Storage and Warehousing

Transcription

1 Warehouse Fuctios Storage ad Warehousig Chapter 0 Provide temporary storage of goods Put together customer orders Serve as a customer service facility Protect goods Segregate hazardous or cotamiated materials Perform value-added services Ivetory Elemets of a Warehouse Storage Media Material Hadlig System Buildig Storage Media Block Stackig Stackig frames Stool like frames Portable (collapsible) frames Catilever Racks Storage Media (Cotiued) Selective Racks Sigle-deep Double-deep Multiple-depth Combiatio Drive-i Racks Drive-through Racks Storage Media (Cotiued) Mobile Racks Flow Racks Push-Back Rack

2 Storage Media (Cotiued) Storage ad Retrieval Systems Racks for AS/RS Combiatio Racks Modular drawers (high desity storage) Racks for storage ad buildig support Perso-to-item Item-to-perso Maual S/RS Semi-automated S/RS Automated S/RS Aisle-captive AS/RS Aisle-to-aisle AS/RS Storage ad Retrieval Systems (cot) Storage Carousels Vertical Horizotal Miiload AS/RS Robotic AS/RS High-rise AS/RS (two motors) Phoeix Pharmaceuticals Germa compay fouded i 994 Receives supplies from 9 plats across Germay ad distributes to drugstores $400 millio aual turover Phoeix Pharmaceuticals Phoeix Pharmaceuticals (cot.) 30% market share Fill orders i < 30 miutes 87,000 items 6% pharmaceutical, 39% cosmetic 50-0,000 picks per moth Three levels of automatio Maual pickig via flow-racks Semi-automated usig dispesers Full automatio via robotic AS/RS 2

3 AVS/RS RFID Warehouse Problems Desig Operatioal or Plaig Warehouse Desig Locatio How may? Where? Capacity Overall Layout R C C C C C C EXIT Warehouse Desig Warehouse Desig Layout ad Locatio of Docks Pickup by retail customers? Combie or separate shippig ad receivig? Layout of road/rail etwork Room available for maeuverig trucks? Similar trucks or a variety of them? Dock Face Dock Face Outside buildig wall Caopy Flush dock Ope dock Eclosure Truck Truck Dock berths Dock berths Totally eclosed Straight i, Straight out Truc k Dock Face Sawthoot dock 3

4 Warehouse Desig (cot) Model for Rack Desig Number of Docks Shippig ad receivig combied or separated? Average ad peak umber of trucks or rail cars? Average ad peak umber of items per order? Seasoal highs ad lows Types of load hadled? Sizes? Shapes? Cartos? Cases? Pallets? Protectio from weather elemets Miimize Subject to x,y x(a ) y(b ) 2 xyz it eger x, y are # of colums, rows of rack spaces a, b are aisle space multipliers i x, y directios Model for Rack Desig I the relaxed problem, xyz= x=/yz The ucostraied objective is (a )/ yz y(b ) 2 Model for Rack Desig Takig derivative with respect to y, settig equatio to zero ad solvig, we get (a ) 2y 2 z x (b ) z(a ) b 2 0 ad y (a ) z(b ) Rack Desig Example Cosider warehouse show i figure 0.29 Assume travel origiates at lower left corer Assume reasoable values for the aisle space multipliers a, b Rack Desig Example Example : Determie legth ad width of the warehouse so as to accommodate 2000 square storage spaces of equal area i: 3 levels 4 levels 5 levels 4

5 Rack Desig Example Solutio Reasoable values for a, b are 0.5, 0.2 For the 3-level case, x y 2000(0.2 ) 3(0.5 ) 2000(0.5 ) 3(0.2 ) Rack Desig Example Solutio Previous solutio gives a total storage of 24x29x3=2088 Due to roudig, we get 88 more spaces If iadequate to cover the area required for louge, customer etrace/exit ad other areas, the aisle space multipliers a, b must be icreased appropriately ad the x, y values recalculated Rack Desig Example Solutio For the 4 level ad 5 level case, the buildig dimesios are 25x20 uits ad 8x23 uits, respectively Easy to calculate the average distace traveled - simply substitute a, b, x ad y values i the objective fuctio For 3-level case, average oe-way distace = 35.4 uits Warehouse Desig Model Model Assumptios. The available total storage space is kow. 2. The expected time a product speds o the shelves is kow. This is referred to as the dwell time throughout this paper. 3. The cost of hadlig each product i each flow is kow. 4. The dwell time ad cost have a liear relatioship. 5. The aual product demad rates are kow. 6. The storage policies ad material hadlig equipmet are kow ad these affect the uit hadlig ad storage costs. Model Notatio Parameters i : Number of products i =, 2,,. j : Type of material flow; j=,2,3,4 i : Aual demad rate of product i i uit loads A i : Order cost for product i P i : Price per uit load of product i p i : Average percetage of time a uit load of product i speds i reserve area if product is assiged to material flow 3 q ij : whe product i is assiged to material flow j=, 2 or 4; d i whe product i is assiged to flow j=3, where d i is the ratio of the size of the uit load i reserve area to that i forward area ad d i is the largest iteger greater tha or equal to d i 5

