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Facility Location

ISyE 4803Pinar Keskinocak

Graphics Source: “Geography globe design template.” Microsoft Office Online. Microsoft Corporation. January 20, 2009. <http://office.microsoft.com/en-us/templates/CT101427471033.aspx>

Acknowledgement: Thanks to Amanda Meija for her help in preparing these slides

What is a “Facility”?

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Facilities are used to…

• Hold finished products and ship to customers (distribution center)(distribution center)

• Manufacture products (manufacturing plant)• Sort and ship products (cross-dock)• Display and sell products to customers (retail)

P id h lth i

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• Provide healthcare services – Doctor’s office– Hospital– Vaccination center

Why does a facility’s location matter?

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Why does a facility’s location matter?

• Wal-Mart retail locations within 50 mi of Atlanta

• Wal-Mart distribution f

5

centers within 50 mi of Atlanta (Monroe, GA)

Factors influencing location decision

• Locations of customers

• Locations of suppliers

• Expansion capability

• Local political conditions• Locations of suppliers

• Transportation access

• Real estate costs

• Material costs

• Cost of labor

• Local political conditions

• Climate

• Weather events

• Insurance costs

• Locations of competitors

6Adapted from Johnson, James C., et al. Contemporary Logistics. 7th ed. Upper Saddle River, NJ: Prentice Hall, 1999.

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Simple Example

• Two customers:– Customer A– Customer A

• Demands 100 units/year• Located at mile 10 on a highway

– Customer B • Demands 100 units/year• Located at mile 40 on the same highway

L b k t d t t t if l th

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• Labor market and tax structure uniform along the highway

10 40

A B

• Customer locations and demand:

Why do we need optimization models to locate facilities?

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– A: (2.0, 2.9), 520 units– B: (3.1, 2.5), 800 units– C: (1.8, 2.2), 540 units– D: (2.4, 1.7), 1,550 units– E: (0.5, 1.6), 790 units– F: (1.7, 0.6), 1,260 units– G: (3.3, 1.4), 2,050 units

B

A

C

E D

23

8

• Locate a distribution center to minimize the weighted distance from customers (center-of mass)

F

GE

0 1 2 3 4

01

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Measuring Distances

• “Manhattan distance” (1-norm)– Best for cities with perpendicular streets

2121L1 yyxx −+−=p p– N-S distance + E-W distance

• “As the crow flies” (2-norm)– Best for long distances and highways

• Larger of N-S and E-W distances (∞-norm)

2121L1 yyxx +

( ) ( )2 221

221L2 yyxx −+−=

9

Larger of N S and E W distances ( norm)– Useful for automatic warehouses

• Arc Distance on a Sphere– Used in air travel– Complex calculation

( ) ( )( )2121

2121

,max yyxx

yyxxL

−−=

−+−=∞ ∞ ∞∞

Center-of-mass

• Center-of-mass location = The weighted average location of a set of populationsp p

• Weights can be:– Population size– Demand– Importance– Severity of need

Population1

Pop2

Population3( )∑

10

3Population

5

Pop4

( )( )

∑∑

=

ii

iiii

w

yxwyx

,,

wi = weight of each population ixi = x-coordinate of each population iyi = y-coordinate of each population i

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Example

• Center of Mass:dxdxdxdxdxdxdx ++++++

GFEDCBA

GGFFEEDDCCBBAA

ddddddddxdxdxdxdxdxdxx

++++++++++++

=

( ) ( ) ( ) ( ) ( ) ( ) ( )205012607901550540800520

2050*3.31260*7.1790*5.01550*4.2540*8.1800*1.3520*0.2++++++

++++++=

3.2=

( ) ( ) ( ) ( ) ( ) ( ) ( )

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( ) ( ) ( ) ( ) ( ) ( ) ( )205012607901550540800520

2050*4.11260*6.0790*6.11550*7.1540*2.2800*5.2520*9.2++++++

++++++=y

6.1=

Example (cont.)

• Least-distance facility located at (2 3 1 6)

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located at (2.3, 1.6)

• What could be wrong with this location?– Cost of land

Availability of land

B

A

C

E D

23

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– Availability of land– Traffic congestion

to/from facility– No information about

tax structure– Zoning

F

GE

0 1 2 3 4

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Example (cont.)

