1. Logistics cost

2. Componenti del costo logistico
Costi di trasporto (in ingresso, intra-company. in uscita) Costi di magazzino (immagazinamento e deposito dei prodotti, localizzazione del deposito e della azienda) Costo di mantenimento scorte (capitale per materie prime, WIP, prodotti finiti, scarti) Costi di acquisto/produzione lotti
(movimentazione materiali, messa a punto linee)
Costo degli imballi per trasporto e movimentazione
(pallet, materiali da imballaggio, altri) Costi amministrativi/ di struttura (direzione, personale amministrativo, sistemi informativi, elaborazione ordini)

3. Time space diagrams Consider a simple Product to Consumption Network
Primary Tradeoffs:
Moving (Handle, Transport)/Holding (Rent, Inventory)
Consumption
Storage
Production

4. Cumulative count curve
Shipments Arrivals
Horizontal distances
Transit time tm
Headway hi nd
Cumulative Count of Items nw nt D hi
Vertical distances
Items waiting transport nw
Items in transit nt
Items at destination nd tm D
Time

5. Logistics cost LC function
LC (per item) = Holding Costs + Moving Costs
Holding Costs = Rent Cost + Inventory Cost
Moving Costs = Transport Cost + Handling Cost
LC (per item) = (R + I) + (T + H)

6. AnnualRentCost = cr Dhmax
Rent cost R (€/item)
Function of max inventory
Captures facility costs
Not affected by the item by item flow
AnnualRentCost = cr Dhmax
R = cr hmax
D = Annual demand (items/time)
cr = Rent cost (€/item-time)
hmax = Maximum headway (time)

7. Inventory cost I (€/item)
Function of delay to items.
Captures capital costs, insurance, etc.
Implies that each time period matters equally
Constant to each time period (linear)
Annual Inventory Cost = ci D(1/2ħ + tm)
I = ci(1/2ħ + tm)
ci = Inventory cost (€/item-time)
ħ = average headway
tm =Transit time

8. Transport cost T (€/item)
Shipment size u has a shipment cost s
s = cf + cvv
the cost of n shipments is
ns =Sn(cf + cvvi) =ncf + cvSnvi =ncf + cvD
The transport cost T is
T = cf (n/D) + cv = cf /ū+ cv
T decreases with increase in average shipment size ū.

9. Nomenclature cf = Fixed transport cost (€/shipment)
cv = Variable transport cost (€/item)
u = Shipment size (items)
D = Annual number of items shipped = Snu
ū = D/n average shipment size
n = number of annual shipments

10. Transport costs T with headways h
Number of items in an average shipment
ū = Dħ
T decreases with increase in average ħ
I increases with ħ
T = cf /Dħ + cv = cf /ū + cv
cf = Fixed transport cost (€/shipment)
cv = Variable transport cost (€/item)

11. Transport costs and distance
“Fixed” costs cf are variable with number of delivery stops ns and distance d
Variable costs have a fixed component and a linear component variable with distance d
cf = cfs (1 + ns ) + cfdd
cv = cvs + cvdd
T = cf /ū + cv
T = cfs (1 + ns )/ū + cfdd/ū + cvs + cvdd
Marginal cost of an additional item ignore cvd

12. Transport costs in relation to capacity
ft(u) is sub-additive: ft(u1+u2)≤ ft(u1)+ft(u2)
cvuMAX<<cf
Transport Cost per Shipment
Shipment Size
ft(v) cv cf
CMAX CMAX 3CMAX
CMAX= Maximum capacity for transport mode (items)

13. Transport costs with multiple modes
Transport Cost per Shipment
Shipment Size
Mode 1
Mode 2
Mode 3 cf cv

14. Handling costs H (€/item)
Loading items into boxes, pallets, containers, etc.
If handled individually – linear with each item
If handled in batches – fixed & variable components
H = cfh + cvhCh
Ch= capacity (items) for load unit h

15. Moving costs T + H (€/item)
T + H = cf/ū + (cv + cfh /ChMAX + cvh)
T + H= cfs(1+ ns)/ū + cfdd/ū + cvs+ cfh/ChMAX+ cvh
Generally handling costs are subsumed within transport costs as long as D>>ChMAX (total shipment size is much greater than pallet)

16. Total logistics cost LC function
LC(€/item) = crhmax+ 1/2ciħ+ citm + cfs(1+ns)/ū + cfdd/ū + cvs
cr = Rent cost (€/item-time)
ci= Inventory cost (€/item-time)
tm =Transit time
hMAX= Maximum headway (time)
cfs= Fixed cost per stop (€/stop)
cfd= Cost per distance (€/distance)
cvs= Marginal cost per item
ns= Number of delivery stops
d = distance
ū = D/n = Dħ average shipment size

17. Total logistics cost function
VMAX = Maximum capacity for transport mode (items)

18. Optimum shipment and headway
LC(€/item) = crhmax+ 1/2ciħ+ citm + cfs(1+ns)/ū + cfdd/ū + cvs
ū = D/n = Dħ average shipment size
LC(€/item) = crhmax+ 1/2ciū/D+ citm + cfs(1+ns)/ū + cfdd/ū + cvs
dLC/dv = 1/2ci/D - cfs[(1+ns) + cfdd]/ū2 = 0
ū* = {2cfsD[(1+ns) + cfdd]/ci}1/2
The size ū* increases with respect to the square root of D, and the square root of d.
The optimum delivery frequency D/ū* will increase in proportion to the square root of D and will fall with respect to the square root of d.