LSA_8_chp4_one-to-many -inventory-routing.pdf

# For l dispatches l kne d v t nd r e d n t d v v t d l

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for L dispatches: ( ) ( ) ( ) ( ) [ ] ( ) [ ] L kNE d v t ND r E d N t D V v t D L N n n v N t D tours detour linehaul s 2 1 max max max max max max max 2 1 # = = = + = + = δ Constant Constant LSA: Chapter 4 Identical customers Holding costs i. Very cheap items: c i << c r (Rent cost dominates: minimize maximum accumulation) ( ) L t D v max = Equal lot sizes at each customer ( ) max max : t c L t D N cost holding min r (minimize waiting time) t l T ( ) max t D 0 t 1 t 2 t max t time Cum’l items ( ) t D ii. General case To apply the methods for one-to-one problem in Chapter 3, note that The fixed cost for one dispatch now becomes (after omitting constants) c s N + c d k E( δ –1/2 ) N distance cost per dispatch stopping cost per dispatch time Dynamic programming and/or shooting methods from Chapter 3 then would apply.

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4 LSA: Chapter 4 Identical customers Pipeline inventory cost Location-dependent limits on route length (if line haul pipeline inventory cost is not negligible) ( ) + + s s m t n s d t 1 2 1 d: length of tour s: vehicle speed t m : motion time t s : stopping time ( ) ( ) ( ) total pipeline # # # 2 cost i i s i s c c t item dist item c t item stops s + + [ ] ( ) [ ] ( ) max max 2 1 2 1 v kNLE t D r E N 2 + δ ( ) ( ) ( ) + + = + + 2 ) 1 ( 2 1 2 2 1 2 1 d 2 1 s s i s i i s i s s i i m i n t c t c d s c t c t n c s c t c ( ) max t ND ( ) max 2 1 NLv Observation: On average, an item stays in the vehicle half of the entire tour time [ ] ( ) [ ] ( ) L Nv t c t ND t c L s v kNE c s t D r E N c s i s i i i + + + 2 2 2 max max max max 2 1 δ s s i i s i i p n t c s k c t c s r c z + + + 2 2 2 2 1 δ Pipeline inv. cost per item (in terms of r and δ): Sum across dispatches, tours, and items… Total number of items = ND(t max ), and D(t max ) = L v max / n s , then Cost per item Assume further that in the time-space continuum, system parameters are slow-varying. Considers the area around point (t, x): D’(t) D’ δ (x) δ r(x) r Define three decision variables/functions load per stop stop per tour n s (t, x) LSA: Chapter 4 Identical customers; loads not given Continuum Approximation headway H(t, x) ( ) ( ) ( ) x t H t D x t v , ' , = Only two decision variables are independent. Let V be the vehicle capacity, then D’n s H V and L=D(t max )/(D’H).
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• Fall '16
• Trigraph, #, 1 L, 2 L, 0 min, slow-varying.

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