xo).
where
ox[zX~-z~]<~qbma×-ep(N(Xo), Xo)
and
OX[ZLI--ZAI ] >dPmax--dp(N(Xo), Xo).
3. Consider all layouts X that have
rx(X)<~oJx(Xo)
VX~A..
Let q be the total number of such layouts that are nondominated by X0. If q = s, then the s alternative configurations ( N ( X ) , X) are the s best solutions to the problem. Stop. If q > s, then rank all q layouts X (for easier ranking use easily identifiable dominance relationships) and pick the s best ( N ( X ) , X) as the solution to the problem. Stop. Otherwise, select another layout X ' with
+(N(X'), X') < ~(N(Xo), Xo). Set ( N ( X 0), Xo) = ( N ( X ' ) , X ' ) . Go to step 2. Corollary 7.
Algorithm A~ provides the s best solutions for formulations (MQAP-S).
Proof. The algorithm was constructed in such a way that X0 dominates all layouts except q of them. If q = s we accomplished our task. If q > s by ranking the q layouts we obtain the s best; if q < s we need to choose another X0 to accomplish our task. [] If the number of solutions at each period becomes too many to be effectively considered by the DP, a simpler heuristic approach could be used. For each period generate only a best configurations ( a ~< 5) and then apply the dynamic programming procedure to the resulting state space. Another heuristic approach in generating an appropriate state space is to consider the layouts that are optimal for at least one period.
6. Conclusions We have considered the layout decisions in the case of uncertainties that are pertaining to part demands, process plans and operation assignments. Such uncertainties are very c o m m o n during the design phase of automated manufacturing systems. In fact, the term 'flexible' has been used for some manufacturing systems because of the capabilities of such systems to be profitable in a wide range of operating conditions. We have considered two cases. First, a static case where once a layout design has been made, if cannot be changed. We have developed some dominance rules that enable us to determine efficient (i.e., non-dominated) alternative layouts. This new concept has been used in an algorithmic structure to determine the optimal solution(s). We have also considered the case when there are multiple planning periods with stochastic demands. A DP formulation has been developed in this case, to determine the optimal solution(s). Our numerical experiments indicate that the models are realistic. They can be implemented and used for small to medium sized (i.e., 4 - 8 stations) problems. The work reported here can be extended to cover even more comprehensive models. The authors have already undertaken work to consider selection of both number and location of machines and M H S design parameters.
Acknowledgment We would like to thank anonymous referees for their valuable comments and suggestions. This research is partially supported by N S F under Grant DMC-87-09171.
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References [1] Armour, G.C., and Buffa, E.S., "A heuristic algorithm and simulation approach to the relative location of facilities", Management Science 9/2 (1963) 294-309. [2] Bazaraa, M.S., and Elshafei, A.N., "An exact branch and bound procedure for quadratic assignment problems", Naval Research Logistics Quaterly, 26 (1979) 109-121. [3] Bazaraa, M.S., and Sherali, M.D., "Benders' partitioning scheme applied to a new formulation of the quadratic assignment problem", Naval Research Logistics Quarterly 27/1 (1980) 29-41. [4] Buffa, E.S., Armour, G.C., and Vollmann, T.E., "Allocating facilities with CRAFT", Harvard Business Review 42 (1964) 136-159. [5] Burkard, R.E., "Locations with spatial interaction-quadratic assignment problem", in: R.L. Francis and P.B. Mirchandani (eds.), Discrete Location Theory, Academic Press, New York, 1984. [6] Burkard, R.E., and Bonninger, T., "A heuristic for quadratic Boolean program with applications to quadratic assignment problems", European Journal of Operational Research 13 (1983) 374-386. [7] Burkard, R.E., and Stratman, K.H., "Numerical investigations on quadratic assignment problems", Naval Research Logistics Quarterly 25 (1978) 129-144. [8] Drezner, Z., "DISCON: A new method for the layout problem", Operations Research 28 6 (1980) 1375-1384. [9] Gilmore, P.C., "Optimal and suboptimal algorithms for the quadratic assignment problem", S l A M Journal 10/2 (1963) 305-313. [10] Golany, B., and Rosenblatt, M.J., "A heuristic algorithm for the quadratic assignment formulation for the plant layout problem", International Journal of Production Research 27/2 (1989) 293-308. [11] Gross, D., and Harris, C.M., Fundamentals of Queueing Theory, 2nd edition, John Wiley & Sons, New York, 1985. [12] Heragu, S.S., and Kusiak, A., "Machine layout problem in flexible manufacturing systems", Operations Research 36/2 (1988) 258-268. [13] Hillier, F.S., "Quantitative tools for plant layout analysis", Journal of Industrial Engineering 14/1 (1963) 33-40. [14] Hillier, F.S., and Connors, M.M., "Quadratic assignment problem algorithms and the location of indivisible facilities", Management Science 13 (1966) 42-57. [15] Kaku, B.K., and Thompson, G.L., "An exact algorithm for the general quadratic assignment problem", European Journal of Operational Research 23 (1986) 382-390. [16] Kelly, F.P., Reversibility and Stochastic Networks, John Wiley & Sons, New York, 1979. [17] Kouvelis, P., and Kiran, A.S., "The plant layout problem in automated manufacturing systems", Annals of Operations Research 26 (1990) 397-412. [18] Kouvelis, P., and Kiran, A.S., "Single and multiple layout models for automated manufacturing systems", Working Paper 89-2, USC, Dept. of ISE (1989). [19] Kusiak, A., and Heragu, S.S., "The facility layout problem", European Journal of Operational Research 29 (1987) 229-251. [20] Lawler, E.L., "The quadratic assignment problem", Management Science 9/4 (1963) 586-599. [21] Lee, R., and Moore, J.M., "CORELAP--Computerized relationship layout planning", Journal of Industrial Engineering 18 (1967) 195-200. [22] Levany, P.R., and Kalchik, S., "Facilities layout-A survey of solution procedures", Computers and Industrial Engineering 9/2 (1985) 141-148. [23] Lewis, W.P., and Block, T.E., "On the application of computer aids to plant layout", International Journal of Production Research 18/1 (1980) 11-20. [24] Neghabat, F., "An efficient equipment layout algorithm", Operations Research 22 (1974) 622-628. [25] Rosenblatt, M.J., "The dynamics of plant layout", Management Science 32/1 (1986) 76-86. [26] Sahni, S., and Gonzalez, T., "P-complete approximation problems", Journal of the A C M 23/3 (1976) 555-565. [27] Seehof, J.M., and Evans, W.R., "Automated layout design program, Journal of Industrial Engineering 18/12 (1967) 690-695. [28] Sweeney, D.S., and Tatham, R.L., "An improved long-run model for multiple warehouse location", Management Science 22/7 (1976) 748-758. [29] Tompkins, J.A., and Reed, Jr., R., "An applied model for the facilities design problem", International Journal of Production Research 14/5 (1976) 583-595. [30] Zoller, K., and Adendorff, K., "Layout planning by computer simulation", A I I E Transactions 4/2 (1972) 116-125.
