Location and layout are tightly interlaced and complementary. This section introduces two design optimi-zation models which should formalize the relationship between layout and location. The models are part of the global optimization trend described earlier. They both deal explicitly with dynamics and uncer-tainty. The first is a dynamic probabilistic discrete location model, whereas the second is a dynamic proba-bilistic discrete location and continuous layout model. The exposition of these models aims to counterbalance the design issue orientation of Sections 5.2 to 5.11 and the solution methodology orienta-tion of Secorienta-tion 5.2 by taking a formal mathematical modeling orientaorienta-tion. In no way should these two models be perceived as the models. They must, rather, be understood as two examples of a vast continuum of potential models to formalize the design issues described in Sections 5.2 through 5.11.
5.13.1 Dynamic Probabilistic Discrete Location Model
This model optimizes the dynamic assignment of a set of centers (or facilities) to a set of discrete locations.
The model supports a number of predefined future scenarios, each with a number of successive periods covering the planning horizon. The occurrence of a future is probabilistic. For each period in each future, each center has specific space requirements and pairwise unitary flow travel (or proximity relationship) costs are defined. Each center has also a fixed cost for being assigned in a specific location during a period of a future. Each location can be dynamically made available, expanded or contracted through time, with associated costs. Centers can be moved from a period to another, incurring a moving cost. The model recognizes that decisions relative to the first period are the only rigid ones, as all others will be revisable later on based on further information, as future scenarios will either become past, present, or nearer future scenarios. Below are first exposed the objective function and the constraints, followed by defini-tions for variables, parameters, and sets. Then the model is described in detail.
Minimize
f f
cltf cltf cl t
clc l tf cltf c l t
p a A c A
∑ ∑
∀ >+
,
(
1
′ ′ ′ ′ff clc lt
cll tf cll t
cl t f cl tf
A ) m A , , A
∀ ∀
∑
+∑
−′ ′
′
′
1 ′
+
∑
( + +l l + − −l l)+∑
+l
cl cl
cl f
f
e E e E l L
∑ ∑ ∑
p + + ++ − −∀ >
( )
, ,
ltf ltf ltf ltf ltf ltf l t f
s S s S s S
1
(5.4)
Subject to
cltf L
A
l∈ c
∑
=1 ∀c,t,f (5.5)cl f cl
A 1 = L ∀c,l,f (5.6)
ctf cltf C
ltf lm
r A S s
c∈ l
∑
≤ ≤ ∀l,t,f (5.7)ltf l t f ltf ltf
S =S,−1, +S+−S− ∀l,t 1,f> (5.8)
l f l l l l
S1 =E = +s0 E+−E− ∀l,f (5.9)
where the variables are
Acltf Binary variable equal to 1 when center c is assigned to location l in period t of future f, or 0 otherwise.
E E El, ,l+ l− Continuous non-negative variables deciding the space availability, expansion, and con-traction of location l in period 1.
Lcl Binary variable equal to 1 when center c is assigned in location l in period 1, or 0 otherwise.
S S Sltf, ,ltf+ ltf− Continuous non-negative variable deciding space availability, expansion, and contrac-tion of locacontrac-tion l in period t of future f.
while the parameters and sets are
acltf Cost of assigning center c to location l in period t of future f.
cclc′l′tf Cost of concurrently assigning center c to location l and center c′ to location l′ in period t of future f.
Cl Set of centers allowed to be located in location l.
e e el l, ,+ l− Unit space availability, expansion, and contraction costs for location l in period 1.
lcl Cost of assigning center c to location l in period 1.
Lc Set of locations in which center c is allowed to be located.
mcll′tf Cost of moving center c from location l to location l′ in period t of future f.
Pf Probability of occurrence of future f.
rctf Space requirements for center c in period t of future f.
S0l Initial space availability at location l.
S S Sltf, ,ltf+ ltf− Unit space availability, expansion, and contraction costs for location l in period t of future f.
