1

Facility Location

Facility location is the determination of the geographical site(s) in which to locate a firm’s operations.

• Globalisation• Factors to consider• Quantitative tools for analysis

– locating a single facility– locating within a network of

facilities

• Location decisions must be co-ordinated with production planning and distribution strategies

2

Location Theory - Early History Weber’s classification of

industries (1909)• Weight-losing process

– locate close to raw materials– e.g. steel making

• Weight-gaining process– incorporate “ubiquitous” raw

materials e.g. air, water– locate near markets

Hoover’s (1957) tapered transportation rates

• tapered transportation rates• minimum costs at either

production point or market point

3

Factors - Location Related• Land/Construction costs

• Community receptivity

• Local business climate

• Quality of life

• Government incentives– tax breaks– free trade zone

• Government barriers– currency controls– trading blocs– local content– environmental regulations

• Political Risks

4

Factors - Resource/Cost Related• Proximity to suppliers

• Quality/Availability of labour

• Transportation/Energy infrastructure

• Proximity to customers

• Inbound/Outbound distribution costs

• Other (company-owned) facilities

Competitive Advantage

5

Quantitative Tools

• Center of Gravity Method

• Mixed Integer Programming

• Simulation

• Heuristics

• Other methods

• Single facility location

• Multi-facility location

• Supply chain network design

• Dynamic location models

6

Single Facility LocationGiven a set of demand points,

each located at (xi,yi) with a specified volume Vi to be moved to a facility (at transportation rate Ri ),

locate a single facility to minimise total transportation costs.

Find (X,Y) to

Minimise Vi Ri di

where

di = [(Xi - X)2 + (Yi - Y)2]1/2

7

Centre of Gravity Method

• Grid method

• centroid method

Locate facility at:

i i

ii

i i

iii

i i

ii

i i

iii

d

RVd

YRV

Y

d

RVd

XRV

X

8

Concerns/Assumptions of Centre-of-Gravity Model • continuous

• demand concentrated at a point

• transportation costs proportional to Euclidean distance

• fixed cost of establishing facility ignored

• static

• simple

• useful “first-cut” solution

9

Multi-Facility Location

• How many sites?

• Where to locate each?

• Capacities?

• Which customers assigned to each site?

• Which products to stock/produce at each site?

10

Multiple Centre-of-Gravity Approach

• Pre-assign demand points to each facility (i.e. cluster customers that are closest together).

• For each cluster, locate one facility at centre of gravity.

• With facility locations fixed, re-assign customers to closest facility.

• Find centres of gravity for new clusters.

• Repeat cluster-assign until no further change.

11

Linear and Mixed Integer Programming

• LP useful in calculating distribution costs

• Mixed Integer Programming can `optimize’ site selection and distribution plan simultaneously

• Detailed cost estimates needed

12

p-median Problem

Locate p facilities so as to minimise the sum of fixed cost for establishing facilities and transportation costs from demand points to assigned facility.

Ballou (Logware):• demand point co-ordinates

given• assume out-and-back along

Euclidean distance

p-median problem --MIP formulation

Let if location chosen

otherwise

if customer assigned to location

otherwise

s.t.

yj

xi j

c x f y

x i

x y i j

y j

j

ij

ij ij j jjji

ijj

ij j

j

1

0

1

0

1

0

0 1

min

,

{ , }

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Heuristic Methods = Rules of Thumb

Kuehn & Hamburger (1963) ADD• No facilities open initially• For each facility not currently used:

evaluate the savings in total cost if opened (reduced transportation costs less fixed cost)

• Add facility that gives maximum (positive) savings

DROP• All (Selected set of) facilities open

initially• For each facility currently used:

evaluate the savings in total cost if closed (fixed cost less increased transportation costs)

• Drop facility that gives maximum savings

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Multi-Facility Multi-Product Location-Allocation Problem

Find the number and location of the facilities to minimise the total (fixed and variable) costs of moving all products through the logistics network, subject to:

• available supply at each plant cannot be exceeded for each product

• demand for all products met• throughput of each facility cannot

exceed its capacity• minimum throughput of a facility

must be achieved before it can be opened

• all products from same customers must be met from one facility.