European Journal of Operational Research 52 (1991) 300-314 North-Holland
Theory and Methodology
Single and multiple period layout models for automated manufacturing systems Panagiotis Kouvelis
Management Department, University of Texas, Austin, TX 78712, USA A l i S. K i r a n *
Department of Industrial and Systems Engineering, Unioersity of Southern California, Los Angeles, CA 90089, USA Abstract: Layout models for automated manufacturing systems are considered. The changes in product mix, part routings and process plans are incorporated into a stochastic single period layout model. A solution method, based on generating non-dominated layouts, is developed and illustrated. A dynamic programming formulation is also developed to capture the dynamic layout decisions in an automated manufacturing environment. A state space reduction approach is proposed for solving this multi-period model. Keywords: Machine layout, quadratic assignment, flexible manufacturing
I. I n t r o d u c t i o n
The complex interplay of engineering design and analysis is nowhere more apparent than in the design of automated manufacturing systems. The designer of such a system faces the difficult task of developing a system that is capable of handling a variety of products with variable demands at a reasonable cost. Alternate and probabilistic routings, and schedule and inventory constraints further complicate the design task. The designer uses analysis tools to ensure that the selected resource levels are adequate enough to provide the output requirements in a wide range of operating conditions. Making the designer's task even more difficult is the fact that decisions regarding the different subsystems cannot be made in isolation. This has been illustrated by Kouvelis and Kiran (1990) as the interactions between the layout, material handling system and storage requirements of an automated manufacturing system. Optimal design of the physical layout is one of the most important issues that must be resolved in the early stages of the manufacturing system design. Cost consequences of the decisions related to layout of the machines can be observed not only during the implementation but also in the operation of the system. Good solutions to these problems provide a necessary foundation for effective utilization of the system. Tompkins and White (1984) emphasized the importance of layout decisions for effective material handling by pointing out that 20% to 50% of the total operating expenses in manufacturing are attributed to material handling and layout related costs. * Present address: Department of Industrial Engineering, Dai-Yeh Institute of Technology,Taiwan, ROC Received March 1989; revisedAugust 1989 0377-2217/91/$03.50 © 1991 - ElsevierSciencePublishers B.V. (North-Holland)
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In this paper we address the layout decisions of an automated manufacturing system in realistic scenarios. Our work is motivated by the fact that the product mix that will be operational, is highly uncertain during the design phase. Usually at the time layout decisions are made, the products and the process plans have not been completely determined. Furthermore, the product mix (may be given as part production rates) is subject to changes because of forecasting errors and fluctuations. The designer's task is to develop a satisfactory layout design in this highly nondeterministic environment. We consider this problem in two different scenarios: First, we consider the cases where once chosen, a layout cannot be changed in the foreseeable future. During the layout design phase the product mix is uncertain. However once realized, the product mix is expected to remain stable over the planning horizon. Such a scenario is applicable in automated environments where layout changes are not feasible because of very high changeover costs and uncontrollable changeover times. Automated paint lines, coordinate measurement machines, automated washing and deburring stations are examples of the facilities with high installation costs and times. In the second case, we consider layout decisions in a dynamic way. The planning horizon for the automated manufacturing system design is divided into smaller planning periods. The product mix for each planning period, though uncertain during the design phase, remains stable over that period. The designer is allowed to specify different layouts, one for each planning period, over the planning horizon. This second case is motivated by Flexible Manufacturing System (FMS) installations. Such systems consist of cells of physically identical machines that perform a variety of operations. At the beginning of each planning period (week, month), operations allocation decisions are made. These decisions cause a significant change in the operational product mix of the cell in each period, and in most cases imply sub-optimality of a static layout configuration. The paper is organized as follows. The next section introduces the notation and provides a brief background on the previous work in the area. In Section 3, we present the effects of product mix changes on layout decisions. Section 4 deals with the above scenario 1: A single period layout decision model handling the product mix uncertainty is developed and a solution approach for the model is also described. In Section 5, we present a multiperiod dynamic layout model, able to handle the uncertainty of product mix at each planning period (i.e., scenario 2). In Section 6, we summarize our conclusions.