The objective function 5.4 minimizes the overall actualized marginal cost, here described along two lines. The first line includes the sum of three cost components over all probable futures, weighted by the probability of occurrence of each future f. The first sums, over all allowed combinations, the cost of assign-ing a center c to a location l in period t > 1, independent of where other centers are located in this period t and of where center c was located in the previous period. The second sums, over all allowed combina-tions, the cost associated with concurrently locating center c in location l and center c′ in location l′ in period t. This is generically the cost associated with relations, interactions, and flows between centers. The third sums over all allowed combinations, the cost of moving center c from its location l in period t – 1 to location l′ in period t. This is generically the dynamic center relocation cost.
The second line of the objective function includes three cost components. The first two add up all immediate transition costs from the actual state to the proposed state in period one. First is the cost asso-ciated with the space of each location as proposed for the first period. It includes the cost of making this space available and the cost of either expanding or contracting the location from its actual state. Second is the cost of implementing each center in its proposed location. These two components do not explicitly refer to specific futures, as they are common to all futures since they are a direct result of the location deci-sions and will be incurred in all futures. The third component is similarly the space availability, expansion, and contraction cost for all locations in all later periods of all futures.
Constraint set 5.5 makes sure that a center c is located in a single location l in each period t of each future f. Constraint set 5.6 attaches the location decisions made for time period 1 over all probable futures.
So in the first period, each center c is assigned to the same location l in all futures. These are the decisions that have to be taken now that will definitely lead to implementation. These decisions cannot be altered afterward. In all later periods, the location decisions are allowed to vary from one future to another. They define a probabilistic plan that will be alterable subsequently, in light of further information availability, until they are associated with the first period in the revised model and become the hard location decision leading to immediate implementation.
Constraint set 5.7 ensures that the space availability constraint of each location l is respected at each period t of each future f, with the constraint that the sum of the required spaces of each center assigned to a location l does not exceed its space availability at that time. This availability is bounded for each location l to a specified maximum. The space availability of a location l can vary from one period to the next. For each future, constraint set 5.8 keeps an account of planned expansions and contractions of each location at all periods except the first. As constraint set 5.6 does for the location assignments of period one, con-straint set 5.9 deals with the incumbent expansion or contraction of each location in the forthcoming first period, common to all futures for each location.
When the space requirement and availability parameters are restrained to one, there is a single time period and a single future, and no location expansion or contraction is allowed, then this model reduces to the well known QAP.
5.13.2 Dynamic Probabilistic Discrete Location and Continuous Layout Model
This model generalizes the above model by allowing treatment of each discrete location as a facility within which its assigned centers have to be laid out. The model thus explicitly deals with center shaping and location within facilities and with avoidance of spatial interference between centers. Centers are restricted to rectangular shapes. They are allowed to be moved between facilities and within facility from a period to the next in a future. Below are first exposed the objective function and the constraints, followed by defini-tions for variables, parameters, and sets. Then the model is described in detail.