• Candidate locations, costs:– 1: (2.3, 1.6), $30M + $50/unit

4

3( )

– 2: (1.3, 2.5), $15M + $40/unit

– 3: (1.9, 3.7), $12M + $30/unit

– 4: (3.7, 3.2), $15M + $35/unit

– 5: (0.8, 0.3), $10M + $40/unitB

A

C

E D

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2

1

4

13

• Assumptions:– Travel costs are $1/mile-unit

– Demand fulfilled weekly

– Each unit square is 8x8 milesF

GE

0 1 2 3 4

01

1

5

Developing the optimization model

• Decisions to make:Whether to open each candidate location 51}10{ =∈ jw– Whether to open each candidate location

– How much to ship to each customer from each opened candidate location

5,...,1,}1,0{ =∈ jw j

locations candidate,customers, == jizij

• Optimization model:– Minimize cost

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– Subject to constraints:• All demand must be fulfilled• Limit number of facilities to be opened• Only ship from open facilities • Decision variable wj is binary and zij ≥ 0

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Calculating costs

• Opening cost = ( Cost of opening facility ) * wj

( )5121513$1 000 000

• Operating cost = ( Cost per unit ) * (Units produced )

• Transport cost =

( )54321 5.12.15.13$1,000,000 wwwww ++++=

∑∑∑∑∑=====

++++=7

15

7

14

7

13

7

12

7

11 4035304050$

ii

ii

ii

ii

ii zzzzz

⎟⎟⎞

⎜⎜⎛

×⎟⎟⎞

⎜⎜⎛

×⎟⎟⎞

⎜⎜⎛

shippedUnitstraveledMilesunitpermileperCost

15

Transport cost

( ) ( )22

7

155

7

144

7

133

7

122

7

111

10$

jijiij

iii

iii

iii

iii

iii

yyxx

zzzzz

−+−=Δ

⎟⎠

⎞⎜⎝

⎛Δ+Δ+Δ+Δ+Δ= ∑∑∑∑∑

=====

⎟⎟⎠

⎜⎜⎝ ∑

×⎟⎟⎠

⎜⎜⎝ ∑

×⎟⎠

⎜⎝ Δ

443442144 344 21444 3444 21i iji ij z

shippedUnitstraveledMilesunitper mileper Cost $10

Creating the optimization model

Min ( )

+++++

++++

∑∑∑∑∑77777

54321

4035304050$

5.12.15.13$1,000,000

zzzzz

wwwww

Subject to

⎟⎠

⎞⎜⎝

⎛Δ+Δ+Δ+Δ+Δ+

+++++

∑∑∑∑∑

∑∑∑∑∑

=====

=====

7

155

7

144

7

133

7

122

7

111

15

14

13

12

11

10$

4035304050$

iii

iii

iii

iii

iii

ii

ii

ii

ii

ii

zzzzz

zzzzz

{ }7,...,1

5

5

1∈∀≥∑

=jiij iDz Weekly demand fulfilled

Only one facility opened

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{ }{ }1,0

5,...,1

15

1

∈

∈∀≤

≤∑=

j

jij

jj

w

jMwz

w Only one facility opened

Only open facilities serve customers

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Why do we need optimization models to locate facilities?

• Solve the model and compare optimization with distance-only modeldistance only model

• Compare 1 facility with 2 or more facilities

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Facility Location Integer Programming (IP) Sample Formulation (1)

• Objective Function – Can be minimize cost minimize delivery time maximize– Can be minimize cost, minimize delivery time, maximize

“impact”, etc.• Decision Variables:

– Binary variables indicating whether each facility i should be opened

– Amount of each item to be held at each facility

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– Amount of each item to be received from each supplier to each facility

– Amount of each item to be sent from each facility to each customer

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Facility Location IP Sample Formulation (2)

• Inputs:– Set of possible locations to open facilities

– Set of suppliers

– Set of customers

– Cost or time along each transport route

S t f d t t b li d t t

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– Set of products to be supplied to customers

– Demand of each customer for each product

– Max number of facilities to open

– Max inventory space available at each facility

Facility Location IP Sample Formulation (3)

