One to Many Distribution

Transcription

1 CTL.SC1x -Supply Chain & Logistics Fundamentals One to Many Distribution MIT Center for Transportation & Logistics

2 How can I distribute products? One-to-One direct or point to point movements from origin to destination DC One-to-Many multi-stop moves from a single origin to many destinations DC Many-to-Many moving from multiple origins to multiple destinations usually with a hub or terminal DC T DC DC 2

3 Example: OfficeMin Your firm delivers office supplies to firms within the I-95 highway loop around Boston from your distribution center located in Newton. You want to estimate: 1. Expected cost per day, 2. Expected truck fleet size, and 3. Sensitivity of the solution. What information do I need? What methodology should I use? 3

4 Defining Delivery Districts 4

5 One to Many System Single Distribution Center: Products originate from one origin Products are demanded at many destinations All destinations are within a specified Service Region Ignore inventory (same day delivery) Assumptions: Vehicles are homogenous Same capacity, Q MAX Fleet size is constant Case adapted from Hernandez Lopez, J.J. (2003) Evaluation of Bulk and Packaged Distribution Strategies in a Specialty Chemical Company, MIT Supply Chain Management Program Thesis. 5

6 One to Many System Finding the estimated total distance: Divide the Service Region into Delivery Districts Estimate the distance required to service each district 6

7 One to Many System 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 d 2d + d TOUR LineHaul Local d LineHaul = Distance from origin to center of gravity (centroid) of delivery district d Local = Local delivery between c customers in one district How do we estimate distances? 1. Point to Point 2. Routing or within a Tour 7

8 Estimating Point to Point Distances 8

9 Distance Estimation: Point to Point Why bother? How to do it? Depends on the topography of the underlying region w Euclidean Space: d A-B = [(x A -x B ) 2 +(y A -y B ) 2 ] w Grid: d A-B = x A -x B + y A -y B w Random Network: different approach Euclidean Space (L 2 Metric) Manhattan Metric / Grid (L 1 Metric) Random Network 9

10 Distance Estimation: Point to Point For Random (real) Networks use: D A-B = k CF d A-B Find d A-B - the as crow flies distance. Euclidean: for really short distances w d A-B = SQRT((x A -x B )2+(y A -y B )2) Great Circle: for locations within the same hemisphere w d A-B = 3959(arccos[sin[LAT A ]sin[lat B ] +cos[lat A ] cos[lat B ]cos[long A -LONG B ]]) Where: w LAT i = Latitude of point i in radians w LONG i = Longitude of point i in radians w Radians = (Angle in Degrees)(π/180 o ) Apply an appropriate circuity factor (k CF ) How do you get this value? What do you think the ranges are? What are some cautions for this approach? 10

11 Selected Values of k CF Country k CF StdDev Argentina Australia Belarus Brazil Canada China Egypt Europe England France Germany Italy Spain Hungary India Indonesia Country k CF StdDev Japan Mexico New Zealand Poland Russia Saudi Arabia South Africa Thailand Turkey Ukraine United States Alaska US East US West Source: Ballou, R. (2002) Selected country circuity factors for road travel distance estimation, Transportation Research Part A, p

12 Estimating Route Distances 12

13 Distance Estimation: Routing Traveling Salesman Problem Starting from an origin, find the minimum distance required to visit each destination once and only once and return to origin. The expected TSP distance, d TSP, is proportional to (na) where n= number of stops and A=area of district The factor (k TSP ) is a function of the topology d TSP =k TSP (na) Euclidean Space (L 2 Metric) Manhattan Metric / Grid (L 1 Metric) Random Network 13

14 One to Many System What can we say about the expected TSP distance to cover n stops in district with an area of A? A good approximation, assuming a "fairly compact and fairly convex" area, is: E! " d TSP What values of k TSP should we use? # $ k & na = k TSP TSP n n ) ( + = k n TSP ' δ * δ A=Area of district n=number of stops in district δ=density (# stops/area) k=vrp network factor (unitless) d TSP =Traveling Salesman Distance d stop =Average distance per stop Lots of research on this for L 1 and L 2 networks - depends on district shape, approach to routing, etc. Euclidean (L 2 ) Networks k TSP = 0.57 to 0.99 depending on clustering & size of N (MAPE~4%, MPE~-1%) k TSP =0.765 commonly used Grid (L 1 ) Networks k TSP = 0.97 to 1.15 depending on clustering and partitioning of district References: Daganzo, C.. (2010) Logistics Systems Analysis, 4th Edition Springer-Verlag. Larson, R. and Odoni, A. (1981) Urban Operations Research, 14