Minimize
( ) csc csc
csc
4 + f
∑
s tf s tf+s t
cs cstf cs
p d ′ ′ D ′ ′ m M
′ ′ tt
f
∑
∑
(5.10)subject to (5.5) to (5.9) and
ctfu ctfl
ctfu ctfl
X −X Y Y rctf
( ) (
−)
= ∀c,t,f (5.11)ltfu ltfl
ltfu ltfl
X −X Y Y Sltf
( ) (
−)
= ∀l,t,f (5.12)etfu etfl etf
etfu etfl
etf uetf e
Y Y
f− X X f Y
( )
≤(
−)
≤(
−Ylttf)
∀ ∈ ∪e C L( ),t,f (5.13)ltfl
cltf ctfl ctfu
ltfu
X −m
(
1−A)
≤X ≤X ≤X +m((
1−Acltf)
∀c,l,t,f (5.14)ltfl
cltf ctfl ctfu
ltfu
Y −m
(
1−A)
≤Y ≤Y ≤Y +m((
1−Acltf)
∀l,t,f (5.15)ll ltfl
ltfu lu
x ≤X ≤X ≤x ∀l,t,f (5.16)
l l ltfl
ltfu l
y Y≤ ≤Y ≤yu ∀l,t,f (5.17)
ctfl cstfs
uctf
X ≤X ≤X ∀c,s,t,f (5.18)
ctfl cstfs
ctfu
Y ≤Y ≤Y ∀c,s,t,f (5.19)
c tfl ctfu
cc tf
X ′ −X m Px′ ′
( )
≥(
−1)
∀c,c t,f (5.20)c tfl ctfu
cc tfy
Y ′ −Y m P ′ ′
( )
≥(
−1)
∀c,c t,f (5.21)cc tf
x c ctfx
cc tfy c ctfy
cltf c ltf
P ′ +P′ +P ′ +P′ ≥A + AA −′ 1 ∀c c l,t,f< ′ (5.22)
e fl el
X1 =X ∀ ∈ ∪e C L( ),f (5.23)
e fl el
Y1 =Y ∀ ∈ ∪e C L( ),f (5.24)
e fu eu
X1 =X ∀ ∈ ∪e C L( ),f (5.25)
e fu ue
Y 1 =Y ∀ ∈ ∪e C L( ),f (5.26)
cstfs c s tf
s x s tf
x s tf
X −X ′ ′ =Dcsc+′ ′ −Dcsc−′ ′ ∀c s, ,, ,t f (5.27)
cstfs c s tf
s y s tf
y s tf
Y −Y ′ ′ =Dcsc+′ ′ −Dcsc−′ ′ ∀c s, ,, ,t f (5.28) Dcsc′ ′s tf≥Dcscx+′ ′s tf+Dcscx−′ ′s tf+Dcscy++′ ′s tf+Dcscy−′ ′s tf−m
(
2−Acltf−Ac ltf′)
∀c s l t f, , , , (5.29)0 5.
( (
Xctfl +Xctfu)
−(
Xc tl,−1,f+Xc tu,−1,f) )
=Mctx+ff −Mctfx− ∀c t f, , (5.30)0 5.
( (
Yctfl +Yctfu)
−(
Yc tl,−1,f+Yc tu,−1,f) )
=Mcty+ff −Mctfy− ∀c t f, , (5.31)Mctf Mctfx M M M m A
ctfx ctfy
ctfy
cltf c
++ −+ ++ −−
(
2− − All t,−1,f)
∀c l t f, , , (5.32)where new variables are
X le , X ue , Y le , Y ue Continuous variables for the coordinates of the lower and upper boundaries of the sides of entity e along the X and Y axes in period 1 for all futures, where an entity is either a center or a location.
X letf , X uetf , Y etfl , Y uetf Continuous variables for the coordinates of the lower and upper boundaries of the sides of entity e along the X and Y axes in period t of future f, where an entity is either a center or a location.
X scstf , X scstf Continuous variables for the X and Y coordinates of I/O station s of center c in period t of future f.
Dcsc′s′tf Continuous non-negative variables for the rectilinear distance between
station s of center c and station s′ of center c′ in period t of future f.
Dcscx+′ ′s tf,Dcscx s tf,Dcscy s tf,Dcscs
−′ ′ +′ ′
′ ′ttf
y− Continuous non-negative variables for the positive and negative com-ponents along the X and Y axes of the rectilinear distance between station s of center c and station s′ of center c′ in period t of future f.
Mcxtf+,M Mcxtf−, cytf+,Mcytf− Continuous non-negative variables for the positive and negative compo-nents along the X and Y axes of the rectilinear move of center c in period t of future f from its coordinates in the previous period of future f.
Mctf Continuous non-negative variables for the rectilinear move of center c in period t of future f from its coordinates in the previous period of the same future, whenever center c is assigned to the same location in periods t and t – 1.