MIP formulation

Plant P1

Production = $4/cwt.Capacity = 60,000 cwt.

Plant P2

Production = $4/cwt.Capacity = unrestricted

Product 1

Customer C1

50,000 cwt.

Customer C2

100,000 cwt.

Customer C3

50,000 cwt.

Handling = $2/cwt.Warehouse W1

Handling = $1/cwt.Warehouse W2

$0/cwt.$5/cwt.

$4/c

wt.

$2/cwt.

$3/cwt.

$2/cwt.

$4/cwt.

$1/cwt.

$2/c

wt.

$5/cwt.

Plant P1

Production = $3/cwt.Capacity = 50,000 cwt.

Plant P2

Production = $2/cwt.Capacity = unrestricted

Product 2

Customer C1

20,000 cwt.

Customer C2

30,000 cwt.

Customer C3

60,000 cwt.

Handling = $2/cwt.Warehouse W1

Handling = $1/cwt.Warehouse W2

$0/cwt.$5/cwt.

$4/c

wt.

$2/cwt.

$2/cwt.

$3/cwt.

$3/cwt.

$2/cwt.

$3/c

wt.

$4/cwt.

Fixed = $100,000Capacity = 110,000 cwt.

Fixed = $500,000Capacity = unrestricted

FIGURE 13.5 A small Multiproduct Warehouse LocationProblem for Mixed Integer Linear Programming

Technical Supplement

This is the model formulation to the problem shown

in Figure 13-5.

Minimize

subject to the following:

Available production capacity cannot be exceeded:

for all

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where

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Relevant Costs for Relevant Costs for Location DecisionLocation Decision

• production/purchase costs

• warehouse storage and handling

• warehouse fixed costs

• cost for carrying inventory

• stock order and customer ordering costs

• warehouse inbound and outbound transportation costs

Tradeoffs? Tradeoffs? (Figure 13-8, Ballou)(Figure 13-8, Ballou)

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Selective Evaluation

• Modified multiple centre-of-gravity to include inventory and fixed costs– form clusters of `markets’– find centres of gravity– re-assign `markets’– evaluate total costs (including

transportation, inventory and fixed costs)

• Can be used to determine the number of warehouses that best tradeoffs transportation, inventory and fixed costs

(See Ballou, p. 506-507)

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Guided Linear Programming

• Relevant costs include both fixed and variable costs

• Accurate model requires a mixed-integer program

• Computationally intensive

• Approximation: distribute the fixed cost over the throughput (unknown until problem solved)

• Problem then becomes a linear programming which is much easier to solve

• Allocate fixed costs according to approximate throughput, solve LP, re-adjust fixed cost allocation, re-solve LP, etc.

(See Ballou, p. 508-510 )

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Simulation Methods

Optimisation models are often “approximation” of “real-world” problems

• accurate problem description• incorporate time-related aspects• integrate inventory and

geographical concerns

• only evaluative• candidate solutions must be

provided• no optimality guarantee

Sub-optimal solution to accurately described problem

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Appraisal of Multi-Location Methods

• Mathematical Programming based methods gaining popularity

• inexpensive and robust decision support tool

Extensions:• non-linear cost structure • discontinuous cost structure• integrated inventory and

transportation issues• revenue effects

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Other methods

• Regression analysis

• Factor rating system

• Analytic Hierarchy Process (AHP)

• Covering models

• Game theory

• Location-allocation models– goal programming– mixed integer programming

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Logistics Network (Supply Chain) Planning

(Multi) product flow from source to demand points

• number, size and location of production facilities

• number, size and location of distribution centres

• assignment of products and customers to DC’s

• assignment of DC’s to production sites

• choice of transportation modes

• inventory policies: – frequency of replenishment– order size

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Complexities

• Spatial and temporal aspects

• Data collection and aggregation

• Costs allocation and approximation– fixed– storage (related to inventory

levels)

– handling (related to throughput)

• transportation cost non-linear

• inventory-throughput relationship non-linear

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Integrated Decisions

• Location

• Transportation (Allocation)

• Inventory

Iterative approach• Solve approximations of

each problem in sequence

• Update approximations and iterate