ECONOMIC LOT SCHEDULING

Transcription

1 ECONOMIC LOT SCHEDULING JS, FFS ad ELS Job Shop (JS) - Each ob ca be differet from others - Make to order, low volume - Each ob has its ow sequece Fleible Flow Shop (FFS) - Limited umber of product types - Make to stock, mass productio - Movemet of obs cotrolled by hadlig system Ecoomic Lot Schedulig (ELS) - Large umber of idetical obs/items (ru) - Make to stock for log periods of time - Applicatios to cotiuous maufacturig (paper, chemistry) Miimizig makespa Maimizig throughput Miimizig productio costs 09

2 Ecoomic lot sizig Large umber of idetical obs Cotiuous maufacturig i moths or eve years Log rus to make-to-stock, implyig ivetory holdig costs Setup time ad setup costs are sigificat Setup may be sequece depedet Termiology obs = items sequece of idetical obs = ru 0 ELS: Schedulig Obective: miimize total cost setup costs ivetory holdig costs Optimal schedule Trade-off betwee the two obectives Cyclic schedules are used ofte Algorithm Determie the legth of the rus (i.e. lot sizes) Determie the order of the rus (i.e. sequece to miimize setup cost)

3 Ecoomic Lot Schedulig : models. Oe type of item / oe machie with ad without setup time. Several types of items / oe machie rotatio schedules arbitrary schedules with / without sequece depedet setup times / cost 3. Geeralizatios to multiple machies Oe type of item/oe machie Assumptios type of item machie Productio rate: = /p (p processig time of ob ) Demad rate: D Machie capacity/utilizatio: = D/ Machie capacity is sufficiet to meet demad, i.e. >D Machie is idle util ivetory is depleted Setup costs (but o setup time) Problem Determie ru legth (cycle time): 3 3

4 Ivetory evolutio Demad over a cycle = D Legth of productio ru eeded = D/ = Maimum ivetory level = ( D)D/ Ivetory average ivetory level: ( D) D D D idle time Time 4 Ivetory costs Setup cost is c ad ivetory holdig cost per item per uit time is h. Average setup cost is c/ Average ivetory holdig cost: D h D Total cost (ivetory + setup): D h D c 5 4

5 Optimizig cost Solve hd D c mi Derivative the total cost with respect to : hd hd d d D c D c Solvig the miimizatio problem: D c hd 0 6 Optimal cycle legth hd D c c hd( D) c hd( D) 7 5

6 Optimal lot size The lot size is Dc D h( D) If productio rate is very high (p 0, ): Dc Dc Dc h( D) h h Ecoomic Lot Size (ELS) or Ecoomic Order uatity (EO): Dc D h 8 Setup time Setup time s Idle time of a machie durig a cycle: ( D/) If s ( ) solutio is still optimal Otherwise cycle legth machie is ever idle. s is optimal, i.e. 9 6

7 Eample 7.. No setup time Productio = 90/week Demad D = 50/week Setup cost c = 000 Holdig cost h = 0 /item c hd( D) (90 50) Optimal schedule Cycle time = 3 weeks Lot size = D = 50 items Idle time = 3( 5/9) =.33 weeks Ivetory Time (weeks) 7

8 Eample with setup times Now assume setup time If s <.33 weeks (about 9 days) the 3 weeks cycle is still optimal s Otherwise the cycle time must be: If setup last weeks (maiteace ad cleaig): Ivetory 00 = /( 5/9) = 4.5 weeks Time (weeks) Eample 7.. = , D = 0.0, c = 90, h = 5 determie, lot size ( ).678 Lot size: D =.678. What happes i a discrete settig? 3 8

9 Eample 7.. (discrete) Time to produce oe item is p = / = 3 days. Demad rate is item every 0 days. Lot size of k has to be produced every 0k days. Total cost per day of lot size of every 0 days is 90/0 = 9 Total cost of lot size of every 0 days is: ( )/0 = 6.5 Total cost of lot size of 3 every 30 days is: ( )/30 = 6.5 So the optimal is to produce every 0 days a lot of. 4 Multiple items ad rotatio schedules Assumptios differet items ad oe machie Productio rate for item is = /p Demad rate for item is D Setup cost per item s idepedet of the sequece Legth of productio ru of item is D / Problem Determie the best rotatio schedule that cotais a sigle ru of each item Cycle legth must be idetical to all items The order of sequece does ot matter 5 9

10 0 Ivetory ad costs Cycle legth determies the ru legth for each item Legth of productio ru eeded: D / Average ivetory level of item : With cost c, the total average cost per uit time is 6 D D c D D h Optimal cycle legth Solvig as i the previous case: Machie idle time durig a cycle: With productio capabilities ulimited ( ): 7 c D D h ) ( D hd c

11 Eample 7.3. Productio rates, demad rates, holdig costs ad setup costs items 3 4 _D _ _h _c Optimal cycle legth hd( D) moths c 9

12 Solutio Idle time is 0.48 = moths. The total average cost per time uit is: Ivetory h D D c Time (moths) 30 With setup times With sequece idepedet setup costs ad o setup times the sequece withi each lot does ot matter Oly a lot sizig problem Eve with setup times, if they are ot ob depedet the still oly lot sizig 3

13 Sequece idepedet setup times If sum of setup times < idle time the our optimal cycle legth remais optimal Otherwise we take it as small as possible. By icreasig util setup time is equal to idle time: s The optimal * is give by: * s 3 Sequece depedet setup times Now there is a sequecig problem Obective: miimize sum of setup times Problem is NP-hard If the sequece solutio has setup times < idle time optimal lot size ad sequece are optimal Else Optimal cycle legth has to be larger 33 3

14 Rotatio schedules with parallel machies m idetical machies i parallel There are setup cost but o setup time Item process o oly oe of the m machies For item, utilizatio factor is agai = D /. Coditio for a feasible solutio is: m 34 Decisio variables Assume rotatio schedule equal cycle for all machies Same as previous multi-item problem Additio: assigmet of items to machies Obective: balace the load Use heuristic LPT with as processig times. 35 4

15 Differet cycle legths Allow differet cycle legths for machies Ituitio: should be able to reduce cost Obective: assig items to machies to balace the load Complicatio: should ot assig items that favor short cycle to the same machie as items that favor log cycle. 36 Heuristic balacig Compute cycle legth for each item Rak i decreasig order Allocate obs sequetially to the machies util capacity of each machie is reached Adust balace if ecessary 37 5

16 Rotatio schedules with machies i series Flow shop Machies cofigured i series Assume o setup time Assume productio rate of each item is idetical for every machie Ca be sychroized Problem is reduced to sigle machie problem with setup cost: m c ci i 38 Variable productio rates Productio rate for each item ot equal for every machie Difficult problem Little research Fleible flow shop: eed eve more striget coditios 39 6

17 Discussio Lot sizig models demad assumed kow, which determies throughput make-to-stock systems: due date of little importace/ot available Obective: miimize ivetory ad setup costs (time). Practical problems are a combiatio of make-to-stock ad make-to-order. I these problems facilities are set up i series. This area of research is kow as: Supply Chai Maagemet 40 7