P xcc′tf , P ycc′tf Binary variables stating whether or not center c is to position lower
than center c′ along axes X and Y whenever both centers are assigned to the same location in period t of future f.
while new parameters are
dcsc′tf Unitary positive interaction cost associated with the rectilinear distance between station s of center c and station s′ of center c′ whenever both centers are assigned to the same location in period t of future f.
fetf Maximum allowed ratio between the longest and shortest sides of rectangular entity e, which is either a location or a center; this ratio can be distinct for each period of each future except for the first period, when it has to be the same for all futures.
mctf Unitary positive move cost associated with the rectilinear displacement of center c in period t of future f from its coordinates in the previous period of the same future, when-ever center c is assigned to the same location in periods t and t – 1.
m A very large number.
x ll , x ul , y ll , y ul Lower and upper limits for location l along the X and Y axes.
The objective function 5.10 minimizes the sum of objective function 5.4 and the overall expected actualized interaction and move costs. These costs result from the summation over all futures, weighted by their prob-ability of occurrence, of their future-specific costs. When laid out in the same location (site, building, etc.), pairs of centers having significant interactions (flows, relationships) incur a cost when their involved I/O stations are positioned a positive distance from each other. For example, if there is flow from the output station of center A to the input station of center B, then a unitary cost is specified for this pair. Then the interaction cost associated with the pair is the product of their unitary interaction cost and their rectilinear
distance. The move cost for a center is computed over all periods of a future, multiplying the rectilinear displacement of its centroid from a period to the next by the unitary move cost specified for this center.
The constraint set includes previously defined constraints 5.5 to 5.9. The new constraints 5.11 to 5.32 are associated with the actual layout of centers assigned to the same location, where they have to share space without interfering with each other while satisfying their shape requirements. Each center and location is restricted to have a rectangular shape and to be orthogonally laid out relative to each other. Each is defined through the positioning of its lower X and Y axis corner and its upper X and Y axis corner.
Constraints 5.11 and 5.12, respectively, enforce that each center and location respect its specified area requirements. These quadratic constraints can be linearized using a set of linear approximation variables and constraints (e.g., Sherali et al. 2003). Constraints 5.13 impose a maximal form ratio between the lon-gest and smallest sides of each center and location.
Constraints 5.14 and 5.15 ensure that whenever a center is assigned to a location, then it is to be laid out within the rectangular area of the location. Constraints 5.16 and 5.17 guarantee that each location is itself located within its maximal allowed coordinates. For example, a building cannot be extended beyond its site boundaries. Similarly, constraints 5.18 and 5.19 impose that each I/O station of a center be positioned within the center’s rectangular area.
Constraints 5.20 to 5.22 ensure no physical overlap between centers assigned to the same location in a specific period of a future. They do so by imposing that for any two such centers, the former is either lower or upper along the X axis, or lower or upper along the Y axis.
Similar to constraint 5.6, constraints 5.23 to 5.26 recognize that the first layout decisions are imposed to all futures, to be immediately implemented while all other layout decisions can be subsequently altered depending on future information.
Constraints 5.27 to 5.29 compute the rectilinear distance between any two I/O stations of centers having positive interactions. The first two constraints linearize the computation of the rectilinear distance by adding its positive and negative components along the X and Y axes respectively, while the latter adds up all these components to get the overall rectilinear distance. Constraints 5.30 to 5.32 similarly compute the rectilinear displacement of the centroid of each center from its previous position to its current position. Constraints 5.27 to 5.32 assume positive unitary interaction and move costs. When negative unitary costs are involved, such as when one wants two centers to be far from each other, then the con-straints have to be altered using binary variables to adequately compute the rectilinear distances and displacements.
When all centers are a priori assigned to the same location and the layout is to be fixed over the entire planning horizon, then the model simplifies to the static continuous block layout model introduced by Montreuil (1991).