2. Background A plant layout problem in a conventional manufacturing setting is often defined as determination of the relative location of a given number of machines among the candidate locations. A typical objective is the total material handling cost, which is defined as the product of material movement frequencies and distances between the machines. The problem is then modelled as a Quadratic Assignment Problem (QAP), which is NP-complete (Sahni and Gonzalez, 1976). Gilmore (1963) and Lawler (1963) were the first to develop optimal branch and bound solution procedures for the problem. Bazaraa and Elshafei (1979) proposed a branch and bound algorithm for the QAP which is based on the stage by stage assignments of single facilities to unoccupied locations. Another branch and bound algorithm was recently suggested by Kaku and Thompson (1986). Cutting plane methods for the solution of the QAP are presented in Bazaraa and Sherali (1980) and Burkard and Bonninger (1983). Armour and Buffa (1963), Hillier (1963), Hillier and Connors (1963), Zoller and Adendorff (1972), Neghabat (1974), Lewis and Block (1980), Heragu and Kusiak (1988), and Golany and Rosenblatt (1989) proposed heuristic procedures. Computerized packages that solve the plant layout problem are available, to mention a few, CRAFT (Buffa et al., 1964), COFAD (Tompkins and Reed, 1976), ALDEP (Seehof and Evans, 1967), and CORELAP (Lee and Moore, 1967). Surveys of algorithms on the quadratic assignment problem can be found in Burkard (1984) and Kusiak and Heragu (1987). Extensive computational results comparing the various algorithms are reported in Burkard and Stratman (1978), Burkard (1984), and Levany and Kalchik (1985). Rosenblatt (1988) developed a dynamic programming formulation for a multiperiod plant layout problem. Kouvelis and Kiran (1990) define the
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layout p r o b l e m for a u t o m a t e d m a n u f a c t u r i n g systems as the selection of locations for M stations subject to a given part p r o d u c t i o n rate requirement. Their modified f o r m u l a t i o n of the Q u a d r a t i c A s s i g n m e n t P r o b l e m ( M Q A P ) is given below: M
(MQAP)
M
z(N, X) = E
Min N,X
M
M
Y'~ Y] E ajlkmXjlXkm + h ( N ) ,
j=l
l=1 k=l
(1)
m=l
M
s.t.
~
x/=
1,
l=1 ..... M,
(2)
j=l
(3)
j=l M
Y] x / t = 1,
. . . . . M,
1=1
T H ( N , X ) > V, x/={O,
1},
(4)
Ninteger,
where
= [ F ( j , k ) d ( l , m) a/kt'~ [ F ( j , j ) d ( j , j)+Cjl
ifj=gkorl--/=m, ifj=kandl=m,
(5)
P
V= E Vp,
(6)
p=l
x/=
1
if workstation j is assigned to location l,
0
otherwise.
All relevant n o t a t i o n is explained in T a b l e 1. M o d i f i c a t i o n of the G i l m o r e - L a w l e r b r a n c h a n d b o u n d procedure h a n d l e s the above p r o b l e m successfully for usual sizes of a u t o m a t e d m a n u f a c t u r i n g systems Table 1 Notation P:
p:
s~: M: j:
eje: #j:
s/ = (s+)~=1: N: d(I, m): Op(i):
R?(i, i + 1): Tp(i, i +1): Wpi: X: q: Qp.
F(j, k): h(N):
number of different part types part type index, p =1,..., P number of unit operations required to produce part type p required production rate of parts of type p number of stations station index j = 1,..., M visit frequency of part type p to station j service rate of station j number of servers at station j vector representation of the station configuration number of jobs in the system at any point in time transportation cost (based on some metric of distance per unit work flow between location l and location m) i-th operation of part type p (random variable) time required to perform refixturing on part p between the successive operation Op(i) and Op(+ 1) (random variable) time required to transport parts between the work stations Wei and Wml+ 1 workstation where operation i of part type p takes place layout decision matrix, binary matrix of assignment of workstations to locations index for process plans, q = 1.... ' Qe number of different process plans for part type p cost per unit time associated directly with assigning workstation j to location l workflow per unit time between workstation j and workstation k inventory and resource allocation costs of job population N
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303
Table 2 Computational results of the branch and bound solution procedure for MQAP Number of Stations
(CPU sec on VAX 11/780) "
<5 5 6 7 8 > 8
<5 6.5 16.9 65.2 410.5 > 1112.7
Reported average performance over 5 randomly generated examples in each case.