6 Model Notatio Model Notatio a,b,c : Levels of space available i the vertical dimesio i each fuctioal area, a - cross-dockig, b - reserve, c forward r: Ivetory carryig cost rate H ij : Cost of hadlig a uit load of product i i material flow j C ij : Cost of storig a uit load of product i i material flow j per year S i : Space required for storig a uit load of product i TS : Total available storage space Q i : Order quatity for product i (i uit loads) T i : Dwell time (i years) per uit load of product i LL CD, UL : Lower ad upper storage space limit for cross-dockig area CD LL F,UL F : Lower ad upper storage space limit for forward area LL R,UL R : Lower ad upper storage space limit for reserve area a,b,c : Levels of space available i the vertical dimesio i each fuctioal area, a - cross-dockig, b - reserve, c forward r: Ivetory carryig cost rate H ij : Cost of hadlig a uit load of product i i material flow j C ij : Cost of storig a uit load of product i i material flow j per year S i : Space required for storig a uit load of product i TS : Total available storage space Q i : Order quatity for product i (i uit loads) T i : Dwell time (i years) per uit load of product i LL CD, UL : Lower ad upper storage space limit for cross-dockig area CD LL F,UL F : Lower ad upper storage space limit for forward area LL R,UL R : Lower ad upper storage space limit for reserve area Decisio Variables X = if product i is assiged to flow type j ; 0 otherwise ij,, : Proportio of available space assiged to each fuctioal area, - crossdockig, - reserve, - forward Model Model Model Miimize q ij H ij i X ij + q ij C ij Q i X ij /2 () 4 j X ij i j i j i (2) Q i X i /2 ats (3) i Q i X i2 /2 p i Q i X i3 bts (4) i i ( p i )Q i X i3 /2 Q i X i4 /2 cts (5) i i (6) LL CD a TS UL CD (7) LL R b TS UL R (8) LL F c TS UL F (9),, 0 (0) X ij 0or i, j () Spreadsheet Based AS/RS Desig Tool Spreadsheet Based AS/RS Desig Tool 6

7 Block Stackig Simple formula to determie a ear-optimal lae depth assumig goods are allocated to storage spaces usig the radom storage operatig policy istataeous repleishmet i predetermied lot sizes repleishmet doe oly whe ivetory excludig safety stock has bee fully depleted lots are rotated o a FIFO basis Block Stackig withdrawal of lots takes place at a costat rate empty lot is available for use immediately Let Q, w ad z deote lot size i pallet loads, width of aisle (i pallet stacks) ad stack height i pallet loads, respectively Block Stackig Kid s (975) formula for ear-optimal lae depth, d d Qw z w 2 Block Stackig E.g., if lot size is 60 pallets, pallets are stacked 3 pallets high ad aisle width is.7 pallet stacks, the d 60(.7) pallets Verify optimality by checkig the utilizatio for all possible lae depths (a fiite umber) Block Stackig Several issues omitted i Kid s formula. Some examples What if pallets withdraw ot at a costat rate but i batches of varyig sizes? What if lots are relocated to cosolidate pallets cotaiig similar items? Storage Policies Radom I practice, ot purely radom Dedicated Requires more storage space tha radom, but throughput rate is higher because o time is lost i searchig for items Cube-per-order idex (COI) policy Class-based storage policy 7

8 Storage Policies Shared storage policy Class based ad shared storage policies are betwee the two extreme policies - radom ad dedicated Class based policy variatios if each item is a class, we have dedicated policy if all items i oe class, we have radom policy Desig Model for Dedicated Policy Warehouse has p I/O poits m items are stored i oe of storage spaces or locatios Each locatio requires the same storage space Item i requires storage spaces Desig Model for Dedicated Policy Ideally, we would like However, if LHS < RHS, add a dummy product (m+) to take up remaiig spaces m i m i Desig Model for Dedicated Policy So, assume that the above equality holds But, if RHS < LHS, o feasible solutio Model Parameters f ik trips of item i through I/O poit k cost of movig a uit load of item i to/from I/O poit k is c ik distace of storage space j from I/O poit k is d kj Desig Model for Dedicated Policy Model Variable biary decisio variable x ij specifyig whether or ot item i is assiged to storage space j Desig Model for Dedicated Policy Miimize Subject to m i j p k c ik f ik d kj x ij x ij i,2,...,m j 8

