Transportation II. Lecture 16 ESD.260 Fall Caplice

Transcription

1 Transportation II Lecture 16 ESD.260 Fall 2003 Caplice

2 One to One System 1+ ns d LC($ / item) = c H + ch + ct + c + c + c r MAX i MAX i m s d vs Mode 1 v v Cost per Item c i t m v MAX 2 2v MAX Shipment Size

3 Single Distribution Center: Products originate from one origin Products are demanded at many destinations All destinations are within a specified Service Region Assumptions: Vehicles are homogenous Same capacity, v MAX Fleet size is constant 3

4 LCF for Cost Function Storage (Rent) Holding Costs Inventory Holding Costs Transportation Costs Handling Costs Key Cost Drivers? 4

5 Finding the estimated total distance: Divide the Service Region into Delivery Districts Estimate the distance required to service each district 5

6 Route to serve a specific district: Line haul from origin to the 1 st customer in the district Local delivery from 1 st to last customer in the district Back haul (empty) from the last customer to the origin 6

7 An Aside: Routing & Scheduling Problem: How do I route vehicle(s) from origin(s) to destination(s) at a minimum cost? A HUGE literature and area of research One type of classification (by methodology) 1. One origin, one destination, multiple paths Shortest Path Problem 2. Single path to reach all the destinations Minimum Spanning Tree 3. Many origins, many destinations, constrained supply Transportation Method (LP) 4. One origin, many destinations, sequential stops - Traveling Salesman Problem Vehicle Routing Problem Stops may require delivery & pick up Vehicles have different capacity (capacitated) Stops have time windows Driving rules restricting length of tour, time, number of stops 7

8 Find the estimated distance for each tour, d TOUR Capacitated Vehicle Routing Problem Cluster-first, Route-second Heuristic d 2d + d TOUR LineHaul Local d LineHaul = Distance from origin to center of gravity of delivery district d Local = Local delivery between c customers in district (TSP) 8

9 What can we say about the expected TSP distance to cover n stops in district of area A? Hard bound and some network specific estimates: [ ] E d TSP [ ] TSP 1.15 na E d k na For n>25 over Euclidean space, k=.7124 For straight line (Manhattan Metric), k=.7650 Density, δ, number of stops per area Average distance per stop, d stop 9 d stop δ = n/ A d na k k n n δ TSP = = =

10 Length of local tours Number of customer stops, c, times d stop over entire region Exploits property of TSP being sub-divided TSP of disjoint sub-regions TSP over entire region 10

11 Finding the total distance traveled on all, l, tours: [ ] E d TOUR [ ] [ ] = 2d + E d = le d = 2ld + 11 LineHaul ck δ AllTours TOUR LineHaul The more tours I have, the shorter the line haul distance Minimize number of tours by maximizing vehicle capacity [ ] 2 E d AllTours l D = vmax Q = dlinehaul + vmax + + nk δ nk δ [x] + is lowest integer value greater than x a step function Estimate this with continuous function: E([x] + ) ~ E(x) + ½

12 So that expected distance is: [ ] E d AllTours [ ] 1 [ ] E D = 2 + dlinehaul + vmax 2 E n k Note that if each delivery district has a different density, then: E[ Di] 1 E[ ni] E[ dalltours ] = 2 i + dlinehaul + k i i vmax 2 δi For identical districts, the transportation cost becomes: [ ] 1 [ ] 1 [ ] E D E D E n k TransportCost = cs E[ n] cd 2 + dlinehaul + + cvse D vmax 2 vmax 2 δ δ [ ] 12

13 Fleet Size Find minimum number of vehicles required, M Base on, W, amount of required work time t w = available worktime for each vehicle per period s = average vehicle speed l = number of shipments per period t l =loading time per shipment t s = unloading time per stop d Mt W = + lt + nt s AllTours w l s [ ] 2dLineHaul E D 1 k W = + tl + + E[ n] + ts s vmax 2 δ 13

14 Note that W is a linear combination of two random variables, n and D EaX [ + by] = aex [ ] + bey [ ] Var ax by a Var X b Var Y abcov X Y 2 2 [ + ] = [ ] + [ ] + 2 [, ] Substituting in, we can find E[W] and Var[W] a 2dLineHaul 1 = + tl s vmax k b= + t δ s Given a service level, SL P[W<Mt w ]=SL Thus, M= (E[W] + k(sl) StDev[W])/t w 14