(6-10 machines grouped into 5 or fewer stations, as reported in a large scale survey on American and Japanese manufacturing systems by Jaikumar (1986)). Computational results demonstrating the ability of such a method to handle real size manufacturing layout problems are presented in Table 2. In order to apply the MQAP formulation, an appropriate throughput evaluation model for the FMS is needed. Kouvelis and Kiran (1990) proposed a simple Closed Queueing Network (CQN) model for that purpose. The model captures the effects of layout decisions on the throughput of the manufacturing systems. Here we summarize the model for convenience. Let us assume that our FMS has M - 1 machining stations and a load/unload ( L / U L ) station. For reasons of convenience, assume that the L / U L station performs only refixturing operations with random refixturing Rp(i, i + 1) that follow a general distribution with mean time R m. The material handling (transportation) times of parts between stations are modelled with the random variables Tp(i, i + 1) that follow a general distribution with mean time Tp,, which is a function of the layout decision and can be estimated as
d(wp,. .I..,+,) T,, =
(7)
U~t
where UM is the average speed of the material handling system (MHS). In the Kouvelis and Kiran (1990) paper the operation of the MHS and the L / U L station is modelled as a single node, node M, of the proposed CQN model. The service time /z~t1 at node M includes both transportation delays and the refixturing operations at the L / U L station: sp-i
t~Ml= /--, --~ E ( Rp,+ p=l
r,,).
(8)
i=1
The average service time is an adequate descriptive parameter for node M. It is modelled as an M / G / ~ node of the CQN which still has a product form solution (Kelly, 1979). The other important input quantities to the proposed CQN model are the visit frequencies of parts to a station. The visit frequency of part type p to station j is given by
up s. e,, = -~ Z lj ( Wm ).
(9)
~=1
where
II(WP')={Io
if i-th°perati°n°fparttype
i takesplaceatstati°n J'i'e"
(10)
Visit frequency of all parts to station j is given by P
e,= E ejp. p=l
(11)
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304
Using standard results from CQN theory (e.g., Gross and Harris, 1985) the steady state probability of having n~, n 2. . . . . nM number of jobs at stations 1, 2 . . . . . M, respectively, is given by
1 P"" n2 . . . . . n i - -
(e'/"J) °'
G(N)
~ ( n j ) = t n j! Is('J S,)s t
fj(n,)
j=l
for nj ~
'
j=l
(12)
. . . . . M,
(13)
M (ejlzj)'.
G(N) =
.,+.2+
E
I--I +.M=NJ=' ( ( ' J )
(14)
The throughput function of station j is given by (see e.g., Solberg, 1977) THj=e/
G ( N - 1) G(N)
(15)
Assuming e i = 1, the throughput of the system is given by TH = TH/-
G ( N - 1) G(N)
(16)
Since the normalizing constant and throughput rate are functions of the layout decisions, we use the notation G(N, X ) and T H ( N , X) respectively to express the relation between layout decisions, X, and these functions.
3. Effects of product mix changes on layout decisions In this section we explore the interactions of product mix and other uncertain input factors (e.g., operation allocation policies, alternate process plans for part types) with the optimal layout decisions. The understanding of such interactions is a step towards the development of design models that will capture the relations between the components of automated manufacturing systems. Using these models, integration of the various subsystems into an optimum system design could be possible. Usually the designer starts with a set of part types and a rough operation process chart for each part type. In modelling the part processing operations, we will use 'unit operation' as a basic unit. A 'unit operation' is defined as an elementary transformation of an input object into an output object with no interruption. Since unit operations are defined to be specific to a part type, we order the index of unit operations to be such that the first n I unit operations correspond to part type 1, the subsequent n 2 unit operations correspond to part type 2, and so on. For a particular part type P ' , let Np, be the cumulative number of unit operations for part types 1 to P ' - 1, i.e., P'--I
Np, = ~
n p,
and
Nl=0.
P=I
The unit operations of part type p are organized as an operations graph Qp (Ip, Ap) where the node set lp corresponds to unit operations of the part type p and the directed arc s e t Ap captures all technological precedence relations between the unit operations. We denote a process plan by Sqp, q = 1 . . . . . Qp, and the associated probability of utilizing process plan q by 0qp with ~Q£10qp = 1. The product mix to be processed in the automated manufacturing system can be described by production volumes cp, p = 1 . . . . . P, over a planning horizon. Let yj, be the percentage of time (during the
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305
planning horizon) that unit operation i is allocated to machine j. A long term operation allocation policy can be summarized by matrix Y = [y,]. Then the workflow over the planning horizon between stations j and k is given by ?
F(j, k)=
9.
(17)
E '9 E Oq; E Yj,Yk,,+, p-1
q~l
i~S,;'
and the visit frequencies of parts to workstations are given by Up Qp
(18) p=l
p=l
q=l
i~S~'
From equation (1), (4), (5), (16), (17), and (18) the dependency of layout decisions on the product mix, selection of process plans and long term allocation policy of operations to machines is apparent. To indicate such dependency, we introduce the following parametric notation:
a,kt,,,=a,k,,,(v, O, Y ) ,
T H ( N , X; v, O, Y)
and
z ( N , X; v, O, Y ) ,
where
v = ( v e , p = l ..... P),
O=(OP, q = l ..... Qp, p = l ..... P ) .