9 Desig Model for Dedicated Policy m x ij j,2,..., i x ij 0 or, i,2,...,m, j,2,..., Desig Model for Dedicated Policy Substitutig w ij Miimize m i j w ij x ij p k c ik f ik d kj, the obj f.is Desig Model for Dedicated Policy Model is geeralized QAP Ca be solved via trasportatio algorithm No eed for biary restrictios i the model Desig Model for Dedicated Policy - Example WH Layout Desig Model for Dedicated Policy - Example 3 I/O poits located i middle of south, west ad orth walls 4 items Desig Model for Dedicated Policy Example [f ik (c ik )] (5) 25(5) 88(5) (7) 200(3) 50(6) (4) 5(7) 85(9) (5) 35(8) 90(2) 6 9

10 Desig Model for Dedicated Policy Example Solutio (d kj ) Desig Model for Dedicated Policy Example Solutio (w ij ) Desig Model for Dedicated Policy - Example Solutio Desig Model for COI Policy Cosider special case of dedicated storage policy model All items use I/O poits i same proportio Cost of movig a uit load of item i is idepedet of I/O poit Defie P k as % trips through I/O poit k No eed for the first subscript i f ik as well as c ik Desig Model for COI Policy Miimize Subject to m i j p k c i f i d kj x ij x ij i,2,...,m j Desig Model for COI Policy m x ij j,2,..., i x ij 0 or, i,2,...,m, j,2,..., 0

11 Desig Model for COI Policy Substitutig w j Miimize m i j c i f i p k P k d kj, the obj f.is w j x ij Desig Model for COI Policy - Solutio COI model easier tha Dedicated Model Rearrage cost, distace terms (c i f i / ), w j i o-icreasig ad o-decreasig order Match Item correspodig to st elemet i ordered cost list with storage spaces correspodig to st elemets i ordered distace list Desig Model for COI Policy - Solutio Secod item with storage spaces correspodig to ext S l elemets, ad so o COI policy calculates iverse of the cost term ad orders elemets i o-decreasig order, of their COI values, thereby producig the same result as above Desig Model for COI Policy - Solutio Arragig cost ad distace vectors i oicreasig ad o-decreasig order ad takig their product provides a lower boud o cost fuctio Above algorithm is optimal Desig Model for COI Policy - Example Cosider dedicated policy example Igore c ik ad f ik data Assume all 4 items use 3 I/O poits i same proportio pallets moved/time period are 00, 80, 20 ad 90 cost to move uit load through uit distace is $.00 Determie optimal assigmet of items to storage spaces Desig Model for COI Policy Example Solutio

12 Desig Model for COI Policy - Example Solutio Sort [c i f i / ] values i o-icreasig order [60, 33.33, 6, 5], correspodig to items 3,, 2 ad 4 Optimal storage space assigmet Item to Storage Spaces 2, 5, 7 Item 2 to Storage Spaces, 3, 9,, 4 Item 3 to Storage Spaces 6, 0 Item 4 to Storage Spaces 4, 8, 2, 3, 5, 6 Desig Model for COI Policy Example Solutio Desig Model for Radom Policy Items stored radomly i empty ad available storage spaces Each empty space has a equal probability of beig selected Storage or retrieval may ot be purely radom, but we assume so for model Desig Model for Radom Policy Problem Defiitio Determie storage space layout so total expected travel distace betwee each of storage spaces ad p I/O poits is miimized Sum of distaces of each storage space from each I/O poit is p d kj k Desig Model for Radom Policy- Solutio Arrage spaces i o-decreasig order of the sum of above distaces Pick the closest storage spaces depeds upo ivetory levels of all items is less tha that required uder dedicated policy Desig Model for Radom Policy - Example Determie storage space layout for 56 storage spaces i a 40x70 feet warehouse Radom storage policy Miimize total distace traveled Each storage space is a 0x0 feet square I/O poit located i middle of south wall 2

13 Desig Model for Radom Policy - Example Desig Model for Radom Policy - Example Solutio Calculate distace of all potetial storage spaces to the I/O poit Arrage them i o-decreasig order Desig Model for Radom Policy - Example Solutio Largest distace traveled is 70 feet Sum total distace traveled (2800) by umber of storage spaces (56) to get average distace traveled = 50 feet Desig Model for Radom Policy - Example Solutio Travel Time Models For radom policy, average distace traveled p d kj k j Whe umber of storage spaces are large, calculatig average distace ca be tedious Travel Time Models If storage spaces are small relative to total area, approximate average distace traveled assume spaces are cotiuous poits o a plae X Y use the itegral 0 0 (x y)dxdy A 3

