Module #3a: Basic Location Models Location analysis and SC system design Single-echelon single-commodity location models – Fixed Charge Model – P-median Model – (Demand Allocation – Transportation Problem)

Location models in the public sector – Coverage Model – P-center Model

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Location analysis and SC system The main issue of the logistics system design: the number, location, (equipment/type) and size of new facilities. Cost vs. Service Level Related Issues: – – – –

(p. 73-74)

Location and allocation decisions are intertwined. Location decisions may affect demand. When are location decisions needed? Location analysis may be strategic or tactical.

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Example of SC system design Suppliers

(p. 75)

Plants

DCs

3

Classification of location analysis Time Horizon – Single-period problem

Facility typology – Type (e.g., RDC vs. CDC)

Material flows – Single-commodity

Interaction among facilities – e.g. location problems with interaction

Demand divisibility – Unit sourcing

Influence of transportation on location decisions – e.g. location routing problem

Retail location – e.g. competitive location models

Dominant material flows – Inbound, Outbound, and echelon (p. 74-76)

4

Four common location models Classification from Network and Discrete Location (Daskin, M., 1995) – – – –

Fixed Charge Model P-median Problem Coverage Problem P-center Problem

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Single-Echelon Location Model Homogeneous facilities Either Inbound or outbound neglected Homogeneous material flows (Single commodity) Linear transportation Cost Fixed (linear or concave) facility cost

(p. 77)

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Illustration of SESC Problem Facilities, V1 (index i)

Customers, V2 (index j)

operation (level), ui facility cost, Fi (ui) Shipment (volume), sij transportation cost, Cij(sij)

capacity, qi

(p. 78)

demand, dj

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General Model of SESC Problem Minimize

∑ ∑C i∈V1

subject to

j∈V2

∑s

ij

j∈v2

∑s i∈V1

(p. 78)

ij

ij

( sij ) + ∑ Fi (ui )

(3.1)

i∈V1

= ui , i ∈ V1 ,

(3.2)

= d j,

(3.3)

j ∈ V2 ,

ui ≤ qi , i ∈ V1 ,

(3.4)

sij ≥ 0, i ∈ V1 , j ∈ V2 ,

(3.5)

ui ≥ 0, i ∈ V1

(3.6) 8

Fixed Charge Model • Fixed facility cost Fi (ui ) = f i New decision variable, decision to set up facility i yi • Linear transportation cost Minimize

∑ ∑c x + ∑ f y i∈V1 j∈V2

subject to

∑x i∈V1

∑d j∈v2

(p. 79-80)

ij

ij ij

=1

i∈V1

i

∀ j ∈ V2

x ≤ qi yi

j ij

i

∀i ∈ V1

(3.9) (3.10) (3.11)

xij ≥ 0 ∀i ∈ V1 , j ∈ V2

(3.13)

yi ∈ {0, 1} ∀i ∈ V

(3.14)

Better to use this? 0 ≤ xij ≤ 1 ∀i ∈ V1 , j ∈ V2 (3.13) 9

Goutte Example - Plants and Markets (DCs)

Mascouche Terrebonne

Sainte-Julie

Montreal Verdun LaSalle

Brossard

Granby Sherbrooke

Valleyfield

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Goutte Example - Cost and Capacity dij Brossard Granby LaSalle Mascouche Montreal Sainte-Julie Sherbrooke Terrebonne Valleyfield Verdun

Brossard Granby Sainte-Julie Sherbrooke Valleyfield Verdun Per-km cost 0.0 76.1 30.4 139.4 72.6 11.7 0.92 76.1 0.0 71.0 77.2 144.5 83.7 Truck Load 20.8 92.9 47.2 156.1 47.5 11.7 150 54.7 113.3 52.9 187.2 93.0 45.2 Round-trip 13.5 85.5 28.0 148.7 67.3 9.3 2 30.4 71.0 0.0 138.2 94.5 38.1 139.4 77.2 138.2 0.0 207.9 146.9 47.8 106.5 46.2 180.2 86.7 38.9 72.6 144.5 94.5 207.9 0.0 63.4 11.7 83.7 38.1 146.9 63.4 0.0

Cij-bar Brossard Granby LaSalle Mascouche Montreal Sainte-Julie Sherbrooke Terrebonne Valleyfield Verdun Demand

Brossard Granby Sainte-Julie Sherbrooke Valleyfield Verdun 0.000 0.933 0.373 1.710 0.891 0.144 0.933 0.000 0.871 0.947 1.773 1.027 0.255 1.140 0.579 1.915 0.583 0.144 0.671 1.390 0.649 2.296 1.141 0.554 0.166 1.049 0.343 1.824 0.826 0.114 0.373 0.871 0.000 1.695 1.159 0.467 1.710 0.947 1.695 0.000 2.550 1.802 0.586 1.306 0.567 2.210 1.064 0.477 0.891 1.773 1.159 2.550 0.000 0.778 0.144 1.027 0.467 1.802 0.778 0.000 14000 10000 8000 12000 10000 9000

