Facility location
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xi1 =
1 if facility i is assigned to locationj, and 0 otherwise.
See Verification, validation, and testing of models.
Then the QAP is to
I subject to I I Minimize
FACILITIES LAYOUT Bharat K. Kaku American University, Washington, DC In both manufacturing and service operations, the relative location of facilities is a critical decision affecting costs and efficiency of operations. The facility layout problem (FLP) deals with the design of layouts wherein a given number of discrete entities are to be located in a given space. The definitions of entities and spaces can vary considerably, making solution techniques applicable in a wide variety of settings, as can be seen from the examples given below.
ciixii
+
I
IJipdjqXijXpq
i,p
i,j
xi1 =
1 Vi
Xij =
1
(2)
vj
Xij E {0, 1)
(1)
j,q
(3) \;/ i,j
Alternatively, we can define p(i) as the location to which facility i is assigned, leading to an equivalent but more compact statement of the problem. The QAP is then to find a mapping of the set of facilities into the set of locations so as to Minimize
I
ci,p(i)
+
I
dp(i),p(pJ·
(4)
·i,p
Entities
Space
Objective
Departments Departments Departments Interdependent plants Indicators and controls Components Keys
Office building Factory floor Hospital Geographical market Control panel Electronic boards Typewriter keyboard
Minimize cost of interactions Minimize cost of material handling Minimize movement of patients and medical staff Maximize profit Minimize eye/hand movement Minimize cost of connections Minimize typing time
We first discuss approaches used to model the FLP, followed by optimal algorithms and heuristic approaches to solving these problems, and end with some remarks concerning directions for future research. THE QuADRATic AssiGNMENT FoRMULATION: The FLP is most often treated in the OR/MS literature as the Quadratic Assignment Problem (QAP), which is a special case requiring identical area and shape requirements for the locations of all facilities. This allows pre-definition of the locations and calculation of the distances between them (typically center-to-center, either rectilinear or Euclidean). Suppose there are N facilities to be assigned to N locations. Define four N X N matrices whose elements are, respectively: ci1 =
fixed cost of assigning facility ito locationj
fii =
level of interaction between facilities i andj
dii
= cost of one unit of interaction (e.g., the distance) between locations i andj
The quadratic assignment problem was first formulated by Koopmans and Beckmann (1957) in the context of the location of interdependent plants. The cii elements represent the expected revenue of operating plant i at location j independent of other plant locations, the fii elements represent the required commodity flows from plant i to plant j, and the d;i elements represent the transportation costs per unit between location i and locationj. The objective function maximizes the net revenue, that is, the excess of expected revenue over the transportation costs. It is the interdepende
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