It is easy to observe that a new part type introduction or volume changes for existing parts will affect (17) and (18). Any technological changes or changes in process plans may change Qp and 0qp, q = 1 Qp, hence affecting (17) and (18). In the layout model developed in the next section, these interactions are captured in the values of the visit frequencies that are input to CQN representation of the system. The CQN model of the system is an integral part of the MQAP formulation through the production requirement constraint. The methodology developed in the remainder of the paper explicitly considers the effect of product mix changes in layout decisions. It is, however, easy to apply this methodology to any of the previously mentioned interactions between layout decisions and other operational decisions (e.g., process selection, product introduction, operation allocation policies, etc.). . . . . .
4. Single period model
A major advantage of an automated manufacturing system is its capability to process highly variable product mixes and consequently respond effectively to demand changes. Our purpose is twofold in addressing the single-period stochastic case. First, for the systems with very high relocation costs, the single-period stochastic model may be sufficient. Secondly, the properties of the single-period model will be used in subsequent sections for the development of a multi-period model. Assume that the manufacturing system will produce a variety of products in different product mixes. Let M ~ ~ A, be the product mix index where A is the set of all possible product mixes. A specific product mix can be defined as vx = (rex, p = 1. . . . . P) where vpx is the volume of part type p to be produced over the planning period. Let o x be the probability that product mix vx will be the operational product mix. Let o = (ox, ~ = 1, ..., ~,) be the probability vector and vx = Ep=lV)~. e x Alternatively o x can be considered as a percentage of time that vx be operational during the planning period. Assuming each part is associated with a revenue, Rp, a modified QAP formulation for the static (i.e., single-period, stochastic) layout
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306
problem is given as follows: (MQAP-S)
Max
if(N, X ) = E {
RpVXp-FC(X; vX)-h(N))
~
(19)
p=l M
E Xjz=l,
s.t.
l = 1 . . . . . M,
(20)
j=l
(21)
j=l M
Y~Xj,=I,
. . . . . M,
/=l
T H ( N , X; v X l > v x,
Xjl =
{0, 1 },
X~A,={X~A:
ox>0},
(22)
N integer,
where M
FC(X; vx) = E
M
M
M
E
E
E
aj*,m(vX)Xj,X*m•
j = l 1=1 k = l m = l
Also to be consistent with the previous notation call,
z(N, X;
vX)= FC(X;
vX)+h(N).
Then
p=l
z'NX v '1
Equation (19) is equivalent to a cost minimization objective: min z(N, X; vX). In our presentation we preferred to follow a profit maximization objective. If there is only a single product mix, i.e., A a = (%0}, then (MQAP-S) reduces to (MQAP) (to be r~ p ~0 is a referred to as (MQAP-%o) to show the dependency to ?%). This is obvious by recalling that ~7'P Z~p=11XpO constant, E{ z(N, X; vx° } = z(N, X; vxo) and the only active throughput constraint is T H ( N , X; 'vx°) > vx°. Let z rXo denote the objective function value of the r-th best layout for the given volume vector vxo. The optimal objective value of (MQAP-h0) can also be defined as zlx°= z(N*x0, X*'xo, vX°) where (Nx0, XX*o)is the optimal solution (MQAP-~0). Let (N* *, X* *) be an optimal solution to (MQAP-S). Obviously, if a solution is optimal with respect to each X, it is the optimal solution to (MQAP-S), i.e., if 3X: X = Xx* V~, the X = X* * and N * * = N x VX. Furthermore, P
,(N**, X**)= E XEA
Z
Z
p=l
XEA
If ~IX: X = Xx* V% then we define an upper bound on the objective value of the optimal solution as P (/)max =
Z Oh Z A~A p=l
RpOXp- Z ~,~A
ohZ~l >~q ~ ( g * * ,
X**).
P. Kouvelis, A.S. Kiran / Layout models for automated manufacturing systems
307
For a specific layout X, we define the following quantities of interest: N x ( X ) = { m i n N : T H ( N , X; v X ) > v x}
and
N(X)=
max{Nx(X)}, X~A,,
R(X)
=
{r,(X),..
. ,
rx(X ) , . . .,rlx,,l(X)} = {rx(X ) , ) ~ A , , : z ( N x ( X ) , X; vX) -- -' a~ r x ( . ~ ) } , and
Definition 1: Layout X~ dominates layout X2 iff
eP(N( X1), X , ) > * ( N ( X 2 ) ,
X2)-
Proposition 2: Layout X 1 dominates layout X 2 iff E X~ ~,,/~(
O)~[Zrhx(X2)--Z~I] > ¢ m a x - ¢ ( N ( X , )
" X1)-
X 2)
Proof. First observe that (N(Xa),)(1) is a feasible solution to the (MQAP-S) since if satisfies (20), (21) and (22). Then observe that N(X) for a specific layout X is the minimum N that will satisfy constraints (22) and consequently ~(N(X), X) is the best objective value for layout X. But P
E
~b(N(X2), )(2)= y" o x y~ RpvXp-
O).Zrx(X2)
,k~A
p=l
X~ A
;k~ A
p = 1
~ A ( X2)
XcA
p=l
=
~ ~ A ,,/A ( X 2 )
A ~ A , , / A ( X2)
X~A(X2)
X ~ A ~ / A ( X2)
+
Y'X ~ ,4 ,,/A
( X 2)
P
E ~,~A
E RA- zoo:, X~A,,
p=l
x
Zx
XEX./A(X2)
So
+( N(X2), X 2 ) : + .... -
2
°x[z,X~
X ~ ) % / A ( X 2)
From the definition of a dominance relationship between two layouts we have O(N(X2), X2)
X,) ¢~
Y'.