15 Estimating Tour Distances 15

16 Estimating Tour Distance Finding the total distance traveled on all tours, where: l = number of tours c = number of customer stops per tour and n=total number of stops = c*l E! d # " TOUR $ = 2d + ck TSP LineHaul δ E! d # " AllTours $ = le! d # " TOUR$ = 2ld + nk TSP LineHaul δ Minimize number of tours by maximizing vehicle capacity! l = # " E! d # " AllTours $ = 2! D # " D Q MAX Q MAX $ & % + $ & % + d LineHaul + nk TSP δ [x] + = lowest integer value > x. This is a step function Estimate this with continuous function: E([x] + ) ~ E(x) + ½ 16

17 Continuous Approximation In this example, Q MAX =20. The number of tours, l, would be [D/Q MAX ] + which is a step function. Step functions are not continuous lets create a continuous approximation of this function that we can use. 5.0 Number of Tours (D/Qmax) D/Qmax [D/Qmax]+ (D/Qmax)+1/ Values of Demand (D) 17

18 One to Many System So that expected distance for all tours becomes:! E! " d # AllTours $ = 2 E! " D # $ # $ & "# Q MAX %& + d LineHaul + E! " n# $ k TSP δ! = 2 E! " D # $ + 1 $ # & "# Q MAX 2 %& d + E! " n# $ k TSP LineHaul δ Note that if each delivery district has a different density, then:! E! " d # AllTours $ = 2 E! " D # i $ + 1 # % & "% Q MAX 2 $ & d E! " n # i $ + k i LineHaul i TSP i δ i 18

19 Putting it all together For identical districts, the approximate transportation cost to deliver to each customer becomes: Expected number of loads/unloads (customer + origin) Expected number of linehaul moves or tours Expected distance for the linehaul portion. How do I get this? Expected local delivery distance regardless of number of tours! TransportCost = c s E! " n # $ + E! " D # $ + 1 # % & "% Q MAX 2 $ & + c d '! 2 E! " D # $ + 1 # % & "% Q MAX 2 $ & d + E! ) " n# $ k TSP LineHaul ) ( δ *,, + c E! D # vs " $ + E[n]=Expected number of stops in district E[D]=Expected demand in district Q MAX =Capacity of each truck c s =Cost per stop ($/stop) c d =Cost per distance ($/mile) c vs =Cost per unit per stop ($/item-stop) δ=density (# stops/area) k TSP =TSP network factor (unitless) d TSP =Traveling Salesman Distance d stop =Average distance per stop Cost per item per stop 19

20 Solution OfficeMin 20

21 OfficeMin Problem You deliver office supplies to firms within the I95 loop around Boston from your DC in Newton. This region is about 8 miles by 14 miles. You expect ~ 100 customer orders per day for about 1 to 2 pallets of product each. Local vans can handle 5 pallets at most. You estimate it costs about $10 per stop (to load or unload), about $5 per pallet to deliver to end customer, and about $1 a mile for driving. 14 Miles What is the expected daily transportation cost? 8 Miles 21

22 OfficeMin! TransportCost = c s E! " n # $ + E! " D # $ + 1 # % & "% Q MAX 2 $ & + c d '! 2 E! " D # $ + 1 # % & "% Q MAX 2 $ & d + E! ) " n# $ k TSP LineHaul ) ( δ *,, + c E! D # vs " $ + What do we know? c s = 10 $/stop c d = 1 $/mile c vs = 5 $/pallet E[n] = 100 E[D] = 150 Q MAX = 5 pallets What do we need to find? k = 1.15 (estimate) δ = 100/(8)(14) = 0.89 ~ 1 d linehaul =?? 14 Miles Avg Y-distance ~ 3.5 miles Avg X-distance ~ 4 miles 8 Miles Avg Distance: ( ) d linehaul ~ 5.3 miles 22

23 OfficeMin c s = 10 $/stop c d = 1 $/mile c vs = 5 $/pallet E[n] = 100 E[D] = 150 Q MAX = 5 pallets k = 1.15 δ = 1 d linehaul = 5! TransportCost = c s E! " n # $ + E! " D # $ + 1 # '! % & "% Q MAX 2 $ & + c 2 E! " D # $ + 1 # % & d "% Q MAX 2 $ & d + E! " n# $ k * ) TSP, LineHaul ) ( δ, + c E! D # vs " $ + Estimated number of tours per day: Estimated stop (load/unload) cost per day: l = E! " D # $ + 1 Q MAX 2 = = c s! " E! " n # $ + E! " l# # $ $ Estimated distance (driving) cost per day: % c d 2E! " l # $ d + E! " n# $ k ( % TSP 100(1.15) =1 2(30.5)(5) + ' LineHaul & δ * ' ) & 1 Estimated stop-pallet costs per day: Estimated total daily cost ~ $2400 to $2500 =10( ) = $1305 ( * = = $420 ) c vs E! " D# $ = 5(150) = $750 23