Fixed Cost Capacity 81400 22000 83800 24000 88600 28000 91000 30000 79000 20000 86200 26000 88600 28000 91000 30000 79000 20000 80200 21000

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Goutte Example - Solution xij

Brossard Granby LaSalle Mascouche Montreal Sainte-Julie Sherbrooke Terrebonne Valleyfield Verdun Demand met?

sij

Brossard Granby LaSalle Mascouche Montreal Sainte-Julie Sherbrooke Terrebonne Valleyfield Verdun Total Cost =

Brossard

Granby 1 0 0 0 0 0 0 0 0 0 1 1

0 1 0 0 0 0 0 0 0 0 1 1

Sainte-Julie Sherbrooke Valleyfield Verdun 0.75 0 0 0.22222222 0.25 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.77777778 0 0 0 0 1 1 1 1 1 1 1 1

yi 1 1 0 0 0 0 0 0 1 0

Outbound Brossard Granby Sainte-Julie Sherbrooke Valleyfield Verdun 14000 0 6000 0 0 2000 22000 0 10000 2000 12000 0 0 24000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10000 7000 17000 0 0 0 0 0 0 0 265274 Variable Cost =

21074 Fixed Charge Cost =

What happens if there is no capacity constraint? What happens if there is a service limitation? 22000.0 24000.0 0.0 0.0 0.0 0.0 0.0 0.0 20000.0 0.0

244200

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Solution w/ service constraint xij

Brossard Granby LaSalle Mascouche Montreal Sainte-Julie Sherbrooke Terrebonne Valleyfield Verdun Demand met?

sij

Brossard Granby LaSalle Mascouche Montreal Sainte-Julie Sherbrooke Terrebonne Valleyfield Verdun Total Cost =

Brossard

Granby 1 0 0 0 0 0 0 0 0 0 1 1

0 1 0 0 0 0 0 0 0 0 1 1

Sainte-Julie Sherbrooke Valleyfield Verdun 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1

yi 0 0 0 0 0 0 0 0 1 0 1 1

1 1 0 0 0 0 1 0 1 0

Outbound Brossard Granby Sainte-Julie Sherbrooke Valleyfield Verdun 14000 0 8000 0 0 0 22000 0 10000 0 0 0 0 10000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12000 0 0 12000 0 0 0 0 0 0 0 0 0 0 0 10000 9000 19000 0 0 0 0 0 0 0 342783 Variable Cost =

9983 Fixed Charge Cost =

If the distance is larger than 70km, it is unacceptable.

22000.0 24000.0 0.0 0.0 0.0 0.0 28000.0 0.0 20000.0 0.0

332800

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Unit (Single) Sourcing Considertion Non-divisible demand Minimize ∑ ∑ cij xij + ∑ fi yi i∈V1 j∈V 2

subject to

∑x

i∈V1

ij

i∈V1

=1

∑d x j∈v 2

(3.9)

i ij

≤ qi yi

∀ j ∈ V2 ∀i ∈ V1

(3.10) (3.11)

xij ≥ 0 ∀i ∈ V1 , j ∈ V2

(3.13)

yi ∈ {0, 1}

(3.14)

xij ∈{0, 1}

∀i ∈ V1 , j ∈ V2 14

P-median model Minimizing transportation / variable cost only Why? (no facility cost)

Minimize

∑ ∑c x

ij ij

i∈V1 j∈V2

subject to

∑x

=1

∑y

=p

ij

i∈V1

i∈V1

i

∀i ∈ V1

xij ≤ yi

∀i ∈ V1 , j ∈ V2

xij ≥ 0

∀i ∈ V1 , j ∈ V2

yi ∈ {0, 1} (p. 80, slightly different definition)

∀ j ∈ V2

(usually, no facility capacity limit)

∀i ∈ V1

What is the solution if p=1? 15

Demand Allocation Model (A Typical Transportation Problem) • Linear transportation cost

• No facility cost

Fi (ui ) = 0

Unit transportation cost cij from i to j New decision variable, fraction of demand j served by i xij Cij ( sij ) = cij sij , sij = d j xij , cij = d j cij Minimize

∑ ∑c x i∈V1 j∈V2

subject to

∑x i∈V1

∑d j∈v2

ij

(3.16)

ij ij

=1 x ≤ qi

j ij

∀ j ∈ V2

(3.17)

∀i ∈ V1

(3.18)

xij ≥ 0 ∀i ∈ V1 , j ∈ V2 (3.19) (p. 83)

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Fixed and Variable Facility Cost

slope: variable component

intercept: fixed component

(p. 92)

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Fixed and Variable Facility Cost • A more general facility cost function

• Linear transportation cost Minimize

∑ ∑c x + ∑ f y ij ij

i∈V1 j∈V2

subject to

∑x i∈V1

ij

=1

∑d x j∈v2

i∈V1

i ij

≤ qi yi

i

i

∀ j ∈ V2 ∀i ∈ V1

(3.9) (3.10) (3.11)

xij ≥ 0 ∀i ∈ V1 , j ∈ V2

(3.13)

yi ∈ {0, 1} ∀i ∈ V

(3.14)

Minimize

∑ ∑t i∈V1 j∈V2

 f i if ui > 0 Fi (ui ) =   0 if ui = 0

 f i + g i ui Fi (ui ) =  0 ui =

x + ∑ f i yi

ij ij

j∈V2

j

xij

(3.29)

i∈V1

tij = cij + g i d j (p. 91)

∑d

(3.30) 18

if ui > 0 if ui = 0

Min. and Max. facility operation

(p. 93)

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Concave facility cost

How to link the level and cost?