OX[ZrXx(X2)--ZXl]> ~ma×--dp(N(X,), )(1). []
X ~ A . / A ( X 2)
An easy case in detecting dominance to two layouts is given in Figure 1. We refer to that case as
absolute dominance or 'path non-crossing' (see Figure 1 for motivation of the terminology). Definition 3. Layout X1 absolutely dominates layout X2 iff x~
~-r;~ ( .vl ) ~
z rx(x2) x2
V X ~ A.
The previously stated proposition supports the development of the following algorithm for the (MQAP-S).
P. Kouvelis,A.S. Kiran / Layoutmodelsfor automatedmanufacturingsystems
308
Product Mix
Rank Order Solutions (High to Low Profit)
7,.2
%,I
v
V
V
(N:x, JAal (Xl),Xl) /
(N~.I(X,),X1)
/
/ (Nx2(X1),Xl)
/ ( N x I(X2),
(N;k2 (X2),X2).
/
(NxIAaI(X2),X2)
/
2)
Figure 1. An exampleof absolutedominance('Paths non-crossing')
Algorithm A1. Step 1. Select an initial good feasible solution (N(X0), X0). Step 2. For each A ~ h a determine ~0x(X0):=r,
where
ox[zX~-zX] <~q~max-q~(N(Xo), Xo)
and
ox[zX~- zX~] > ~max-- q~(N(X0), X0).
Step 3. Compare all layouts X that have rx(X)<~o~x(Xo)
V~cA~.
Eliminate absolutely dominated layouts using Definition 3. For the remaining pairs of layouts apply Proposition 2. The remaining non-dominated layout(s) are the optimal layouts. Let X* * be a non-dominated layout given by Algorithm A1. We define
N ( X * * ) = max {min N: T H ( N , X**; vx) > vX}. Theorem 4. Algorithm A1 provides an optimal solution to ( MQAP-S).
Proof. First observe that for all layouts
qXo ~ A,:
rxo(X) <~tOxo(Xo)
and the following holds even if A a / A ( X ) = { h0 }: E
°h[ zhrx(X) -zlx] ~Oho[Zhrx°tX)--2#°] >~max-d~(N(Xo), Xo)
)~eA./A( X)
Based on Proposition 2, we can imply that layout Xo dominates such layouts and consequently layouts eliminated in Step 3 are clearly dominated.
P. Kouvelis, A.S. Kiran / Layout models for automated manufacturing systems
309
Table 3 Part routings and processing times Part type
1
Operations
Req'd station
Proc. time
II Req'd station
Proc time
Req'd station
Proc time
Req'd station
Proc. time
1 2 3 4
1 2 .
3 1 -
2 1 4
1 3 1
3 2 4
2 1 2
4 2 1 3
1 2 1 1
.
.
III
.
.
IV
.
Then observe that N(X**) is the minimum number of parts that satisfies the constraints (22) for layout X**. Due to the monotonicity of the objective function with respect to N, we can conclude that N ( X * *) is the optimal N and ( N * * , X* *) is the optimal solution to (MQAP-S). []
We have implemented Algorithm A1 on a multi-user VAX 750 system. The following example illustrates the model and the solution procedure. The computational requirements for solving such an example, is less than 120 CPU seconds. For manufacturing systems up to 8 workstations, the computational requirements of our procedure do not exceed 25 CPU minutes. Such a requirement is reasonable during the initial design phase of a manufacturing system. Example. We have considered a system with 4 workstations and a load/unload/refixturing station. We assumed that four part types are produced in the system. Part routing and processing information is given in Table 3, and location costs, distances and workflow in Table 4. Product mixes considered are given in Table 5. The best six layouts for each product mix are shown in Figure 2. As can be seen from Figure 2, the algorithm eliminates layout 2 / 4 / 3 / 5 / 1 and 2 / 4 / 5 / 3 / 1 since they are dominated by layout 5 / 4 / 1 / 2 / 3 .