• Minimize or maximize

• Subject to: – Customer demand is satisfied

– Flow out of each facility = Flow into each facility

– Max number of facilities is not exceeded

– Max inventory space is not exceeded

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Max inventory space is not exceeded

– Inventory only held at open facilities

– Decision variables are non-negative

– Binary decision variable equal to 0 or 1

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Many Different Objective Functions

• Maximize profit (common in for-profit problems)• Minimize cost under a minimum acceptable service level

(number of facilities a certain percentage of demand met)(number of facilities, a certain percentage of demand met)• Maximize service level under some constraint (i.e. budget)• Minimize average response time• Common healthcare objectives:

– Minimize cost (common in for-profit healthcare)– Maximize benefit to health (can be minimize negative health outcomes

and maximize positive health outcomes)Maximize equitability or fairness

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– Maximize equitability or fairness• Multiple objectives (i.e. min unmet demand and cost)• Long-term strategic factors may also be included in for-profit

and non-profit problems

Facility Location Example Problems

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Location of Preventive Health Care Facilities (1)

• What is preventive health?– Reduces likelihood of life-threatening diseasesg– Aims at diagnosis of serious medical conditions– Includes immunizations, blood tests, and cancer screenings– Recovery is often more likely if an illness is diagnosed at an early

stage– Can improve overall efficiency and effectiveness of a health care

systemAn increasingly integral part of many health care reform programs

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– An increasingly integral part of many health care reform programs within the past decade

• Participation in preventive services– Accessibility of facilities is very important, since healthy people are

willing to travel much less than sick people for medical care– Unless services offered at convenient locations, participation not likely

Based on V. Verter and S.D. LaPierre. Location of preventive health care facilities. Annals of Operations Research, 110(1-4):123-132, 2002.

Location of Preventive Health Care Facilities (2)

• The Problem– Given a set of populations centers and potential facility sitesG e a set o popu at o s ce te s a d pote t a ac ty s tes– Want to identify the optimal number and location of facilities that

maximize participation– Uses Maximal Covering Location Problem to maximize total coverage– Total coverage defined as the number of people within a threshold

distance from a facility

• Assumptions

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Assumptions– Each participating individual seeks services of the closest preventive

health care facility– Probability of participation decreases with distance– Each open facility needs to have a minimum number of clients (FDA

requirements)Based on V. Verter and S.D. LaPierre. Location of preventive health care facilities. Annals of Operations Research, 110(1-4):123-132, 2002.

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Location of Preventive Health Care Facilities (3)

• I = the set of population centers (indexed by i)• J = the set of potential facility sites (indexed by j)• h = the number of residents in population center i• hi = the number of residents in population center i• dij = the distance between population center i and facility site j• P = probability of participation when travel distance is negligible• D = maximum distance an individual would travel for preventive care• Wmin= minimum required workload at a facility

• cij = expected number of i residents who would seek services of facility j

• Sij = the set of potential facility sites that are closer to population center ithan site j, indexed by l

25Based on V. Verter and S.D. LaPierre. Location of preventive health care facilities. Annals of Operations Research, 110(1-4):123-132, 2002.

Decision Variables:yj = 1 if a facility is opened at site j, 0 otherwisexij = 1 if a population center i is served by a facility at site j, 0 otherwise

( )⎪⎩

⎪⎨⎧

>

≤−=

Dd

DdhDPdPc

ij

ijiijij ,0

,/ij p y j

Location of Preventive Health Care Facilities (4)

Maximize ∑∑i j

ijij xc

∑

Expected number of participants

At most one facility can serve asubject to Iixj

ij ∈≤∑ 1

{ } JjIiJjIiSlyx

JjIiyx

JjyWxc

ijlij

jij

ji

ijij

∈∈∈−≤∈∈≤

∈≥∑

10,,1

,min

At most one facility can serve a population center

Minimum workload requirement

Only open facilities operate

Pops served by closest facility

26Based on V. Verter and S.D. LaPierre. Location of preventive health care facilities. Annals of Operations Research, 110(1-4):123-132, 2002.

{ }{ } Jjy

JjIix

j

ij

∈∈∈∈∈

1,0,1,0 y

Integrality constraints