(p. 92)

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Set covering model • Fixed facility cost Fi (ui ) = f i

decision variable, decision to set up facility i yi coverage, 0-1 parameters, facility i covering customer j aij

Minimize

∑fy i∈V1

subject to

i

∑a i∈V1

ij

i

yi ≥ 1

yi ∈ {0, 1}

∀ j ∈ V2 ∀i ∈ V1

What happens if fixed cost is site-independent?

(p. 114)

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Coverage vs. Number of facilities

In case, there is a budget constraint limiting the number of stations. (Daskin, Figure 1.1)

22

Variations of set-covering problems

Alternatively, set a penalty as in GLM, p.112.

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Maximum coverage model decision variable to indicate the coverage of customer j rj number of facilities limited by budget p Maximize

∑d r

j j

j∈V2

subject to

rj ≤ ∑ aij yi

∀ j ∈ V2

∑y

=p

∀i ∈ V1

yi ∈ {0, 1}

∀i ∈ V1

i∈V1

i∈V1

i

rj ≤ 1

∀ j ∈ V2

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Coverage model with penalty decision variable to indicate that customer j is NOT covered z j Minimize

∑f y +∑p z i∈V1

subject to

i

∑a i∈V1

ij

i

j∈V2

j

yi + z j ≥ 1

yi ∈ {0, 1} zj ≤1

j

∀ j ∈V2 ∀i ∈ V1

∀ j ∈ V2

What happens if pj is fairly large? (p.112)

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25

One variation of the coverage model a big constant

travel time form i to j

What is the model about? (p.114)

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Fire station example in Portugal

(p.114)

27

Fire station example in Portugal

(p.114)

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Results of the fire station example aij Almada Azenha Carregosa Corroios Lavradio Macau Moita Montijo Palmela Pinhal Novo

xij Almada Azenha Carregosa Corroios Lavradio Macau Moita Montijo Palmela Pinhal Novo

Covered?

Azenha 1 0 0 1 0 0 0 0 0 0

Corroios

0 1 1 0 1 0 1 0 0 0

Carregosa 0 1 1 0 1 1 1 1 0 1

Corroios

0 0 0 0 0 0 1 0 0 0 1 1

Carregosa 0 0 0 0 0 0 1 0 0 0 1 1

Azenha 1 0 0 0 0 0 0 0 0 0 1 1

Lavradio 1 0 0 1 0 0 0 0 0 0

Macau 0 1 1 0 1 0 1 0 0 1

Lavradio 1 0 0 0 0 0 0 0 0 0 1 1

Moita 0 0 1 0 0 1 1 1 1 1

Macau 0 0 0 0 0 0 1 0 0 0 1 1

Montijo 0 1 1 0 1 1 1 1 1 1

Moita 0 0 0 0 0 0 1 0 0 0 1 1

0 0 1 0 0 1 1 1 0 1

Montijo 0 0 0 0 0 0 1 0 0 0 1 1

0 0 0 0 0 1 1 0 1 1

Pinhal Novo M (fixed) 0 1000 0 1000 1 1000 0 1000 1 1000 1 1000 1 1000 1 1000 1 1000 1 1000

0 0 0 0 0 0 1 0 0 0 1 1

Pinhal Novo 0 0 0 0 0 0 1 0 0 0 1 1

Palmela

Palmela 0 0 0 0 0 0 1 0 0 0 1 1

yi 1 0 0 0 0 0 1 0 0 0

The facilities are located in Almada and Moita, and the former only serves Corroios and itself.

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2 4 7 2 5 6 8 5 4 7

-8 0 0 0 0 0 -2 0 0 0

0 0 0 0 0 0 0 0 0 0

P-center model (Minimize Max. Model) decision variable to indicate maximum response time W response time (distance) from i to j tij Minimize

W

subject to

∑x

ij

=1

∑y

i

=p

i∈V1

Does xij have to be binary?

i∈V1

xij ≤ yi

∀ j ∈V2 ∀i ∈V1 ∀i ∈V1 , j ∈V2

W ≥ ∑ tij xij i∈v1

xij ≥ 0 (Daskin, 1995, slightly different from, p. 108)

∀ j ∈V2

∀i ∈V1 , j ∈V2

yi ∈{0, 1}

∀i ∈V1 30

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