Multiple Single-Facility Location 9 Distribution 3 6 - - PowerPoint PPT Presentation

Multiple Single-Facility Location

Suppliers Manufacturing Customers

10 9 11 12 3 4 5 1 2 6 7 8

Distribution

EFs EFs NFs

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Best Retail Warehouse Locations

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Optimal Number of NFs

1 5 2 3 4 6

Transport Cost

TC Number of NFs

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Fixed Cost and Economies of Scale

• How to estimate facility fixed cost?

– Cost data from existing facilities can be used to fit linear estimate

• y-intercept is fixed cost, k

– Economies of scale in production

 k > 0 and β < 1

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max

act min est act 1 act est

max , 0.62, Hand tool mfg. 0.48, Construction 0.41, Chemical processing 0.23, Medical centers fixed cost

f f p p p

f TPC TPC TPC f TPC c f TPC TPC APC f f f k APC c f k k c

β β β

β

< −

      =              =     = + = = = + =

min max MES

constant unit production cost / min/max feasible scale / base cost/rate f f f Minimum Efficient Scale TPC f = = = =

fmin fMES f0 fmax

Production Rate (ton/yr)

TPCmin

k

TPC0

cp

TPCact ( = 0.5) TPCest Actual EF cost APCact APCest

MILP

{ }

LP: max ' s.t. MILP: some integer ILP: integer BLP: 0,1

i

x ≤ ≥ ∈ c x Ax b x x x

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1 4 2 6 3 5 1 2 3 4

1

x

2

x

1 2 1 2 1 1 2

max 6 8 s.t. 2 3 11 2 7 , x x x x x x x + + ≤ ≤ ≥

[ ]

6 8 2 3 11 , 2 7 =     = =         c A b

* *

1 3 2 2 , 31 1 3 13     ′ = =       x c x

Branch and Bound

1 4 2 6 3 5 1 2 3 4

2 313 1 2 1 2 1 1 2 1 2

max 6 8 s.t. 2 3 11 2 7 , , integer x x x x x x x x x + + ≤ ≤ ≥

1

x

[ ]

6 8 2 3 11 , 2 7 =     = =         c A b

2

x

1 313 26 31 2 303 28 30 2 31 , 3 UB LB = =

1

3 x ≤

1

4 x ≥

2

1 x ≤

2

2 x ≥

1

2 x ≤

1

3 x ≥

2

2 x ≤

2

3 x ≥

1 31 , 3 UB LB = = 1 31 , 26 3 UB LB = =

31, 26 UB LB = =

2 30 , 26 3 UB LB = = 2 30 , 30 3 2 30 30 1 3 UB LB gap = = = − <  2 30 , 28 3 UB LB = =

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MILP Formulation of UFL

{ }

min s.t. 1, , , 1, , 0,1 ,

i i ij ij i N i N j M ij i N i ij ij i

k y c x x j M y x i N j M x i N j M y i N

∈ ∈ ∈ ∈

+ = ∈ ≥ ∈ ∈ ≤ ≤ ∈ ∈ ∈ ∈

   

{ } { }

where fixed cost of NF at site 1,..., variable cost from to serve EF 1,..., 1, if NF established at site 0,

• therwise

fraction of EF demand served from NF at site .

i ij i ij

k i N n c i j M m i y x j i = ∈ = = ∈ =  =   =

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MILP Formulation of p-Median

{ }

min s.t. 1, , , 1, , 0,1 ,

ij ij i N j M i i N ij i N i ij ij i

c x y p x j M y x i N j M x i N j M y i N

∈ ∈ ∈ ∈

= = ∈ ≥ ∈ ∈ ≤ ≤ ∈ ∈ ∈ ∈

   

{ }

where number of NF to establish variable cost from to serve EF 1,..., 1, if NF established at site 0,

• therwise

fraction of EF demand served from NF at site .

ij i ij

p c i j M m i y x j i = = ∈ =  =   =

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