Inventory Aggregation and Discounting

Transcription

1 Inventory Aggregaton and Dscountng Matchng Supply and Demand utdallas.edu/~metn 1

2 Outlne Jont fxed costs for multple products Long term quantty dscounts utdallas.edu/~metn

3 Example: Lot Szng wth Multple Products Shppng multple products over the same route to the same retaler Demand per year R L = 1,000; R M = 1,00; R H = 10 Common transportaton cost per delvery, S = $4,000 Product specfc order cost per product n each delvery s L = $1,000; s M = $1,000; s H = $1,000 Holdng cost, h = 0. Unt cost C L = $500; C M = $500; C H = $500 utdallas.edu/~metn 3

4 Delvery Optons No Aggregaton: Each product ordered separately Complete Aggregaton: All products delvered on each truck Talored Aggregaton: Selected subsets of products for each truck utdallas.edu/~metn 4

5 No Aggregaton: Order each product ndependently Ltepro Medpro Heavypro Demand per year 1,000 1,00 10 Fxed cost / order $5,000 $5,000 $5,000 Optmal order sze 1, Order frequency 11.0 / year 3.5 / year 1.1 / year Annual cost $109,544 $34,64 $0,954 Total cost = $155,140 utdallas.edu/~metn 5

6 Complete Aggregaton: Order jontly All Products n All Trucks Total orderng cost S*=S+s L +s M +s H = $7,000 n: common orderng frequency Annual orderng cost = n S* Total holdng cost: Total cost: utdallas.edu/~metn TC( n) n * L Rn hc M Rn hc RH n hc L M * h S n n R C R C R C h R C R C R C S L L M M H H L L M M H H * H 6

7 Complete Aggregaton: Order all products jontly Ltepro Medpro Heavypro Demand per year 1,000 1,00 10 Order frequency 9.75/year 9.75/year 9.75/year Optmal order sze 1, Annual holdng cost $61,51 $6,151 $615 Annual order cost = 9.75 $7,000 = $68,50 Annual total cost = $136,58 Orderng hgh and low volume tems at the same frequency cannot be a good dea. utdallas.edu/~metn 7

8 Talored Aggregaton: Orderng Selected Subsets Example: Orders may look lke (L,M); (L,H); (L,M); (L,H). Most frequently ordered product: L M and H are ordered n every other delvery. We can assocate fxed order cost S wth product L because t s ordered every tme there s an order. Products other than L, the rest are assocated only wth ther ncremental order costs (s values). An Algorthm: Step 1: Identfy most frequently ordered product Step : Identfy frequency of other products as a relatve multple Step 3: Recalculate orderng frequency of most frequently ordered product Step 4: Identfy orderng frequency of all products utdallas.edu/~metn 8

9 Talored Aggregaton: Orderng Selected Subsets s the generc ndex for tems, s L, M or H. Step 1: Fnd most frequently ordered tem: hc R n n max{ n} ( S s ) The frequency of the most frequently ordered tem wll be modfed later. Ths s an approxmate computaton. Step : Relatve order frequency of other tems, m m are relatve order frequences, they must be ntegers. They do not change n the remander. utdallas.edu/~metn n hc R n m s n 9

10 Talored Aggregaton: Orderng Selected Subsets Step 3: Recompute the frequency of the most frequently ordered tem. Ths tem s ordered n every order whereas others are ordered n every m orders. The average fxed orderng cost s: s n * utdallas.edu/~metn S m Annual orderng cost Annual holdng cost S R m hc s m s n( S ) m R n / m hc formula (10.9) on p.74of the textbook 10

11 Talored Aggregaton: Orderng Selected Subsets Step 4: Recompute the orderng frequency n of other products: n n m Total Annual orderng cost: ns+n H s H +n M s M +n L s L n (the frequency of the most frequently ordered product) s one of the followng values n H, n M, n L Total Holdng cost: utdallas.edu/~metn RL n hc RM n hc R L M n L M H H hc H 11

12 Step 1: n L Talored Aggregaton: Orderng Selected Subsets hclrl ( S s ) L = 11, n = 3.5, n = 1.1 n max{ n } 11 M H Step : hcm RM n nm nh mm mh sm = 7.7, =.4;, 5 nm Item L s ordered most frequently. Every other L order contans one M order. Every 5 L orders contan one H order. At ths step we only now relatve frequences, not the actual frequences. utdallas.edu/~metn 1

13 Step 3: Step 4: =( )100 utdallas.edu/~metn Talored Aggregaton: Orderng Selected Subsets n Total orderng cost: M ns+n H s H +n M s M +n L s L =11.47(4000)+11.47(1000)+5.73(1000)+.9(1000) Total holdng cost n m * M 5.73 =45,880+11,470+5,730+,90=65370 RL n hc RM n hc R L M n L n * hcrm (0.)500(1000*1100* 10*5) s ( /11000 / 1000 / 5) S m M H H hc (. ) (. ) (. ) (. ) (. 9) ( 0. ) 500 H n H n m * H.9 13

