Logistics cost

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)

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

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

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)

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)

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

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.

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

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)

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

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 2CMAX 3CMAX CMAX= Maximum capacity for transport mode (items)

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

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

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)

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

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

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.