Table 4 Cost of locating station i to location j 400 500 500 500 600
700 700 800 700 500
600 500 600 600 400
800 400 400 600 500
600 500 700 800 600
60 110 0 40 90
80 20 40 0 40
60 90 90 40 0
40 80 0 0 80
60 120 0 0 40
20 60 80 40 0
Distance b / w location i and location j 0 130 60 80 60
130 0 110 20 40
Workflow b / w station i and j per unit time 0 120 40 60 20
120 0 80 120 60
Desired throughput Velocity of transportation system h(N ) Refixturing time
17.00 900.00 1000N 0
P. Kouvelis, A.S. Kiran / Layout models for automated manufacturingsystems
310 Table 5 Product mixes
Part type t [ox]
I
II
III
IV
1 (0.3) 2 (0.3) 3 (0.1) 4 (0.3) Rp
20 60 30 30 250
60 20 50 60 250
80 80 70 70 250
40 40 50 40 250
Layout 3 / 4 / 5 / 2 / 1 is dominated by layout 5 / 4 / 3 / 2 / 1 . Applying Proposition 2 to the remaining 3 layouts, layout 5 / 4 / 1 / 2 / 3 is chosen as the optimal layout. We have generated the best 6 solutions by modifying the branch-and-bound procedure given in Kouvelis and Kiran (1990). This can be done via the introduction of cutting planes that eliminate certain feasible points. Let us consider (MQAP-S). After finding z rx and defining
At= {(;, J')" X,j=l),
B,= {(;,
X,.j=0},
the following cutting plane is introduced as a constraint in the formulation: ~j-
~
(i, j ) E A r
~j=M-2.
(i, j ) ~
The programming implementation of such a step is significantly easier than its explanation in optimization terms.
Product
Rank Order Solutions (High to Low Profit)
i
Mix
2
5/4/I1213 8600 \
3
/4/3/2/1 12100 %
4
5/4/1/2/3 __5/4/1/2/3 8300
8900
2/4/3/5/1\ / 3/4/5/2/1 8100 ~ ; l 11600
3/4/1/2/5 8100
3141112/5 8300
3
3/4/i/2/5~ 8000 /
~ 3/4/1/2/5 [~ 10400 /
2/4/5/3/1 7700
2/4/3/5/1 8000
4
2/4/5/3/1 / 7800
5/4/1/2/3 10200
2/4/3/5/1 7600
5
5/4/3/2/1 7700
2/4/5/3/1 10200
\5/4/3/2/1 1 7200
214151311 7700
6
3/4/5/2/1 6400
2/4/3/5/1 9700
3/4/5/2/1 6100
3/4/s/2/1
1
/
\
\
Figure 2. Solution of example problem
//
s14131211 7900
6600
P. Kouvelis, A, S. Kiran / Layout models for automated manufacturing systems
311
5. A dynamic programming model for multi-period problems In this section, we will consider product mix changes from one period to another. For example in a real manufacturing setting, a favorite product mix in one year may not be run for even a small percentage of time another year because of the introduction of new products and market dynamics. We have formulated this and similar situations into a dynamic programming model. Recall that for each possible product mix we defined an associated probability o x. Here we extend the definition as a time dependent one, i.e.,
where o k is the prior probability that product mix v x will be operational during period t. Furthermore, we update these probabilities at the end of each period. The transition mechanism to update the probabilities is denoted by ~'x(° ''+1, 0') where 0 ''+1 is the updated probability vector for period t + 1, updated at the end of period t. Note that the previous notation will be used, with appropriate modifications throughout the remainder of the paper. Additionally, we define: T = number of planning periods (planning horizon), fl = discount factor, t = planning period index t = 1 . . . . . T, A,(N '-~, X' 1, N', X ' ) = c o s t of changing the configuration of the manufacturing system from ( N ' 1 X , - I ) to (N', X') at the end of period t - 1/beginning of period t; cost elements included are: additional buffer spaces, additional pallets, layout changeover costs, etc., Ft(N, 1, X,-1; at) = optimal cost function for periods t through T given that ( N ' - I , X,-1) is the layout at the end of period t - 1 and o' is the probability vector for period t, (N'*, X'*) = optimal solution to (MQAP-S) for period t. Then, our functional equation is
[ (POMDP)
max |q~t(N', X ' ) + A t ( N '-1, X '-l, N', X t)
F,(Nt 1, X ' - ' ; o ' ) =
( N', X')
[
A
+fl ~
'
,+1 .
o')F,+,(N',
X'; o ''+l )
] ,
X=I
(23)
FT+,(, -, ") =0. The formulation (POMDP) is extremely helpful in understanding the nature of the problem. Unfortunately, such models, though extremely powerful in accurately depicting the nature of the realistic problem, are computationally intractable as they face the curse of dimensionality of state space. In the literature, structural results for the optimal policy of the above functional equation do not exist. Also, available computational approaches for partially observed Markov Processes fail to address realistic problems. For reasons of completeness in our presentation we have outlined the above general formulation. Now we discuss a special case of the formulation (POMDP) where
o , ) = 1{o ,'+, = o , + , ) with 1{. } an indicator function. In this case we get the following functional equation (for fl = 1): (DP)
F,(N'-', X'-')=max[q,,(N', F T + , ( ' , ") = 0 .