14 Travel Time Models We assume i previous itegral that warehouse is i st quadrat oly oe I/O poit (at origi ad SW corer) distace metric of iterest is rectiliear Previous itegral ca be easily modified if two or more I/O poits distace metric is ot rectiliear o restrictios o locatio of warehouse Travel Time Models Suppose desiger iterested i shape that miimizes travel time The, depedig upo umber ad locatio of I/O poits, distace metric, warehouse shape ca rage from diamod to circle to trapezium!!! Travel Time Models Models miimizig costructio costs ad travel distace Cosider followig assumptios Warehouse shape is fixed - rectagle Warehouse area = A Costructio cost is fuctio of warehouse perimeter - r[2(a+b)] r is uit (perimeter) distace costructio cost a ad b are warehouse dimesios Travel Time Models Oe I/O poit at origi ad SW corer coordiates are (p, q) cost for each uit distace traveled = c Model 2r(a b) c p a p q b q A ( x y )dxdy Travel Time Models Optimal value of a ad b, give that a I/O poit must be o or outside exterior walls, i.e., p $ 0 warehouse area must be A square uits A c 8r 2c 8r ad b A 2c 8r c 8r Travel Time Models Sigle commad cycle Dual or multiple commad cycles 4

15 Warehouse Operatios Warehouse operatioal problems Sequece i which orders to be picked How frequetly orders picked from high-rise storage area? Batch pickig or pick whe order comes i? Limit o umber of items picked? If so, what is the limit? Operator assigmet to stacker craes Warehouse Operatios How to balace pickig operator s workload? Release items from stacker crae ito sortig statios i batches or as soo as items are picked? Order pickig cosumes over 50% of the activities i warehouse Warehouse Operatios Not surprisig that order pickig is the sigle largest expese i warehouse operatios Sice costructio ad operatio of AS/RS are very high,maagers iterested i maximizig throughput capacity Order Pickig Sequece Two basic pickig methods Order pickig Zoe pickig Cosider this: A AS/R machie has two idepedet motors Movemet i horizotal ad vertical directios simultaeously Order Pickig Sequece Time to travel from (x i, y i ) to (x j, y j ) max x x i j h, y i y j v Order Pickig Sequece Model Miimize Subject to w ij j, ji i j, j d ij w ij w ij i,i j j,2,..., i,2,..., u i u j w ij 2 i j w ij 0 or i, j,2,..., u i arbirary real umbers 5

16 Order Pickig Sequece Algorithms Costructio Improvemet Hybrid Order Pickig Sequece Algorithms 2-opt 3-opt Brach-ad-boud Simulated Aealig Covex Hull Covex Hull Algorithm - Phase Covex Hull Algorithm - Phase Fid x max ad y max Delete poits iside polygo formed by x max, y max ad origi For each regio, costruct covex path betwee extreme poits Sort poits i regios ad 2 i ascedig order of x-coordiate Sort regio 3 poits i descedig order Startig with st extreme poit, compute V for three cosecutive poits i, i+, i+2 V= (y i+ -y i )(x i+2 -x i+ )+(x i -x i+ )(y i+2 -y i+ ). Repeat util other extreme poit is reached If V # 0, o covex hull with i, i+, i+2 Otherwise, covex hull possible Covex Hull Algorithm - Phase Covex Hull Algorithm - Phase Regio y max Regio 2 Usig some or all of the sorted poits i regios, 2, ad 3, three at a time, geerate covex hull (sub-tour) Poits ot i sub-tour are cosidered i phases 2 ad 3. x max If x max = y max followig explaatio still good Regio 3 0 6

17 Covex Hull Algorithm - Phase 2 Covex Hull Algorithm - Phase 3 Isert poits that maybe icluded i sub-tour without icreasig cost Such free isertio poits lie o a parallelogram with two adjacet poits i the sub-tour as its corer Isert poits ot icluded i the sub-tour i phases ad 2 usig miimal isertio cost criteria greedy hull steepest descet hull If o poits left for isertio i phase 2 or 3, phase sub-tour is optimal Simulated Aealig Algorithm Set S, z, r, T i, T= T i ; T fi = 0.T i Radomly select poits i ad j i S ad exchage their positios If ew solutio S' has z < z, set S = S', ad z = z Otherwise, set S= S' with probability e -d/t Simulated Aealig Algorithm Repeat Step util umber of ew solutios = 6 times the umber of eighbors Set T= rt. If T > T fi, go to Step Otherwise retur S, ad STOP TSP Software Routig Problem 7

18 Multimedia CD for Distributio Ceter Desig 8