This document discusses models for facility location and capacity allocation. It describes network optimization models which aim to design networks that maximize profitability and responsiveness by optimizing facilities, capacity, transportation costs and inventory costs. Gravity location models are also discussed, which identify optimal facility locations by minimizing total transportation costs calculated based on distances between facilities and markets/suppliers. The document provides an example of a capacitated plant location model and outlines steps for allocation of demand to production facilities and locating plants and warehouses simultaneously.

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Models For
Facility Location
And
Capacity Allocation

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Contents:-
1. Need
2. Types of Models
3. Network Optimization Model
4. Gravity Location Model
5. Allocation and location
6. References

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Total
Revenues
Product
Sale
Facilities
Labor
Transportation
Materials
Inventories
Profit + Taxes
Increasing Facilities and capacity
decreases transportation costs but
increases inventory costs.
1. Need:-
Aim is to design a network
which increases Profitability
with apt. Responsiveness.

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Phases in network design:-
Phase I Phase II Phase III Phase
IV
A. Network Optimization Model
B. Gravity Location Models
2. Types of Models:-

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3. Network Optimization Model.

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Steps:-

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Example:- Capacitated Plant location model
Capacities of plants are known.
n = Number of Potential Plant Locations
m = Number of markets or demand points
Dj = annual demand from market j
Ki = potential capacity of plant i
Fi = annualized fixed costs for keeping factory i open
Cij = cost of producing and shipping one unit from factory
i to market j

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Yi = 1 if plant i is open , else 0.
Xij = quantity shipped fron plant i to market j.
So,
Min ∑Fi*Yi + ∑∑ Cij*Xij
Subject to:
∑Xij = Dj For j= 1 to m
∑Xij ≤ Ki * Yi For i = 1 to n
Yi є {0,1} For i = 1 to n, Xij ≥ 0
n mn
i=1 i=1 j=1
m
j=1
n
i=1


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4. Gravity Location Model.

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Basic Model:
F1
S1
S2
S3
M2
M3
M1
(Xn,Yn) = Co ordinate location of either market or supply
source n
Fn = Cost of Shipping One unit for one mile between
facility and either market or supply source n
Dn = Quantity to be shipped between facility and either
market or supply source n

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Let,
(X,Y) = location selected for the facility
dn = distance between the facility at location (X,Y)
and Supply source or market n is given by:-
dn = √(X-Xn) + (Y-Yn)
The total Transportation cost TC:-
TC = ∑ (dn * Dn * Fn)
The optimal location is one that minimizes TC.
2 2
k
n=1

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5. Allocation And Location:-
a. Allocating Demand to production facilities:
Find Xij s
b. Locating Plants
Find Yi=1s
c. Locating Plants and Warehouses Simultaneously
Introducing new decision variables
Introducing new constraints
Introducing certain changes in the objective
function

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6. References:-
a. Supply Chain Management By Sunil Chopra ,
Third Edition.
b. citeseerx.ist.psu.edu
c. www.dms.umontreal.ca

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