X ' ) + A , ( N '-1, X ' - ' , N', X ' ) + F , + , ( N ' ,
X')],(24)
312
P. Kouvelis,A.S. Kiran / Layout modelsfor automatedmanufacturingsystems
The computational feasibility of the dynamic programming formulation (DP) is dependent on the number of planning periods (stages in our dynamic programming formulation) and the number of alternative solutions (potential layout configurations) considered at each planning period. The computational time required by any dynamic programming procedure increases linearly with the number of stages and exponentially with the number of states (alternative layouts). Therefore, the number of layouts to be considered at each planning period is the most critical factor from a computational point of view. The state space reduction approach about to be presented in this section tries to minimize the number of alternative layouts that are considered at each planning period. Let us consider the formulation (MQAP-S) for planning period t. Let Utr denote the objective value of the r-th best layout in period t. Then T Umax =
E Utl t=l
is an upper bound on the optimal solution of the multiperiod problem. Let L t = ( N t, X t) be a layout that satisfies (22) for period t. We define
L={L,,t=I
. . . . . t}.
Then L is a feasible solution for (DP), and T U(L)
=
E [gPt(Nt, X t ) + A t ( N t - 1 ,
x t - l , Nt, Xt)]
t=l
is a lower bound on the optimal solution of (DP).
Proposition 5. Let L be a feasible solution for (DP). Call S(t)
= Urea x - U ( L ) .
Let Rt( L ) be the number of best layouts in period t that are to be considered by (DP) for finding an optimum solution. R ( L ) satisfies U~x - Uta,(z.) <~S ( L )
and
Vtl -- UtR,(L)+I ~" S ( L ) .
Proof. Consider a period t and a layout configuration (N ~, X ~) such that Uir = ~i(N ~, X ~) and r > R i ( L ). Then the value of the best multiperiod solution containing the configuration (N i, X i) in period t is bounded by
E Utl~- Uir= E U t l - Uilq- U~r< V m a x - - a ( t ) = t~?
U(L)
t=l
so that such a configuration need not be considered for the solution of (DP).
[]
Corollary 6. For any period t and a potentially optimal layout X t we need to consider in our (DP) procedure only ( Nt( Xt), Xt) (where Nt( X t) corresponds to the definition N( X ) given in Section 4). Proposition 5 contributes to a significant reduction of the state space. After the state space reduction, by applying the standardized dynamic programming methodology on the recursive relationship, we obtain the optimal solution to the dynamic layout problem. In order for the dynamic programming procedure to be implemented we need to have an efficient algorithm for generating the s-th best solution for the formulation (MQAP-S). Below, we present a modification of Algorithm A1 to generate the s best solutions for the (MQAP-S) formulation (refer to Section 4). We call the modified algorithm A s.
P. Kouvelis, A.S. Kiran / Layout models"for automatedmanufacturingsystems Algorithm A,. Step 1. Select an initial feasible solution (N(,go), Step 2. For each X ~ A a determine o~x(X0) := r
Step
313
Xo).
where
ox[zX~-z~]<~qbma×-ep(N(Xo), Xo)
and
OX[ZLI--ZAI ] >dPmax--dp(N(Xo), Xo).
3. Consider all layouts X that have
rx(X)<~oJx(Xo)
VX~A..
Let q be the total number of such layouts that are nondominated by X0. If q = s, then the s alternative configurations ( N ( X ) , X) are the s best solutions to the problem. Stop. If q > s, then rank all q layouts X (for easier ranking use easily identifiable dominance relationships) and pick the s best ( N ( X ) , X) as the solution to the problem. Stop. Otherwise, select another layout X ' with
+(N(X'), X') < ~(N(Xo), Xo). Set ( N ( X 0), Xo) = ( N ( X ' ) , X ' ) . Go to step 2. Corollary 7.
Algorithm A~ provides the s best solutions for formulations (MQAP-S).
Proof. The algorithm was constructed in such a way that X0 dominates all layouts except q of them. If q = s we accomplished our task. If q > s by ranking the q layouts we obtain the s best; if q < s we need to choose another X0 to accomplish our task. [] If the number of solutions at each period becomes too many to be effectively considered by the DP, a simpler heuristic approach could be used. For each period generate only a best configurations ( a ~< 5) and then apply the dynamic programming procedure to the resulting state space. Another heuristic approach in generating an appropriate state space is to consider the layouts that are optimal for at least one period.
6. Conclusions We have considered the layout decisions in the case of uncertainties that are pertaining to part demands, process plans and operation assignments. Such uncertainties are very c o m m o n during the design phase of automated manufacturing systems. In fact, the term 'flexible' has been used for some manufacturing systems because of the capabilities of such systems to be profitable in a wide range of operating conditions. We have considered two cases. First, a static case where once a layout design has been made, if cannot be changed. We have developed some dominance rules that enable us to determine efficient (i.e., non-dominated) alternative layouts. This new concept has been used in an algorithmic structure to determine the optimal solution(s). We have also considered the case when there are multiple planning periods with stochastic demands. A DP formulation has been developed in this case, to determine the optimal solution(s). Our numerical experiments indicate that the models are realistic. They can be implemented and used for small to medium sized (i.e., 4 - 8 stations) problems. The work reported here can be extended to cover even more comprehensive models. The authors have already undertaken work to consider selection of both number and location of machines and M H S design parameters.
Acknowledgment We would like to thank anonymous referees for their valuable comments and suggestions. This research is partially supported by N S F under Grant DMC-87-09171.
314
P. Kouvelis, A.S. Kiran / Layout models for automated manufacturing systems
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