24 Estimating Fleet Size [OPTIONAL MORE ADVANCED] 24

25 Estimating the Fleet Size Find minimum number of vehicles required based on the amount of required work time each day where M = minimum number of vehicles needed in fleet t w = available worktime for each vehicle per period W = required amount of work time each day s = average vehicle speed Mt w W l = number of shipments per period t l = loading time per shipment t s = unloading time per stop W = % 2E! " l # $ d + E! " n# $ k ( TSP ' LineHaul & δ * ) s + E! l " # $ t + E! n l " # $ t s W = d AllTours s + lt l + nt s! W = 2d LineHaul # " s $ + t l &E' ( l ) * % + E ' ( n)! * k TSP s δ + t $ # & s " % 25

26 Fleet Size Note that W is a linear combination of two random variables, n and D. But, they are not independent, in fact, D = nd c where D c is the number of pallets per customer! W = 2d LineHaul # " s c s = 10 $/stop c d = 1 $/mile c vs = 5 $/pallet $ ' D + t l & + 1 *! ),+ n k TSP %( Q MAX 2+ s δ + t $ # & s " %! a = 2d $ ' LineHaul 1 * # + t s l &), = 0.144hrs " %( Q MAX +! b = k TSP s δ + t $ # & = hrs s " % c = 1! 2d $ # Linehaul + t 2 s l & = 0.361hrs " % E[n] = 100 E[D] = 150 Q MAX = 5 pallets W = Xn + c k = 1.15 δ = 1 d linehaul = 5 W = ad + bn + c E! D # " c $ =1.5 Var[D c ] = W = ad c n + bn + c W = (ad c + b)n + c Setting X=aD c +b, W is now a function of random variables, X and n. E[W ] = E[X ]E[n]+ c t w = 10 hrs s = 45 mph t l = 0.5 hr t s = 0.5 hr n x i x i=1( ) 2 = 0.25 n Var[W ] = E[n]Var[X ]+ E[X ] 2 Var[n] 26

27 Fleet Size c s = 10 $/stop c d = 1 $/mile c vs = 5 $/pallet E[n] = 100 E[D] = 150 Q MAX = 5 pallets k = 1.15 δ = 1 d linehaul = 5 t w = 10 hrs s = 45 mph t l = 0.5 hr t s = 0.5 hr! W = 2d LineHaul # " s $ ' D + t l & + 1 *! ),+ n k TSP %( Q MAX 2+ s δ + t $ # & s " % a = 0.144hrs b = hrs c = 0.361hrs E[D c ] =1.5 pallets Var[D c ] = 0.25 W = (ad c + b)n + c = ( 0.144D c )n = Xn + c E! " X # $ = ae! D # " c$ + b = 0.144(1.5) = Var[X ] = a 2 Var[D c ] = 0.021(0.25) = E[n] = 100 customers Var[n] = 400 (assume σ n = 20) E[W ] = E! " X # $ E! " n# $ + c E[W ] = (0.741)(100) = Var[W ] = E! " n # $ Var! " X # $ + E! " X # $2 Var! " n # $ Var[W ] = (100)( ) + (0.741) 2 (400) = 220 Distribution of required daily work hours: µ W ~ 75 hrs σ W ~ 15 hrs

28 Fleet Size Daily distribution of required work time ~N(75, 15) Set the fleet size (M) to match our level of risk how? Note: This is not the TSP k! Select a cycle service level (CSL) equal to P[W<Mt w ] Set M= (µ W + k CSL σ W ) / t w M(80%) = (75 hrs (15 hrs))/ (10 hrs/veh) = 8.00 = 8 M(90%) = (75 hrs (15 hrs))/ (10 hrs/veh) = 9.42 = 10 M(95%) = (75 hrs (15 hrs))/ (10 hrs/veh) = 9.96 = 10 M(99%) = (75 hrs (15 hrs))/ (10 hrs/veh) = = 11 Using very few, very rough estimates of input values, we can get a feel for the trade-offs between costs and service. Approximations can be used for sensitivity analysis. 28

29 Key Take Aways

30 Key Take Aways Many forms of product flow: One-to-One, One-to-Many, Many-to-One, and Many-to-Many Approximations are good first steps Require minimal data Allow for fast sensitivity analysis Enables quick scoping of the solution space Optimal methods usually require tremendous amounts of detailed information Planning problems usually have lots of uncertainty the actual conditions are unknown Before deciding to spend the time and energy to find an optimal solution, it is helpful to see if it is worth it. 30

31 CTL.SC1x -Supply Chain & Logistics Fundamentals Questions, Comments, Suggestions? Use the Discussion! MIT Center for Transportation & Logistics Dexter continuously approximating Yankee Golden Retriever Rescued Dog (