14 Talored Aggregaton: Order selected subsets Ltepro Medpro Heavypro Demand per year 1,000 1,00 10 Order frequency 11.47/year 5.73/year.9/year Optmal order sze Annual holdng cost $5,810 $10,470 $,630 utdallas.edu/~metn Annual order cost = $65,370 Total annual cost = $130,650 Compare wth $136K of total aggregaton and wth $155K of no aggregaton 14

15 Lessons From Aggregaton Aggregaton allows a frm to lower lot sze wthout ncreasng cost Order frequences wthout aggregaton and wth talored aggregaton» (11; 3.5; 1.1) vs. (11.47; 5.73;.9)» More frequent orderng mples smaller order szes Talored aggregaton s effectve f product specfc fxed cost s a large fracton of jont fxed cost Complete aggregaton s effectve f product specfc fxed cost s a small fracton of jont fxed cost Informaton technology can decrease product specfc orderng costs. utdallas.edu/~metn 15

16 The word of the moment: Retal Retal: The sale of goods n small quanttes drectly to the customer. Opposte of the word wholesale. Retal s a very flexble word. It can be used as a Noun: I work n retal. Verb: Albertson retals varous groceres. Adjectve: Retal margns are too narrow. Adverb: Wal-mart sells everythng retal. Etymology: A varant of Old French retalle "pece cut off" from retaller "to cut up" from re- "repeat" + taller "cut." Akn to "talor" whch comes from Old French talleor from taller "to cut" gong back to Late Latn talare "cut." utdallas.edu/~metn 16

17 Quantty Dscounts Lot sze based All unts Margnal unt at the end of these fle Volume based How should buyer react? What are approprate dscountng schemes? utdallas.edu/~metn 17

18 All-Unt Quantty Dscounts Cost/Unt Total Materal Cost $3 $.96 $.9 5,000 10,000 q 1 q Order Quantty 5,000 10,000 Order Quantty utdallas.edu/~metn 18

19 All-Unt Quantty Dscounts {0,q 1,q } are prce break quanttes Fnd EOQ for prce n range q to q +1 If q EOQ < q +1,» Canddate n ths range s EOQ, evaluate cost of orderng EOQ If EOQ < q,» Canddate n ths range s q, evaluate cost of orderng q If EOQ q +1,» Canddate n ths range s q +1, evaluate cost of orderng q +1 Warnng: Do not gnore purchase cost The annual materal cost of buyng n lot szes of q Q < q +1 s C R. Fnd mnmum cost over all canddates utdallas.edu/~metn 19

20 Total Cost Fndng Q wth all unts dscount Q 1 RS hc 1 Q RS hc Q 3 RS hc 3 Quantty utdallas.edu/~metn 0

21 Total Cost Fndng Q wth all unts dscount Q 1 RS hc 1 Q 3 RS hc 3 utdallas.edu/~metn Q RS hc Quantty 1

22 Total Cost Fndng Q wth all unts dscount 1 Quantty utdallas.edu/~metn

23 Summary Aggregaton: Jont fxed costs for multple products Dscounts: All unt quantty dscounts over a long term utdallas.edu/~metn 3

24 Margnal Unt Quantty Dscounts Cost/Unt Total Materal Cost c 0 c 1 c $3 $.96 $.9 V V 1 5,000 10,000 q 1 q Order Quantty 5,000 10,000 Order Quantty utdallas.edu/~metn 4

25 Margnal Unt Quantty Dscounts utdallas.edu/~metn V Cost of buyng exactly q. V 0. V c ( q q ) c ( q q )... c ( q q ) If q Q q 1 Annual order cost = Total cost( Q) Q EOQ R Q S Annual holdng cost = V Q q c h ( ) R Annual materal cost = Q V Q q c ( ) For range,, 0 R Q S c h R Q V q c R S V q c hc 0 5

26 Margnal-Unt Quantty Dscounts Fnd EOQ for prce n range q to q +1 If q EOQ < q +1,» Canddate n ths range s EOQ, evaluate cost of orderng EOQ If EOQ < q,» Canddate n ths range s q, evaluate cost of orderng q If EOQ q +1,» Canddate n ths range s q +1, evaluate cost of orderng q +1 Fnd mnmum cost over all canddates utdallas.edu/~metn 6

27 Margnal Unt Quantty Dscounts Total cost EOQ 1 EOQ 3 utdallas.edu/~metn q 1 q Lot sze Compare ths total cost graph wth that of all unt quantty dscounts. Here the cost graph s contnuous whereas that of all unt quantty dscounts has breaks. 7

28 Margnal Unt Quantty Dscounts Total cost EOQ 1 EOQ EOQ 3 q 1 q Lot sze utdallas.edu/~metn 8