CEE
LSA_8_chp4_one-to-many -inventory-routing.pdf

# The feasible solution to formula 420 vv max n s

• 8

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The feasible solution to formula 4.20: v=v max /n s , yields: 3 4 2 4 2 3 if 1 otherwise u l V V z z α α α α α α 4 3 2 3 * 2 4 2 2 2 u l V V z z z α α α α ε α α α = = 3 2 2 V α ε α = 3 2 pipeline inventory per delivery detour 2 2*vehicle motion cost per delivery detour V α α = = If ε <<1, then full vehicles are near optimal If ε >>1, full vehicle condition does not hold inventory motion Inventory v.s. motion 1 2 0 2 4 * 4 1 2 1 0 4 0 2 4 2 2 therwise l if V V z z V o V V V α α α α α α α α α α α α α α + + + + + + + Therefore, the (percentage) gap is where

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7 LSA: Chapter 4 Claim2: When ~ 1, should not use full vehicles The vehicle capacity constraint is no longer binding. Express eq. 4.20 as an unconstrained minimization ( ) v n v v n n z s s s 4 3 2 1 0 α α α α α + + + + = v n s 3 1 α α Unconstrained minimum ( ) v v v v z 3 1 4 2 0 * 2 α α α α α + + + = • achievable if n s >1 • convex ( ) 2 3 3 1 4 2 2 * + = v v dv v dz α α α α ( ) ( ) ( ) ( ) ( ) 4 4 2 3 1 2 3 1 3 2 4 2 3 1 4 2 3 1 2 3 1 3 2 4 2 2 4 2 2 4 2 3 1 2 2 4 3 1 2 2 ' ' 1 2 1 2 1 2 1 2 1 2 3 v v v v 1 v v since v v 1 v by multiply v v β α α α α α α α α α α α α α α α α α α α α α α α α α α α α α = + = + = = + = + ( ) 2 3 1 3 2 4 α α α α β = ( ) 2 3 1 2 1 ' α α α v v = ( ) ' 1 ' 4 v v + = β 3 2 2 V α ε α = where LSA: Chapter 4 Claim2: When ~ 1, should not use full vehicles ( ) 2 3 1 3 2 4 α α α α β = ( ) ( ) ( ) = 3 1 4 1 2 1 2 3 1 3 2 4 2 3 1 3 2 4 2 3 1 , ' α α α α α α α α α α α v v [ ] 3 1 4 1 , ' β β v = 3 2 3 1 ' α α α α v v v v n : load optimal s ( ) ' ' ' 2 1 2 2 3 1 3 2 v v v n : stops of number optimal s α α α α α α α = = ( ) 3 1 2 2 ' α α α v v = If n s v V & n s 1, then the solution is: 3 2 2 V α ε α = ( ) v n v v n n z s s s 4 3 2 1 0 α α α α α + + + + = Finally, plug this back into the cost function (cost per item for a customer near time-location ( t , x ) )
8 LSA: Chapter 4 Identical customers; loads not given Continuum Approximation Final Remarks •The optimal objective value z* obtained from the above represents the cost per item near ( t , x ); it depends on D’ ( t ) , δ ( x ) , r ( x ); •To estimate total cost, one needs to average the above solution z* ( t , x ) across t , x ; i.e.
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• Fall '16
• Trigraph, #, 1 L, 2 L, 0 min, slow-varying.

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